Let $\triangle ABC$ be a nondegenerate isosceles triangle with $AB=AC$, and let $D, E, F$ be the midpoints of $BC, CA, AB$ respectively. $BE$ intersects the circumcircle of $\triangle ABC$ again at $G$, and $H$ is the midpoint of minor arc $BC$. $CF\cap DG=I, BI\cap AC=J$. Prove that $\angle BJH=\angle ADG$ if and only if $\angle BID=\angle GBC$. [i]Proposed by David Stoner[/i]

Let $ABC$ be a triangle, and erect three rectangles $ABB_1A_2$, $BCC_1B_2$, $CAA_1C_2$ externally on its sides $AB$, $BC$, $CA$, respectively. Prove that the perpendicular bisectors of the segments $A_1A_2$, $B_1B_2$, $C_1C_2$ are concurrent.

Determine the set of all real numbers $r$ for which there exists an infinite sequence $a_1,a_2,\dots$ of positive integers satisfying the following three properties: (1) No number occurs more than once in the sequence. (2) The sum of two different elements of the sequence is never a power of two. (3) For all positive integers $n$, we have $a_n<r \cdot n$.

Given that the expression $\frac{20^{21}}{20^{20}} +\frac{20^{20}}{20^{21}}$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$. [i]Proposed by Ada Tsui[/i]

Let $\triangle ABC$ be an acute scalene triangle. Denote by $k_A$ the circle with diameter $BC$, and let $B_A,C_A$ be the contact points of the tangents from $A$ to $k_A$, chosen so that $B$ and $B_A$ lie on opposite sides of $AC$ and $C$ and $C_A$ lie on opposite sides of $AB$. Similarly, let $k_B$ be the circle with diameter $CA$, with tangents from $B$ touching at $C_B,A_B$, and $k_C$ the circle with diameter $AB$, with tangents from $C$ touching at $A_C,B_C$. Prove that the lines $B_AC_A, C_BA_B, A_CB_C$ are concurrent.

The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

A circle is contained in a quadrilateral with successive sides of lengths $3,6,5$ and $8$. Prove that the length of its radius is less than $3$. [i]Proposed by K. Kokhas[/i]

There are $n$($\ge 2$) clubs $A_1,A_2,...A_n$ in Korean Mathematical Society. Prove that there exist $n-1$ sets $B_1,B_2,...B_{n-1}$ that satisfy the condition below. (1) $A_1\cup A_2\cup \cdots A_n=B_1\cup B_2\cup \cdots B_{n-1}$ (2) for any $1\le i<j\le n-1$, $B_i\cap B_j=\emptyset, -1\le\left\vert B_i \right\vert -\left\vert B_j \right\vert\le 1$ (3) for any $1\le i \le n-1$, there exist $A_k,A_j $($1\le k\le j\le n$)such that $B_i\subseteq A_k\cup A_j$

A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths $1, 24,$ and $3,$ and the segment of length $24$ is a chord of the circle. Compute the area of the triangle.

Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$

Let $p$ be a prime and let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Assume that the numbers $f(1),f(2),\dots,f(p)$ leave exactly $k$ distinct remainders when divided by $p$, and $1<k<p$. Prove that \[ \frac{p-1}{d}\leq k-1\leq (p-1)\left(1-\frac1d \right) .\] [i] Dániel Domán, Gauls Károlyi, and Emil Kiss [/i]

Let $E$ be the point of intersection of the diagonals of the cyclic quadrilateral $ABCD$, and let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$, respectively. Prove that the radii of the circles circumscribed around the triangles $KLE$ and $MNE$ are equal.

Pedro and Tiago are playing a game with a deck of n cards, numbered from $1$ to $n$. Starting with Pedro, they choose cards alternately, and receive the number of points indicated by the cards. However, whenever the player chooses the card with the highest number among those remaining in the deck, he is forced to pass his next turn, not choosing any card. When the deck runs out, the player with the most points wins. Knowing that Tiago can at least draw, regardless of Pedro's moves, how many cards are in the deck? Indicates all possibilities,

Higher Secondary P5 Let $x>1$ be an integer such that for any two positive integers $a$ and $b$, if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$. Find with proof the number of positive integers that divide $x$.

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent. [i]Proposed by Robin Park[/i]

Square $ABCD$ is shown in the diagram below. Points $E$, $F$, and $G$ are on sides $\overline{AB}$, $\overline{BC}$ and $\overline{DA}$, respectively, so that lengths $\overline{BE}$, $\overline{BF}$, and $\overline{DG}$ are equal. Points $H$ and $I$ are the midpoints of segments $\overline{EF}$ and $\overline{CG}$, respectively. Segment $\overline{GJ}$ is the perpendicular bisector of segment $\overline{HI}$. The ratio of the areas of pentagon $AEHJG$ and quadrilateral $CIHF$ can be written as $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] draw((0,0)--(50,0)--(50,50)--(0,50)--cycle); label("$A$",(0,50),NW); label("$B$",(50,50),NE); label("$C$",(50,0),SE); label("$D$",(0,0),SW); label("$E$",(0,100/3-1),W); label("$F$",(100/3-1,0),S); label("$G$",(20,50),N); label("$H$",((100/3-1)/2,(100/3-1)/2),SW); label("$I$",(35,25),NE); label("$J$",(((100/3-1)/2+35)/2,((100/3-1)/2+25)/2),S); draw((0,100/3-1)--(100/3-1,0)); draw((20,50)--(50,0)); draw((100/6-1/2,100/6-1/2)--(35,25)); draw((((100/3-1)/2+35)/2,((100/3-1)/2+25)/2)--(20,50)); [/asy]

Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that $\bullet$ all of his happy snakes can add $\bullet$ none of his purple snakes can subtract, and $\bullet$ all of his snakes that can’t subtract also can’t add Which of these conclusions can be drawn about Tom’s snakes? $\textbf{(A)}$ Purple snakes can add. $\textbf{(B)}$ Purple snakes are happy. $\textbf{(C)}$ Snakes that can add are purple. $\textbf{(D)}$ Happy snakes are not purple. $\textbf{(E)}$ Happy snakes can't subtract.

Prove the among $16$ consecutive integers it is always possible to find one which is relatively prime to all the rest.

The product of any three of these four natural numbers is a perfect square. Prove that these numbers themselves are perfect squares.

Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?

Let $x = 1 - 2^{-2009}$. Show that $x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010$ for all positive integers $m$.

Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.

Suppose that $m,n$ are integers greater than $1$ such that $m+n-1$ divides $m^2+n^2-1$. Prove that $m+n-1$ cannot be a prime number.

Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that \[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \] and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$. [i]Proposed by Victor Wang[/i]

Let $x_i \in \{0,1\}(i=1,2,\cdots ,n)$,if the value of function $f=f(x_1,x_2, \cdots ,x_n)$ can only be $0$ or $1$,then we call $f$ a $n$-var Boole function,and we denote $D_n(f)=\{(x_1,x_2, \cdots ,x_n)|f(x_1,x_2, \cdots ,x_n)=0\}.$ $(1)$ Find the number of $n$-var Boole function; $(2)$ Let $g$ be a $n$-var Boole function such that $g(x_1,x_2, \cdots ,x_n) \equiv 1+x_1+x_1x_2+x_1x_2x_3 +\cdots +x_1x_2 \cdots x_n \pmod 2$, Find the number of elements of the set $D_n(g)$,and find the maximum of $n \in \mathbb{N}_+$ such that $\sum_{(x_1,x_2, \cdots ,x_n) \in D_n(g)}(x_1+x_2+ \cdots +x_n) \le 2017.$