If $\angle A$ is four times $\angle B$, and the complement of $\angle B$ is four times the complement of $\angle A$, then $\angle B=$ $ \textbf{(A)}\ 10^{\circ} \qquad\textbf{(B)}\ 12^{\circ} \qquad\textbf{(C)}\ 15^{\circ} \qquad\textbf{(D)}\ 18^{\circ} \qquad\textbf{(E)}\ 22.5^{\circ} $

Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite. [i]Proposed by Carl Schildkraut[/i]

Point $P$ is located inside an acute-angled triangle $ABC$. $A_1$, $B_1$, $C_1$ are points symmetric to $P$ with respect to the sides of triangle $ABC$. It turned out that the hexagon $AB_1CA_1BC_1$ is inscribed. Prove that $P$ is the Torricelli point of triangle $ABC$.

Given the quadratic trinomial $ f (x) = x ^ 2 + ax + b $ with integer coefficients, satisfying the inequality $ f (x) \geq - {9 \over 10} $ for any $ x $. Prove that $ f (x) \geq - {1 \over 4} $ for any $ x $.

In a triangle $ABC$, let $x=\cos\frac{A-B}{2},y=\cos\frac{B-C}{2},z=\cos\frac{C-A}{2}$. Prove that $$x^4+y^4+z^2\leq 1+2x^2y^2z^2.$$

In acute-angled triangle $ABC$, altitude of $A$ meets $BC$ at $D$, and $M$ is midpoint of $AC$. Suppose that $X$ is a point such that $\measuredangle AXB = \measuredangle DXM =90^\circ$ (assume that $X$ and $C$ lie on opposite sides of the line $BM$). Show that $\measuredangle XMB = 2\measuredangle MBC$.Proposed by Davood Vakili

Segments $AC$ and $BD$ intersect at point $P$ such that $PA = PD$ and $PB = PC$. Let $E$ be the foot of the perpendicular from $P$ to the line $CD$. Prove that the line $PE$ and the perpendicular bisectors of the segments $PA$ and $PB$ are concurrent.

$S=\{(x,y)|x^2-y^2 \text{is odd},x,y\in\mathbb{R}\},T=\{(x,y)|\sin(2\pi x^2)-\sin(2\pi y^2)=\cos(2\pi x^2)-\cos(2\pi y^2),x,y\in\mathbb{R}\}$, then $\text{(A)}S\subset T\qquad\text{(B)}T\subset S\qquad\text{(C)}S=T\qquad\text{(D)}S\cap T=\varnothing$

Let $ABC$ be an acute-angled triangle with incircle $\omega$ and incenter $I$. Let $\omega$ touch $AB, BC$ and $CA $ at points $D, E, F$ respectively. The circles $\omega_1$ and $\omega_2$ centered at $J_1$ and $J_2$ respectively are inscribed into A$DIF$ and $BDIE$. Let $J_1J_2$ intersect $AB$ at point $M$. Prove that $CD$ is perpendicular to $IM$.

A cube has side length $5$. Let $S$ be its surface area and $V$ its volume. Find $\frac{S^3}{V^2}$ .

In Wonderland, the towns are connected by roads, and whenever there is a direct road between two towns there is also a route between these two towns that does not use that road. (There is at most one direct road between any two towns.) The Queen of Hearts ordered the Spades to provide a list of all ”even” subsystems of the system of roads, that is, systems formed by subsets of the set of roads, where each town is connected to an even number of roads (possibly none). For each such subsystem they should list its roads. If there are totally $n$ roads in Wonderland and $x$ subsystems on the Spades’ list, what is the number of roads on their list when each road is counted as many times as it is listed?

For any positive integer $n$, let $a_n$ be the number of pairs $(x,y)$ of integers satisfying $|x^2-y^2| = n$. (a) Find $a_{1432}$ and $a_{1433}$. (b) Find $a_n$ .

If $A=(a_1,\cdots,a_n)$ , $B=(b_1,\cdots,b_n)$ be $2$ $n-$tuple that $a_i,b_i=0 \ or \ 1$ for $i=1,2,\cdots,n$, we define $f(A,B)$ the number of $1\leq i\leq n$ that $a_i\ne b_i$. For instance, if $A=(0,1,1)$ , $B=(1,1,0)$, then $f(A,B)=2$. Now, let $A=(a_1,\cdots,a_n)$ , $B=(b_1,\cdots,b_n)$ , $C=(c_1,\cdots,c_n)$ be 3 $n-$tuple, such that for $i=1,2,\cdots,n$, $a_i,b_i,c_i=0 \ or \ 1$ and $f(A,B)=f(A,C)=f(B,C)=d$. $a)$ Prove that $d$ is even. $b)$ Prove that there exists a $n-$tuple $D=(d_1,\cdots,d_n)$ that $d_i=0 \ or \ 1$ for $i=1,2,\cdots,n$, such that $f(A,D)=f(B,D)=f(C,D)=\frac{d}{2}$.

Let $ABC$ be a non-degenerate triangle with incenter $I$ and circumcircle $\Gamma$. Denote by $M_a$ the midpoint of the arc $\widehat{BC}$ of $\Gamma$ not containing $A$, and define $M_b$, $M_c$ similarly. Suppose $\triangle ABC$ has inradius $4$ and circumradius $9$. Compute the maximum possible value of \[IM_a^2+IM_b^2+IM_c^2.\][i]Proposed by David Altizio[/i]

Let $ ABCD$ be a cyclic quadrilater. Denote $ P\equal{}AD\cap BC$ and $ Q\equal{}AB \cap CD$. Let $ E$ be the fourth vertex of the parallelogram $ ABCE$ and $ F\equal{}CE\cap PQ$. Prove that $ D,E,F$ and $ Q$ lie on the same circle.

Let $\{a_n\}$ be the sequence such that $a_0=2019$ and $$a_n=-\frac{2020}{n}\sum_{k=0}^{n-1}a_k.$$ Compute the last three digits of $\sum_{n=1}^{2020}2020^na_nn$.

$O$ is the circumcenter of $ABC$, and $H$ is the orthocenter of $ABC$. If $D$ is a midpoint of $AC$ and $E$ is the intersection of $BO$ and $ABC$'s circumcircle not $B$, show that three points $H, D, E$ are collinear.

For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.

Consider real numbers $A$, $B$, \dots, $Z$ such that \[ EVIL = \frac{5}{31}, \; LOVE = \frac{6}{29}, \text{ and } IMO = \frac{7}{3}. \] If $OMO = \tfrac mn$ for relatively prime positive integers $m$ and $n$, find the value of $m+n$. [i]Proposed by Evan Chen[/i]

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

In trapezoid $ABCD$ with $AB || CD$, $\omega_1$ and $\omega_2$ are two circles with diameters $AD$ and $BC$, respectively. Let $X$ and $Y$ be two arbitrary points on $\omega_1$ and $\omega_2$, respectively. Show that the length of segment $XY$ is not more than half the perimeter of $ABCD$. [i]Proposed by Mahdi Etesami Fard[/i]

A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.

Let $P$ be a point inside a quadrilateral $ABCD$ such that $\angle APB+\angle CPD=180^{\circ}$. Points $P_{a}$, $P_{b}$, $P_{c},$ $P_{d}$ are isogonally conjugated to $P$ with respect to the triangles $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the diagonals of the quadrilaterals $ABCD$ and $P_{a}P_{b}P_{c}P_{d}$ concur. Proposed by: G.Galyapin

Each face of a cube has been divided into four equal quarters and each quarter is painted with one of three available colours. Quarters with common sides are painted with different colours . Prove that each of the available colours was used in painting $8$ quarters.

Prove that in a plane, arbitrary $ n$ points can be overlapped by discs that the sum of all the diameters is less than $ n$, and the distances between arbitrary two are greater than $ 1$. (where the distances between two discs that have no common points are defined as that the distances between its centers subtract the sum of its radii; the distances between two discs that have common points are zero)