A parabola with equation $ y \equal{} x^2 \plus{} bx \plus{} c$ passes through the points $ (2,3)$ and $ (4,3)$. What is $ c$? $ \textbf{(A) } 2 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11$

Let $f(x)$ be a periodic function of period $T > 0$ defined over $\mathbb R$. Its first derivative is continuous on $\mathbb R$. Prove that there exist $x, y \in [0, T )$ such that $x \neq y$ and \[f(x)f'(y)=f'(x)f(y).\]

Prove that the product of all integers from $2^{1917} +1$ up to $2^{1991} -1$ is not the square of an integer. (V. Senderov, Moscow)

Alex starts with a rooted tree with one vertex (the root). For a vertex $v$, let the size of the subtree of $v$ be $S(v)$. Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex $v_1$). The expected value of $S(v_1)$ is of the form $\tfrac{m}{n}$ for relatively prime positive integers $m, n$. Find $m+n$. [b]Note:[/b] In a rooted tree, the subtree of $v$ consists of its indirect or direct descendants (including $v$ itself). [i]Proposed by Yang Liu[/i]

Joyce made $12$ of her first $30$ shots in the first three games of this basketball game, so her seasonal shooting average was $40\% $. In her next game, she took $10$ shots and raised her seasonal shooting average to $50\% $. How many of these $10$ shots did she make? $\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$

A [i]uniform covering[/i] of the integers $1,2,...,n$ is a finite multiset of subsets of $\{1,2,...,n\}$, so that each number lies in the same amount of sets from the covering. A covering may contain the same subset multiple times, it must contain at least one subset, and it may contain the empty subset. For example, $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})$ is a uniform covering of $1,2,3,4$ (every number occurs in two sets). The covering containing only the empty set is also uniform (every number occurs in zero sets). Given two uniform coverings, we define a new uniform covering, their [i]sum[/i] (denoted by $\oplus$), by adding the sets from both coverings. For example: $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})\oplus(\{1\},\{2\},\{3\},\{4\})=$ $(\{1\},\{1\},\{1\},\{2\},\{3\},\{4\},\{2,3\},\{3,4\},\{2,4\})$ A uniform covering is called [i]non-composite[/i] if it's not a sum of two uniform coverings. Prove that for any $n\geq1$, there are only finitely many non-composite uniform coverings of $1,2,...,n$.

A rectangular checkered board is painted black and white as a chessboard, and is tiled by dominoes $1 \times 2$. If a horizontal and a vertical dominoes have common segment, it has a door which has the color of the adjoining cell of the domino adjacent by a short side. Is it necessarily true that the number of white doors equals the number of black doors?

Let $ABC$ be an acute-angled triangle with circumcenter $O$ and incenter $I$. The midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$ is denoted $S$. Points $E, F$ were chosen on line $OI$ for which $BE$ and $CF$ are both perpendicular to $OI$. Point $X$ was chosen so that $XE\perp AC$ and $XF\perp AB$. Point $Y$ was chosen so that $YE\perp SC$ and $YF\perp SB$. $D$ was chosen on $BC$ so that $DI\perp BC$. Prove that $X$, $Y$, and $D$ are collinear.

Find the sum of the decimal digits of the number \[ 5\sum_{k=1}^{99} k(k + 1)(k^2 + k + 1). \] [i]Proposed by Robin Park[/i]

Ana and Bojan are playing a game: Ana chooses positive integers $a$ and $b$ and each one gets $2016$ pieces of paper, visible to both - Ana gets the pieces with the numbers $a+1$, $a+2$, $\ldots$, $a+2016$ and Bojan gets the pieces with the numbers $b+1$, $b+2$, $\ldots$, $b+2016$ on them. Afterwards, one of them writes the number $a+b$ on the board. In every move, Ana chooses one of her pieces of paper and hands it to Bojan who chooses one of his own, writes their sum on the board and removes them both from the game. When they run out of pieces, they multiply the numbers on the board together. If the result has the same remainder than $a+b$ when divided by $2017$, Bojan wins, otherwise, Ana wins. Who has the winning strategy?

Two perpendicular lines pass through the orthocenter of an acute-angled triangle. The sidelines of the triangle cut on each of these lines two segments: one lying inside the triangle and another one lying outside it. Prove that the product of two internal segments is equal to the product of two external segments. [i]Nikolai Beluhov and Emil Kolev[/i]

The angle bisector of $\angle ABC$ of triangle $ABC$ ($AB>BC$) cuts the circumcircle of that triangle in $K$. The foot of the perpendicular from $K$ to $AB$ is $N$, and $P$ is the midpoint of $BN$. The line through $P$ parallel to $BC$ cuts line $BK$ in $T$. Prove that the line $NT$ passes through the midpoint of $AC$.

A $\textit{divisibility chain}$ is a sequence of positive integers $(a_1, a_2, \ldots, a_n)$ such that $a_k$ divides $a_{k+1}$ for all $1 \le k < n $. Compute the number of divisibility chains of the form $(a, b, a^2, c, a^3, 360^9)$. [i]Proposed by Michael Tang[/i]

Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$, $60^\circ$, and $60.001^\circ$. For each positive integer $n$ define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\triangle A_nB_nC_n$ is obtuse? $\phantom{}$ $\textbf{(A) } 10 \qquad \textbf{(B) }11 \qquad \textbf{(C) } 13\qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15$

A square, with an area of $40$, is inscribed in a semicircle. The area of a square that could be inscribed in the entire circle with the same radius, is: $ \textbf{(A)}\ 80 \qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 120\qquad\textbf{(D)}\ 160\qquad\textbf{(E)}\ 200 $

$a,b,c$ are nonnegative integers that satisfy $a^2+b^2+c^2=3$. Find the minimum and maximum value the sum $$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a}$$ may achieve and find all $a,b,c$ for which equality occurs.\\ \\ [i](Andrei Bâra)[/i]

Let $s_1$ be the sum of the first $n$ terms of the arithmetic sequence $8,12,\cdots$ and let $s_2$ be the sum of the first $n$ terms of the arithmetic sequence $17,19\cdots$. Assume $n\ne 0$. Then $s_1=s_2$ for: $\text{(A)} \ \text{no value of n} \qquad \text{(B)} \ \text{one value of n} \qquad \text{(C)} \ \text{two values of n}$ $\text{(D)} \ \text{four values of n} \qquad \text{(E)} \ \text{more than four values of n}$

Circles ω[size=35]1[/size] and ω[size=35]2[/size] have no common point. Where is outerior tangents a and b, interior tangent c. Lines a, b and c touches circle ω[size=35]1[/size] respectively on points A[size=35]1[/size], B[size=35]1[/size] and C[size=35]1[/size], and circle ω[size=35]2[/size] – respectively on points A[size=35]2[/size], B[size=35]2[/size] and C[size=35]2[/size]. Prove that triangles A[size=35]1[/size]B[size=35]1[/size]C[size=35]1[/size] and A[size=35]2[/size]B[size=35]2[/size]C[size=35]2[/size] area ratio is the same as ratio of ω[size=35]1[/size] and ω[size=35]2[/size] radii.

Let $BD$ and $CE$ be the altitudes of triangle $ABC$ that intersect at point $H$. Let $F$ be a point on side $AC$ such that $FH\perp CE$. The segment $FE$ intersects the circumcircle of triangle $CDE$ at the point $K$. Prove that $HK\perp EF$ . (Matthew Kurskyi)

Is it possible to cut a square of side $1$ into two parts and rearrange them so that one can cover a circle having diameter greater than $1$? (Note: any circle with diameter greater than $1$ suffices) [i](A. Shapovalov)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

Choose nine of the digits from $0$ to $9$ and place them in the boxes in the figure so that there are no repeated digits and the indicated sum is correct. [img]https://cdn.artofproblemsolving.com/attachments/6/2/7f06575ec70eb9ddd58c6cf9dd3cb60d306e7c.png[/img] Which digit was not used? You can fill in the boxes so that the unused digit is other?

There are some players in a Ping Pong tournament, where every $2$ players play with each other at most once. Given: \\(1) Each player wins at least $a$ players, and loses to at least $b$ players. ($a,b\geq 1$) \\(2) For any two players $A,B$, there exist some players $P_1,...,P_k$ ($k\geq 2$) (where $P_1=A$,$P_k=B$), such that $P_i$ wins $P_{i+1}$ ($i=1,2...,k-1$). \\Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$, such that $Q_i$ wins $Q_{i+1}$ ($i=1,...,a+b$)

At 12 o'clock in the afternoon, "Zaporozhets" and "Moskvich" were at a distance of 90 km and began to move towards each other at a constant speed. Two hours later they were again at a distance of 90 km. Dunno claims that ''Zaporozhets'' before meeting with ''Moskvich'' and ''Moskvich'' after the meeting with ''Zaporozhets'' , have drove a total of 60 km. Prove that he is wrong. [hide=original wording]В 12 часов дня ''Запорожец'' и ''Москвич'' находилисьна расстоянии 90 км и начали двигаться навстречу друг другу с постоянной скоростью. Через два часа они снова оказались на расстоянии 90 км. Незнайка утверждает, что ''Запорожец'' до встречи с ''Москвичом'' и ''Москвич'' после встречи с ''Запорожцем'' проехали в сумме 60 км. Докажите, что он не прав. [/hide]

Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation \[ f(f(f(n))) = f(n+1 ) +1 \] for all $ n\in \mathbb{Z}_{\ge 0}$.

Given three non-negative real numbers $a$, $b$ and $c$. The sum of the modules of their pairwise differences is equal to $1$, i.e. $|a- b| + |b -c| + |c -a| = 1$. What can the sum $a + b + c$ be equal to?