Let $k$ be a positive integer. Some of the $2k$-element subsets of a given set are marked. Suppose that for any subset of cardinality less than or equal to $(k+1)^2$ all the marked subsets contained in it (if any) have a common element. Show that all the marked subsets have a common element.

In $\vartriangle ABC$, $AB = 2019$, $BC = 2020$, and $CA = 2021$. Yannick draws three regular $n$-gons in the plane of $\vartriangle ABC$ so that each $n$-gon shares a side with a distinct side of $\vartriangle ABC$ and no two of the $n$-gons overlap. What is the maximum possible value of $n$?

Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

Circles $\mathcal{C}_1$, $\mathcal{C}_2$, and $\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\mathcal{C}_2$ and $\mathcal{C}_3$ and has a negative slope. Given that lines $t_1$ and $t_2$ intersect at $(x,y)$, and that $x=p-q\sqrt{r}$, where $p$, $q$, and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.

Let the sequence $\{a_i\}^\infty_{i=0}$ be defined by $a_0 =\frac12$ and $a_n = 1 + (a_{n-1} - 1)^2$. Find the product $$\prod_{i=0}^\infty a_i=a_0a_1a_2\ldots$$

The grid on the right has $12$ boxes and $15$ edges connecting boxes. In each box, place one of the six integers from $1$ to $6$ such that the following conditions hold: [list] [*]For each possible pair of distinct numbers from $1$ to $6$, there is exactly one edge connecting two boxes with that pair of numbers. [*]If an edge has an arrow, then it points from a box with a smaller number to a box with a larger number.[/list] You do not need to prove that your conguration is the only one possible; you merely need to find a conguration that satises the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justication acceptable.) [asy] size(200); defaultpen(linewidth(0.8)); int i,j; for(i=0;i<4;i=i+1) { for(j=0;j<3;j=j+1) { draw((i,j)--(i,j+1/2)--(i+1/2,j+1/2)--(i+1/2,j)--cycle); } } draw((1/2,1/4)--(1,1/4)^^(1/2,5/4)--(1,5/4)^^(3/2,5/4)--(2,5/4)^^(5/2,5/4)--(3,5/4)^^(5/2,9/4)--(3,9/4)); draw((1/4,1)--(1/4,1/2),Arrow(5)); draw((5/4,1)--(5/4,1/2),Arrow(5)); draw((1/4,2)--(1/4,3/2),Arrow(5)); draw((9/4,2)--(9/4,3/2),Arrow(5)); draw((13/4,2)--(13/4,3/2),Arrow(5)); draw((13/4,1)--(13/4,1/2),Arrow(5)); draw((2,1/4)--(3/2,1/4),Arrow(5)); draw((1,9/4)--(1/2,9/4),Arrow(5)); draw((5/2,1/4)--(3,1/4),Arrow(5)); draw((3/2,9/4)--(2,9/4),Arrow(5)); [/asy]

Maisy and Jeff are playing a game with a deck of cards with $4$ $0$’s, $4$ $1$’s, $4$ $2$’s, all the way up to $4$ $9$’s. You cannot tell apart cards of the same number. After shuffling the deck, Maisy and Jeff each take $4$ cards, make the largest $4$-digit integer they can, and then compare. The person with the larger $4$-digit integer wins. Jeff goes first and draws the cards $2,0,2,1$ from the deck. Find the number of hands Maisy can draw to beat that, if the order in which she draws the cards matters. [i]Proposed by Richard Chen[/i]

Prove that the arithmetic progression $3,7,11,15,...$. contains infinitely many prime numbers.

Let $\mathcal{R}$ be the region in the plane bounded by the graphs of $y=x$ and $y=x^2$. Compute the volume of the region formed by revolving $\mathcal{R}$ around the line $y=x$.

We are given six points in space with distinct distances, no three of them collinear. Consider all triangles with vertices among these points. Show that among these triangles there is one such that its longest side is the shortest side in one of the other triangles.

Let $C$ and $D$ be points on the semicircle with center $O$ and diameter $AB$ such that $ABCD$ is a convex quadrilateral. Let $Q$ be the intersection of the diagonals $[AC]$ and $[BD]$, and $P$ be the intersection of the lines tangent to the semicircle at $C$ and $D$. If $m(\widehat{AQB})=2m(\widehat{COD})$ and $|AB|=2$, then what is $|PO|$? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ \sqrt 3 \qquad\textbf{(C)}\ \frac{1+\sqrt 3} 2 \qquad\textbf{(D)}\ \frac{1+\sqrt 3}{2\sqrt 2} \qquad\textbf{(E)}\ \frac{2\sqrt 3} 3 $

A museum wants to protect its most valuable piece by maintaining constant surveillance. To do this, he wants to place guards to watch the place, in shifts of $7$ consecutive hours. Each guard starts his shift at the same time every day. A guard is essential if there is any time during the day when you are alone to watch the item. Indicates all possibilities for the number of guards guarding the piece, so that everyone is indispensable.

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

Let $ABCD$ be a parallelogram and a circle $k$ passes through $A, C$ and meets rays $AB, AD$ at $E, F$. If $BD, EF$ and the tangent at $C$ concur, show that $AC$ is diameter of $k$.

Find the smallest possible value of the expression $$\frac{(a+b) (b + c)}{a + 2b+c}$$ where $a, b, c$ are arbitrary numbers from the interval $[1,2]$.

Each segment whose endpoints are the vertices of a given regular $100$-gon is colored red, if the number of vertices between its endpoints is even, and blue otherwise. (For example, all sides of the $100$-gon are red.) A number is placed in every vertex so that the sum of their squares is equal to $1$. On each segment the product of the numbers at its endpoints is written. The sum of the numbers on the blue segments is subtracted from the sum of the numbers on the red segments. What is the greatest possible result? (Ilya Bogdanov)

Smallville is populated by unmarried men and women, some of them are acquainted. Two city’s matchmakers are aware of all acquaintances. Once, one of matchmakers claimed: “I could arrange that every brunette man would marry a woman he was acquainted with”. The other matchmaker claimed “I could arrange that every blonde woman would marry a man she was acquainted with”. An amateur mathematician overheard their conversation and said “Then both arrangements could be done at the same time! ” Is he right?

$60$ children participate in a summer camp. Among any $10$ of the children there are three or more who live in the same block. Prove that there must be $15$ or more children from the same block. (Folklore)

There were three times as many red candies as blue candies on a table. After Darrel took the same number of red candies and blue candies, there were four times as many red candies as blue candies left on the table. Then after Cloe took $12$ red candies and $12$ blue candies, there were five times as many red candies as blue candies left on the table. Find the total number of candies that Darrel took.

Find all positive integers $n$ such that $2021^n$ can be expressed in the form $x^4-4y^4$ for some integers $x,y$.

Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$, as shown in the figure. Let $d$ be the distance between the midpoints of edges $AB$ and $CD$. Find $d^{2}$. [asy] pair C=origin, D=(4,11), A=(8,-5), B=(16,0); draw(A--B--C--D--B^^D--A--C); draw(midpoint(A--B)--midpoint(C--D), dashed); label("27", B--D, NE); label("41", A--B, SE); label("7", A--C, SW); label("$d$", midpoint(A--B)--midpoint(C--D), NE); label("18", (7,8), SW); label("13", (3,9), SW); pair point=(7,0); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D));[/asy]

Let $n$ be a natural number greater than 6. $X$ is a set such that $|X| = n$. $A_1, A_2, \ldots, A_m$ are distinct 5-element subsets of $X$. If $m > \frac{n(n - 1)(n - 2)(n - 3)(4n - 15)}{600}$, prove that there exists $A_{i_1}, A_{i_2}, \ldots, A_{i_6}$ $(1 \leq i_1 < i_2 < \cdots, i_6 \leq m)$, such that $\bigcup_{k = 1}^6 A_{i_k} = 6$.

Consider a quarter-circle with center $O,$ arc $\widehat{AB},$ and radius $2.$ Draw a semicircle with diameter $\overline{OA}$ lying inside the quarter-circle. Points $P$ and $Q$ lie on the semicircle and segment $\overline{OB},$ respectively, such that line $PQ$ is tangent to the semicircle. As $P$ and $Q$ vary, compute the maximum possible area of triangle $BQP.$

Find all functions $f: \mathbb{R}\to\mathbb{R}$ that satisfy $y^2f(x)(f(x)-2x)\le (1-xy)(1+xy) $ for any $x,y \in\mathbb{R}$.

Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]