Let $ ABCD$ be a trapezoid such that $ AB \parallel CD$ and assume that there are points $ E$ on the line outside the segment $ BC$ and $ F$ on the segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Let $ I,J,K$ respectively be the intersection of line $ EF$ and line $ CD$, the intersection of line $ EF$ and line $ AB$, and the midpoint of segment $ EF$. Prove that $ K$ is on the circumcircle of triangle $ CDJ$ if and only if $ I$ is on the circumcircle of triangle $ ABK$. [i]Utari Wijayanti, Bandung[/i]

Anja has to write $2014$ integers on the board such that arithmetic mean of any of the three numbers is among those $2014$ numbers. Show that this is possible only when she writes nothing but $2014$ equal integers.

Let $A_1,A_2,\ldots,A_{2n}$ are the vertices of a regular $2n$-gon and $P$ is a point from the incircle of the polygon. If $\alpha_i=\angle A_iPA_{i+n}$, $i=1,2,\ldots,n$. Prove the equality $$\sum_{i=1}^n\tan^2\alpha_i=2n\frac{\cos^2\frac\pi{2n}}{\sin^4\frac\pi{2n}}.$$

At least $3$ players take part in a tennis tournament. Each participant plays exactly one match against each other participant. After the tournament has ended, we find out that each player has won at least one match. (There are no ties in tennis). Show that in the tournament, there was at least one trio of players $A,B,C$ such that $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$.

Let $ f : \mathbb{R}\to \mathbb{R}$ be a continuous function. Suppose that for any $ c > 0$, the graph of $ f$ can be moved to the graph of $ cf$ using only a translation or a rotation. Does this imply that $ f(x) = ax+b$ for some real numbers $ a$ and $ b$?

Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\tfrac{12}5\sqrt2$. What is the volume of the tetrahedron? $\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$

All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center? $\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$

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.

Given $a$, $b$ are positive integers greater than $1$, and for every positive integer $n$, $b^{n}-1$ divides $a^{n}-1$. Define the polynomial $p_{n}(x)$ as follows: $p_0{x}=-1$, $p_{n+1}(x)=b^{n+1}(x-1)p_{n}(bx)-a(b^{n+1}-1)p_{n}(x)$, for $n \ge 0$. Prove that there exist integers $C$ and positive integer $k$ such that $p_{k}(x)=Cx^k$.

In triangle $\Delta ADC$ we got $AD=DC$ and $D=100^\circ$. In triangle $\Delta CAB$ we got $CA=AB$ and $A=20^\circ$. Prove that $AB=BC+CD$.

Let $n$ be a given positive integer. Determine all positive divisors $d$ of $3n^2$ such that $n^2 + d$ is the square of an integer.

[b]p1.[/b] Farmer John wants to bring some cows to a pasture with grass that grows at a constant rate. Initially, the pasture has some nonzero amount of grass and it will stop growing if there is no grass left. The pasture sustains $100$ cows for ten days. The pasture can also sustain $100$ cows for five days, and then $120$ cows for three more days. If cows eat at a constant rate, fund the maximum number of cows Farmer John can bring to the pasture so that they can be sustained indefinitely. [b]p2.[/b] Sam is learning basic arithmetic. He may place either the operation $+$ or $-$ in each of the blank spots between the numbers below: $$5\,\, \_ \,\, 8\,\, \_ \,\,9\,\, \_ \,\,7\,\,\_ \,\,2\,\,\_ \,\,3$$ In how many ways can he place the operations so the result is divisible by $3$? [b]p3.[/b] Will loves the color blue, but he despises the color red. In the $5\times 6$ rectangular grid below, how many rectangles are there containing at most one red square and with sides contained in the gridlines? [img]https://cdn.artofproblemsolving.com/attachments/1/7/7ce55bdc9e05c7c514dddc7f8194f3031b93c4.png[/img] [b]p4.[/b] Let $r_1, r_2, r_3$ be the three roots of a cubic polynomial $P(x)$. Suppose that $$\frac{P(2) + P(-2)}{P(0)}= 200.$$ If $\frac{1}{r_1r_2}+ \frac{1}{r_2r_3}+\frac{1}{r_3r_1}= \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m + n$. [b]p5.[/b] Consider a rectangle $ABCD$ with $AB = 3$ and $BC = 1$. Let $O$ be the intersection of diagonals $AC$ and $BD$. Suppose that the circumcircle of $ \vartriangle ADO$ intersects line $AB$ again at $E \ne A$. Then, the length $BE$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p6.[/b] Let $ABCD$ be a square with side length $100$ and $M$ be the midpoint of side $AB$. The circle with center $M$ and radius $50$ intersects the circle with center $D$ and radius $100$ at point $E$. $CE$ intersects $AB$ at $F$. If $AF = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. [b]p7.[/b] How many pairs of real numbers $(x, y)$, with $0 < x, y < 1$ satisfy the property that both $3x + 5y$ and $5x + 2y$ are integers? [b]p8.[/b] Sebastian is coloring a circular spinner with $4$ congruent sections. He randomly chooses one of four colors for each of the sections. If two or more adjacent sections have the same color, he fuses them and considers them as one section. (Sections meeting at only one point are not adjacent.) Suppose that the expected number of sections in the final colored spinner is equal to $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p9.[/b] Let $ABC$ be a triangle and $D$ be a point on the extension of segment $BC$ past $C$. Let the line through $A$ perpendicular to $BC$ be $\ell$. The line through $B$ perpendicular to $AD$ and the line through $C$ perpendicular to $AD$ intersect $\ell$ at $H_1$ and $H_2$, respectively. If $AB = 13$, $BC = 14$, $CA = 15$, and $H_1H_2 = 1001$, find $CD$. [b]p10.[/b] Find the sum of all positive integers $k$ such that $$\frac21 -\frac{3}{2 \times 1}+\frac{4}{3\times 2\times 1} + ...+ (-1)^{k+1} \frac{k+1}{k\times (k - 1)\times ... \times 2\times 1} \ge 1 + \frac{1}{700^3}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Inside the $7\times8$ rectangle below, one point is chosen a distance $\sqrt2$ from the left side and a distance $\sqrt7$ from the bottom side. The line segments from that point to the four vertices of the rectangle are drawn. Find the area of the shaded region. [asy] import graph; size(4cm); pair A = (0,0); pair B = (9,0); pair C = (9,7); pair D = (0,7); pair P = (1.5,3); draw(A--B--C--D--cycle,linewidth(1.5)); filldraw(A--B--P--cycle,rgb(.76,.76,.76),linewidth(1.5)); filldraw(C--D--P--cycle,rgb(.76,.76,.76),linewidth(1.5)); [/asy]

Prove that the polynomial \[ f(x) \equal{} \frac {x^n \plus{} x^m \minus{} 2}{x^{\gcd(m,n)} \minus{} 1}\] is irreducible over $ \mathbb{Q}$ for all integers $ n > m > 0$.

Let $ f$ be a continuous, nonconstant, real function, and assume the existence of an $ F$ such that $ f(x\plus{}y)\equal{}F[f(x),f(y)]$ for all real $ x$ and $ y$. Prove that $ f$ is strictly monotone.

A robot initially at position $0$ along a number line has a [i]movement function[/i] $f(u, v)$. It rolls a fair $26$-sided die repeatedly, with the $k$-th roll having value $r_k$. For $k \ge 2$, if $r_k > r_{k-1}$, it moves $f(r_k, r_{k-1})$ units in the positive direction. If $r_k < r_{k-1}$, it moves $f(r_k, r_{k-1})$ units in the negative direction. If $r_k = r_{k-1}$, all movement and die-rolling stops and the robot remains at its final position $x$. If $f(u, v) = (u^2 - v^2)^2 + (u - 1)(v + 1)$, compute the expected value of $x$.