Points $A_1$, $A_2$, $A_3$, $A_4$ are the vertices of a regular tetrahedron of edge length $1$. The points $B_1$ and $B_2$ lie inside the figure bounded by the plane $A_1A_2A_3$ and the spheres of radius $1$ and centres $A_1$, $A_2$, $A_3$. Prove that $B_1B_2 < \max\{B_1A_1, B_1A_2, B_1A_3, B_1A_4\}$. [i] A. Kupavsky [/i]

For any pair $(x,y)$ of real numbers, a sequence $(a_{n}(x,y))$ is defined as follows: $$a_{0}(x,y)=x, \;\;\;\; a_{n+1}(x,y) =\frac{a_{n}(x,y)^{2} +y^2 }{2} \;\, \text{for}\, n\geq 0$$ Find the area of the region $\{(x,y)\in \mathbb{R}^{2} \, |\, (a_{n}(x,y)) \,\, \text{converges} \}$.

Given three points $H_1, H_2, H_3$ on a plane. The points are the reflections of the intersection point of the heights of the triangle $\vartriangle ABC$ through its sides. Construct $\vartriangle ABC$.

What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$ ${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$

For what values of the variable $x$ does the following inequality hold: \[ \dfrac{4x^2}{(1-\sqrt{2x+1})^2}<2x+9 \ ? \]

Given that \[ \frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!} \] find the greatest integer that is less than $\frac N{100}.$

Let $ ABC$ be an acute triangle. Take points $ P$ and $ Q$ inside $ AB$ and $ AC$, respectively, such that $ BPQC$ is cyclic. The circumcircle of $ ABQ$ intersects $ BC$ again in $ S$ and the circumcircle of $ APC$ intersects $ BC$ again in $ R$, $ PR$ and $ QS$ intersect again in $ L$. Prove that the intersection of $ AL$ and $ BC$ does not depend on the selection of $ P$ and $ Q$.

Prove that for any odd $ n $ there exists a unique polynomial $ P (x) $ $ n $ -th degree satisfying the equation $ P \left (x- \frac {1} {x} \right) = x ^ n- \frac {1} {x ^ n}. $ Is this true for any natural number $ n $?

There are $n$($n\geq 3$) circles in the plane, all with radius $1$. In among any three circles, at least two have common point(s), then the total area covered by these $n$ circles is less than $35$.

Inside triangle $ABC$ there are three circles $k_1, k_2, k_3$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_1, k_2, k_3$ are $1, 4$, and $9$. Determine the radius of $k.$

Consider the sytem of equations \[ a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0 \]\[a_{21}x_1+a_{22}x_2+a_{23}x_3 =0\]\[a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0 \] with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions: a) $a_{11}, a_{22}, a_{33}$ are positive numbers; b) the remaining coefficients are negative numbers; c) in each equation, the sum ofthe coefficients is positive. Prove that the given system has only the solution $x_1=x_2=x_3=0$.

For $100$ straight lines on a plane, let $T$ be the set of all right-angled triangles bounded by some $3$ lines. Determine, with proof, the maximum value of $|T|$.

Sonya the frog chooses a point uniformly at random lying within the square $[0, 6] \times [0, 6]$ in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from $[0, 1]$ and a direction uniformly at random from {north, south east, west}. All he choices are independent. She now hops the distance in the chosen direction. What is the probability that she lands outside the square? $\textbf{(A) } \frac{1}{6} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{4} \qquad \textbf{(D) } \frac{1}{10} \qquad \textbf{(E) } \frac{1}{9}$

Does there exist a bijective map $f:\mathbb{N} \rightarrow \mathbb{N}$ so that $\sum^{\infty}_{n=1}\frac{f(n)}{n^2}$ is finite?

Let $n$ be a positive integer. In a village, $n$ boys and $n$ girls are living. For the yearly ball, $n$ dancing couples need to be formed, each of which consists of one boy and one girl. Every girl submits a list, which consists of the name of the boy with whom she wants to dance the most, together with zero or more names of other boys with whom she wants to dance. It turns out that $n$ dancing couples can be formed in such a way that every girl is paired with a boy who is on her list. Show that it is possible to form $n$ dancing couples in such a way that every girl is paired with a boy who is on her list, and at least one girl is paired with the boy with whom she wants to dance the most.

We have $n$ keys, each of them belonging to exactly one of $n$ locked chests. Our goal is to decide which key opens which chest. In one try we may choose a key and a chest, and check whether the chest can be opened with the key. Find the minimal number $p(n)$ with the property that using $p(n)$ tries, we can surely discover which key belongs to which chest.

Let $ABC$ be a non-right-angled triangle, with $AC\ne BC$. Let $F$ be the midpoint of side $BC$. Let $D$ be a point on line $AB$ satisfying$CA=CD$,and let $E$ be a point on line $BC$ satisfying $EB = ED$. The line passing through $A$ and parallel to $ED$ meets line $FD$ at point $I$. Line $AF$ meets line $ED$ at point $J$. Prove that points $C$, $I$ and $J$ are collinear.

Let $k$ and $n$ be positive integers. Show that if $x_j$ ($1\leqslant j\leqslant n$) are real numbers with $\sum_{j=1}^n\frac{1}{x_j^{2^k}+k}=\frac{1}{k}$, then \[\sum_{j=1}^n\frac{1}{x_j^{2^{k+1}}+k+2}\leqslant\frac{1}{k+1}\mbox{.}\]

Let $ABCDEF$ be a hexagon inscribed in a circle (with vertices in that order) with $\angle B + \angle C > 180^o$ and $\angle E + \angle F > 180^o$. Let the lines $AB$ and $CD$ intersect at $X$ and the lines $AF$ and $DE$ intersect at $S$. Let $XY$ and $ST$ be the diameters of the circumcircles of $\vartriangle BCX$ and $\vartriangle EFS$ respectively. If $U$ is the intersection point of the lines $BX$ and $ES$ and $V$ is the intersection point of the lines $BY$ and $ET,$ prove that the lines $UV, XY$ and $ST$ are all parallel.

Let $\triangle ABC$ and a point $D$ on $AC$ such that $BD = DC = 3$. If $AD = 6$ and $\angle ACB = 30^\circ$, calculate $\angle ABD$.

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [u] Set 4[/u] [b]C.16 / G.6[/b] Let $a, b$, and $c$ be real numbers. If $a^3 + b^3 + c^3 = 64$ and $a + b = 0$, what is the value of $c$? [b]C.17 / G.8[/b] Bender always turns $60$ degrees clockwise. He walks $3$ meters, turns, walks $2$ meters, turns, walks $1$ meter, turns, walks $4$ meters, turns, walks $1$ meter, and turns. How many meters does Bender have to walk to get back to his original position? [b]C.18 / G.13[/b] Guang has $4$ identical packs of gummies, and each pack has a red, a blue, and a green gummy. He eats all the gummies so that he finishes one pack before going on to the next pack, but he never eats two gummies of the same color in a row. How many different ways can Guang eat the gummies? [b]C.19[/b] Find the sum of all digits $q$ such that there exists a perfect square that ends in $q$. [b]C.20 / G.14[/b] The numbers $5$ and $7$ are written on a whiteboard. Every minute Stev replaces the two numbers on the board with their sum and difference. After $2017$ minutes the product of the numbers on the board is $m$. Find the number of factors of $m$. [u]Set 5[/u] [b]C.21 / G.10[/b] On the planet Alletas, $\frac{32}{33}$ of the people with silver hair have purple eyes and $\frac{8}{11}$ of the people with purple eyes have silver hair. On Alletas, what is the ratio of the number of people with purple eyes to the number of people with silver hair? [b]C.22 / G.15[/b] Let $P$ be a point on $y = -1$. Let the clockwise rotation of $P$ by $60^o$ about $(0, 0)$ be $P'$. Find the minimum possible distance between $P'$ and $(0, -1)$. [b]C.23 / G.18[/b] How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct. [b]C.24[/b] Jeremy and Kevin are arguing about how cool a sweater is on a scale of $1-5$. Jeremy says “$3$”, and Kevin says “$4$”. Jeremy angrily responds “$3.5$”, to which Kevin replies “$3.75$”. The two keep going at it, responding with the average of the previous two ratings. What rating will they converge to (and settle on as the coolness of the sweater)? [b]C.25 / G.20[/b] An even positive integer $n$ has an [i]odd factorization[/i] if the largest odd divisor of $n$ is also the smallest odd divisor of $n$ greater than $1$. Compute the number of even integers $n$ less than $50$ with an odd factorization. [u]Set 6[/u] [b]C.26 / G.26[/b] When $2018! = 2018 \times 2017 \times ... \times 1$ is multiplied out and written as an integer, find the number of $4$’s. If the correct answer is $A$ and your answer is $E$, you will receive $12 \min\, \, (A/E, E/A)^3$points. [b]C.27 / G.27[/b] A circle of radius $10$ is cut into three pieces of equal area with two parallel cuts. Find the width of the center piece. [img]https://cdn.artofproblemsolving.com/attachments/e/2/e0ab4a2d51052ee364dd14336677b053a40352.png[/img] If the correct answer is $A$ and your answer is $E$, you will receive $\max \, \,(0, 12 - 6|A - E|)$points. [b]C.28 / G.28[/b] An equilateral triangle of side length $1$ is randomly thrown onto an infinite set of lines, spaced $1$ apart. On average, how many times will the boundary of the triangle intersect one of the lines? [img]https://cdn.artofproblemsolving.com/attachments/0/1/773c3d3e0dfc1df54945824e822feaa9c07eb7.png[/img] For example, in the above diagram, the boundary of the triangle intersects the lines in $2$ places. If the correct answer is $A$ and your answer is $E$, you will receive $\max\, \,(0, 12-120|A-E|/A)$ points. [b]C.29 / G.29[/b] Call an ordered triple of integers $(a, b, c)$ nice if there exists an integer $x$ such that $ax^2 + bx + c = 0$. How many nice triples are there such that $-100 \le a, b, c \le 100$? If the correct answer is $A$ and your answer is $E$, you will receive $12 \min\, \,(A/E, E/A)$ points. [b]C.30 / G.30[/b] Let $f(i)$ denote the number of MBMT volunteers to be born in the $i$th state to join the United States. Find the value of $1f(1) + 2f(2) + 3f(3) + ... + 50f(50)$. Note 1: Maryland was the $7$th state to join the US. Note 2: Last year’s MBMT competition had $42$ volunteers. If the correct answer is $A$ and your answer is $E$, you will receive $\max\, \,(0, 12 - 500(|A -E|/A)^2)$ points. PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here [/url] and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The side lengths $a,b,c$ of a triangle $ABC$ are integers with $\gcd(a,b,c)=1$. The bisector of angle $BAC$ meets $BC$ at $D$. (a) show that if triangles $DBA$ and $ABC$ are similar then $c$ is a square. (b) If $c=n^2$ is a square $(n\ge 2)$, find a triangle $ABC$ satisfying (a).

In Greifswald there are three schools called $A,B$ and $C$, each of which is attended by at least one student. Among any three students, one from $A$, one from $B$ and one from $C$, there are two knowing each other and two not knowing each other. Prove that at least one of the following holds: [list] [*]Some student from $A$ knows all students from $B$. [*]Some student from $B$ knows all students from $C$. [*] Some student from $C$ knows all students from $A$.[/list]

In preparation for a game of Fish, Carl must deal $48$ cards to $6$ players. For each card that he deals, he runs through the entirety of the following process: $1$. He gives a card to a random player. $2$. A player $Z$ is randomly chosen from the set of players who have at least as many cards as every other player (i.e. $Z$ has the most cards or is tied for having the most cards). $3$. A player $D$ is randomly chosen from the set of players other than $Z$ who have at most as many cards as every other player (i.e. $D$ has the fewest cards or is tied for having the fewest cards). $4$. $Z$ gives one card to $D$. He repeats steps $1-4$ for each card dealt, including the last card. After all the cards have been dealt, what is the probability that each player has exactly $8$ cards?

Let $ABC$ be a triangle and $D$ be a point on segment $BC$. If $\triangle ABD$ is equilateral and $\angle ACB = 14^{\circ}$, what is $\angle{DAC}$? $\textbf{(A) }26^{\circ}\qquad\textbf{(B) }34^{\circ}\qquad\textbf{(C) }46^{\circ}\qquad\textbf{(D) }50^{\circ}\qquad\textbf{(E) }54^{\circ}$