An equilateral triangle side $10$ is divided into $100$ equilateral triangles of side $1$ by lines parallel to its sides. There are m equilateral tiles of $4$ unit triangles and $25 - m$ straight tiles of $4$ unit triangles (as shown below). For which values of $m$ can they be used to tile the original triangle. [The straight tiles may be turned over.]

For each integer $ n$ of the form $ n\equal{}p_1 p_2 p_3 p_4$, where $ p_1,p_2,p_3,p_4$ are distinct primes, let $ 1\equal{}d_1<d_2<...<d_{15}<d_{16}\equal{}n$ be the divisors of $ n$. Prove that if $ n<1995$, then $ d_9\minus{}d_8 \not\equal{} 22$.

Given a rectangle paper of size $15$ cm $\times$  $20$ cm, fold it along a diagonal. Determine the area of the common part of two halfs of the paper?

Show that there exist (at least) a rearrangement $a_0, a_1, a_2,..., a_{63}$ of the numbers $0,1, 2,..., 63$, such that $a_i - a_j \ne a_j - a_k$, for any $i < j < k \in \{0,1, 2,..., 63\}$.

If $(x^2-x-1)^n=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0$, where $a_i,i\in\{0,1,2,..,2n\}$, find $a_1+a_3+...+a_{2n-1}$ and $a_0+a_2+a_4+...+a_{2n}$.

Find all pairs $(x,y)$ of real numbers such that $ |x|+ |y|=1340$ and $x^{3}+y^{3}+2010xy= 670^{3}$ .

Suppose $ABCD$ is a square piece of cardboard with side length $a$. On a plane are two parallel lines $\ell_1$ and $\ell_2$, which are also $a$ units apart. The square $ABCD$ is placed on the plane so that sides $AB$ and $AD$ intersect $\ell_1$ at $E$ and $F$ respectively. Also, sides $CB$ and $CD$ intersect $\ell_2$ at $G$ and $H$ respectively. Let the perimeters of $\triangle AEF$ and $\triangle CGH$ be $m_1$ and $m_2$ respectively. Prove that no matter how the square was placed, $m_1+m_2$ remains constant.

Let $n$ be a natural number. Find all solutions $x$ of the system of equations $$\left\{\begin{matrix} sinx+cosx=\frac{\sqrt{n}}{2}\\tg\frac{x}{2}=\frac{\sqrt{n}-2}{3}\end{matrix}\right.$$ On interval $\left[0,\frac{\pi}{4}\right).$

On each of the 2006 cards a natural number is written. Cards are placed arbitrarily in a row. 2 players take in turns a card from any end of the row until all the cards are taken. After that each player calculates sum of the numbers written of his cards. If the sum of the first player is not less then the sum of the second one then the first player wins. Show that there's a winning strategy for the first player.

On the legs of $\angle AOB$, the segments $OA$ and $OB$ lie, $OA > OB$. Points $M$ and $N$ on lines $OA$ and $OB$, respectively, are such that $AM = BN = x$. Find $x$ for which the length of $MN$ is minimal.

The sides of $\triangle ABC$ have lengths $6, 8$ and $10$. A circle with center $P$ and radius $1$ rolls around the inside of $\triangle ABC$, always remaining tangent to at least one side of the triangle. When $P$ first returns to its original position, through what distance has $P$ traveled? [asy] draw((0,0)--(8,0)--(8,6)--(0,0)); draw(Circle((4.5,1),1)); draw((4.5,2.5)..(5.55,2.05)..(6,1), EndArrow); dot((0,0)); dot((8,0)); dot((8,6)); dot((4.5,1)); label("A", (0,0), SW); label("B", (8,0), SE); label("C", (8,6), NE); label("8", (4,0), S); label("6", (8,3), E); label("10", (4,3), NW); label("P", (4.5,1), NW); [/asy] $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 17 $

A polynomial $P(x)$ has degree at most $2k$, where $k = 0, 1,2,\cdots$. Given that for an integer $i$, the inequality $-k \le i \le k$ implies $|P(i)| \le 1$, prove that for all real numbers $x$, with $-k \le x \le k$, the following inequality holds: \[|P(x)| < (2k + 1)\dbinom{2k}{k}\]

Does there exist a convex polyhedron having equal number of edges and diagonals? [i](A diagonal of a polyhedron is a segment through two vertices not lying on the same face) [/i]

Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2\times 2$ matrix $M$ with real entries satisfying $\mathrm{trace}(M)=a$ and $\mathrm{det}(M)=b$. (Proposed by Stephan Wagner, Stellenbosch University)

Fix a point $O$ in the plane and an integer $n\geq 3$. Consider a finite family $\mathcal{D}$ of closed unit discs in the plane such that: (a) No disc in $\mathcal{D}$ contains the point $O$; and (b) For each positive integer $k < n$, the closed disc of radius $k + 1$ centred at $O$ contains the centres of at least $k$ discs in $\mathcal{D}$. Show that some line through $O$ stabs at least $\frac{2}{\pi} \log \frac{n+1}{2}$ discs in $\mathcal{D}$.

Let $A$ be an infinite set of positive integers such that every $n \in A$ is the product of at most $1987$ prime numbers. Prove that there is an infinite set $B \subset A$ and a number $p$ such that the greatest common divisor of any two distinct numbers in $B$ is $p.$

In space $4$ points are given. How many planes equidistant from these points are there? Consider separately (a) the generic case (the points given do not lie on a single plane) and (b) the degenerate cases.

find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \] for every natural numbers $ a $ and $ b $

Given a regular pentagon, consider the convex pentagon limited by its diagonals. You are asked to calculate: a) The similarity relation between the two convex pentagons. b) The relationship of their areas. c) The ratio of the homothety that transforms the first into the second.

Convex figure $\mathcal{K}$ has the following property: if one looks at $\mathcal{K}$ from any point of the certain circle $\mathcal{Y}$, then $\mathcal{K}$ is seen in the right angle. Show that the figure is symmetric with respect to the centre of $\mathcal{Y.}$

A set of positive integers is said to be [i]pilak[/i] if it can be partitioned into 2 disjoint subsets $F$ and $T$, each with at least $2$ elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers. Find all positive integers $n$ such that the set containing all the positive divisors of $n$ except $n$ itself is pilak.

Let $ABCD$ be a trapezium with $AC = BC$. Let $H$ be the midpoint of the base $AB$ and let $\ell$ be a line passing through $H$. Let $\ell$ meet $AD$ at $P$ and $BD$ at $Q$. Prove that the angles $ACP$ and $QCB$ are either equal or have a sum of $180^o$. (I. Sharygin, Moscow)

A game works as follows: the player pays $2$ tokens to enter the game. Then, a fair coin is flipped. If the coin lands on heads, they receive $3$ tokens; if the coin lands on tails, they receive nothing. A player starts with $2$ tokens and keeps playing this game until they do not have enough tokens to play again. What is the expected value of the number of tokens they have left at the end? [i]2020 CCA Math Bonanza Team Round #9[/i]

Show that for any positive real numbers $a, b, c$ such that $a + b + c = ab + bc + ca$, the following inequality holds $3 + \sqrt[3]{\frac{a^3+1}{2}}+\sqrt[3]{\frac{b^3+1}{2}}+\sqrt[3]{\frac{c^3+1}{2}}\leq 2(a+b+c)$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

Let $n=3k+1$, where $k$ is a positive integer. A triangular arrangement of side $n$ is formed using circles with the same radius, as is shown in the figure for $n=7$. Determine, for each $k$, the largest number of circles that can be colored red in such a way that there are no two mutually tangent circles that are both colored red.