You are given $n$ not necessarily distinct real numbers $a_1, a_2, \ldots, a_n$. Let's consider all $2^n-1$ ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to $1$? For example, if $a = [-1, 2, 2]$, then we got $3$ once, $4$ once, $2$ twice, $-1$ once, $1$ twice, so the total number of ones here is $2$. [i](Proposed by Anton Trygub)[/i]

Given a triangle. A variable poin $ D$ is chosen on side $ BC$. Points $ K$ and $ L$ are the incenters of triangles $ ABD$ and $ ACD$, respectively. Prove that the second intersection point of the circumcircles of triangles $ BKD$ and $ CLD$ moves along on a fixed circle (while $ D$ moves along segment $ BC$).

Let $ab^2=126, bc^2=14, cd^2=128, da^2=12.$ Find $\tfrac{bd}{ac}.$

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.

Let $x$ and $y$ be real numbers such that $x^2+y^2-22x-16y+113=0.$ Determine the smallest possible value of $x.$

For a positive integer number $n$ we denote $d(n)$ as the greatest common divisor of the binomial coefficients $\dbinom{n+1}{n} , \dbinom{n+2}{n} ,..., \dbinom{2n}{n}$. Find all possible values of $d(n)$

Consider positive integers $m <n,p < q$ such that $(m, n) = 1, (p, q) = 1$ and satisfy the condition that if $\frac{m}{n}= tan\alpha$ and $\frac{p}{q} = tan\beta$, then $\alpha + \beta = 45^o$. i) Given $m, n$, find $p, q$. ii) Given $n, q$, find $m, p$. ii) Given $m, q$, find $n, p$.

Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ \minus{} 1$. Prove that if \[ a_{0} \plus{} 2a_{1} \plus{} 2^{2}a_{2} \plus{} \cdots \plus{} 2^{n}a_{n} \equal{} 0,\] then $ a_{0} \plus{} a_{1} \plus{} \cdots \plus{} a_{n} > 0$.

For coprime positive integers $a,b$,denote $(a^{-1}\bmod{b})$ by the only integer $0\leq m<b$ such that $am\equiv 1\pmod{b}$ (1)Prove that for pairwise coprime integers $a,b,c$, $1<a<b<c$,we have\[(a^{-1}\bmod{b})+(b^{-1}\bmod{c})+(c^{-1}\bmod{a})>\sqrt a.\] (2)Prove that for any positive integer $M$,there exists pairwise coprime integers $a,b,c$, $M<a<b<c$ such that \[(a^{-1}\bmod{b})+(b^{-1}\bmod{c})+(c^{-1}\bmod{a})< 100\sqrt a.\]

The right prism $ABCA'B'C'$, with $AB = AC = BC = a$, has the property that there exists an unique point $M \in (BB')$ so that $AM \perp MC'$. Find the measure of the angle of the straight line $AM$ and the plane $(ACC')$ .

Let $ f:[2,\infty )\rightarrow\mathbb{R} $ be a differentiable function satisfying $ f(2)=0 $ and $$ \frac{df}{dx}=\frac{2}{x^2+f^4{x}} , $$ for any $ x\in [2,\infty ) . $ Show that there exists $ \lim_{x\to\infty } f(x) $ and is at most $ \ln 3. $ [i]Gabriel Daniilescu[/i]

Let $a, b$, and $c$ be integers such that $a+b+c$ divides $a^2 +b^2 +c^2$. Prove that there are infinitely many positive integers $n$ such that $a+b+c$ divides $a^n +b^n +c^n$.

Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$. [i]Proposed by Evan Chen[/i]

Right triangle $\vartriangle ABC$ with $\angle A = 30^o$ and $\angle B = 90^o$ is inscribed in a circle $\omega_1$ with radius $4$. Circle $\omega_2$ is drawn to be the largest circle outside of $\vartriangle ABC$ that is tangent to both $\overline{BC}$ and $\omega_1$, and circles $\omega_3$ and $\omega_4$ are drawn this same way for sides $\overline{AC}$ and $\overline{AB}$, respectively. Suppose that the intersection points of these smaller circles with the bigger circle are noted as points $D$, $E$, and $F$. Compute the area of triangle $\vartriangle DEF$.

The polynomials $P(z)$ and $Q(z)$ with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true for the polynomials $$P(z)+1 \;\; \text{and} \;\; Q(z)+1.$$ Prove that $P(z)=Q(z).$

Let $n>4$ be a natural number. Prove that \[\sum_{k=2}^n\sqrt[k]{\frac{k}{k-1}}<n.\]

For the simultaneous equations \[ 2x\minus{}3y\equal{}8\] \[ 6y\minus{}4x\equal{}9\] $\textbf{(A)}\ x=4,y=0 \qquad \textbf{(B)}\ x=0,y=\dfrac{3}{2}\qquad \textbf{(C)}\ x=0,y=0 \qquad\\ \textbf{(D)}\ \text{There is no solution} \qquad \textbf{(E)}\ \text{There are an infinite number of solutions}$

For all primes $p \geq 3,$ define $F(p) = \sum^{\frac{p-1}{2}}_{k=1}k^{120}$ and $f(p) = \frac{1}{2} - \left\{ \frac{F(p)}{p} \right\}$, where $\{x\} = x - [x],$ find the value of $f(p).$

Given a pyramid whose base is an $n$-gon inscribable in a circle, let $H$ be the projection of the top vertex of the pyramid to its base. Prove that the projections of $H$ to the lateral edges of the pyramid lie on a circle.

Prove that for any natural number $n{}$ the numbers $1,2,\ldots,n$ can be divided into several groups so that the sum of the numbers in each group is equal to a power of three. [i]Proposed by V. Novikov[/i]

Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

In the country of Princetonia, there are an infinite number of cities, connected by roads. For every two distinct cities, there is a unique sequence of roads that leads from one city to the other. Moreover, there are exactly three roads from every city. On a sunny morning in early July, n tourists have arrived at the capital of Princetonia. They repeat the following process every day: in every city that contains three or more tourists, three tourists are picked and one moves to each of the three cities connected to the original one by roads. If there are $2$ or fewer tourists in the city, they do nothing. After some time, all tourists will settle and there will be no more changing cities. For how many values of n from $1$ to $2020$ will the tourists end in a configuration in which no two of them are in the same city?

All segments joining $n$ points (no three of which are collinear) are coloured in one of $k$ colours. What is the smallest $k$ for which there always exists a closed polygonal line with the vertices at some of the $n$ points, whose sides are all of the same colour?

Find the number of integers $n$ from $1$ to $2020$ inclusive such that there exists a multiple of $n$ that consists of only $5$'s. [i]Proposed by Ephram Chun and Taiki Aiba[/i]