Let $ABC$ be an acute triangle such that $AB<AC$. Denote by $A'$ the reflection of $A$ with respect to $BC$. The circumcircle of $A'BC$ meets the rays $AB$ and $AC$ at $D$ and $E$ respectively, such that $B$ is between $A$ and $D$, and $E$ is between $A$ and $C$. Denote by $P$ and $Q$ the midpoints of the segments $CD$ and $BE$, and let $S$ be the midpoint of $BC$. Show that the lines $BC$ and $AA'$ meet on the circumcircle of $PQS$. [i] Authored by Nikola Velov[/i]

At lunch, the seven members of the Kubik family sit down to eat lunch together at a round table. In how many distinct ways can the family sit at the table if Alexis refuses to sit next to Joshua? (Two arrangements are not considered distinct if one is a rotation of the other.)

Nam spent $20$ dollars for $20$ stationery items consisting of books, pens and pencils. Each book, pen, and pencil cost $3$ dollars, $1.5$ dollars and $0.5$ dollar respectively. How many dollars did Nam spend for books?

Determine all positive integers which cannot be represented as $\frac{a}{b}+\frac{a+1}{b+1}$ with $a,b$ being positive integers.

Elmo has 2023 cookie jars, all initially empty. Every day, he chooses two distinct jars and places a cookie in each. Every night, Cookie Monster finds a jar with the most cookies and eats all of them. If this process continues indefinitely, what is the maximum possible number of cookies that the Cookie Monster could eat in one night? [i]Proposed by Espen Slettnes[/i]

Is it possible to find $p,q,r\in\mathbb Q$ such that $p+q+r=0$ and $pqr=1$? [i]Proposed by Máté Weisz, Cambridge[/i]

We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets. For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$. [i]Proposed by Victor Wang[/i]

A machine takes $6$ seconds to make $4$ coins. How long does it take for the machine to make $22$ coins? The machine makes coins at the same constant rate. $\textbf{(A) }30\qquad\textbf{(B) }33\qquad\textbf{(C) }36\qquad\textbf{(D) }39\qquad\textbf{(E) }42$

Let $ x,y,z$ be positive real numbers such that $ xy \plus{} yz \plus{} zx \equal{} x \plus{} y \plus{} z$. Prove the inequality $ \frac1{x^2 \plus{} y \plus{} 1} \plus{} \frac1{y^2 \plus{} z \plus{} 1} \plus{} \frac1{z^2 \plus{} x \plus{} 1}\le1$ When does the equality hold?

Let $n$ be an integer and $p$ a prime greater than 2. Show that: $$(p-1)^nn!|(p^n-1)(p^n-p)(p^n-p^2)\cdots(p^n-p^{n-1}).$$

A ball is launched upward from the ground at an initial vertical speed of $v_\text{0}$ and begins bouncing vertically. Every time it rebounds, it loses a proportion of the magnitude of its velocity due to the inelastic nature of the collision, such that if the speed just before hitting the ground on a bounce is $v$, then the speed just after the bounce is $rv$, where $r < 1$ is a constant. Calculate the total length of time that the ball remains bouncing, assuming that any time associated with the actual contact of the ball with the ground is negligible. (a) $\frac{2v_\text{0}}{g}\frac{1}{1-r}$ (b) $\frac{v_\text{0}}{g}\frac{r}{1-r}$ (c) $\frac{2v_\text{0}}{g}\frac{1-r}{r}$ (d) $\frac{2v_\text{0}}{g}\frac{1}{1-r^2}$ (e) $\frac{2v_\text{0}}{g}\frac{1}{1+(1-r)^2}$

$30$ same balls are put into four boxes $ A$, $ B$, $ C$, $ D$ in such a way that sum of number of balls in $ A$ and $ B$ is greater than sum of in $ C$ and $ D$. How many possible ways are there? $\textbf{(A)}\ 2472 \qquad\textbf{(B)}\ 2600 \qquad\textbf{(C)}\ 2728 \qquad\textbf{(D)}\ 2856 \qquad\textbf{(E)}\ \text{None}$

13 LHS Students attend the LHS Math Team tryouts. The students are numbered $1, 2, .. 13$. Their scores are $s_1,s_2, ... s_{13}$, respectively. There are 5 problems on the tryout, each of which is given a weight, labeled $w_1, w_2, ... w_5$. Each score $s_i$ is equal to the sums of the weights of all problems solved by student $i$. On the other hand, each weight $w_j$ is assigned to be $\frac{1}{\sum_ {s_i} }$, where the sum is over all the scores of students who solved problem $j$. (If nobody solved a problem, the score doesn't matter). If the largest possible average score of the students can be expressed in the form $\frac{\sqrt{a}}{b}$, where $a$ is square-free, find $a+b$. [i]Proposed by Jeff Lin[/i]

Find all pairs $(a, b)$ of real numbers such that $a \cdot \lfloor b \cdot n\rfloor = b \cdot \lfloor a \cdot n \rfloor$ applies to all positive integers$ n$. (For a real number $x, \lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.)

The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = 0$, $a_n = a_{[n/2]} + (-1)^{n(n+1)/2}$. Show that for any positive integer $k$ we can find $n$ in the range $2^k \leq n < 2^{k+1}$ such that $a_n = 0$.

Given two positive integers $n$ and $m$ and a function $f : \mathbb{Z} \times \mathbb{Z} \to \left\{0,1\right\}$ with the property that \begin{align*} f\left(i, j\right) = f\left(i+n, j\right) = f\left(i, j+m\right) \qquad \text{for all } \left(i, j\right) \in \mathbb{Z} \times \mathbb{Z} . \end{align*} Let $\left[k\right] = \left\{1,2,\ldots,k\right\}$ for each positive integer $k$. Let $a$ be the number of all $\left(i, j\right) \in \left[n\right] \times \left[m\right]$ satisfying \begin{align*} f\left(i, j\right) = f\left(i+1, j\right) = f\left(i, j+1\right) . \end{align*} Let $b$ be the number of all $\left(i, j\right) \in \left[n\right] \times \left[m\right]$ satisfying \begin{align*} f\left(i, j\right) = f\left(i-1, j\right) = f\left(i, j-1\right) . \end{align*} Prove that $a = b$.

For every $ n \ge 2$, $ a_n \equal{} \sqrt [3]{n^3 \plus{} n^2 \minus{} n \minus{} 1}/n$. What is the least value of positive integer $ k$ satisfying $ a_2a_3\cdots a_k > 3$ ? $\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 102 \qquad\textbf{(C)}\ 104 \qquad\textbf{(D)}\ 106 \qquad\textbf{(E)}\ \text{None}$

$(HUN 4)$IMO2 If $a_1, a_2, . . . , a_n$ are real constants, and if $y = \cos(a_1 + x) +2\cos(a_2+x)+ \cdots+ n \cos(a_n + x)$ has two zeros $x_1$ and $x_2$ whose difference is not a multiple of $\pi$, prove that $y = 0.$

Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB=\angle BFC=\angle CFA$. Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that \[ AB+AC\geq4DE. \]

Let $ k$ be a positive integer. Prove that the number $ (4 \cdot k^2 \minus{} 1)^2$ has a positive divisor of the form $ 8kn \minus{} 1$ if and only if $ k$ is even. [url=http://www.mathlinks.ro/viewtopic.php?p=894656#894656]Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above.[/url] [i]Author: Kevin Buzzard and Edward Crane, United Kingdom [/i]

A classroom has $30$ students and $30$ desks arranged in $5$ rows of $6$. If the class has $15$ boys and $15$ girls, in how many ways can the students be placed in the chairs such that no boy is sitting in front of, behind, or next to another boy, and no girl is sitting in front of, behind, or next to another girl?

Show that for every prime number $p$ and every positive integer $n\geq2$ there exists a positive integer $k$ such that the decimal representation of $p^k$ contains $n$ consecutive equal digits.

Let $k$ be a positive integer. Determine the least positive integer $n$, with $n\geq k+1$, for which the game below can be played indefinitely: Consider $n$ boxes, labelled $b_1,b_2,...,b_n$. For each index $i$, box $b_i$ contains exactly $i$ coins. At each step, the following three substeps are performed in order: [b](1)[/b] Choose $k+1$ boxes; [b](2)[/b] Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add to the remaining box, if labelled $b_i$, a number of $i$ coins. [b](3)[/b] If one of the boxes is left empty, the game ends; otherwise, go to the next step. [i]Proposed by Demetres Christofides, Cyprus[/i]

Let $ABCD$ be a quadrilateral with $AB>AD$ such that the inscribed circle $k_1$ of $\triangle ABC$ with center $O_1$ and the inscribed circle $k_2$ of $\triangle ADC$ with center $O_2$ have a common point on $AC$. If $k_1$ is tangent to $AB$ at $M$ and $k_2$ is tangent to $AD$ at $L$, prove that the lines $BD$, $LM$ and $O_1O_2$ pass through a common point.

Determine all real numbers $a$ and $b$ such that $a+b\in\mathbb{Z}$ and $a^2+b^2=2$.