Let $p$ be a prime such that $p\mid 2a^2-1$ for some integer $a$. Show that there exist integers $b,c$ such that $p=2b^2-c^2$.

Suppose $ABCD$ is a convex quadrilateral.Points $P,Q,R$ and $S$ are four points on the line segments $AB,BC,CD$ and $DA$ respectively.The line segments $PR$ and $QS$ meet at $T$.Suppose that each of the quadrilaterals $APTS,BQTP,CRTQ$ and $DSTR$ have an incircle.Prove that the quadrilateral $ABCD$ also has an incircle.

Find all real numbers $p$ for which the equation $x^3+3px^2+(4p-1)x+p=0$ has two real roots with difference $1$.

Given arbitrary complex numbers $w_1,w_2,\ldots,w_n$, show that there exists a positive integer $k\le 2n+1$ for which $\text{Re} (w_1^k+w_2^k+\cdots+w_n^k)\ge 0$.

Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$.

Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$.Let $A$ be a point on $\omega_2$ not lying on the line through the centres of the two circles.Let $AB$ and $AC$ be the tangents to $\omega_1$.Lines $BP$ and $CP$ meet $\omega_2$ for the second time at points $E$ and $F$.Prove that the line $EF$,the tangent to $\omega_2$ at $A$ and the common tangent at $P$ concur.

Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.

Find all ordered triplets of positive integers $(a,\ b,\ c)$ such that $2^a+3^b+1=6^c$.

Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.

Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.

Suppose $p,q,r$ are three distinct primes such that $rp^3+p^2+p=2rq^2+q^2+q$. Find all possible values of $pqr$.

Let $O$ be the centre of the square $ABCD$. Let $P,Q,R$ be respectively on the segments $OA,OB,OC$ such that $OP=3,OQ=5,OR=4$. Suppose $S$ is on $OD$ such that $X=AB\cap PQ,Y=BC\cap QR$ and $Z=CD\cap RS$ are collinear. Find $OS$.

Let $ABC$ and $PQR$ be two triangles such that (a) $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$. (b) $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$ Prove that $AB+AC=PQ+PR$.

Let $p>3$ be a prime and let $1+\frac 12 +\frac 13 +\cdots+\frac 1p=\frac ab$.Prove that $p^4$ divides $ap-b$.

Let $d(n)$ denote the number of positive divisors of the positive integer $n$.Determine those numbers $n$ for which $d(n^3)=5d(n)$.

Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$.

The circles $\mathcal{K}_1,\mathcal{K}_2$ and $\mathcal{K}_3$ are pairwise externally tangent to each other; the point of tangency betwwen $\mathcal{K}_1$ and $\mathcal{K}_2$ is $T$. One of the external common tangents of $\mathcal{K}_1$ and $\mathcal{K}_2$ meets $\mathcal{K}_3$ at points $P$ and $Q$. Prove that the internal common tangent of $\mathcal{K}_1$ and $\mathcal{K}_2$ bisects the arc $PQ$ of $\mathcal{K}_3$ which is closer to $T$.

Denote by $F_n$ the $n^{\text{th}}$ Fibonacci number $(F_1=F_2=1)$.Prove that if $a,b,c$ are positive integers such that $a| F_b,b|F_c,c|F_a$,then either $5$ divides each of $a,b,c$ or $12$ divides each of $a,b,c$.

Determine all polynomials $f$ with integer coefficients with the property that for any two distinct primes $p$ and $q$, $f(p)$ and $f(q)$ are relatively prime.

Let $ABC$ be a triangle in which $\angle B$ is obtuse.Let $\Gamma$ be its circumcircle and $O$ be the centre of $\Gamma$.Let the tangent to $\Gamma$ at $C$ intersect the line $AB$ in $B_1$.Let $O_1$ be the circumcentre of the circumcircle $\Gamma_1$ of $\triangle AB_1 C$.Take any point $B_2$ on the segment $BB_1$ different from $B,B_1$.Let $C_1$ be the point of contact of the tangent from $B_2$ to $\Gamma$ which is closer to $C$.Let $O_2$ be the circumcentre of $\triangle AB_2 C_1$.Prove that $O,O_2,O_1,C_1,C$ are concyclic if $OO_2\perp AO_1$.

Let $d(n)$ be the number of positive divisors of a natural number $n$.Find all $k\in \mathbb{N}$ such that there exists $n\in \mathbb{N}$ with $d(n^2)/d(n)=k$.

Consider a regular triangular array of $n(n+1)/2$ points.Let $f(n)$ denote the number of equilateral triangles formed by taking some $3$ points in the array as vertices.Prove that $f(n)=\frac{(n-1)n(n+1)(n+2)}{24}$.

Let $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$ be two partitions of a set $M$ such that $|A_j\cup B_k|\ge n$ for any $j,k\in\{1,2,\ldots,n\}$. Prove that $|M|\ge n^2/2$.

Let $(x_j,y_j)$, $1\le j\le 2n$, be $2n$ points on the half-circle in the upper half-plane. Suppose $\sum_{j=1}^{2n}x_j$ is an odd integer. Prove that $\displaystyle{\sum_{j=1}^{2n}y_j \ge 1}$.

(a) Let $k,n\ge 1$.Find the number of sequences $\phi=S_0,S_1,\ldots,S_k$ of subsets of $[n]=\{1,2,3,\ldots,n\}$ if for all $1\le i\le k$ we have either (i)$S_{i-1}\subset S_i$ and $|S_i-S_{i-1}|$,or (ii)$S_i\subset S_{i-1}$ and $|S_{i-1}-S_i|=1$. (b) Suppose that we add the additional condition that $S_k=\phi$.Show that now the number $f_k(n)$ of sequences is given by$f_k(n)=\frac{1}{2^n}\sum_{i=0}^n\binom ni (n-2i)^k$. Note that $f_k(n)=0$ if $k$ is odd.

Fix positive integers $n,j,k$.How many integer sequences are there of the form $1\le a_1<a_2<\ldots<a_k\le n$,where $a_{i+1}-a_i\ge j$ for all $1\le i\le k-1$.

Fix positive integers $k$ and $n$.Derive a simple expression involving Fibonacci numbers for the number of sequences $(T_1,T_2,\ldots,T_k)$ of subsets $T_i$ of $[n]$ such that $T_1\subseteq T_2\supseteq T_3\subseteq T_4\supseteq\ldots$. Moderator says: and the original source for this one is Richard Stanley, Enumerative Combinatorics vol. 1 (1st edition), exercise 1.15.

Show that the number of ordered pairs $(S,T)$ of subsets of $[n]$ satisfying $s>|T|$ for all $s\in S$ and $t>|S|$ for all $t\in T$ is equal to the Fibonacci number $F_{2n+2}$. Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=296007#p296007 http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=515970&hilit=Putnam+1990