In $\triangle ABC$, $AB>AC$, the bisectors of $\angle ABC, \angle ACB$ meet sides $AC,AB$ at $D,E$ respectively. The tangent at $A$ to the circumcircle of $\triangle ABC$ intersects $ED$ extended at $P$. Suppose that $AP=BC$. Prove that $BD\parallel CP$.

If $\log_{2x}216 = x$, where $x$ is real, then $x$ is: $ \textbf{(A)}\ \text{A non-square, non-cube integer} \qquad$ $\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad$ $\textbf{(C)}\ \text{An irrational number} \qquad$ $\textbf{(D)}\ \text{A perfect square}\qquad$ $\textbf{(E)}\ \text{A perfect cube} $

A clock has a second, minute, and hour hand. A fly initially rides on the second hand of the clock starting at noon. Every time the hand the fly is currently riding crosses with another, the fly will then switch to riding the other hand. Once the clock strikes midnight, how many revolutions has the fly taken? $\emph{(Observe that no three hands of a clock coincide between noon and midnight.)}$

A convex set $S$ on a plane, not lying on a line, is painted in $p$ colors. Prove that for every $n \ge 3$ there exist infinitely many congruent $n$-gons whose vertices are of the same color.

Let $ABCD$ be a convex quadrilateral with $AB=2, AD=7,$ and $CD=3$ such that the bisectors of acute angles $\angle{DAB}$ and $\angle{ADC}$ intersect at the midpoint of $\overline{BC}.$ Find the square of the area of $ABCD.$

$\ddot{x}=3x-x^3-1$. In which of the potential wells is the period of oscillation greater (in the shallower or the deeper) with equal values of the total energy?

For each positive integer $n$ denote: \[n!=1\cdot 2\cdot 3\dots n\] Find all positive integers $n$ for which $1!+2!+3!+\cdots+n!$ is a perfect square.

Say a real number $r$ is \emph{repetitive} if there exist two distinct complex numbers $z_1,z_2$ with $|z_1|=|z_2|=1$ and $\{z_1,z_2\}\neq\{-i,i\}$ such that \[ z_1(z_1^3+z_1^2+rz_1+1)=z_2(z_2^3+z_2^2+rz_2+1). \] There exist real numbers $a,b$ such that a real number $r$ is \emph{repetitive} if and only if $a < r\le b$. If the value of $|a|+|b|$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $100p+q$. [i]Proposed by James Lin[/i]

Given are two polyominos, the first one is an L-shape consisting of three squares, the other one contains at least two squares. Prove that if $n$ and $m$ are coprime then at most one of the $n\times n$ and $m\times m$ boards can be tiled by translated copies of the two polyominos. [i]Proposed by: András Imolay, Dávid Matolcsi, Ádám Schweitzer and Kristóf Szabó, Budapest[/i]

Initially, a pair of numbers $(1,1)$ is written on the board. If for some $x$ and $y$ one of the pairs $(x, y-1)$ and $(x+y, y+1)$ is written on the board, then you can add the other one. Similarly for $(x, xy)$ and $(\frac {1} {x}, y)$. Prove that for each pair that appears on the board, its first number will be positive.

$a,b$ and $c$ are positive real numbers so that $\sum_{\text{cyc}} (a+b)^2=2\sum_{\text{cyc}} a +6abc$. Prove that $$\sum_{\text{cyc}} (a-b)^2\leq\left|2\sum_{\text{cyc}} a -6abc\right|.$$

Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that 1) $f(0,x)$ is non-decreasing ; 2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ; 3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ; 4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ .

There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.

In acute angled triangle $ABC$, $BC=a$,$CA=b$,$AB=c$, and $a>b>c$. $I,O,H$ are the incentre, circumcentre and orthocentre of $\triangle{ABC}$ respectively. Point $D \in BC$, $E \in CA$ and $AE=BD$, $CD+CE=AB$. Let the intersectionf of $BE$ and $AD$ be $K$. Prove that $KH \parallel IO$ and $KH = 2IO$.

On each cell of a $200 \times 200$ grid, we place a car, which faces in one of the four cardinal directions.In a move, one chooses a car that does not have a car immediately in front of it, and slides it one cellforward. If a move would cause a car to exit the grid, the car is removed instead. The cars are placedso that there exists a sequence of moves that eventually removes all the cars from the grid. Across all such starting configurations, determine the maximum possible number of moves to do so.

Given are positive integers $a, b$ for which $5a + 3b$ is divisible by $a + b$. Prove that $a = b$.

Find the largest $n$ such that the last nonzero digit of $n!$ is $1$.

In a contest, there are $11$ contestants to solve $9$ math problems. After the end of the contest, it was found that any two contestants solved no more than $ 1$ problem together. Find the largest positive integer $k$ such that each problem can be solved by at least $k$ candidates.

Denote $f_n(X) \in \Bbb Z [X]$ the polynomial $\Pi_{j=1}^n ( X + j -1)$. Show that if the numbers $\alpha$ and $\beta$ satisfy $f'_{1997} (\alpha) = f'_{1999} (\beta) = 0$ , then $f_{1997} (\alpha ) \neq f_{1999} (\beta)$ .

Do either (1) or (2): (1) Show that any solution $f(t)$ of the functional equation $$f(x+y)f(x-y)=f(x)^{2} +f(y)^{2} -1$$ for $x,y\in \mathbb{R}$ satisfies $$f''(t)= \pm c^{2} f(t)$$ for a constant $c$, assuming the existence and continuity of the second derivative. Deduce that $f(t)$ is one of the functions $$ \pm \cos ct, \;\;\; \pm \cosh ct.$$ (2) Let $(a_{i})_{i=1,...,n}$ and $(b_{i})_{i=1,...,n}$ be real numbers. Define an $(n+1)\times (n+1)$-matrix $A=(c_{ij})$ by $$ c_{i1}=1, \; \; c_{1j}= x^{j-1} \; \text{for} \; j\leq n,\; \; c_{1n+1}=p(x), \;\; c_{ij}=a_{i-1}^{j-1} \; \text{for}\; i>1, j\leq n,\;\; c_{in+1}=b_{i-1}\; \text{for}\; i>1.$$ The polynomial $p(x)$ is defined by the equation $\det A=0$. Let $f$ be a polynomial and replace $(b_{i})$ with $(f(b_{i}))$. Then $\det A=0$ defines another polynomial $q(x)$. Prove that $f(p(x))-q(x)$ is a multiple of $$\prod_{i=1}^{n} (x-a_{i}).$$

We define $N$ as the set of natural numbers $n<10^6$ with the following property: There exists an integer exponent $k$ with $1\le k \le 43$, such that $2012|n^k-1$. Find $|N|$.

Let $y=(x-a)^2+(x-b)^2, a, b$ constants. For what value of $x$ is $y$ a minimum? $ \textbf{(A)}\ \frac{a+b}{2} \qquad\textbf{(B)}\ a+b \qquad\textbf{(C)}\ \sqrt{ab} \qquad\textbf{(D)}\ \sqrt{\frac{a^2+b^2}{2}}\qquad\textbf{(E)}\ \frac{a+b}{2ab} $

Find all pairs $(m,p)$ of natural numbers , such that $p$ is a prime and \[2^mp^2+27\] is the third power of a natural numbers

Let $N$ be the set of positive integers. Find all functions $f : N \to N$ such that both $\bullet$ $f(f(m)f(n)) = mn$ $\bullet$ $f(2022a + 1) = 2022a + 1$ hold for all positive integers $m, n$ and $a$.

[b]p1.[/b] David runs at $3$ times the speed of Alice. If Alice runs $2$ miles in $30$ minutes, determine how many minutes it takes for David to run a mile. [b]p2.[/b] Al has $2019$ red jelly beans. Bob has $2018$ green jelly beans. Carl has $x$ blue jelly beans. The minimum number of jelly beans that must be drawn in order to guarantee $2$ jelly beans of each color is $4041$. Compute $x$. [b]p3.[/b] Find the $7$-digit palindrome which is divisible by $7$ and whose first three digits are all $2$. [b]p4.[/b] Determine the number of ways to put $5$ indistinguishable balls in $6$ distinguishable boxes. [b]p5.[/b] A certain reduced fraction $\frac{a}{b}$ (with $a,b > 1$) has the property that when $2$ is subtracted from the numerator and added to the denominator, the resulting fraction has $\frac16$ of its original value. Find this fraction. [b]p6.[/b] Find the smallest positive integer $n$ such that $|\tau(n +1)-\tau(n)| = 7$. Here, $\tau(n)$ denotes the number of divisors of $n$. [b]p7.[/b] Let $\vartriangle ABC$ be the triangle such that $AB = 3$, $AC = 6$ and $\angle BAC = 120^o$. Let $D$ be the point on $BC$ such that $AD$ bisect $\angle BAC$. Compute the length of $AD$. [b]p8.[/b] $26$ points are evenly spaced around a circle and are labeled $A$ through $Z$ in alphabetical order. Triangle $\vartriangle LMT$ is drawn. Three more points, each distinct from $L, M$, and $T$ , are chosen to form a second triangle. Compute the probability that the two triangles do not overlap. [b]p9.[/b] Given the three equations $a +b +c = 0$ $a^2 +b^2 +c^2 = 2$ $a^3 +b^3 +c^3 = 19$ find $abc$. [b]p10.[/b] Circle $\omega$ is inscribed in convex quadrilateral $ABCD$ and tangent to $AB$ and $CD$ at $P$ and $Q$, respectively. Given that $AP = 175$, $BP = 147$,$CQ = 75$, and $AB \parallel CD$, find the length of $DQ$. [b]p11. [/b]Let $p$ be a prime and m be a positive integer such that $157p = m^4 +2m^3 +m^2 +3$. Find the ordered pair $(p,m)$. [b]p12.[/b] Find the number of possible functions $f : \{-2,-1, 0, 1, 2\} \to \{-2,-1, 0, 1, 2\}$ that satisfy the following conditions. (1) $f (x) \ne f (y)$ when $x \ne y$ (2) There exists some $x$ such that $f (x)^2 = x^2$ [b]p13.[/b] Let $p$ be a prime number such that there exists positive integer $n$ such that $41pn -42p^2 = n^3$. Find the sum of all possible values of $p$. [b]p14.[/b] An equilateral triangle with side length $ 1$ is rotated $60$ degrees around its center. Compute the area of the region swept out by the interior of the triangle. [b]p15.[/b] Let $\sigma (n)$ denote the number of positive integer divisors of $n$. Find the sum of all $n$ that satisfy the equation $\sigma (n) =\frac{n}{3}$. [b]p16[/b]. Let $C$ be the set of points $\{a,b,c\} \in Z$ for $0 \le a,b,c \le 10$. Alice starts at $(0,0,0)$. Every second she randomly moves to one of the other points in $C$ that is on one of the lines parallel to the $x, y$, and $z$ axes through the point she is currently at, each point with equal probability. Determine the expected number of seconds it will take her to reach $(10,10,10)$. [b]p17.[/b] Find the maximum possible value of $$abc \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^3$$ where $a,b,c$ are real such that $a +b +c = 0$. [b]p18.[/b] Circle $\omega$ with radius $6$ is inscribed within quadrilateral $ABCD$. $\omega$ is tangent to $AB$, $BC$, $CD$, and $DA$ at $E, F, G$, and $H$ respectively. If $AE = 3$, $BF = 4$ and $CG = 5$, find the length of $DH$. [b]p19.[/b] Find the maximum integer $p$ less than $1000$ for which there exists a positive integer $q$ such that the cubic equation $$x^3 - px^2 + q x -(p^2 -4q +4) = 0$$ has three roots which are all positive integers. [b]p20.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle ABC = 60^o$,$\angle ACB = 20^o$. Let $P$ be the point such that $CP$ bisects $\angle ACB$ and $\angle PAC = 30^o$. Find $\angle PBC$. PS. You had better use hide for answers.