Canadian National Mathematical Olympiad 1980 Solutions

  1. If $a679b$ is the decimal expansion of a number in base $10$, such that it is divisible by $72$, determine $a,b$.
  2. The numbers from $1$ to $50$ are printed on cards. The cards are shuffled and then laid out face up in $5$ rows of $10$ cards each. The cards in each row are rearranged to make them increase from left to right. The cards in each column are then rearranged to make them increase from top to bottom. In the final arrangement, do the cards in the rows still increase from left to right?
  3. Among all triangles having (i) a fixed angle $A$ and (ii) an inscribed circle of fixed radius $r$, determine which triangle has the least minimum perimeter.
  4. A gambling student tosses a fair coin. She gains $1$ point for each head that turns up, and gains $2$ points for each tail that turns up. Prove that the probability of the student scoring exactly $n$ points is $\frac{1}{3}\cdot\left(2+\left(-\frac{1}{2}\right)^{n}\right)$.
  5. A parallelepiped has the property that all cross sections, which are parallel to any fixed face $F$, have the same perimeter as $F$. Determine whether or not any other polyhedron has this property.

Canadian National Mathematical Olympiad 1979 Solutions

  1. Given:
    (i) $a$, $b > 0$;
    (ii) $a$, $A_1$, $A_2$, $b$ is an arithmetic progression;
    (iii) $a$, $G_1$, $G_2$, $b$ is a geometric progression.
    Show that \[A_1 A_2 \ge G_1 G_2.\]
  2. It is known in Euclidean geometry that the sum of the angles of a triangle is constant. Prove, however, that the sum of the dihedral angles of a tetrahedron is not constant.
  3. Let $a$, $b$, $c$, $d$, $e$ be integers such that $1 \le a < b < c < d < e$. Prove that \[\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]} \le \frac{15}{16},\] where $[m,n]$ denotes the least common multiple of $m$ and $n$ (e.g. $[4,6] = 12$).
  4. A dog standing at the centre of a circular arena sees a rabbit at the wall. The rabbit runs round the wall and the dog pursues it along a unique path which is determined by running at the same speed and staying on the radial line joining the centre of the arena to the rabbit. Show that the dog overtakes the rabbit just as it reaches a point one-quarter of the way around the arena.
  5. A walk consists of a sequence of steps of length 1 taken in the directions north, south, east, or west. A walk is self-avoiding if it never passes through the same point twice. Let $f(n)$ be the number of $n$-step self-avoiding walks which begin at the origin. Compute $f(1)$, $f(2)$, $f(3)$, $f(4)$, and show that \[2^n < f(n) \le 4 \cdot 3^{n - 1}.\]

Canadian National Mathematical Olympiad 1978 Solutions

  1. Let $n$ be an integer. If the tens digit of $n^2$ is 7, what is the units digit of $n^2$?
  2. Find all pairs of $a$, $b$ of positive integers satisfying the equation $2a^2 = 3b^3$.
  3. Determine the largest real number $z$ such that $$\begin{align*} x + y + z = 5 \\ xy + yz + xz = 3 \end{align*}$$ and $x$, $y$ are also real.
  4. The sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are extended to meet at $E$. Let $H$ and $G$ be the midpoints of $BD$ and $AC$, respectively. Find the ratio of the area of the triangle $EHG$ to that of the quadrilateral $ABCD$.
  5. Eve and Odette play a game on a $3\times 3$ checkerboard, with black checkers and white checkers. The rules are as follows:
    i) They play alternately.
    ii) A turn consists of placing one checker on an unoccupied square of the board.
    iii) In her turn, a player may select either a white checker or a black checker and need not always use the same colour.
    iv) When the board is full, Eve obtains one point for every row, column or diagonal that has an even number of black checkers, and Odette obtains one point for very row, column or diagonal that has an odd number of black checkers.
    v) The player obtaining at least five of the eight points WINS.
    a) Is a $4-4$ tie possible? Explain.
    b) Describe a winning strategy for the girl who is first to play.
  6. Sketch the graph of $x^3 + xy + y^3 = 3$.

Canadian National Mathematical Olympiad 1977 Solutions

  1. If $f(x) = x^2 + x$, prove that the equation $4f(a) = f(b)$ has no solutions in positive integers $a$ and $b$.
  2. Let $O$ be the centre of a circle and $A$ a fixed interior point of the circle different from $O$. Determine all points $P$ on the circumference of the circle such that the angle $OPA$ is a maximum.
  3. Let $N$ is an integer whose representation in base $b$ is $777$. Find the smallest integer $b$ for which $N$ is the fourth power of an integer.
  4. Let \[p(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0\] and \[q(x) = b_m x^m + a_{m - 1} x^{m - 1} + \dots + b_1 x + b_0\] be two polynomials with integer coefficients. Suppose that all the coefficients of the product $p(x) \cdot q(x)$ are even but not all of them are divisible by 4. Show that one of $p(x)$ and $q(x)$ has all even coefficients and the other has at least one odd coefficient.
  5. A right circular cone has base radius 1 cm and slant height 3 cm is given. $P$ is a point on the circumference of the base and the shortest path from $P$ around the cone and back to $P$ is drawn (see diagram). What is the minimum distance from the vertex $V$ to this path?
import graph;

unitsize(1 cm);

draw(((1.5,0) + (0.3,0.1))--((0,4) + (0.3,0.1)),Arrows);

label("$V$", (0,4), N);
label("1 cm", (0.75,-0.5), N);
label("$P$", (-1.5,0), SW);
label("3 cm", (1.7,2));
  6. Let $0 < u < 1$ and define \[u_1 = 1 + u, \quad u_2 = \frac{1}{u_1} + u, \quad \dots, \quad u_{n + 1} = \frac{1}{u_n} + u, \quad n \ge 1.\] Show that $u_n > 1$ for all values of $n = 1$, 2, 3, $\dots$.
  7. A rectangular city is exactly $m$ blocks long and $n$ blocks wide (see diagram). A woman lives in the southwest corner of the city and works in the northeast corner. She walks to work each day but, on any given trip, she makes sure that her path does not include any intersection twice. Show that the number $f(m,n)$ of different paths she can take to work satisfies $f(m,n) \le 2^{mn}$.

Canadian National Mathematical Olympiad 1976 Solutions

  1. Given four weights in geometric progression and an equal arm balance, show how to find the heaviest weight using the balance only twice.
  2. Suppose \[ n(n+1)a_{n+1}=n(n-1)a_n-(n-2)a_{n-1} \] for every positive integer $ n\ge1$. Given that $ a_0=1,a_1=2$, find \[ \frac{a_0}{a_1}+\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots+\frac{a_{50}}{a_{51}}. \]
  3. Two grade seven students were allowed to enter a chess tournament otherwise composed of grade eight students. Each contestant played once with each other contestant and received one point for a win, one half point for a tie and zero for a loss. The two grade seven students together gained a total of eight points and each grade eight student scored the same number of points as his classmates. How many students for grade eight participated in the chess tournament? Is the solution unique?
  4. Let $ AB$ be a diameter of a circle, $ C$ be any fixed point between $ A$ and $ B$ on this diameter, and $ Q$ be a variable point on the circumference of the circle. Let $ P$ be the point on the line determined by $ Q$ and $ C$ for which $ \frac{AC}{CB}=\frac{QC}{CP}$. Describe, with proof, the locus of the point $ P$.
  5. Prove that a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of two.
  6. If $ A,B,C,D$ are four points in space, such that \[ \angle ABC=\angle BCD=\angle CDA=\angle DAB=\pi/2, \] prove that $ A,B,C,D$ lie in a plane.
  7. Let $ P(x,y)$ be a polynomial in two variables $ x,y$ such that $ P(x,y)=P(y,x)$ for every $ x,y$ (for example, the polynomial $ x^2-2xy+y^2$ satisfies this condition). Given that $ (x-y)$ is a factor of $ P(x,y)$, show that $ (x-y)^2$ is a factor of $ P(x,y)$.
  8. Each of the 36 line segments joining 9 distinct points on a circle is coloured either red or blue. Suppose that each triangle determined by 3 of the 9 points contains at least one red side. Prove that there are four points such that the 6 segments connecting them are all red.

Canadian National Mathematical Olympiad 1975 Solutions

  1. Simplify $$ \left(\frac {1 \cdot 2 \cdot 4 + 2 \cdot 4 \cdot 8 + \cdots + n \cdot 2n \cdot 4n}{1 \cdot 3 \cdot 9 + 2 \cdot 6 \cdot 18 + \cdots + n \cdot 3n \cdot 9n}\right)^{\frac {1}{3}}$$
  2. A sequence of numbers $ a_1, a_2, a_3, ...$ satisfies
    (i) $ a_1 = \frac{1}{2}$
    (ii) $ a_1+a_2 + \cdots + a_n = n^2 a_n \ (n \geq 1)$
    Determine the value of $ a_n \ (n \geq 1)$.
  3. For each real number $ r$, $ [r]$ denotes the largest integer less than or equal to $ r$, e.g. $ [6] = 6, [\pi] = 3, [-1.5] = -2$. Indicate on the $ (x,y)$-plane the set of all points $ (x,y)$ for which $ [x]^2 + [y]^2 = 4$.
  4. For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.
  5. $ A,B,C,D$ are four "consecutive" points on the circumference of a circle and $ P, Q, R, S$ are points on the circumference which are respectively the midpoints of the arcs $ AB,BC,CD,DA$. Prove that $ PR$ is perpendicular to $ QS$.
  6. (i) 15 chairs are equally placed around a circular table on which are name cards for 15 guests. The guests fail to notice these cards until after they have sat down, and it turns out that no one is sitting in the correct seat. Prove that the table can be rotated so that at least two of the guests are simultaneously correctly seated.
    (ii) Give an example of an arrangement in which just one of the 15 guests is correctly seated and for which no rotation correctly places more than one person.
  7. A function $ f(x)$ is periodic if there is a positive number $ p$ such that $ f(x+p) = f(x)$ for all $ x$. For example, $ \sin x$ is periodic with period $ 2 \pi$. Is the function $ \sin(x^2)$ periodic? Prove your assertion.
  8. Let $ k$ be a positive integer. Find all polynomials \[ P(x) = a_0 + a_1 x + \cdots + a_n x^n,\] where the $ a_i$ are real, which satisfy the equation \[ P(P(x)) = \{ P(x) \}^k\]

Canadian National Mathematical Olympiad 1974 Solutions

  1. i) If $x = \left(1+\frac{1}{n}\right)^{n}$ and $y=\left(1+\frac{1}{n}\right)^{n+1}$, show that $y^{x}= x^{y}$.
    ii) Show that, for all positive integers $n$, \[1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).\]
  2. Let $ABCD$ be a rectangle with $BC=3AB$. Show that if $P,Q$ are the points on side $BC$ with $BP = PQ = QC$, then \[\angle DBC+\angle DPC = \angle DQC.\]
  3. Let \[f(x) = a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}\] be a polynomial with coefficients satisfying the conditions: \[0\le a_{i}\le a_{0},\quad i=1,2,\ldots,n.\] Let $b_{0},b_{1},\ldots,b_{2n}$ be the coefficients of the polynomial $$\begin{align*}\left(f(x)\right)^{2}&= \left(a_{0}+a_{1}x+a_{2}x^{2}+\cdots a_{n}x^{n}\right)\\ &= b_{0}+b_{1}x+b_{2}x^{2}+\cdots+b_{2n}x^{2n}. \end{align*}$$ Prove that $b_{n+1}\le \frac{1}{2}\left(f(1)\right)^{2}$.
  4. Let $n$ be a fixed positive integer. To any choice of real numbers satisfying \[0\le x_{i}\le 1,\quad i=1,2,\ldots, n,\] there corresponds the sum \[\sum_{1\le i<j\le n}|x_{i}-x_{j}|.\] Let $S(n)$ denote the largest possible value of this sum. Find $S(n)$.
  5. Given a circle with diameter $AB$ and a point $X$ on the circle different from $A$ and $B$, let $t_{a}$, $t_{b}$ and $t_{x}$ be the tangents to the circle at $A$, $B$ and $X$ respectively. Let $Z$ be the point where line $AX$ meets $t_{b}$ and $Y$ the point where line $BX$ meets $t_{a}$. Show that the three lines $YZ$, $t_{x}$ and $AB$ are either concurrent (i.e., all pass through the same point) or parallel.
  6. An unlimited supply of 8-cent and 15-cent stamps is available. Some amounts of postage cannot be made up exactly, e.g., 7 cents, 29 cents. What is the largest unattainable amount, i.e., the amount, say $n$, of postage which is unattainable while all amounts larger than $n$ are attainable? (Justify your answer.)
  7. A bus route consists of a circular road of circumference 10 miles and a straight road of length 1 mile which runs from a terminus to the point $Q$ on the circular road. It is served by two buses, each of which requires 20 minutes for the round trip. Bus No. 1, upon leaving the terminus, travels along the straight road, once around the circle clockwise and returns along the straight road to the terminus. Bus No. 2, reaching the terminus 10 minutes after Bus No. 1, has a similar route except that it proceeds counterclockwise around the circle. Both buses run continuously and do not wait at any point on the route except for a negligible amount of time to pick up and discharge passengers. A man plans to wait at a point $P$ which is $x$ miles ($0\le x < 12$) from the terminus along the route of Bus No. 1 and travel to the terminus on one of the buses. Assuming that he chooses to board that bus which will bring him to his destination at the earliest moment, there is a maximum time $w(x)$ that his journey (waiting plus travel time) could take. Find $w(2)$; find $w(4)$. For what value of $x$ will the time $w(x)$ be the longest?

Canadian National Mathematical Olympiad 1973 Solutions

  1. (i) Solve the simultaneous inequalities, $x<\frac{1}{4x}$ and $x 16$.
    (iii) Give a rational number between $11/24$ and $6/13$.
    (iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10.
    (v) Without the use of logarithm tables evaluate \[\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.\]
  2. Find all real numbers that satisfy the equation $|x+3|-|x-1|=x+1$. (Note: $|a| = a$ if $a\ge 0$; $|a|=-a$ if $a>0$.)
  3. Prove that if $p$ and $p+2$ are prime integers greater than 3, then 6 is a factor of $p+1$.
  4. The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: $P_{0}P_{1}P_{3}$, $P_{0}P_{3}P_{6}$, $P_{0}P_{6}P_{7}$, $P_{0}P_{7}P_{8}$, $P_{1}P_{2}P_{3}$, $P_{3}P_{4}P_{6}$, $P_{4}P_{5}P_{6}$. In how many ways can these triangles be labeled with the names $\triangle_{1}$, $\triangle_{2}$, $\triangle_{3}$, $\triangle_{4}$, $\triangle_{5}$, $\triangle_{6}$, $\triangle_{7}$ so that $P_{i}$ is a vertex of triangle $\triangle_{i}$ for $i = 1, 2, 3, 4, 5, 6, 7$? Justify your answer.
  5. For every positive integer $n$, let \[h(n) = 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}.\] For example, $h(1) = 1$, $h(2) = 1+\frac{1}{2}$, $h(3) = 1+\frac{1}{2}+\frac{1}{3}$. Prove that for $n=2,3,4,\ldots$ \[n+h(1)+h(2)+h(3)+\cdots+h(n-1) = nh(n).\]
  6. If $A$ and $B$ are fixed points on a given circle not collinear with centre $O$ of the circle, and if $XY$ is a variable diameter, find the locus of $P$ (the intersection of the line through $A$ and $X$ and the line through $B$ and $Y$).
  7. Observe that \[\frac{1}{1}= \frac{1}{2}+\frac{1}{2};\quad \frac{1}{2}=\frac{1}{3}+\frac{1}{6};\quad \frac{1}{3}=\frac{1}{4}+\frac{1}{12};\quad \frac{1}{4}= \frac{1}{5}+\frac{1}{20}. \] State a general law suggested by these examples, and prove it. Prove that for any integer $n$ greater than 1 there exist positive integers $i$ and $j$ such that \[\frac{1}{n}= \frac{1}{i(i+1)}+\frac{1}{(i+1)(i+2)}+\frac{1}{(i+2)(i+3)}+\cdots+\frac{1}{j(j+1)}. \]

Canadian National Mathematical Olympiad 1972 Solutions

  1. Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.
  2. Let $a_1,a_2,\ldots,a_n$ be non-negative real numbers. Define $M$ to be the sum of all products of pairs $a_ia_j$ $(i<j)$, i.e \[ M = a_1(a_2+a_3+\cdots+a_n)+a_2(a_3+a_4+\cdots+a_n)+\cdots+a_{n-1}a_n. \] Prove that the square of at least one of the numbers $a_1,a_2,\ldots,a_n$ does not exceed $2M/n(n-1)$.
  3. a) Prove that $10201$ is composite in all bases greater than 2.
    b) Prove that $10101$ is composite in all bases.
  4. Describe a construction of quadrilateral $ABCD$ given:
    (i) the lengths of all four sides;
    (ii) that $AB$ and $CD$ are parallel;
    (iii) that $BC$ and $DA$ do not intersect.
  5. Prove that the equation $x^3+11^3=y^3$ has no solution in positive integers $x$ and $y$.
  6. Let $a$ and $b$ be distinct real numbers. Prove that there exist integers $m$ and $n$ such that $am+bn<0$, $bm+an>0$.
  7. a) Prove that the values of $x$ for which $x=(x^2+1)/198$ lie between $1/198$ and $197.99494949\cdots$.
    b) Use the result of problem a) to prove that $\sqrt{2}<1.41\overline{421356}$.
    c) Is it true that $\sqrt{2}<1.41421356$?
  8. During a certain election campaign, $p$ different kinds of promises are made by the different political parties ($p>0$). While several political parties may make the same promise, any two parties have at least one promise in common; no two parties have exactly the same set of promises. Prove that there are no more than $2^{p-1}$ parties.
  9. Four distinct lines $L_1,L_2,L_3,L_4$ are given in the plane: $L_1$ and $L_2$ are respectively parallel to $L_3$ and $L_4$. Find the locus of a point moving so that the sum of its perpendicular distances from the four lines is constant.
  10. What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?

Canadian National Mathematical Olympiad 1971 Solutions

  1. $DEB$ is a chord of a circle such that $DE=3$ and $EB=5$. Let $O$ be the centre of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$. (See diagram). Given $EC=1$, find the radius of the circle.

pair O = (0,0), B = dir(110), D = dir(30), E = 0.4 * B + 0.6 * D, C = intersectionpoint(O--2*E, unitcircle);



label("$B$", B, B);
label("$C$", C, C);
label("$D$", D, D);
label("$E$", E, dir(280));
label("$O$", O, dir(270));
  2. Let $x$ and $y$ be positive real numbers such that $x+y=1$. Show that \[ \left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)\ge 9. \]
  3. $ABCD$ is a quadrilateral with $AD=BC$. If $\angle ADC$ is greater than $\angle BCD$, prove that $AC>BD$.
  4. Determine all real numbers $a$ such that the two polynomials $x^2+ax+1$ and $x^2+x+a$ have at least one root in common.
  5. Let \[ p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0, \] where the coefficients $a_i$ are integers. If $p(0)$ and $p(1)$ are both odd, show that $p(x)$ has no integral roots.
  6. Show that, for all integers $n$, $n^2+2n+12$ is not a multiple of 121.
  7. Let $n$ be a five digit number (whose first digit is non-zero) and let $m$ be the four digit number formed from $n$ by removing its middle digit. Determine all $n$ such that $n/m$ is an integer.
  8. A regular pentagon is inscribed in a circle of radius $r$. $P$ is any point inside the pentagon. Perpendiculars are dropped from $P$ to the sides, or the sides produced, of the pentagon.
    a) Prove that the sum of the lengths of these perpendiculars is constant.
    b) Express this constant in terms of the radius $r$.
  9. Two flag poles of height $h$ and $k$ are situated $2a$ units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.
  10. Suppose that $n$ people each know exactly one piece of information, and all $n$ pieces are different. Every time person $A$ phones person $B$, $A$ tells $B$ everything that $A$ knows, while $B$ tells $A$ nothing. What is the minimum number of phone calls between pairs of people needed for everyone to know everything? Prove your answer is a minimum.

Canadian National Mathematical Olympiad 1969 Solutions

  1. If $a_1/b_1=a_2/b_2=a_3/b_3$ and $p_1,p_2,p_3$ are not all zero, show that for all $n\in\mathbb{N}$, \[ \left(\frac{a_1}{b_1}\right)^n = \frac{p_1a_1^n+p_2a_2^n+p_3a_3^n}{p_1b_1^n+p_2b_2^n+p_3b_3^n}. \]
  2. Determine which of the two numbers $\sqrt{c+1}-\sqrt{c}$, $\sqrt{c}-\sqrt{c-1}$ is greater for any $c\ge 1$.
  3. Let $c$ be the length of the hypotenuse of a right angle triangle whose two other sides have lengths $a$ and $b$. Prove that $a+b\le c\sqrt{2}$. When does the equality hold?
  4. Let $ABC$ be an equilateral triangle, and $P$ be an arbitrary point within the triangle. Perpendiculars $PD,PE,PF$ are drawn to the three sides of the triangle. Show that, no matter where $P$ is chosen, \[ \frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}. \]
  5. Let $ABC$ be a triangle with sides of length $a$, $b$ and $c$. Let the bisector of the angle $C$ cut $AB$ in $D$. Prove that the length of $CD$ is \[ \frac{2ab\cos \frac{C}{2}}{a+b}. \]
  6. Find the sum of $1\cdot 1!+2\cdot 2!+3\cdot 3!+\cdots+(n-1)(n-1)!+n\cdot n!$, where $n!=n(n-1)(n-2)\cdots2\cdot1$.
  7. Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.
  8. Let $f$ be a function with the following properties:
    1) $f(n)$ is defined for every positive integer $n$;
    2) $f(n)$ is an integer;
    3) $f(2)=2$;
    4) $f(mn)=f(m)f(n)$ for all $m$ and $n$;
    5) $f(m)>f(n)$ whenever $m>n$.
    Prove that $f(n)=n$.
  9. Show that for any quadrilateral inscribed in a circle of radius 1, the length of the shortest side is less than or equal to $\sqrt{2}$.
  10. Let $ABC$ be the right-angled isosceles triangle whose equal sides have length 1. $P$ is a point on the hypotenuse, and the feet of the perpendiculars from $P$ to the other sides are $Q$ and $R$. Consider the areas of the triangles $APQ$ and $PBR$, and the area of the rectangle $QCRP$. Prove that regardless of how $P$ is chosen, the largest of these three areas is at least $2/9$.

Canadian National Mathematical Olympiad 1970 Solutions

  1. Find all number triples $(x,y,z)$ such that when any of these numbers is added to the product of the other two, the result is 2.
  2. Given a triangle $ABC$ with angle $A$ obtuse and with altitudes of length $h$ and $k$ as shown in the diagram, prove that $a+h\ge b+k$. Find under what conditions $a+h=b+k$.

pair A = dir(105), C = dir(170), B = dir(10), D = foot(B, A, C), E = foot(A, B, C);


dot(A); dot(B); dot(C); dot(D); dot(E);

label("$A$", A, dir(110));
label("$B$", B, B);
label("$C$", C, C);
label("$D$", D, D);
label("$E$", E, dir(45));

label("$h$", A--E, dir(0));
label("$k$", B--D, dir(45));
  3. A set of balls is given. Each ball is coloured red or blue, and there is at least one of each colour. Each ball weighs either 1 pound or 2 pounds, and there is at least one of each weight. Prove that there are two balls having different weights and different colours.
  4. a) Find all positive integers with initial digit 6 such that the integer formed by deleting 6 is $1/25$ of the original integer.
    b) Show that there is no integer such that the deletion of the first digit produces a result that is $1/35$ of the original integer.
  5. A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths $a$, $b$, $c$ and $d$ of the sides of the quadrilateral satisfy the inequalities \[ 2\le a^2+b^2+c^2+d^2\le 4. \]
  6. Given three non-collinear points $A,B,C$, construct a circle with centre $C$ such that the tangents from $A$ and $B$ are parallel.
  7. Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.
  8. Consider all line segments of length 4 with one end-point on the line $y=x$ and the other end-point on the line $y=2x$. Find the equation of the locus of the midpoints of these line segments.
  9. Let $f(n)$ be the sum of the first $n$ terms of the sequence \[ 0, 1,1, 2,2, 3,3, 4,4, 5,5, 6,6, \ldots\, . \]
    a) Give a formula for $f(n)$.
    b) Prove that $f(s+t)-f(s-t)=st$ where $s$ and $t$ are positive integers and $s>t$.
  10. Given the polynomial \[ f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n \] with integer coefficients $a_1,a_2,\ldots,a_n$, and given also that there exist four distinct integers $a$, $b$, $c$ and $d$ such that \[ f(a)=f(b)=f(c)=f(d)=5, \] show that there is no integer $k$ such that $f(k)=8$.

Nguyễn Hữu Điển - Tuyển tập Đề thi và Lời giải Olympic Toán các nước qua các năm (5 tập)

Dưới đây là tuyển tập Đề thi và Lời giải Olympic Toán các nước qua nhiều năm. Tuyển tập này bao gồm 5 tập.
Để thử gói lệnh lamdethi.sty tôi biên soạn một số đề toán thi Olympic, mà các học trò của tôi đã làm bài tập khi học tập $\LaTeX$. Để phụ vụ các bạn ham học toán tôi thu thập và gom lại thành các sách điện tử, các bạn có thể tham khảo. Mỗi tập tôi sẽ gom khoảng 50 bài với lời giải. Tập này có sự đóng góp của Bùi Thế Anh, Vũ Thị Hồng Hạnh, Cao Thị Mai Len, Tạ Xuân Hòa, Nguyễn Thị Loan, Nguyễn Thị Quý Sửu, Nguyễn Thị Định, Nguyễn ngọc Long.

Rất nhiều bài toán dịch không được chuẩn, nhiều điểm không hoàn toàn chính xác vậy mong bạn đọc tự ngẫm nghĩ và tìm hiểu lấy. Nhưng đây là nguồn tài liệu tiếng Việt về chủ đề này, tôi đã có xem qua và người dịch là chuyên về ngành Toán phổ thông.

Nguyễn Hữu Điển

Tạp chí Nhóm Toán số 3 năm 2015

Đề ra kỳ này

Phạm Anh Vũ-"Sự tương đương trong chứng minh bất đẳng thức."

Nguyễn Minh Tiến-"Hình chữ nhật và các dạng toán cơ bản."

Hướng đến kỳ thi THPT 2015-2016 - Đề số 3 - Hướng dẫn giải.

Nguyễn Thành Hiển-"100 tính chất hình phẳng trong Oxy."

Tạp chí Nhóm Toán số 2 năm 2015

Đề ra kỳ này

Hướng dẫn giải bài kỳ trước

Bài viết các chuyên đề luyện thi

1. Lê Đức Bin-"Lượng liên hợp trong giải hệ phương trình."
2. Nguyễn Thành Hiển-"Tổng hợp Oxy trong các đề thi thử THPT 2015."
3. Đỗ Viết Lân-"Phép thế lượng giác trong chứng minh bất đẳng thức."

Hướng đến kỳ thi THPT 2015-2016

– Đề số 2

Đấu Trường

1. Lời giải bài thách đấu 03.
2. Trần Quốc Việt-Bài thách đấu số 04
3. Ngô Minh Ngọc Bảo-Bài thách đấu số 05

Tự học : "Phương trình - Bất phương trình"

Tạp chí Nhóm Toán số 1 năm 2015

Đề ra kỳ này

Bài viết các chuyên đề luyện thi

1. Nguyễn Thành Hiển-"Hệ trục tọa độ và chứng minh các yếu tố hình phẳng."
2. Nguyễn Văn Phú - THPT Mỹ Đức A, Hà Nội-"Một vài mẹo nhỏ trong phương pháp liên hợp giải hệ phương trình."
3. Công Dân Lương Thiện-"Sáng tạo từ một bất đẳng thức cơ bản."
4. Nguyễn Thành Hiển-"Phương trình - Bất phương trình trong các đề thi thử 2015 (Phần 1)."
5. Ẩn Danh-"Dự đoán tính chất hình học trong bài toán Oxy."
6. Ẩn Danh-"Hệ phương trình giải bằng phương pháp đánh giá."
7. Ngô Đình Tuấn-"Thứ tự biến để giải bài toán cực trị."

Diễn đàn dạy và học toán

1. Trần Minh Quang-"Tính đơn điệu của hàm số và một số sai lầm khi giải toán."
2. Đỗ Viết Lân-"Phép thế Ravi và bất đẳng thức Padoa."
3. Nguyễn Minh Tuấn-"Sử dụng công thức lượng giác để xây dựng một sột phương trình lượng giác dạng tích."
4. Trương Công Việt-"Ứng dụng tính đơn điệu của hàm số trong giải phương trình."
5. Nguyễn Thành Hiển-"Ứng dụng dãy tỷ số trong sáng tạo và giải hệ phương trình (Phần 1)."

Hướng đến kỳ thi THPT 2015-2016

– Đề số 1

Đấu Trường

1. Ngô Minh Ngọc Bảo-Bài thách đấu số 1
2. Trần Hưng-Bài thách đấu số 2
3. Trần Quốc Việt-Bài thách đấu số 3

Các chuyên đề chọn lọc của k2pi lần 1

Nhân ngày hạnh phúc của thế giới, BBT của k2pi gởi đến các bạn tập tài liệu chọn lọc lần 1, sau một thời gian khá dài biên soạn. Tài liệu bao gồm:


  • THỂ TÍCH VÀ KHOẢNG CÁCH (Nguyễn Trung Kiên)


  • BẤT PHƯƠNG TRÌNH (hthtb22)
  • HỆ PHƯƠNG TRÌNH (Lê Nhất Duy)
  • MINH BẤT ĐẲNG THỨC (Hoàng Trung Hiếu)


William Lowell Putnam Mathematical Competition 2010 Solutions

  1. Given a positive integer $n,$ what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8,$ the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is at least 3.]
  2. Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f'(x)=\frac{f(x+n)-f(x)}n\] for all real numbers $x$ and all positive integers $n.$
  3. Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.
  4. Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.
  5. Let $G$ be a group, with operation $*$. Suppose that
    (i) $G$ is a subset of $\mathbb{R}^3$ (but $*$ need not be related to addition of vectors);
    (ii) For each $\mathbf{a},\mathbf{b}\in G,$ either $\mathbf{a}\times\mathbf{b}=\mathbf{a}*\mathbf{b}$ or $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ (or both), where $\times$ is the usual cross product in $\mathbb{R}^3.$
    Prove that $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ for all $\mathbf{a},\mathbf{b}\in G.$
  6. Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $$\lim_{x\to\infty}f(x)=0.$$ Prove that $$\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$$ diverges.
  7. Is there an infinite sequence of real numbers $a_1,a_2,a_3,\dots$ such that \[a_1^m+a_2^m+a_3^m+\cdots=m\] for every positive integer $m?$
  8. Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$
  9. There are 2010 boxes labeled $B_1,B_2,\dots,B_{2010},$ and $2010n$ balls have been distributed among them, for some positive integer $n.$ You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving exactly $i$ balls from box $B_i$ into any one other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls?
  10. Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which \[p(x)q(x+1)-p(x+1)q(x)=1.\]
  11. Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$
  12. Let $A$ be an $n\times n$ matrix of real numbers for some $n\ge 1.$ For each positive integer $k,$ let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\cdots,n+1,$ then $A^k=A^{[k]}$ for all $k\ge 1.$

Phạm Quốc Sang - Một số kỹ thuật AM-GM

Bài viết của bạn Phạm Quốc Sang - Học sinh K9 chuyên Nguyễn Bỉnh Khiêm Quảng Nam. Tài liệu bao gồm các mục chính

I. Kỹ thuật chọn điểm rơi:

Đây là kỹ thuật rất quan trọng trong quá trình sử dụng bất đẳng thức Cauchy để giải toán. Dựa vào việc xác định điều kiện xảy ra đẳng thức, ta sẽ thêm bớt hay tách biểu thức cần chứng minh thành các nhóm hay biểu thức phù hợp.

II. Kỹ thuật đồng bậc hóa:

Đồng bậc là kỹ thuật giải toán dựa vào việc tạo ra các biểu thức đồng bậc nhau ở hai vế củ bất đẳng thức cần chứng minh. Ở chương này, ta sẽ tập trung đi vào giải các bài toán gồm ba biến số.

III. Kỹ thuật Cauchy ngược dấu: ( Kỹ thuật đánh giá phủ định của phủ định)

Trong quá trình chứng minh bất đẳng thức, ta thường sử dụng các bất đẳng thức phụ để đánh giá từng vế. Tuy nhiên trong một số trường hợp, việc sử dụng quá mạnh tay bất đẳng thức phụ sẽ dẫn đến sự đổi chiều của của bất đẳng thức ban đầu.

IV. Kỹ thuật cân bằng hệ số:

Ở các kỹ thuật trước, ta thường giải các bài toán bất đẳng thức dựa vào điều kiện xảy ra đẳng thức. Tuy nhiên, các bài tập được đưa vào đều là các bất đẳng thức đối xứng nên việc tìm điều kiện xảy ra đẳng thức khá đơn giản. Trong quá trình giải toán, ta thường xuyên gặp các bất đẳng thức không đối xứng. Do đó, việc dựa vào trực quan để tìm ra điều kiện xảy ra đẳng thức là điều không thể. Vì thế, kỹ thuật cân bằng hệ số được đưa vào nhằm khắc phục các hạn chế trong quá trình giải các bất đẳng thức không đối xứng. Với kỹ thuật này, ta sẽ đưa vào bài toán các tham số giả định để sử dụng bất đẳng thức Cauchy. Khi đó, điều kiện xảy ra đẳng thức cũng chính là điều kiện để tìm các tham số.

IMO 2015 Shortlist (bản tiếng việt)

C1. Ở Lineland có $n \geq 1$ thị trấn, được sắp xếp dọc một con đường từ trái sang phải. Mỗi thị trấn có một xe ủi trái (đặt bên trái của thị trấn và hướng sang trái) và một xe ủi phải (đặt bên phải của thị trấn và hướng sang phải). Kích thước của $2n$ xe ủi là đôi một khác nhau. Tại mỗi thời điểm khi một xe ủi trái đối diện một xe ủi phải, xe lớn hơn sẽ đẩy xe nhỏ hơn ra khỏi đường. Mặt khác, các xe ủi sẽ không được bảo vệ đằng sau; vì vậy, nếu một xe ủi húc vào đuôi của xe khác thì nó sẽ đẩy xe bị húc ra khỏi đường. Cho $A$ và $B$ là hai thị trấn, với $B$ nằm bên phải $A$. Ta nói $A$ có thể quét $B$ biến mất nếu xe ủi phải của $A$ có thể di chuyển đến $B$ và đẩy tất cả xe ủi mà nó gặp ra khỏi đường. Tương tự, $B$ có thể quét $A$ biến mất nếu xe ủi trái của $B$ có thể di chuyển tới $A$ và đẩy tất cả xe ủi mà nó gặp ra khỏi đường. Chứng minh rằng có đúng một thị trấn không bị quét biến mất bởi mỗi thì trấn còn lại.

C2. Ta nói tập hữu hạn $S$ các điểm trong mặt phẳng là cân bằng nếu với mỗi hai điểm khác nhau $A$ và $B$ trong $S$, tồn tại $C$ trong $S$ sao cho $AC = BC$. Ta nói $S$ là không tâm nếu với mỗi ba điểm phân biệt $A, B$ và $C$ của $S$, không tồn tại $P$ trong $S$ sao cho $P A = P B = P C$.
(a) Chứng minh rằng với mỗi $n \geq 3$, tồn tại tập cân bằng chứa $n$ điểm.
(b) Xác định tất cả $n \geq 3$ sao cho tồn tại tập cân bằng và không tâm chứa $n$ điểm.

C3. Với tập hữu hạn các số nguyên dương $A$, một phân hoạch của $A$ thành hai tập con khác rỗng $A_1 , A_2$ được gọi là tốt nếu bội chung nhỏ nhất của các phần tử trong $A_1$ bằng ước chung lớn nhất của các phần tử trong $A_2$. Tìm số nguyên dương $n$ nhỏ nhất sao cho tồn tại tập gồm $n$ số nguyên dương với đúng 2015 phân hoạch tốt.

C4. Cho số nguyên dương $n$. Hai người chơi $A$ và $B$ chơi một trò chơi chọn các số nguyên dương $k \leq n$. Luật chơi là: 
(i) Người chơi không được chọn số đã được chọn ở các bước trước.
(ii) Người chơi không được chọn số liên tiếp với các số đã được người đó chọn ở các bước trước.
(iii) Trò chơi sẽ kết thúc với kết quả hòa nếu không còn số nào để chọn; trong trường hợp còn lại, ai không chọn được sẽ thua.
$A$ đi trước. Xác định kết quả của trò chơi, giả sử rằng cả hai cùng chơi giỏi.

C5. Cho dãy các số nguyên $a_1 , a_2 , . . .$ thỏa mãn đồng thời hai điều kiện:
(i) $1 \leq a_j \leq 2015$ với mỗi $j \geq 1$,
(ii) $k + a_k \ne l +a_l$ với mỗi $1 \leq k < l$. 
Chứng minh rằng tồn tại hai số nguyên dương $b$ và $N$ sao cho $$\left|\sum_{j=m+1}^{n}(a_j - b)\right| \leq 1007^2$$ với mỗi hai số m và n thỏa mãn $n > m \geq N$.

C6. Cho $S$ là tập khác rỗng các số nguyên dương. Một số nguyên dương được gọi là dọn dẹp nếu nó có thể biểu diễn duy nhất thành tổng của một số lẻ các phần tử khác nhau của $S$. Chứng minh rằng tồn tại vô hạn số nguyên dương không dọn dẹp.

C7. Trong một công ty có một số cặp là kẻ thù của nhau. Một nhóm người được gọi là không ưa giao tiếp nếu số thành viên trong nhóm là số lẻ lớn hơn 1, và có thể sắp xếp tất cả các thành viên của nhóm xung quanh một bàn tròn sao cho mỗi cặp ngồi cạnh nhau đều là kẻ thù của nhau. Biết có nhiều nhất 2015 nhóm không ưa giao tiếp, chứng minh rằng có thể chia công ty thành 11 phần sao cho trong mỗi phần không có hai người nào là kẻ thù của nhau.

William Lowell Putnam Mathematical Competition 2009 Solutions

  1. Let $ f$ be a real-valued function on the plane such that for every square $ ABCD$ in the plane, $$ f(A)+f(B)+f(C)+f(D)=0.$$ Does it follow that $ f(P)=0$ for all points $ P$ in the plane.
  2. Functions $ f,g,h$ are differentiable on some open interval around $ 0$ and satisfy the equations and initial conditions $$\begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\ g'&=fg^2h+\frac4{fh},\ g(0)=1,\\ h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*}$$ Find an explicit formula for $ f(x),$ valid in some open interval around $ 0.$
  3. Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $$ d_3 = \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\ \cos4 & \cos5 & \cos 6 \\ \cos7 & \cos8 & \cos 9\end{vmatrix}.$$ The argument of $ \cos$ is always in radians, not degrees). Evaluate $ \lim_{n\to\infty}d_n.$
  4. Let $ S$ be a set of rational numbers such that (a) $ 0\in S;$ (b) If $ x\in S$ then $ x+1\in S$ and $ x-1\in S;$ and (c) If $ x\in S$ and $ x\notin\{0,1\},$ then $ \frac{1}{x(x-1)}\in S.$ Must $ S$ contain all rational numbers?
  5. Is there a finite abelian group $ G$ such that the product of the orders of all its elements is $ 2^{2009}?$
  6. Let $ f: [0,1]^2\to\mathbb{R}$ be a continuous function on the closed unit square such that $ \frac{\partial f}{\partial x}$ and $ \frac{\partial f}{\partial y}$ exist and are continuous on the interior of $ (0,1)^2.$ Let $$ a=\int_0^1f(0,y)\,dy,\quad b=\int_0^1f(1,y)\,dy$$ $$c=\int_0^1f(x,0)\,dx,\quad d=\int_0^1f(x,1)\,dx.$$ Prove or disprove: There must be a point $ (x_0,y_0)$ in $ (0,1)^2$ such that $$ \frac{\partial f}{\partial x}(x_0,y_0)=b-a$$ and $$ \frac{\partial f}{\partial y}(x_0,y_0)=d-c.$$
  7. Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $ \frac{10}9=\frac{2!\cdot 5!}{3!\cdot 3!\cdot 3!}.$
  8. A game involves jumping to the right on the real number line. If $ a$ and $ b$ are real numbers and $ b>a,$ the cost of jumping from $ a$ to $ b$ is $ b^3-ab^2.$ For what real numbers $ c$ can one travel from $ 0$ to $ 1$ in a finite number of jumps with total cost exactly $ c?$
  9. Call a subset $ S$ of $ \{1,2,\dots,n\}$ mediocre if it has the following property: Whenever $ a$ and $ b$ are elements of $ S$ whose average is an integer, that average is also an element of $ S.$ Let $ A(n)$ be the number of mediocre subsets of $ \{1,2,\dots,n\}.$ [For instance, every subset of $ \{1,2,3\}$ except $ \{1,3\}$ is mediocre, so $ A(3)=7.$] Find all positive integers $ n$ such that $$A(n+2)-2A(n+1)+A(n)=1.$$
  10. Say that a polynomial with real coefficients in two variable, $ x,y,$ is balanced if the average value of the polynomial on each circle centered at the origin is $ 0.$ The balanced polynomials of degree at most $ 2009$ form a vector space $ V$ over $ \mathbb{R}.$ Find the dimension of $ V.$
  11. Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that \[ f'(x)=\frac{x^2-\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2+1\right)}\quad\text{for all }x>1.\] Prove that $ \displaystyle\lim_{x\to\infty}f(x)=\infty.$
  12. Prove that for every positive integer $ n,$ there is a sequence of integers $ a_0,a_1,\dots,a_{2009}$ with $ a_0=0$ and $ a_{2009}=n$ such that each term after $ a_0$ is either an earlier term plus $ 2^k$ for some nonnnegative integer $ k,$ or of the form $ b\mod{c}$ for some earlier positive terms $ b$ and $ c.$ [Here $ b\mod{c}$ denotes the remainder when $ b$ is divided by $ c,$ so $ 0\le(b\mod{c})<c.$]

William Lowell Putnam Mathematical Competition 2008 Solutions

  1. Let $ f: \mathbb{R}^2\to\mathbb{R}$ be a function such that $$ f(x,y)+f(y,z)+f(z,x)=0$$ for real numbers $ x,y,$ and $ z.$ Prove that there exists a function $ g: \mathbb{R}\to\mathbb{R}$ such that $$ f(x,y)=g(x)-g(y)$$ for all real numbers $ x$ and $ y.$
  2. Alan and Barbara play a game in which they take turns filling entries of an initially empty $ 2008\times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?
  3. Start with a finite sequence $ a_1,a_2,\dots,a_n$ of positive integers. If possible, choose two indices $ j < k$ such that $ a_j$ does not divide $ a_k$ and replace $ a_j$ and $ a_k$ by $ \gcd(a_j,a_k)$ and $ \text{lcm}\,(a_j,a_k),$ respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: $ \gcd$ means greatest common divisor and lcm means least common multiple.)
  4. Define $ f: \mathbb{R}\to\mathbb{R}$ by \[ f(x)=\begin{cases}x&\text{if }x\le e\\ xf(\ln x)&\text{if }x>e\end{cases}\] Does $ \displaystyle\sum_{n=1}^{\infty}\frac1{f(n)}$ converge?
  5. Let $ n\ge 3$ be an integer. Let $ f(x)$ and $ g(x)$ be polynomials with real coefficients such that the points $ (f(1),g(1)),(f(2),g(2)),\dots,(f(n),g(n))$ in $ \mathbb{R}^2$ are the vertices of a regular $ n$-gon in counterclockwise order. Prove that at least one of $ f(x)$ and $ g(x)$ has degree greater than or equal to $ n-1.$
  6. Prove that there exists a constant $ c>0$ such that in every nontrivial finite group $ G$ there exists a sequence of length at most $ c\ln |G|$ with the property that each element of $ G$ equals the product of some subsequence. (The elements of $ G$ in the sequence are not required to be distinct. A subsequence of a sequence is obtained by selecting some of the terms, not necessarily consecutive, without reordering them; for example, $ 4,4,2$ is a subesequence of $ 2,4,6,4,2,$ but $ 2,2,4$ is not.)
  7. What is the maximum number of rational points that can lie on a circle in $ \mathbb{R}^2$ whose center is not a rational point? (A rational point is a point both of whose coordinates are rational numbers.)
  8. Let $ F_0=\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n+1}(x)=\int_0^xF_n(t)\,dt.$ Evaluate $$\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$$
  9. What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?
  10. Let $ p$ be a prime number. Let $ h(x)$ be a polynomial with integer coefficients such that $ h(0),h(1),\dots, h(p^2-1)$ are distinct modulo $ p^2.$ Show that $ h(0),h(1),\dots, h(p^3-1)$ are distinct modulo $ p^3.$
  11. Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q=a/b$ for some integer $ a$ with $ \gcd(a,b)=1.$) (Note: $ \gcd$ means greatest common divisor.)
  12. Let $ n$ and $ k$ be positive integers. Say that a permutation $ \sigma$ of $ \{1,2,\dots n\}$ is $ k$-limited if $ |\sigma(i)-i|\le k$ for all $ i.$ Prove that the number of $ k$-limited permutations of $ \{1,2,\dots n\}$ is odd if and only if $ n\equiv 0$ or $ 1\pmod{2k+1}.$

William Lowell Putnam Mathematical Competition 2007 Solutions

  1. Find all values of $ \alpha$ for which the curves $$ y=\alpha x^2+\alpha x+\frac1{24}$$ and $$ x=\alpha y^2+\alpha y+\frac1{24}$$ are tangent to each other.
  2. Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy=1$ and both branches of the hyperbola $ xy=-1.$ (A set $ S$ in the plane is called convex if for any two points in $ S$ the line segment connecting them is contained in $ S.$)
  3. Let $ k$ be a positive integer. Suppose that the integers $ 1,2,3,\dots,3k + 1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $ 3$ ? Your answer should be in closed form, but may include factorials.
  4. A repunit is a positive integer whose digits in base $ 10$ are all ones. Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$
  5. Suppose that a finite group has exactly $ n$ elements of order $ p,$ where $ p$ is a prime. Prove that either $ n=0$ or $ p$ divides $ n+1.$
  6. A triangulation $ \mathcal{T}$ of a polygon $ P$ is a finite collection of triangles whose union is $ P,$ and such that the intersection of any two triangles is either empty, or a shared vertex, or a shared side. Moreover, each side of $ P$ is a side of exactly one triangle in $ \mathcal{T}.$ Say that $ \mathcal{T}$ is admissible if every internal vertex is shared by $ 6$ or more triangles. For example: The included figure shows a heptagon. The first and third vertices are connected by a diagonal. Every vertex except the second (the one "outside" the diagonal) is connected by an edge to a single internal vertex; that internal vertex has 6 edges connected to it. There are 7 triangles in the picture; 6 around the internal vertex and one outside the diagonal.
    Prove that there is an integer $ M_n,$ depending only on $ n,$ such that any admissible triangulation of a polygon $ P$ with $ n$ sides has at most $ M_n$ triangles.
  7. Let $ f$ be a polynomial with positive integer coefficients. Prove that if $ n$ is a positive integer, then $ f(n)$ divides $ f(f(n)+1)$ if and only if $ n=1.$
  8. Suppose that $ f: [0,1]\to\mathbb{R}$ has a continuous derivative and that $ \int_0^1f(x)\,dx=0.$ Prove that for every $ \alpha\in(0,1),$ \[ \left|\int_0^{\alpha}f(x)\,dx\right|\le\frac18\max_{0\le x\le 1}|f'(x)|\]
  9. Let $ x_0 = 1$ and for $ n\ge0,$ let $$ x_{n + 1} = 3x_n + \left\lfloor x_n\sqrt {5}\right\rfloor.$$ In particular, $ x_1 = 5,\ x_2 = 26,\ x_3 = 136,\ x_4 = 712.$ Find a closed-form expression for $ x_{2007}.$ ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)
  10. Let $ n$ be a positive integer. Find the number of pairs $ P,Q$ of polynomials with real coefficients such that \[ (P(X))^2+(Q(X))^2=X^{2n}+1\] and $ \text{deg}P<\text{deg}{Q}.$
  11. Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k-1}(n)$ (which may depend on $ k$) such that for any integer $ n,$ \[ \left\lfloor\frac{n}{k}\right\rfloor^k=P_0(n)+P_1(n)\left\lfloor\frac{n}{k}\right\rfloor+ \cdots+P_{k-1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k-1}.\] ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)
  12. For each positive integer $ n,$ let $ f(n)$ be the number of ways to make $ n!$ cents using an unordered collection of coins, each worth $ k!$ cents for some $ k,\ 1\le k\le n.$ Prove that for some constant $ C,$ independent of $ n,$ \[ n^{n^2/2-Cn}e^{-n^2/4}\le f(n)\le n^{n^2/2+Cn}e^{-n^2/4}.\]

William Lowell Putnam Mathematical Competition 2006 Solutions

  1. Find the volume of the region of points $(x,y,z)$ such that \[\left(x^{2}+y^{2}+z^{2}+8\right)^{2}\le 36\left(x^{2}+y^{2}\right). \]
  2. Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)
  3. Let $1,2,3,\dots,2005,2006,2007,2009,2012,2016,\dots$ be a sequence defined by $x_{k}=k$ for $k=1,2\dots,2006$ and $x_{k+1}=x_{k}+x_{k-2005}$ for $k\ge 2006.$ Show that the sequence has 2005 consecutive terms each divisible by 2006.
  4. Let $S=\{1,2\dots,n\}$ for some integer $n>1.$ Say a permutation $\pi$ of $S$ has a local maximum at $k\in S$ if \[\begin{array}{ccc}\text{(i)}&\pi(k)>\pi(k+1)&\text{for }k=1\\ \text{(ii)}&\pi(k-1)<\pi(k)\text{ and }\pi(k)>\pi(k+1)&\text{for }1<k<n\\ \text{(iii)}&\pi(k-1)M\pi(k)&\text{for }k=n\end{array}\] (For example, if $n=5$ and $\pi$ takes values at $1,2,3,4,5$ of $2,1,4,5,3,$ then $\pi$ has a local maximum of $2$ as $k=1,$ and a local maximum at $k-4.$)
    What is the average number of local maxima of a permutation of $S,$ averaging over all permuatations of $S?$
  5. Let $n$ be a positive odd integer and let $\theta$ be a real number such that $\theta/\pi$ is irrational. Set $a_{k}=\tan(\theta+k\pi/n),\ k=1,2\dots,n.$ Prove that \[\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1}a_{2}\cdots a_{n}}\] is an integer, and determine its value.
  6. Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral.
  7. Show that the curve $x^{3}+3xy+y^{3}=1$ contains only one set of three distinct points, $A,B,$ and $C,$ which are the vertices of an equilateral triangle.
  8. Prove that, for every set $X=\{x_{1},x_{2},\dots,x_{n}\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that \[\left|m+\sum_{s\in S}s\right|\le\frac1{n+1}\]
  9. Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A\cup B=S,\ A\cap B=\emptyset,$ and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_{S}$ be the number of linear partitions of $S.$ For each positive integer $n,$ find the maximum of $L_{S}$ over all sets $S$ of $n$ points.
  10. Let $Z$ denote the set of points in $\mathbb{R}^{n}$ whose coordinates are $0$ or $1.$ (Thus $Z$ has $2^{n}$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^{n}$.) Given a vector subspace $V$ of $\mathbb{R}^{n},$ let $Z(V)$ denote the number of members of $Z$ that lie in $V.$ Let $k$ be given, $0\le k\le n.$ Find the maximum, over all vector subspaces $V\subseteq\mathbb{R}^{n}$ of dimension $k,$ of the number of points in $V\cap Z.$
  11. For each continuous function $f: [0,1]\to\mathbb{R},$ let $I(f)=\int_{0}^{1}x^{2}f(x)\,dx$ and $J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$
  12. Let $k$ be an integer greater than $1.$ Suppose $a_{0}>0$ and define \[a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}\] for $n\ge 0.$ Evaluate \[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]

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