Let $ BC$ of right triangle $ ABC$ be the diameter of a circle intersecting hypotenuse $ AB$ in $ D$. At $ D$ a tangent is drawn cutting leg $ CA$ in $ F$. This information is [u]not[/u] sufficient to prove that $ \textbf{(A)}\ DF \text{ bisects }CA \qquad \textbf{(B)}\ DF \text{ bisects }\angle CDA$ $ \textbf{(C)}\ DF \equal{} FA \qquad \textbf{(D)}\ \angle A \equal{} \angle BCD \qquad \textbf{(E)}\ \angle CFD \equal{} 2\angle A$

Find all positive integers $n$ that are quadratic residues modulo all primes greater than $n$.

The results of a cross-country team's training run are graphed below. Which student has the greatest average speed? [asy] for ( int i = 1; i <= 7; ++i ) { draw((i,0)--(i,6)); } for ( int i = 1; i <= 5; ++i ) { draw((0,i)--(8,i)); } draw((-0.5,0)--(8,0), linewidth(1)); draw((0,-0.5)--(0,6), linewidth(1)); label("$O$", (0,0), SW); label(scale(.85)*rotate(90)*"distance", (0, 3), W); label(scale(.85)*"time", (4, 0), S); dot((1.25, 4.5)); label(scale(.85)*"Evelyn", (1.25, 4.8), N); dot((2.5, 2.2)); label(scale(.85)*"Briana", (2.5, 2.2), S); dot((4.25,5.2)); label(scale(.85)*"Carla", (4.25, 5.2), SE); dot((5.6, 2.8)); label(scale(.85)*"Debra", (5.6, 2.8), N); dot((6.8, 1.4)); label(scale(.85)*"Angela", (6.8, 1.4), E); [/asy] $ \textbf{(A)}\ \text{Angela}\qquad\textbf{(B)}\ \text{Briana}\qquad\textbf{(C)}\ \text{Carla}\qquad\textbf{(D)}\ \text{Debra}\qquad\textbf{(E)}\ \text{Evelyn} $

Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\ ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]

The circle $\omega_2$ passing through the center $O$ of the circle $\omega_1$, is tangent to the circle $\omega_2$ at the point $A$. On the circle $\omega_2$, the point $C$ is taken so that the ray $AC$ intersects the circle $\omega_1$ for second time at point $D$, the ray $OC$ intersects the circle $\omega_1$ at point $E$ and the lines $DE$ and $AO$ are parallel. Find the size of the angle $DAE$.

Let $n$ be an even natural number and let $A$ be the set of all non-zero sequences of length $n$, consisting of numbers $0$ and $1$ (length $n$ binary sequences, except the zero sequence $(0,0,\ldots,0)$). Prove that $A$ can be partitioned into groups of three elements, so that for every triad $\{(a_1,a_2,\ldots,a_n), (b_1,b_2,\ldots,b_n), (c_1,c_2,\ldots,c_n)\}$, and for every $i = 1, 2,\ldots,n$, exactly zero or two of the numbers $a_i, b_i, c_i$ are equal to $1$.

Find the minimum $k>2$ for which there are $k$ consecutive integers such that the sum of their squares is a square.

A board of $6\times 6$ is totally covered by $18$ dominoes (of $2\times 1$), that is, there are no overlaps, gaps, and the tiles do not come off the board. Prove that, regardless of the arrangement of the tiles, there is always a line that divides the board into two non-empty parts, and without cutting tiles.

A pair $(m, n)$ of positive integers is called “[i]linked[/i]” if $m$ divides $3n + 1$ and $n$ divides $3m + 1$. If $a, b, c$ are distinct positive integers such that $(a, b)$ and $( b, c)$ are linked pairs, prove that the number $1$ belongs to the set $\{a, b, c\}$

The quadrilateral $ABCD$ is circumscribed about the circle, touches the sides $AB, BC, CD, DA$ in the points $K, L, M, N,$ respectively. Let $P, Q, R, S$ midpoints of the sides $KL, LM, MN, NK$. Prove that $PR = QS$ if and only if $ABCD$ is inscribed.

Is it possible to paint the cells of a $5\times 5$ board in $4$ colors so that the cells standing at the intersection of any two rows and any two columns were painted in at least $ 3$ colors?

Luke and Carissa are finding the sum of the first $20$ positive integers by adding them one at a time. Luke forgets to add a number and gets an answer of $207$. Carissa adds a number twice by mistake and gets an answer of $225$. What is the sum of the number that Luke forgot and the number that Carissa added twice?

Let $x+y=a$ and $xy=b$. The expression $x^6+y^6$ can be written as a polynomial in terms of $a$ and $b$. What is this polynomial?

The points $D$ and $E$ lie on the side $(BC)$ of the triangle $ABC$ such that $D$ is between $B$ and $E.$ A point $R$ on the segment $(AE)$ is called [i]remarkable[/i] if the lines $PQ$ and $BC$ are parallel, where $\{P\}=DR \cap AC$ and $\{Q\}=CR \cap AB.$ A point $R'$ on the segment $(AD)$ is called [i]remarkable[/i] if the lines $P'Q'$ and $BC$ are parallel, where $\{P'\}=BR' \cap AC$ and $\{Q'\}=ER' \cap AB.$ a) If there exists a remarkable point on the segment $(AE),$ prove that any point of the segment $(AE)$ is remarkable. b) If each of the segments $(AD)$ and $(AE)$ contains a remarkable point, prove that $BD=CE=\varphi \cdot DE,$ where $\varphi= \frac{1+\sqrt{5}}{2}$ is the golden ratio.

Let $a,b,c,$ and $d$ be real numbers such that at least one of $c$ and $d$ is non-zero. Let $ f:\mathbb{R}\to\mathbb{R}$ be a function defined as $f(x)=\frac{ax+b}{cx+d}$. Suppose that for all $x\in\mathbb{R}$, we have $f(x) \neq x$. Prove that if there exists some real number $a$ for which $f^{1387}(a)=a$, then for all $x$ in the domain of $f^{1387}$, we have $f^{1387}(x)=x$. Notice that in this problem, \[f^{1387}(x)=\underbrace{f(f(\cdots(f(x)))\cdots)}_{\text{1387 times}}.\] [i]Hint[/i]. Prove that for every function $g(x)=\frac{sx+t}{ux+v}$, if the equation $g(x)=x$ has more than $2$ roots, then $g(x)=x$ for all $x\in\mathbb{R}-\left\{\frac{-v}{u}\right\}$.

Let $a$, $b$, and $c$ be positive real numbers such that $a^2+b^2+c^2=989$ and $(a+b)^2+(b+c)^2+(c+a)^2=2013$. Find $a+b+c$.

Let $k_0$ be a unit semi-circle with diameter $AB$. Assume that $k_1$ is a circle of radius $r_1=\frac12$ that is tangent to both $k_0$ and $AB$. The circle $k_{n+1}$ of radius $r_{n+1}$ touches $k_n,k_0$, and $AB$. Prove that: (a) For each $n\in\{2,3,\ldots\}$ it holds that $\frac1{r_{n+1}}+\frac1{r_{n-1}}=\frac6{r_n}-4$. (b) $\frac1{r_n}$ is either a square of an even integer, or twice a square of an odd integer.

The number $$\frac{9^9-8^8}{1001}$$ is an integer. Compute the sum of its prime factors.

Let $a,b,c$ positive reals such that $a^2+b^2+c^2=1$. Prove that $$\frac{a+b}{\sqrt{ab+1}}+\frac{b+c}{\sqrt{bc+1}}+\frac{c+a}{\sqrt{ac+1}}\le 3$$

Let $ABCD$ be a convex quadrilateral in which $\angle BAC = 50^{\circ}, \angle CAD = 60^{\circ}$and $\angle BDC = 25^{\circ}$. If $E$ is the point of intersection of $AC$ and $BD$, find $\angle AEB$.

On the party every boy gave $1$ candy to every girl and every girl gave $1$ candy to every boy. Then every boy ate $2$ candies and every girl ate $3$ candies. It is known that $\frac 14$ of all candies was eaten. Find the greatest possible number of children on the party.

Let $\Gamma_1$ and $\Gamma_2$ be circles in the plane with centers $O_1$ and $O_2$ and radii $13$ and $10$, respectively. Assume $O_1O_2=2$. Fix a circle $\Omega$ with radius $2$, internally tangent to $\Gamma_1$ at $P$ and externally tangent to $\Gamma_2$ at $Q$ . Let $\omega$ be a second variable circle internally tangent to $\Gamma_1$ at $X$ and externally tangent to $\Gamma_2$ at $Y$. Line $PQ$ meets $\Gamma_2$ again at $R$, line $XY$ meets $\Gamma_2$ again at $Z$, and lines $PZ$ and $XR$ meet at $M$. As $\omega$ varies, the locus of point $M$ encloses a region of area $\tfrac{p}{q} \pi$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$. [i]Proposed by Michael Kural[/i]

A polynomial $c_dx^d+c_{d-1}x^{d-1}+\dots+c_1x+c_0$ with degree $d$ is [i]reflexive[/i] if there is an integer $n\ge d$ such that $c_i=c_{n-i}$ for every $0\le i\le n$, where $c_i=0$ for $i>d$. Let $\ell\ge 2$ be an integer and $p(x)$ be a polynomial with integer coefficients. Prove that there exist reflexive polynomials $q(x)$, $r(x)$ with integer coefficients such that \[(1+x+x^2+\dots+x^{\ell-1})p(x)=q(x)+x^\ell r(x)\]

At a two-round volleyball tournament participated 99 teams. Each played a match at home and a match away. Each team won exactly half of their home matches and exactly half of their away matches. Prove that one of the teams beat another team twice. [i]Proposed by M. Antipov[/i]

Determine all positive integers $k$ for which there exist positive integers $n$ and $m, m\ge 2$, such that $3^k + 5^k = n^m$