Celeste has an unlimited amount of each type of $n$ types of candy, numerated type 1, type 2, ... type n. Initially she takes $m>0$ candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it: $1.$ She eats a candy of type $k$, and in its position in the row she places one candy type $k-1$ followed by one candy type $k+1$ (we consider type $n+1$ to be type 1, and type 0 to be type $n$). $2.$ She chooses two consecutive candies which are the same type, and eats them. Find all positive integers $n$ for which Celeste can leave the table empty for any value of $m$ and any configuration of candies on the table. $\textit{Proposed by Federico Bach and Santiago Rodriguez, Colombia}$

Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

Let $S$ be the sum of all positive integers that can be expressed in the form $2^a \cdot 3^b \cdot 5^c$, where $a$, $b$, $c$ are positive integers that satisfy $a+b+c=10$. Find the remainder when $S$ is divided by $1001$. [i]Proposed by Michael Ren[/i]

A [i]trifecta[/i] is an ordered triple of positive integers $(a, b, c)$ with $a < b < c$ such that $a$ divides $b$, $b$ divides $c$, and $c$ divides $ab$. What is the largest possible sum $a + b + c$ over all trifectas of three-digit integers?

If $0<a<b$, given two fixed points $A(a,0),B(b,0)$. Draw lines $l$ passes $A$, $m$ passes $B$. They have four different intersections with parabola $y^2=x$. If the four points are concyclic, find the path of $P(P=l\cap m)$.

Nondegenerate $ \triangle ABC$ has integer side lengths, $ BD$ is an angle bisector, $ AD \equal{} 3$, and $ DC \equal{} 8$. What is the smallest possible value of the perimeter? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 37$

In a tetrahedral $ABCD$, given that $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$, $AD=BD=3$, and $CD=2$. Find the radius of the circumsphere of $ABCD$.

(a) For each integer $k\ge 3$, find a positive integer $n$ that can be represented as the sum of exactly $k$ mutually distinct positive divisors of $n$. (b) Suppose that $n$ can be expressed as the sum of exactly $k$ mutually distinct positive divisors of $n$ for some $k\ge 3$. Let $p$ be the smallest prime divisor of $n$. Show that \[\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.\]

A parallelogram has $3$ of its vertices at $(1, 2)$, $(3,8)$, and $(4, 1)$. Compute the sum of the possible $x$-coordinates for the $4$th vertex.

In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.

$32$ knights live in a kingdom. Some of them are servants of others. A servant may have only one master and any master is more wealthy than any of his servants. A knight having not less than $4$ servants is called a baron. What is the maximum number of barons? (The kingdom is ruled by the law: “My servant’s servant is not my servant”. (A. Tolpygo, Kiev)

In an isosceles trapezoid, one of the bases is three times larger than the other. Angle at a greater basis is equal to $45^o$. Show how to cut this trapezium into three parts and make a square with them. Justify your answer.

Define a domino to be a $1\times 2$ rectangular block. A $2023\times 2023$ square grid is filled with non-overlapping dominoes, leaving a single $1\times 1$ gap. John then repeatedly slides dominoes into the gap; each domino is moved at most once. What is the maximum number of times that John could have moved a domino? (Example: In the $3\times 3$ grid shown below, John could move 2 dominoes: $D$, followed by $A$.) [asy] unitsize(18); draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)--cycle); draw((0,1)--(3,1)); draw((2,0)--(2,3)); draw((1,1)--(1,3)); label("A",(0.5,2)); label("B",(1.5,2)); label("C",(2.5,2)); label("D",(1,0.5)); [/asy]

Let $A,B,C$ be three points with integer coordinates in the plane and $K$ a circle with radius $R$ passing through $A,B,C$. Show that $AB\cdot BC\cdot CA\ge 2R$, and if the centre of $K$ is in the origin of the coordinates, show that $AB\cdot BC\cdot CA\ge 4R$.

Find all natural numbers $ n > 1$ such that $ n^{2}$ does $ \text{not}$ divide $ (n \minus{} 2)!$.

Suppose $s,k,t\in \mathbb N$. We've colored each natural number with one of the $k$ colors, such that each color is used infinitely many times. We want to choose a subset $\mathcal A$ of $\mathbb N$ such that it has $t$ disjoint monochromatic $s$-element subsets. What is the minimum number of elements of $A$? [i]Proposed by Navid Adham[/i]

Let's call a polynomial [i]mixed[/i] if it has both positive and negative coefficients ($0$ isn't considered positive or negative). Is the product of two mixed polynomials always mixed? [i]Proposed by Vadym Koval[/i]

Do either $(1)$ or $(2)$ $(1)$ Let $A$ be matrix $(a_{ij}), 1 \leq i,j \leq 4.$ Let $d =$ det$(A),$ and let $A_{ij}$ be the cofactor of $a_{ij}$, that is, the determinant of the $3 \times 3$ matrix formed from $A$ by deleting $a_{ij}$ and other elements in the same row and column. Let $B$ be the $4 \times 4$ matrix $(A_{ij})$ and let $D$ be det $B.$ Prove $D = d^3$. $(2)$ Let $P(x)$ be the quadratic $Ax^2 + Bx + C.$ Suppose that $P(x) = x$ has unequal real roots. Show that the roots are also roots of $P(P(x)) = x.$ Find a quadratic equation for the other two roots of this equation. Hence solve $(y^2 - 3y + 2)2 - 3(y^2 - 3y + 2) + 2 - y = 0.$

Kelvin and $15$ other frogs are in a meeting, for a total of $16$ frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$. Kelvin thinks the situation after the meeting is [I]cool[/I] if for each of the $16$ frogs, the number of friends they made during the meeting is a multiple of $4$. Say that the probability of the situation being cool can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime. Find $a$.

Consider the right prism with the rhombus with side $a$ and acute angle $60^{\circ}$ as a base. This prism was intersected by some plane intersecting its side edges, such that the cross-section of the prism and the plane is a square. Determine all possible lengths of the side of this square.

$ \sqrt{164} $ is $ \text{(A)}\ 42\qquad\text{(B)}\ \text{less than }10\qquad\text{(C)}\ \text{between }10\text{ and }11\qquad\text{(D)}\ \text{between }11\text{ and }12\qquad\text{(E)}\ \text{between }12\text{ and }13 $

Consider all quadratic trinomials $x^2+px+q$ with $p,q\in\{1,\ldots,2001\}$. Which of them has more elements: those having integer roots, or those having no real roots?

Find all positive integers $n$ such that there exist $n$ consecutive positive integers whose sum is a perfect square.

Let $S$ be the set of the positive integers greater than $1$, and let $n$ be from $S$. Does there exist a function $f$ from $S$ to itself such that for all pairwise distinct positive integers $a_1, a_2,...,a_n$ from $S$, we have $f(a_1)f(a_2)...f(a_n)=f(a_1^na_2^n...a_n^n)$?

One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is $ \frac23$ of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is $ \textbf{(A)}\ \frac13 \qquad \textbf{(B)}\ \frac25 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac35 \qquad \textbf{(E)}\ \frac23$