Triangle $ABC$ has circumcenter $O$. $D$ is the foot from $A$ to $BC$, and $P$ is apoint on $AD$. The feet from $P$ to $CA, AB$ are $E, F$, respectively, and the foot from $D$ to $EF$ is $T$. $AO$ meets $(ABC)$ again at $A'$. $A'D$ meets $(ABC)$ again at $R$. If $Q$ is a point on $AO$ satisfying $\angle ABP = \angle QBC$, prove that $D, P, T, R$ lie on acircle and $DQ$ is tangent to it.

Let $m,n \in Z, m\ne n, m \ne 0, n \ne 0$ . Find all $f: Z \to R$ such that $f(mx+ny)=mf(x)+nf(y)$ for all $x,y \in Z$ .

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

With three different digits, six-digit numbers are written, multiples of $3$. One of the the digits are in the unit's place, another in the hundred's place, and the third in the remaining places. If we take out two units from the hundred's digit and add these to the unit's digit, the number is left with all the same digits. Find the numbers.

find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(-f(x)-f(y)) = 1-x-y$ $\quad \forall x,y \in \mathbb{Z}$

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?

A Magician has a deck of $52$ cards. Spectators want to know the order of cards in the deck(without specifying face-up or face-down). They are allowed to ask the questions “How many cards are there between such-and-such card and such-and-such card?” One of the spectators knows the card order. Find the minimal number of questions he needs to ask to be sure that the other spectators can learn the card order. (9)

Eva draws an equilateral triangle and its altitudes. In a first step she draws the center triangle of the equilateral triangle, in a second step the center triangle of this center triangle and so on. After each step Eva counts all triangles whose sides lie completely on drawn lines. What is the minimum number of center triangles she must have drawn so that the figure contains more than 2022 such triangles?

Let $ABC$ be an acute triangle such that $AB$ is the shortest side of the triangle. Let $D$ be the midpoint of the side $AB$ and $P$ be an interior point of the triangle such that $\angle CAP = \angle CBP = \angle ACB$. Denote by M and $N$ the feet of the perpendiculars from $P$ to $BC$ and $AC$, respectively. Let $p$ be the line through $ M$ parallel to $AC$ and $q$ be the line through $N$ parallel to $BC$. If $p$ and $q$ intersect at $K$ prove that $D$ is the circumcenter of triangle $MNK$.

Find all triples $(x,y,z)$ of positive integers such that $$\begin{cases} x + y +z = 2010 \\x^2 + y^2 + z^2 - xy - yz - zx =3 \end{cases}$$

In a round-robin tournament, where any two players play each other exactly once, the fact holds that among every three students $A$, $B$, and $C$, one of the students beats the other two. Given that there are six players in the tournament and Aidan beats Zach but loses to Andrew, find how many ways there are for the tournament to play out. Note: The order in which the matches take place does not matter. [i]Proposed by Kevin Zhao[/i]

A stick of length $10$ is marked with $9$ evenly spaced marks (so each is one unit apart). An ant is placed at every mark and at the endpoints, randomly facing either right or left. Suddenly, all the ants start walking simultaneously at a rate of $ 1$ unit per second. If two ants collide head-on, they immediately reverse direction (assume that turning takes no time). Ants fall off the stick as soon as they walk past the endpoints (so the two on the end don’t fall off immediately unless they are facing outwards). On average, how long (in seconds) will it take until all of the ants fall off of the stick?

$\sqrt{8}+\sqrt{18}=$ $\textbf{(A)}\ \sqrt{20} \qquad \textbf{(B)}\ 2(\sqrt{2}+\sqrt{3}) \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 5\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{13}$

In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$

What is the value of $(2^{-1})^{-2}$? [i]2015 CCA Math Bonanza Lightning Round #1.1[/i]

For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where \[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]

Let $n\geq 3$ and $A=\{1,2,\dots ,n\}$. For any function $f:A\rightarrow A$ we define $$A_f=\{|f(1)-f(2)|,|f(2)-f(3)|,\dots ,|f(n-1)-f(n)|,|f(n)-f(1)|\}.$$ Determine the smallest and greatest value of the cardinal of $A_f$ as $f$ goes through all bijective functions from $A$ to $A$. [i]Silviu Cristea[/i]

Determine all positive integers $n$ such that $5^n - 1$ can be written as a product of an even number of consecutive integers.

Let $ABC$ be triangle such that $|AB| = 5$, $|BC| = 9$ and $|AC| = 8$. The angle bisector of $\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\widehat{CAB}$ meets $BC$ at $Y$. Let $Z$ be the intersection of lines $XY$ and $AC$. What is $|AZ|$? $ \textbf{a)}\ \sqrt{104} \qquad\textbf{b)}\ \sqrt{145} \qquad\textbf{c)}\ \sqrt{89} \qquad\textbf{d)}\ 9 \qquad\textbf{e)}\ 10 $

Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that \[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

[b]p13[/b] Let $ABCD$ be a square. Points $E$ and $F$ lie outside of $ABCD$ such that $ABE$ and $CBF$ are equilateral triangles. If $G$ is the centroid of triangle $DEF$, then find $\angle AGC$, in degrees. [b]p14 [/b]The silent reading $s(n)$ of a positive integer $n$ is the number obtained by dropping the zeros not at the end of the number. For example, $s(1070030) = 1730$. Find the largest $n < 10000$ such that $s(n)$ divides $n$ and $n\ne s(n)$. [b]p15[/b] Let $ABCDEFGH$ be a regular octagon with side length $12$. There exists a region $R$ inside the octagon such that for each point $X$ in $R$, exactly three of the rays $AX$, $BX$, $CX$, $DX$, $GX$, and $HX$ intersect segment $EF$. If the area of region $R$ can be expressed as $a -b\sqrt{c}$ for positive integers $a, b, c$ with $c$ squarefree, find $a + b + c$.

Square $ABCD$ has center $O$, $AB=900$, $E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F$, $m\angle EOF =45^\circ$, and $EF=400$. Given that $BF=p+q\sqrt{r}$, wherer $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.

Find all positive integers n with the following property: There exists a positive integer $k$ and mutually distinct integers $x_1,x_2,\ldots,x_n$ such that the set $\{x_i+x_j\mid1\le i<j\le n\}$ is a set of distinct powers of $k$.

After the final exam, Mr. Liang asked each of his 17 students to guess the average final exam score. David, a very smart student, received a 100 and guessed the average would be 97. Each of the other 16 students guessed $30+\frac{n}{2}$ where $n$ was that student’s score. If the average of the final exam scores was the same as the average of the guesses, what was the average score on the final exam? [i]Proposed by Eric Wang[/i]

$101$ people, sitting at a round table in any order, had $1,2,... , 101$ cards, respectively. A transfer is someone give one card to one of the two people adjacent to him. Find the smallest positive integer $k$ such that there always can through no more than $ k $ times transfer, each person hold cards of the same number, regardless of the sitting order.