In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

Let $H$ be the orthocentre and $O$ be the circumcentre of an acute triangle $ABC$. Let $AD$ and $BE$ be the altitudes of the triangle with $D$ on $BC$ and $E$ on $CA$. Let $K =OD \cap BE, L = OE \cap AD$. Let $X$ be the second point of intersection of the circumcircles of triangles $HKD$ and $HLE$, and let $M$ be the midpoint of side $AB$. Prove that points $K, L, M$ are collinear if and only if $X$ is the circumcentre of triangle $EOD$.

A positive rational number is written on the blackboard. Every minute Vasya replaces the number $ r $ written on the board with $ \sqrt {r + 1} $. Prove that someday he will get an irrational number.

Each point of a circle is colored either red or blue. [b](a)[/b] Prove that there always exists an isosceles triangle inscribed in this circle such that all its vertices are colored the same. [b](b)[/b] Does there always exist an equilateral triangle inscribed in this circle such that all its vertices are colored the same?

Let $n = 2^{2015} - 1$. For any integer $1 \le x < n$, let \[f_n(x) = \sum\limits_p s_p(n-x) + s_p(x) - s_p(n),\] where $s_q(k)$ denotes the sum of the digits of $k$ when written in base $q$ and the summation is over all primes $p$. Let $N$ be the number of values of $x$ such that $4 | f_n(x)$. What is the remainder when $N$ is divided by $1000?$

Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$. (a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$. (b) Show that $ a_{2008} \neq 0$

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

Let squares be constructed on the sides $BC,CA,AB$ of a triangle $ABC$, all to the outside of the triangle, and let $A_1,B_1, C_1$ be their centers. Starting from the triangle $A_1B_1C_1$ one analogously obtains a triangle $A_2B_2C_2$. If $S, S_1, S_2$ denote the areas of triangles$ ABC,A_1B_1C_1,A_2B_2C_2$, respectively, prove that $S = 8S_1 - 4S_2.$

A right triangle with perpendicular sides $a$ and $b$ and hypotenuse $c$ has the following properties: $a = p^m$ and $b = q^n$ with $p$ and $q$ prime numbers and $m$ and $n$ positive integers, $c = 2k +1$ with $k$ a positive integer. Determine all possible values of $c$ and the associated values of $a$ and $b$.

Let $n=p_1p_2\cdots p_s$, where $p_1,\ldots,p_s$ are distinct odd prime numbers. (a) Prove that the expression $$F_n(x)=\prod\left(x^{\frac n{p_{i_1}\cdots p_{i_k}}}-1\right)^{(-1)^k},$$where the product goes over all subsets $\{p_{i_1},\ldots,p_{i_k}\}$ or $\{p_1,\ldots,p_s\}$ (including itself and the empty set), can be written as a polynomial in $x$ with integer coefficients. (b) Prove that if $p$ is a prime divisor of $F_n(2)$, then either $p\mid n$ or $n\mid p-1$.

For which pairs $(m,n)$ does there exist an injective function $f:\mathbb{R}^2\to\mathbb{R}^2$ under which the image of every regular $m$-gon is a regular $n$-gon. (Note that $m,n\geq 3$, and that by a regular $N$-gon we mean the union of the boundary segments, not the closed polygonal region.) [i]Proposed by Sutanay Bhattacharya, Bishnupur, India[/i]

Let $A$ be a set of $n\ge2$ positive integers, and let $\textstyle f(x)=\sum_{a\in A}x^a$. Prove that there exists a complex number $z$ with $\lvert z\rvert=1$ and $\lvert f(z)\rvert=\sqrt{n-2}$.

In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10.$ Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCFG.$ [asy] size(250); pair A=(0,12), E=(0,8), B=origin, C=(24*sqrt(2),0), D=(6*sqrt(2),0), F=A+10*dir(A--C), G=intersectionpoint(E--F, A--D); draw(A--B--C--A--D^^E--F); pair point=G+1*dir(250); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$G$", G, dir(point--G)); markscalefactor=0.1; draw(rightanglemark(A,B,C)); label("10", A--F, dir(90)*dir(A--F)); label("27", F--C, dir(90)*dir(F--C)); label("3", (0,10), W); label("9", (0,4), W);[/asy]

Let $ABCD$ be a regular triangular pyramid with base $ABC$ (this means that $ABC$ is a regular triangle, and edges $AD$, $BD$ and $CD$ are equal) and plane angles at the opposite vertex equal to $a$. A plane parallel to $ABC$ intersects $AD$, $BD$ and $CD$, respectively, at points $A_1$, $B_1$ and $C_1$. The surface of the polyhedron $ABCA_1B_1C_1$ is cut along five edges: $A_1B_1$, $B_1C_1$, $C_1C$, $CA$ and $AB$, after which this surface is turned onto a plane. At what values of $a$ will the resulting scan necessarily cover itself?

Kathryn, for a history project on sports, chronicled the history of college football. When she mentioned that Auburn got cheated out of the NCAA Football championship in the $2004-05$ season due to the many flaws in the BCS system, her teacher just couldn’t contain her applause, and awarded an automatic A to her for the rest of the year. The lecture was so popular, in fact, that many students pressed Kathryn to record the lecture on video and sell DVDs of it. If the function for Kathryn’s profit for selling DVDs of her college football presentation is $y = -x^2 + 14x + 251$, where $y$ is Kathryn’s profit and $x$ is the price per DVD, what price (in dollars) will maximize her profit?

Let $ a,b,c>0$. Prove that: $ \dfrac{a}{2a\plus{}b}\plus{}\dfrac{b}{2b\plus{}c}\plus{}\dfrac{c}{2c\plus{}a}\leq 1$

Jon and Steve ride their bicycles on a path that parallels two side-by-side train tracks running in the east/west direction. Jon rides east at 20 miles per hour, and Steve rides west at 20 miles per hour. Two trains of equal length traveling in opposite directions at constant but different speeds, each pass the two riders. Each train takes exactly 1 minute to go past Jon. The westbound train takes 10 times as long as the eastbound train to go past Steve. The length of each train is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let \( n \) be a positive integer, and let \( A \) and \( B \) be \( n \times n \) matrices with real coefficients such that \[ ABBA - BAAB = A - B. \] (a) Prove that \( \text{Tr}(A) = \text{Tr}(B) \) and that \( \text{Tr}(A^2) = \text{Tr}(B^2) \). (b) If \(BA^2B= A^2B^2\) and \(AB^2A= B^2A^2\), prove that \( \det A = \det B \). Note: \( \text{Tr}(X) \) denotes the trace of \( X \), which is the sum of the elements on its main diagonal, and \( \det X \) denotes the determinant of \( X \).

In how many ways can the ”Nikolaus’ House” (see the picture) be drawn? Edges may not be erased nor duplicated, and no additional edges may be drawn. [img]https://cdn.artofproblemsolving.com/attachments/0/5/33795820e0335686b06255180af698e536a9be.png[/img]

A teacher gave a test to a class in which $ 10\%$ of the students are juniors and $ 90\%$ are seniors. The average score on the test was $ 84$. The juniors all received the same score, and the average score of the seniors was $ 83$. What score did each of the juniors receive on the test? $ \textbf{(A)}\ 85 \qquad \textbf{(B)}\ 88 \qquad \textbf{(C)}\ 93 \qquad \textbf{(D)}\ 94 \qquad \textbf{(E)}\ 98$

Give are the integers $a_{1}=11 , a_{2}=1111, a_{3}=111111, ... , a_{n}= 1111...111$( with $2n$ digits) with $n > 8$ . Let $q_{i}= \frac{a_{i}}{11} , i= 1,2,3, ... , n$ the remainder of the division of $a_{i}$ by$ 11$ . Prove that the sum of nine consecutive quotients: $s_{i}=q_{i}+q_{i+1}+q_{i+2}+ ... +q_{i+8}$ is a multiple of $9$ for any $i= 1,2,3, ... , (n-8)$

Let $ABCD$ be a trapezium inscribed in a circle $\Gamma$ with diameter $AB$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$ . The circle with center $B$ and radius $BE$ meets $\Gamma$ at the points $K$ and $L$ (where $K$ is on the same side of $AB$ as $C$). The line perpendicular to $BD$ at $E$ intersects $CD$ at $M$. Prove that $KM$ is perpendicular to $DL$. [i]Greece - Silouanos Brazitikos[/i]

Let $n$ be an even positive integer. We say that two different cells of a $n \times n$ board are [b]neighboring[/b] if they have a common side. Find the minimal number of cells on the $n \times n$ board that must be marked so that any cell (marked or not marked) has a marked neighboring cell.

Given acute triangle $ABC$. Point $D$ is the foot of the perpendicular from $A$ to $BC$. Point $E$ lies on the segment $AD$ and satisfies the equation \[\frac{AE}{ED}=\frac{CD}{DB}\] Point $F$ is the foot of the perpendicular from $D$ to $BE$. Prove that $\angle AFC=90^{\circ}$.

A circular target with a radius of $12$ cm was hit by $19$ shots. Prove that the distance between two hits is less than $7$ cm.