Eddie has a study block that lasts $1$ hour. It takes Eddie $25$ minutes to do his homework and $5$ minutes to play a game of Clash Royale. He can’t do both at the same time. How many games can he play in this study block while still completing his homework? [i]Proposed by Edwin Zhao[/i] [hide=Solution] [i]Solution.[/i] $\boxed{7}$ Study block lasts 60 minutes, thus he has 35 minutes to play Clash Royale, during which he can play $\frac{35}{5}=\boxed{7}$ games. [/hide]

There are $13$ positive integers greater than $\sqrt{15}$ and less than $\sqrt[3]{B}$. What is the smallest integer value of $B$?

Prove that a triangle $ABC$ is right-angled if and only if \[\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1\]

$20$ numbers are written on the board $1, 2, ... ,20$. Two players are putting signs before the numbers in turn ($+$ or $-$). The first wants to obtain the minimal possible absolute value of the sum. What is the maximal value of the absolute value of the sum that can be achieved by the second player?

Determine the smallest positive number $a$ such that the number of all integers belonging to $(a, 2016a]$ is $2016$.

Let $ a_1, a_2, \ldots, a_{2n}$ be $ 2n$ real numbers satisfying the condition $ \sum_{i \equal{} 1}^{2n \minus{} 1} (a_{i \plus{} 1} \minus{} a_i)^2 \equal{} 1$. Find the greatest possible value of $ (a_{n \plus{} 1} \plus{} a_{n \plus{} 2} \plus{} \ldots \plus{} a_{2n}) \minus{} (a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n)$.

Three machines $ \text{P, Q, and R,}$ working together, can do a job in $ x$ hours. When working alone, $ \text{P}$ needs an additional $ 6$ hours to do the job; $ \text{Q}$, one additional hour; and $ R$, $ x$ additional hours. The value of $ x$ is: $ \textbf{(A)}\ \frac23 \qquad \textbf{(B)}\ \frac{11}{12} \qquad \textbf{(C)}\ \frac32 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

Let $ n$ be an integer, $ n\geq 2$. Find all sets $ A$ with $ n$ integer elements such that the sum of any nonempty subset of $ A$ is not divisible by $ n\plus{}1$.

In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$. [img]https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png[/img]

Find the sum of all positive integers $n$ such that \[ \frac{2n+1}{n(n-1)} \] has a terminating decimal representation. [i]Proposed by Evan Chen[/i]

Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.

Find all polynomials $P$ with real coefficients such that for all $x, y ,z \in R$, $$P(x)+P(y)+P(z)+P(x+y+z)=P(x+y)+P(y+z)+P(z+x)$$

Consider a pool table with the shape of an equilateral triangle. A ball of negligible size is initially placed at the center of the table. After it has been hit, it will keep moving in the direction it was hit towards and bounce off any edges with perfect symmetry. If it eventually reaches the midpoint of any edge, we mark the midpoint of the entire route that the ball has travelled through. Repeating this experiment, how many points can we mark at most? [img]https://cdn.artofproblemsolving.com/attachments/5/d/3ae7aad4271b9a417826f3bc8dd7b43aeca5d4.png[/img]

Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.

Two players in turn paint cells of the $7\times7$ table each using own color. A player can't paint a cell if its row or its column contains a cell painted by the other player. The game stops when one of the players can't make his turn. What maximal number of the cells can remain unpainted when the game stops?

Find all triples $(a, b, c)$ of positive integers for which $\frac{32a + 3b + 48c}{4abc}$ is also an integer.

Find all positive $r{}$ satisfying the following condition: For any $d > 0$, there exist two circles of radius $r{}$ in the plane that do not contain lattice points strictly inside them and such that the distance between their centers is $d{}$.

Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $.

Consider a square $ABCD$ with side length $4$ and label the midpoint of side $BC$ as $M$. Let $X$ be the point along $AM$ obtained by dropping a perpendicular from $D$ onto $AM$. Compute the product of the lengths $XC$ and $MD$.

Let $D$ and $E$ be points on the sides $BC$ and $AC$ of the triangle $\triangle ABC$, respectively. The circumcircle of $\triangle ADC$ meets the circumcircle of $\triangle BCE$ for the second time at $F$. The line $FE$ meets the line $AD$ at $G$, while the line $FD$ meets the line $BE$ at $H$. Prove that the lines $CF$, $AH$ and $BG$ pass through the same point. [i]Authored by Petar Filipovski[/i]

There are $11$ members in the competetion committee. The problem set is kept in a safe having several locks. The committee members have been provided with keys in such a way that every six members can open the safe, but no five members can do that. What is the smallest possible number of locks, and how many keys are needed in that case?

Merlin summons the $n$ knights of Camelot for a conference. Each day, he assigns them to the $n$ seats at the Round Table. From the second day on, any two neighbours may interchange their seats if they were not neighbours on the first day. The knights try to sit in some cyclic order which has already occurred before on an earlier day. If they succeed, then the conference comes to an end when the day is over. What is the maximum number of days for which Merlin can guarantee that the conference will last?

Consider systems of three linear equations with unknowns $x,$ $y,$ and $z,$ \begin{align*} a_1 x + b_1 y + c_1 z = 0 \\ a_2 x + b_2 y + c_2 z = 0 \\ a_3 x + b_3 y + c_3 z = 0 \end{align*} where each of the coefficients is either $0$ or $1$ and the system has a solution other than $x = y = z = 0.$ For example, one such system is $\{1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0\}$ with a nonzero solution of $\{x, y, z\} = \{1, -1, 1\}.$ How many such systems are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.) $\textbf{(A) } 302 \qquad \textbf{(B) } 338 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 343 \qquad \textbf{(E) } 344$

Consider the cube with the vertices at the points $(\pm 1, \pm 1, \pm 1)$. Let $V_1,...,V_{95}$ be arbitrary points within this cube. Denote $v_i = \overrightarrow{OV_i}$, where $O = (0,0,0)$ is the origin. Consider the $2^{95}$ vectors of the form $s_1v_1 + s_2v_2 +...+ s_{95}v_{95}$, where $s_i = \pm 1$. (a) If $d = 48$, prove that among these vectors there is a vector $w = (a, b, c)$ such that $a^2 + b^2 + c^2 \le 48$. (b) Find a smaller $d$ (the smaller, the better) with the same property.

$n>1$ is an integer. Let real number $x>1$ satisfy $$x^{101}-nx^{100}+nx-1=0.$$ Prove that for any real $0<a<b<1$, there exists a positive integer $m$ so that $a<\{x^m\}<b.$ [i]Proposed by Chenjie Yu[/i]