Let $f$ be an increasing mapping from the family of subsets of a given finite set $H$ into itself, i.e. such that for every $X \subseteq Y\subseteq H$ we have $f (X )\subseteq f (Y )\subseteq H .$ Prove that there exists a subset $H_{0}$ of $H$ such that $f (H_{0}) = H_{0}.$

In triangle $\, ABC, \,$ angle $\,A\,$ is twice angle $\,B,\,$ angle $\,C\,$ is obtuse, and the three side lengths $\,a,b,c\,$ are integers. Determine, with proof, the minimum possible perimeter.

Prove that for any sequence $a_1, a_2, \ldots , a_{2002}$ of non-negative integers written in the usual decimal notation with $a_1 > 0$ there exists an integer $n$ such that $n^2$ starts with digits $a_1, a_2, \ldots , a_{2002}$ (in this order).

Determine all integers $s \ge 4$ for which there exist positive integers $a$, $b$, $c$, $d$ such that $s = a+b+c+d$ and $s$ divides $abc+abd+acd+bcd$. [i]Proposed by Ankan Bhattacharya and Michael Ren[/i]

Danielle writes a sign '+' or '-' in each of the next $64$ spaces: $$\_\_1 \_\_2 \_\_3 \_\_4 \text{ }.... \text{ }\_\_63 \_\_64=2024$$ such that the equality holds. Find the largest number of negative signs Danielle can use.

Older television screens have an aspect ratio of $ 4: 3$. That is, the ratio of the width to the height is $ 4: 3$. The aspect ratio of many movies is not $ 4: 3$, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $ 2: 1$ and is shown on an older television screen with a $ 27$-inch diagonal. What is the height, in inches, of each darkened strip? [asy]unitsize(1mm); defaultpen(linewidth(.8pt)); filldraw((0,0)--(21.6,0)--(21.6,2.7)--(0,2.7)--cycle,grey,black); filldraw((0,13.5)--(21.6,13.5)--(21.6,16.2)--(0,16.2)--cycle,grey,black); draw((0,2.7)--(0,13.5)); draw((21.6,2.7)--(21.6,13.5));[/asy]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2.25 \qquad \textbf{(C)}\ 2.5 \qquad \textbf{(D)}\ 2.7 \qquad \textbf{(E)}\ 3$

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(2m+2n)=f(m)f(n)$ for all natural numbers $m,n$.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(f(x)^2+f(y^2))=(x-y)f(x-f(y))$

Construct on the real projective plane a continuous curve, consisting of simple points, which is not a straight line and is intersected in a single point by every tangent and every secant of a given conic. [i]F. Karteszi[/i]

Given a segment $AB$ in the plane, choose on it a point $M$ different from $A$ and $B$. Two equilateral triangles $\triangle AMC$ and $\triangle BMD$ in the plane are constructed on the same side of segment $AB$. The circumcircles of the two triangles intersect in point $M$ and another point $N$. (The [b]circumcircle[/b] of a triangle is the circle that passes through all three of its vertices.) (a) Prove that lines $AD$ and $BC$ pass through point $N$. (b) Prove that no matter where one chooses the point $M$ along segment $AB$, all lines $MN$ will pass through some fixed point $K$ in the plane.

Find all prime numbers $ p,q$ less than 2005 and such that $ q|p^2 \plus{} 4$, $ p|q^2 \plus{} 4$.

Show that there is some rational number in the interval $(0,1)$ that can be expressed as a sum of $2021$ reciprocals of positive integers, but cannot be expressed as a sum of $2020$ reciprocals of positive integers.

Find all primes $p \ge 3$ such that $p- \lfloor p/q \rfloor q$ is a square-free integer for any prime $q<p$.

If permutations of the numbers $2, 3,4,..., 102$ are denoted by $a_i,a_2, a_3,...,a_{101}$, find all such permutations in which $a_k$ is divisible by $k$ for all $k$.

A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition: \[ f(a,b,c)\plus{}f(b,c,d)\plus{}f(c,d,e)\plus{}f(d,e,a)\plus{}f(e,a,b)\equal{}a\plus{}b\plus{}c\plus{}d\plus{}e\] Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds: \[ f(x_1,x_2,x_3)\plus{}f(x_2,x_3,x_4)\plus{}\ldots \plus{}f(x_{n\minus{}1},x_n,x_1)\plus{}f(x_n,x_1,x_2)\equal{}x_1\plus{}x_2\plus{}\ldots\plus{}x_n\]

For integer $n\geq4$, find the smallest integer $f(n)$, such that for any positive integer $m$, in any subset with $f(n)$ elements of the set $\{m, m+1, \cdots, m+n-1\}$ there are at least three elements that are relatively prime .

Consider a finite set of points, $\Phi$, in the $\mathbb{R}^2$, such that they follow the following properties : [list] [*] $\Phi$ doesn't contain the origin $\{(0,0)\}$ and not all points are collinear. [*] If $\alpha \in \Phi$, then $-\alpha \in \Phi$, $c\alpha \notin \Phi $ for $c\neq 1$ or $-1$ [*] If $\alpha, \ \beta$ are in $\Phi$, then the reflection of $\beta$ in the line passing through the origin and perpendicular to the line containing origin and $\alpha$ is in $\Phi$ [*] If $\alpha = (a,b) , \ \beta = (c,d)$, (both $\alpha, \ \beta \in \Phi$) then $\frac{2(ac+bd)}{c^2+d^2} \in \mathbb{Z}$ [/list] Prove that there cannot be 5 collinear points in $\Phi$

Let $n \geq 2$ be a natural number. For any two permutations of $(1,2,\cdots,n)$, say $\alpha = (a_1,a_2,\cdots,a_n)$ and $\beta = (b_1,b_2,\cdots,b_n),$ if there exists a natural number $k \leq n$ such that $$b_i = \begin{cases} a_{k+1-i}, & \text{ }1 \leq i \leq k; \\ a_i, & \text{} k < i \leq n, \end{cases}$$ we call $\alpha$ a friendly permutation of $\beta$. Prove that it is possible to enumerate all possible permutations of $(1,2,\cdots,n)$ as $P_1,P_2,\cdots,P_m$ such that for all $i = 1,2,\cdots,m$, $P_{i+1}$ is a friendly permutation of $P_i$ where $m = n!$ and $P_{m+1} = P_1$.

Let \[N = 69^{5} + 5\cdot 69^{4} + 10\cdot 69^{3} + 10\cdot 69^{2} + 5\cdot 69 + 1.\] How many positive integers are factors of $N$? $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 125\qquad\textbf{(E)}\ 216 $

Santa Clause had $n$ sorts of candies, $k$ candies of each sort. He distributed them at random between $k$ gift bags, $n$ candies per a bag and gave a bag to everyone of $k$ children at Christmas party. The children learned what they had in their bags and decided to trade. Two children trade one candy for one candy in case if each of them gets the candy of the sort which was absent in his/her bag. Prove that they can organize a sequence of trades so that finally every child would have candies of each sort.

Let $a, b, c$ be the roots of $x^3-7x^2-6x+5=0$. Compute $(a+b)(a+c)(b+c)$.

Let $M = \{1, 2, 3, 4, 6, 8,12,16, 24, 48\} .$ Find out which of four-element subsets of $M$ are more: those with product of all elements greater than $2009$ or those with product of all elements less than $2009.$

Source: 2018 Canadian Open Math Challenge Part A Problem 3 ----- Points $(0,0)$ and $(3\sqrt7,7\sqrt3)$ are the endpoints of a diameter of circle $\Gamma.$ Determine the other $x$ intercept of $\Gamma.$

Suppose $ABCD$ is a cyclic quadrilateral. Extend $DA$ and $DC$ to $P$ and $Q$ respectively such that $AP=BC$ and $CQ=AB$. Let $M$ be the midpoint of $PQ$. Show that $MA\perp MC$.

How many lattice points $(v, w, x, y, z)$ does a $5$-sphere centered on the origin, with radius $3$, contain on its surface or in its interior?