Show that for nonnegative real numbers $a,b$ and integers $n\ge 2$, \[\frac{a^n+b^n}{2}\ge\left(\frac{a+b}{2}\right)^n\] When does equality hold?

Let $ABC$ be an acute-angled triangle with $AB< AC$, and let $H$ be its orthocenter. The circumference with diameter $AH$ meets the circumscribed circumference of $ABC$ at $P\neq A$. The tangent to the circumscribed circumference of $ABC$ through $P$ intersects line $BC$ at $Q$. Show that $QP=QH$.

A circular loop with radius $r$ spins with angular frequency $\omega$ in a generated magnetic field of strength $B$. It is hooked to a resistor load $R$. How much work is done by the generator in one revolution? [i]Problem proposed by Ahaan Rungta[/i]

An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles? [asy] for(int a=0; a<12; ++a) { draw((a,0)--(a,6)); } for(int b=0; b<7; ++b) { draw((0,b)--(11,b)); } draw((0,6)--(2,6)--(1,4)--cycle,linewidth(1)); draw((3,4)--(3,6)--(5,4)--cycle,linewidth(1)); draw((0,1)--(3,2)--(6,1)--cycle,linewidth(1)); draw((7,4)--(6,6)--(9,4)--cycle,linewidth(1)); draw((8,1)--(9,3)--(10,0)--cycle,linewidth(1));[/asy] $ \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ 5 $

Let $S$ be a union of finitely many disjoint subintervals of $[0,1]$ such that no two points in $S$ have distance $1/10$. Show that the total length of the intervals comprising $S$ is at most $1/2$.

For what positive value $k$ does the equation $\ln x=kx^2$ have exactly one solution?

Bill bought 13 notebooks, 26 pens, and 19 markers for 25 dollars. Paula bought 27 notebooks, 18 pens, and 31 markers for 31 dollars. How many dollars would it cost Greg to buy 24 notebooks, 120 pens, and 52 markers?

$16$ progamers are playing in a single elimination tournament. Each player has a different skill level and when two play against each other the one with the higher skill level will always win. Each round, each progamer plays a match against another and the loser is eliminated. This continues until only one remains. How many different progamers can reach the round that has $2$ players remaining?

There are $64$ booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered $1$ to $64$ in this order. At the center of the table there are $1996$ light bulbs which are all turned off. Every minute the chips move simultaneously in a circular way (following the numbering sense) as follows: chip $1$ moves one booth, chip $2$ moves two booths, etc., so that more than one chip can be in the same booth. At any minute, for each chip sharing a booth with chip $1$ a bulb is lit. Where is chip $1$ on the first minute in which all bulbs are lit?

We color all the points of the plane with two colors. Prove that there are (at least) two points of the plane having the same color and at distance $1$ among them.

Prove the following theorem: If $n > 2$ is a natural number, $a_1, ..., a_n$ are positive real numbers and becomes $\sum_{i=1}^n a_i = s$, then the following holds $$\sum_{i=1}^n \frac{a_i}{s - a_i} \ge \frac{n}{n - 1}$$

Does there exist a sequence of positive real numbers $\{a_i\}_{i=1}^{\infty}$ satisfying: \[ \sum_{i=1}^{n} a_i \geq n^2 \quad \text{and} \quad \sum_{i=1}^{n} a_i^2 \leq n^3 + 2025n \] for all positive integers $n$.

The area of a circle (in square inches) is numerically larger than its circumference (in inches). What is the smallest possible integral area of the circle, in square inches? [i]Proposed by James Lin[/i]

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the diagonals of all such trapezoids share a common point.

Suppose that $a, b, c$ are real numbers such that for all positive numbers $x_1,x_2,\dots,x_n$ we have $(\frac{1}{n}\sum_{i=1}^nx_i)^a(\frac{1}{n}\sum_{i=1}^nx_i^2)^b(\frac{1}{n}\sum_{i=1}^nx_i^3)^c\ge 1$ Prove that vector $(a, b, c)$ is a nonnegative linear combination of vectors $(-2,1,0)$ and $(-1,2,-1)$.

Let $ABC$ be an acute triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\Omega$. Let $M$ be the midpoint of $AH$ and $N$ the midpoint of $BH$. Assume the points $M$, $N$, $O$, $H$ are distinct and lie on a circle $\omega$. Prove that the circles $\omega$ and $\Omega$ are internally tangent to each other. [i]Dhroova Aiylam and Evan Chen[/i]

The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.

Prove that there exist infinitely many triples of positive integers $(a,b,c)$ so that $a,b,c$ are pairwise coprime and $$\bigg \lfloor \frac{a^2}{2021} \bigg \rfloor + \bigg \lfloor \frac{b^2}{2021} \bigg \rfloor = \bigg \lfloor \frac{c^2}{2021} \bigg \rfloor.$$

A polynomial $P(x)$ of degree $n$ has $n$ different real roots. What is the largest number of its coefficients that can be zero?

Let $ABC$ be an equilateral triangle with side length $8.$ Let $X$ be on side $AB$ so that $AX=5$ and $Y$ be on side $AC$ so that $AY=3.$ Let $Z$ be on side $BC$ so that $AZ,BY,CX$ are concurrent. Let $ZX,ZY$ intersect the circumcircle of $AXY$ again at $P,Q$ respectively. Let $XQ$ and $YP$ intersect at $K.$ Compute $KX\cdot KQ.$

Let $a$ and $b$ be integers such that $a - b = a^2c - b^2d$ for some consecutive integers $c$ and $d$. Prove that $|a - b|$ is a perfect square.

Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get? $\textbf{(A)}\ 200 \qquad \textbf{(B)}\ 202 \qquad \textbf{(C)}\ 220 \qquad \textbf{(D)}\ 380 \qquad \textbf{(E)}\ 398$

A function $f: R \to R$ satisfies the conditions: $f (x + 19) \le f (x) + 19$ and $f (x + 94) \ge f (x) + 94$ for all $x \in R$. Prove that $f (x + 1) = f (x) + 1$ for all $x \in R$.

How many numbers of $ n $ digits formed only with $ 1,9,8 $ and $ 6 $ divide themselves by $ 3 $ ?

Let $n \geq 2$ be an integer. There are $n$ distinct circles in general position, that is, any two of them meet in two distinct points and there are no three of them meeting at one point. Those circles divide the plane in limited regions by circular edges, that meet at vertices (note that each circle have exactly $2n-2$ vertices). For each circle, temporarily color its vertices alternately black and white (note that, doing this, each vertex is colored twice, one for each circle passing through it). If the two temporarily colouring of a vertex coincide, this vertex is definitely colored with this common color; otherwise, it will be colored with gray. Show that if a circle has more than $n-2 + \sqrt{n-2}$ gray points, all vertices of some region are grey. Observation: In this problem, a region cannot contain vertices or circular edges on its interior. Also, the outer region of all circles also counts as a region.