Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.

Dagan has a wooden cube. He paints each of the six faces a different color. He then cuts up the cube to get eight identically-sized smaller cubes, each of which now has three painted faces and three unpainted faces. He then puts the smaller cubes back together into one larger cube such that no unpainted face is visible. Compute the number of different cubes that Dagan can make this way. Two cubes are considered the same if one can be rotated to obtain the other. You may express your answer either as an integer or as a product of prime numbers.

Points $A, B, C$ are chosen on the boundary of a circle with center $O$ so that $\angle BAC$ encloses an arc of $120$ degrees. Let $D$ be chosen on $\overline{BA}$ so that $\angle AOD$ is a right angle. Extend $\overline{CD}$ so that it intersects with $O$ again at point $P$. What is the measure of the arc, in degrees, that is enclosed by $\angle ACP$? Please use the $tan^{-1}$ function to express your answer.

Let $ABCD$ be a convex quadrilateral inscribed in a circle with $AC = 7$, $AB = 3$, $CD = 5$, and $AD - BC = 3$. Then $BD = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

For positive integers $i = 2, 3, \ldots, 2020$, let \[ a_i = \frac{\sqrt{3i^2+2i-1}}{i^3-i}. \]Let $x_2$, $\ldots$, $x_{2020}$ be positive reals such that $x_2^4 + x_3^4 + \cdots + x_{2020}^4 = 1-\frac{1}{1010\cdot 2020\cdot 2021}$. Let $S$ be the maximum possible value of \[ \sum_{i=2}^{2020} a_i x_i (\sqrt{a_i} - 2^{-2.25} x_i) \] and let $m$ be the smallest positive integer such that $S^m$ is rational. When $S^m$ is written as a fraction in lowest terms, let its denominator be $p_1^{\alpha_1} p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ for prime numbers $p_1 < \cdots < p_k$ and positive integers $\alpha_i$. Compute $p_1\alpha_1+p_2\alpha_2 + \cdots + p_k\alpha_k$. [i]Proposed by Edward Wan and Brandon Wang[/i]

Find a set of five different relatively prime natural numbers such, that the sum of an arbitrary subset is a composite number.

Construct a triangle, the base of which lies on the given line, and the feet of the altitudes, drawn on the sides, coincide with the given points.

Consider a semi-circle with diameter $AB$. Let points $C$ and $D$ be on diameter $AB$ such that $CD$ forms the base of a square inscribed in the semicircle. Given that $CD = 2$, compute the length of $AB$.

For 31 years, n (>6) tennis players have records of wins. It turns out that for every two players, there is a third player who has won over them before. Prove that for every integer $k,l$ such that $2^{2^k+1}-1>n, 1<l<2k+1$, there exist $l$ players ($A_1, A_2, ... , A_l$) such that every player $A_{i+1}$ won over $A_i$. ($A_{l+1}$ is same as $A_1$)

Find all pairs of primes $(p, q)$ such that $$p^3 - q^5 = (p + q)^2.$$

How many positive integers less than 500 have exactly 15 positive integer factors?

For real numbers $x_i$, the statement \[ x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0\] is always true. (Prove!) For which $n\geq 4$ integers, the statement \[x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0\] is always true. Justify your answer.

Let $P$ be a polynomial with each root is real and each coefficient is either $1$ or $-1$. The degree of $P$ can be at most ? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.

The number $12$ is written on the whiteboard. Each minute, the number on the board is either multiplied or divided by one of the numbers $2$ or $3$ (a division is possible only if the result is an integer) . Prove that the number that will be written on the board in exactly one hour will not be equal to $54$.

A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river $9$ hours after leaving dock $A.$ How many hours did it take the power boat to go from $A $ to $B$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 3.5 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 4.5 \qquad \textbf{(E)}\ 5 $

Let $m<n$ be two positive integers, let $I$ and $J$ be two index sets such that $|I|=|J|=n$ and $|I\cap J|=m$, and let $u_k$, $k\in I\cup J$ be a collection of vectors in the Euclidean plane such that \[|\sum_{i\in I}u_i|=1=|\sum_{j\in J}u_j|.\] Prove that \[\sum_{k\in I\cup J}|u_k|^2\geq \frac{2}{m+n}\] and find the cases of equality.

In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$

Given is an isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB> CD$. The projection from $D$ on $ AB$ is $E$. The midpoint of the diagonal $BD$ is $M$. Prove that $EM$ is parallel to $AC$. (Karl Czakler)

Two 10-digit integers are called neighbours if they differ in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbours.

Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in R$.

It is known that the merchant’s $n$ clients live in locations laid along the ring road. Of these, $k$ customers have debts to the merchant for $a_1,a_2,...,a_k$ rubles, and the merchant owes the remaining $n-k$ clients, whose debts are $b_1,b_2,...,b_{n-k}$ rubles, moreover, $a_1+a_2+...+a_k=b_1+b_2+...+b_{n-k}$. Prove that a merchant who has no money can pay all his debts and have paid all the customer debts, by starting a customer walk along the road from one of points and not missing any of their customers.

Determine all positive integers $n$ for which $2^n + 1$ is divisible by $3$.

Let $O$ be the center of the circle of the triangle $ABC$ with $AB \ne AC$. Furthermore, let $S$ and $T$ be points on the rays $AB$ and $AC$, such that $\angle ASO = \angle ACO$ and $\angle ATO = \angle ABO$. Show that $ST$ bisects the segment $BC$.

Let $\{x_1, x_2, x_3, ..., x_n\}$ be a set of $n$ distinct positive integers, such that the sum of any $3$ of them is a prime number. What is the maximum value of $n$?