Let $S > 1$ be a real number. The Cartesian plane is partitioned into rectangles whose sides are parallel to the axes of the coordinate system. and whose vertices have integer coordinates. Prove that if the area of each triangle if at most $S$, then for any positive integer $k$ there exist $k$ vertices of these rectangles which lie on a line.

Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and \[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\] [i]Alex Zhu.[/i] [hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]

There are $2019$ points given in the plane. A child wants to draw $k$ (closed) discs in such a manner, that for any two distinct points there exists a disc that contains exactly one of these two points. What is the minimal $k$, such that for any initial configuration of points it is possible to draw $k$ discs with the above property?

A rectangularly paper is divided in polygons areas in the following way: at every step one of the existing surfaces is cut by a straight line, obtaining two new areas. Which is the minimum number of cuts needed such that between the obtained polygons there exists $251$ polygons with $11$ sides?

Let $AB$ and $CD$ be two parallel lines connected by the base $BD$. $AD$ and $BC$ are drawn and they intersect at the point $P$. $A$ line $P Q$ is drawn from the point $P$ to $BD$ such that $\angle P AB = \angle DP Q$. Prove that $\frac{1}{AB} + \frac{1}{CD} =\frac{1}{P Q}$.

Back in 1930, Tillie had to memorize her multiplication tables from $0\times 0$ through $12\times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd? $\textbf{(A) }0.21\qquad\textbf{(B) }0.25\qquad\textbf{(C) }0.46\qquad\textbf{(D) }0.50\qquad\textbf{(E) }0.75$

Compute the quadruple integral $$A=\frac1{\pi^2}\int_{[0,1]^2\times[-\pi,\pi]^2}ab\sqrt{a^2+b^2-2ab\cos(x-y)}dadbdxdy$$ [i]Proposed by Moubinool Omarjee[/i]

Triangle $BAD$ is right-angled at $B$. On $AD$ there is a point $C$ for which $AC=CD$ and $AB=BC$. The magnitude of angle $DAB$, in degrees, is: $\textbf{(A)}\ 67\dfrac{1}{2} \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 22\dfrac{1}{2}$

Let $ABCD$ be a parallelogram. Construct the equilateral triangle $DCE$ on the side $DC$ and outside of parallelogram. Let $P$ be an arbitrary point in plane of $ABCD$. Show that \[PA+PB+AD \geq PE.\]

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.