In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the semicircles. The ratio $A:B$ is $1:18$. What is $N$? [asy] draw((0,0)--(18,0)); draw(arc((9,0),9,0,180)); filldraw(arc((1,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((3,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((5,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((7,0),1,0,180)--cycle,gray(0.8)); label("...",(9,0.5)); filldraw(arc((11,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((13,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((15,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((17,0),1,0,180)--cycle,gray(0.8)); [/asy] $\textbf{(A) } 16 \qquad \textbf{(B) } 17 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 36$

Let $ a $ and $ b $ be different real numbers. Prove that for any real numbers $ c_1, c_2, \ldots,c_n $ there exists a sequence of $ n $-elements $ (x_i) $, each term of which is equal to one of the numbers $ a $ or $ b $ such that $$ |x_1c_1 + x_2c_2 + \ldots + x_nc_n| \geq \frac{|b-a|}{2}(|c_1|+|c_2|+\ldots+|c_n|).$$

Find the best rational approximation $x$ to $\sqrt[3]{2016}$ such that $|x-\sqrt[3]{2016}|$ is as small as possible. You may either find an $x=\tfrac{a}{b}$ where $a,b$ are coprime integers or find a decimal approximation. Let $C$ be the actual answer and $A$ be the answer you submit. Your score will be given by $\lceil10+\tfrac{16.5}{0.1+e^{30|A-C|}}\rceil$, where $\lceil x\rceil$ denote the smallest integer which is $\geq x$.

On the plane, a non-isosceles triangle is given, a circle circumscribed around it and the center of its inscribed circle are marked. Using only a ruler without tick marks and drawing no more than seven lines, construct the diameter of the circumcircle.

Let $X$ be a point inside an equilateral triangle $ABC$ such that $BX+CX <3 AX$. Prove that $$3\sqrt3 \left( \cot \frac{\angle AXC}{2}+ \cot \frac{\angle AXB}{2}\right) +\cot \frac{\angle AXC}{2} \cot \frac{\angle AXB}{2} >5$$

On a circular board there are $19$ squares numbered in order from $1$ to $19$ (to the right of $1$ is $2$, to the right of it is $3$, and so on, until $1$ is to the right of $19$). In each box there is a token. Every minute each checker moves to its right the number of the box it is in at that moment plus one; for example, the piece that is in the $7$th place leaves the first minute $7 + 1$ places to its right until the $15$th square; the second minute that same checker moves to your right $15 + 1$ places, to square $12$, etc. Determine if at some point all the tokens reach the place where they started and, if so, say how many minutes must elapse. [hide=original wording]En un tablero circular hay 19 casillas numeradas en orden del 1 al 19 (a la derecha del 1 está el 2, a la derecha de éste está el 3 y así sucesivamente, hasta el 1 que está a la derecha del 19). En cada casilla hay una ficha. Cada minuto cada ficha se mueve a su derecha el número de la casilla en que se encuentra en ese momento más una; por ejemplo, la ficha que está en el lugar 7 se va el primer minuto 7 + 1 lugares a su derecha hasta la casilla 15; el segundo minuto esa misma ficha se mueve a su derecha 15 + 1 lugares, hasta la casilla 12, etc. Determinar si en algún momento todas las fichas llegan al lugar donde empezaron y, si es así, decir cuántos minutos deben transcurrir.[/hide]

Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly $2012$ planes of symmetry. What is the maximal number of symmetry planes of the union of $A$ and $B$, if $B$ has a) $2012$, b) $2013$ symmetry planes? c) What is the answer to the question of p.b), if the symmetry planes are replaced by the symmetry axes?

Prove that if $ a, b, c $ are non-negative numbers, then $$ a^3 + b^3 + c^3 + 3abc \geq a^2(b + c) + b^2(a + c) + c^2(a + b).$$

Compute $\displaystyle\sum_{k=1}^\infty \dfrac{k^4}{k!}$.

Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to $1.$

A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$. [i]Proposed by Aaron Lin[/i]

An acute scalene triangle $ABC$ with circumcircle $\Omega$ is given. The altitude from $B$ intersects side $AC$ at $B_1$ and circle $\Omega$ at $B_2$. The circle with diameter $B_1B_2$ intersects circle $\Omega$ again at $B_3$. Similarly, the altitude from $C$ intersects side $AB$ at $C_1$ and circle $\Omega$ at $C_2$. The circle with diameter $C_1C_2$ intersects circle $\Omega$ again at $C_3$. Let $X$ be the intersection of lines $B_1B_3$ and $C_1C_3$, and let $Y$ be the intersection of lines $B_3C$ and $C_3B$. Prove that line $XY$ bisects side $BC$.

Hammie is in the $6^\text{th}$ grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds? $\textbf{(A)}\ \text{median, by 60} \qquad \textbf{(B)}\ \text{median, by 20} \qquad \textbf{(C)}\ \text{average, by 5} \qquad \textbf{(D)}\ \text{average, by 15}$ \\ $\textbf{(E)}\ \text{average, by 20}$

Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$ Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $% P^{\prime }.$

The natural numbers $a$ and $b$ are such that $ \frac{a+1}{b}+ \frac{b+1}{a}$ is an integer. Show that the greatest common divisor of a and b is not greater than $\sqrt{a+b}$.

Joao Salta-Pocinhas jumps $1$ meter in the first jump, $2$ meters in the second, $4$ meters in the third, . . ., $2^{n-1}$ meters in jump number $n$. Is there any possibility for Joao to choose the directions of his jumps in order to get back to the starting point?

Ana and Benito play a game which consists of $2020$ turns. Initially, there are $2020$ cards on the table, numbered from $1$ to $2020$, and Ana possesses an extra card with number $0$. In the $k$-th turn, the player that doesn't possess card $k-1$ chooses whether to take the card with number $k$ or to give it to the other player. The number in each card indicates its value in points. At the end of the game whoever has most points wins. Determine whether one player has a winning strategy or whether both players can force a tie, and describe the strategy.

What is the greatest integer $k$ which makes the statement "When we take any $6$ subsets with $5$ elements of the set $\{1,2,\dots, 9\}$, there exist $k$ of them having at least one common element." true? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

Let $\Gamma$ be a circle with radius $r$. Let $A$ and $B$ be distinct points on $\Gamma$ such that $AB < \sqrt{3}r$. Let the circle with centre $B$ and radius $AB$ meet $\Gamma$ again at $C$. Let $P$ be the point inside $\Gamma$ such that triangle $ABP$ is equilateral. Finally, let the line $CP$ meet $\Gamma$ again at $Q$. Prove that $PQ = r$.

Does there exist a second-degree polynomial $p(x, y)$ in two variables such that every non-negative integer $ n $ equals $p(k,m)$ for one and only one ordered pair $(k,m)$ of non-negative integers? [i]Proposed by Finland.[/i]

In triangle $ ABC$,$ \angle BAC \equal{} 120^o$. Let the angle bisectors of angles $ A;B$and $ C$ meet the opposite sides at $ D;E$ and$ F$ respectively. Prove that the circle on diameter $ EF$ passes through $ D.$

In how many ways can we enter numbers from the set $\{1,2,3,4\}$ into a $4\times 4$ array so that all of the following conditions hold? (a) Each row contains all four numbers. (b) Each column contains all four numbers. (c) Each "quadrant" contains all four numbers. (The quadrants are the four corner $2\times 2$ squares.)

How many ways are there to place the integers from $1$ to $8$ on the vertices of a regular octagon such that the sum of the numbers on any $4$ vertices forming a rectangle is even? Rotations and reflections of the same arrangement are considered distinct

Compute the remainder of $3^{2021}$ mod $15$

Knowing that the system \[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\] has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.