Solve the function $f: \Re \to \Re$: \[ f( x^{3} )+ f(y^{3}) = (x+y)(f(x^{2} )+f(y^{2} )-f(xy))\]

Prove that there are no integers $m$ and $n$ such that \[19m^2+95mn+2000n^2=1995.\]

On a $3 \times 3$ board the numbers from $1$ to $9$ are written in some order and without repeating. We say that the arrangement obtained is [i]Isthmian [/i] if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements. Note: Two arrangements are considered equal if one can be obtained from the other by rotating the board.

Walter has exactly one penny, one nickel, one dime and one quarter in his pocket. What percent of one dollar is in his pocket? $\text{(A)}\ 4\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 41\% \qquad \text{(E)}\ 59\%$

Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$th day of January. What is $m$? [i]Based on a proposal by Neelabh Deka[/i]

How many positive whole numbers less than $100$ are divisible by $3$, but not by $2$?

An isosceles trapezoid is divided by each diagonal into two isosceles triangles. Determine the angles of the trapezoid.

Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $ (a,0)$ to $ (3,3)$, divides the entire region into two regions of equal area. What is $ a$? [asy]size(200); defaultpen(linewidth(.8pt)+fontsize(8pt)); fill((2/3,0)--(3,3)--(3,1)--(2,1)--(2,0)--cycle,gray); xaxis("$x$",-0.5,4,EndArrow(HookHead,4)); yaxis("$y$",-0.5,4,EndArrow(4)); draw((0,1)--(3,1)--(3,3)--(2,3)--(2,0)); draw((1,0)--(1,2)--(3,2)); draw((2/3,0)--(3,3)); label("$(a,0)$",(2/3,0),S); label("$(3,3)$",(3,3),NE);[/asy]$ \textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac35\qquad \textbf{(C)}\ \frac23\qquad \textbf{(D)}\ \frac34\qquad \textbf{(E)}\ \frac45$

Jeremy has a magic scale, each side of which holds a positive integer. He plays the following game: each turn, he chooses a positive integer $n$. He then adds $n$ to the number on the left side of the scale, and multiplies by $n$ the number on the right side of the scale. (For example, if the turn starts with $4$ on the left and $6$ on the right, and Jeremy chooses $n = 3$, then the turn ends with $7$ on the left and $18$ on the right.) Jeremy wins if he can make both sides of the scale equal. (a) Show that if the game starts with the left scale holding $17$ and the right scale holding $5$, then Jeremy can win the game in $4$ or fewer turns. (b) Prove that if the game starts with the right scale holding $b$, where $b\geq 2$, then Jeremy can win the game in $b-1$ or fewer turns.

Let $ABCD$ be a right-angled trapezoid and $M$ be the midpoint of its greater lateral side $CD$. Circumcircles $\omega_{1}$ and $\omega_{2}$ of triangles $BCM$ and $AMD$ meet for the second time at point $E$. Let $ED$ meet $\omega_{1}$ at point $F$, and $FB$ meet $AD$ at point $G$. Prove that $GM$ bisects angle $BGD$.

Let $m$ and $n$ be natural numbers. Professor Mubariz has $m$ folders and Professor Nazim has $n$ folders; initially, all folders are empty. Every day, where the day numbers are marked as $d = 1,2,3 ....,$ Prof. Mubariz is given $2018$ blue papers, and Prof. Nazim is given $2018$ orange papers. On day $d ( d = 1, 2, 3, ...),$ they both perform the following operations: [list] [*] If the $2018$ papers given to this professor are not enough to place $d$ papers in each of his folders, then he distributes all the $2018$ papers given to him to his students. If the $2018$ papers given to this professor are enough to place $d$ papers in each of his folders, firstly, he places $d$ papers in each of his folders. [*] If this professor still has papers left after the first step, he places them in the other professor's folders, with the same number in each folder and as many as possible. [*] If this professor still has papers left after the second step, he distributes them to his students. [/list] Prove that after $6$ years, the number of blue papers in one folder of Prof. Nazim will be equal to the number of orange papers in one folder of Prof. Mubariz.

A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$. The rod is suspended from a distance $kd$ from the center, and undergoes small oscillations with an angular frequency $\beta \sqrt{\frac{g}{d}}$. Find the maximum value of $\beta$. $ \textbf{(A)}\ 1$ $ \textbf{(B)}\ \sqrt{2}$ $ \textbf{(C)}\ 1/\sqrt{2}$ $ \textbf{(D)}\ \beta \text{ does not attain a maximum value}$ $ \textbf{(E)}\ \text{none of the above}$

Let $ABCD$ be a cyclic quadrilateral, for which $B$ and $C$ are acute angles. $M$ and $N$ are the projections of the vertex $B$ on the lines $AC$ and $AD$, respectively, $P$ and $T$ are the projections of the vertex $D$ on the lines $AB$ and $AC$ respectively, $Q$ and $S$ are the intersections of the pairs of lines $MN$ and $CD$, and $PT$ and $BC$, respectively. Prove the following statements: a) $NS \parallel PQ \parallel AC$; b) $NP=SQ$; c) $NPQS$ is a rectangle if, and only if, $AC$ is a diamteter of the circumscribed circle of quadrilateral $ABCD$.

Let $J_A$ and $J_B$ be the $A$-excenter and $B$-excenter of $\triangle ABC$. Consider a chord $\overline{PQ}$ of circle $ABC$ which is parallel to $AB$ and intersects segments $\overline{AC}$ and $\overline{BC}$. If lines $AB$ and $CP$ intersect at $R$, prove that $$\angle J_AQJ_B+\angle J_ARJ_B=180^\circ.$$

Given a positive integer $k,$ prove that for any integer $n \geq 20k,$ there exist $n - k$ pairwise distinct positive integers whose squares add up to $n(n + 1)(2n + 1)/6.$ [i]The Problem Selection Committee[/i]

Find all positive integers $n$ such that $n!$ can be written as the product of two Fibonacci numbers.

Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that: i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$ ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.

Given the polynomial sequence $(p_{n}(x))$ satisfying $p_{1}(x)=x^{2}-1$, $p_{2}(x)=2x(x^{2}-1)$, and $p_{n+1}(x)p_{n-1}(x)=(p_{n}(x)^{2}-(x^{2}-1)^{2}$, for $n\geq 2$, let $s_{n}$ be the sum of the absolute values of the coefficients of $p_{n}(x)$. For each $n$, find a non-negative integer $k_{n}$ such that $2^{-k_{n}}s_n$ is odd.

Let $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. A point $D$ is chosen on the internal bisector of $\angle ACB$ so that the points $D$ and $C$ are separated by $AB$. A circle $\omega$ centered at $D$ is tangent to the segment $AB$ at $E$. The tangents to $\omega$ through $C$ meet the segment $AB$ at $K$ and $L$, where $K$ lies on the segment $AL$. A circle $\Omega_1$ is tangent to the segments $AL, CL$, and also to $\Omega$ at point $M$. Similarly, a circle $\Omega_2$ is tangent to the segments $BK, CK$, and also to $\Omega$ at point $N$. The lines $LM$ and $KN$ meet at $P$. Prove that $\angle KCE = \angle LCP$. Poland

Let $P$ be a non-zero polynomial with non-negative real coefficients, let $N$ be a positive integer, and let $\sigma$ be a permutation of the set $\{1,2,...,n\}$. Determine the least value the sum \[\sum_{i=1}^{n}\frac{P(x_i^2)}{P(x_ix_{\sigma(i)})}\] may achieve, as $x_1,x_2,...,x_n$ run through the set of positive real numbers. [i]Fedor Petrov[/i]

Let $g(t)$ and $h(t)$ be real, continuous functions for $t\geq 0.$ Show that any function $v(t)$ satisfying the differential inequality $$\frac{dv}{dt}+g(t)v \geq h(t),\;\; v(t)=c,$$ satisfies the further inequality $v(t)\geq u(t),$ where $$\frac{du}{dt}+g(t)u = h(t),\;\; u(t)=c.$$ From this, conclude that for sufficiently small $t>0,$ the solution of $$\frac{dv}{dt}+g(t)v = v^2 ,\;\; v(t)=c$$ may be written $$v=\max_{w(t)} \left( c e^{- \int_{0}^{t} |g(s)-2w(s)| \, ds} -\int_{0}^{t} e^{-\int_{0}^{t} |g(s')-2w(s')| \, ds'} w(s)^{2} ds \right),$$ where the maximum is over all continuous functions $w(t)$ defined over some $t$-interval $[0,t_0 ].$

The integer sequence $(s_i)$ "having pattern 2016'" is defined as follows:\\ $\circ$ The first member $s_1$ is 2.\\ $\circ$ The second member $s_2$ is the least positive integer exceeding $s_1$ and having digit 0 in its decimal notation.\\ $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 1 in its decimal notation.\\ $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 6 in its decimal notation.\\ The following members are defined in the same way. The required digits change periodically: $2 \rightarrow 0 \rightarrow 1 \rightarrow 6 \rightarrow 2 \rightarrow 0 \rightarrow \ldots$. The first members of this sequence are the following: $2; 10; 11; 16; 20; 30; 31; 36; 42; 50$. What are the 4 numbers that immediately follow $s_k = 2016$ in this sequence?

One morning, each member of Manjul’s family drank an $8$-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank $1/7$-th of the total amount of milk and $2/17$-th of the total amount of coffee. How many people are there in Manjul’s family?

Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression

In rectangle $ ABCD$, $ AB\equal{}5$ and $ BC\equal{}3$. Points $ F$ and $ G$ are on $ \overline{CD}$ so that $ DF\equal{}1$ and $ GC\equal{}2$. Lines $ AF$ and $ BG$ intersect at $ E$. Find the area of $ \triangle{AEB}$. [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair A=(0,0), B=(5,0), C=(5,3), D=(0,3), F=(1,3), G=(3,3); pair E=extension(A,F,B,G); draw(A--B--C--D--A--E--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$E$",E,N); label("$F$",F,SE); label("$G$",G,SW); label("$B$",B,SE); label("1",midpoint(D--F),N); label("2",midpoint(G--C),N); label("3",midpoint(B--C),E); label("3",midpoint(A--D),W); label("5",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ \frac{21}{2} \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ \frac{25}{2} \qquad \textbf{(E)}\ 15$