$a_1,a_2,...,a_{2014}$ is a permutation of $1,2,3,...,2014$. What is the greatest number of perfect squares can have a set ${ a_1^2+a_2,a_2... 1.$a_1,a_2,...,a_{2014}$is a permutation of$1,2,3,...,2014$. What is the greatest number of perfect squares can have a set${ a_1^2+a_2,a_2^2+a_3,a_3^2+a_4,...,a_{2013}^2+a_{2014},a_{2014}^2+a_1 }?$2. Do there exist positive integers$a$and$b$such that$a^n+n^b$and$b^n+n^a$are relatively prime for all natural$n$? 3. The triangle$ABC$is inscribed in a circle$w_1$. Inscribed in a triangle circle touchs the sides$BC$in a point$N$.$w_2$— the circle inscribed in a segment$BAC$circle of$w_1$, and passing through a point$N$. Let points$O$and$J$— the centers of circles$w_2$and an extra inscribed circle (touching side$BC$) respectively. Prove, that lines$AO$and$JN$are parallel. 4. Given a scalene triangle$ABC$. Incircle of$\triangle{ABC{}}$touches the sides$AB$and$BC$at points$C_1$and$A_1$respectively, and excircle of$\triangle{ABC}$(on side$AC$) touches$AB$and$BC$at points$ C_2$and$A_2$respectively.$BN$is bisector of$\angle{ABC}$($N$lies on$BC$). Lines$A_1C_1$and$A_2C_2$intersects the line$AC$at points$K_1$and$K_2$respectively. Let circumcircles of$\triangle{BK_1N}$and$\triangle{BK_2N}$intersect circumcircle of a$\triangle{ABC}$at points$P_1$and$P_2$respectively. Prove that$AP_1$=$CP_2$5.$\mathbb{Q}$is set of all rational numbers. Find all functions$f:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}$such that for all$x$,$y$,$z\in\mathbb{Q}$satisfy $$f(x,y)+f(y,z)+f(z,x)=f(0,x+y+z)$$ 6. Prove that, for all$n\in\mathbb{N}$, on$ [n-4\sqrt{n}, n+4\sqrt{n}]$exists natural number$k=x^3+y^3$where$x$,$y\$ are nonnegative integers.

Name