Michael draws $\triangle ABC$ in the sand such that $\angle ACB=90^\circ$ and $\angle CBA=15^\circ$. He then picks a point at random from within the triangle and labels it point $M$. Next, he draws a segment from $A$ to $BC$ that passes through $M$, hitting $BC$ at a point he labels $D$. Just then, a wave washes over his work, so Michael redraws the exact same diagram farther from the water, labeling all the points the same way as before. If hypotenuse $AB$ is $4$ feet in length, let $p$ be the probability that the number of feet in the length of $AD$ is less than $2\sqrt3-2$. Compute $\lfloor1000p\rfloor$.

Determine all functions $f$ from the real numbers to the real numbers, different from the zero function, such that $f(x)f(y)=f(x-y)$ for all real numbers $x$ and $y$.

A triple $(a,b,c)$ of positive integers is called [i]quasi-Pythagorean[/i] if there exists a triangle with lengths of the sides $a,b,c$ and the angle opposite to the side $c$ equal to $120^{\circ}$. Prove that if $(a,b,c)$ is a quasi-Pythagorean triple then $c$ has a prime divisor bigger than $5$.

The roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$ will be reciprocal if: $ \textbf{(A)}\ a \equal{} b \qquad\textbf{(B)}\ a \equal{} bc \qquad\textbf{(C)}\ c \equal{} a \qquad\textbf{(D)}\ c \equal{} b \qquad\textbf{(E)}\ c \equal{} ab$

Into how many regions do $n$ circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?

$ABCD$ is a convex quadrilateral such that $AB<AD$. The diagonal $\overline{AC}$ bisects $\angle BAD$, and $m\angle ABD=130^\circ$. Let $E$ be a point on the interior of $\overline{AD}$, and $m\angle BAD=40^\circ$. Given that $BC=CD=DE$, determine $m\angle ACE$ in degrees.

Find the remainder when $\prod_{n=3}^{33}2n^4 - 25n^3 + 33n^2$ is divided by $2019$.

Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.

We consider the set of four-digit positive integers $x =\overline{abcd}$ with digits different than zero and pairwise different. We also consider the integers $y = \overline{dcba}$ and we suppose that $x > y$. Find the greatest and the lowest value of the difference $x-y$, as well as the corresponding four-digit integers $x,y$ for which these values are obtained.

A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squares ($p = m^2 +n^2$ , with $m$ and $n$ integers) if and only if $p \equiv 1$ (mod $4$). Assuming this result, find which primes $p > 2$ can be written in each of the following forms, using integers $x$ and $y$: a) $x^2 +16y^2, $ b) $4x^2 +4xy+ 5y^2.$

Let $k$ be a circle with diameter $AB$. Let $C$ be a point on the straight line $AB$, so that $B$ between $A$ and $C$ lies. Let $T$ be a point on $k$ such that $CT$ is a tangent to $k$. Let $l$ be the parallel to $CT$ through $A$ and $D$ the intersection of $l$ and the perpendicular to $AB$ through $T$. Show that the line $DB$ bisects segment $CT$.

Compute \[\frac{5+\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{7+\sqrt{12}}{\sqrt{3}+\sqrt{4}}+\dots+\frac{63+\sqrt{992}}{\sqrt{31}+\sqrt{32}}.\]

Moor has $2016$ white rabbit candies. He and his $n$ friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that $n$ is a positive integer (Moor has at least $1$ friend), how many possible values of $n$ exist?

There is a city with $n$ metro stations, each located at a vertex of a regular n-polygon. Metro Line 1 is a line which only connects two non-neighboring stations $A$ and $B$. Metro Line 2 is a cyclic line which passes through all the stations in a shape of regular n-polygon. For each line metro can run in any direction, and $A$ and $B$ are the stations which one can transfer into other line. The line between two neighboring stations is called 'metro interval'. For each station there is one stationmaster, and there are at least one female stationmaster and one male stationmaster. If $n$ is odd, prove that for any integer $k$ $(0<k<n)$ there is a path that starts from a station with a male stationmaster and ends at a station with a female stationmaster, passing through $k$ metro intervals.

[b]p1A[/b] Positive reals $x$, $y$, and $z$ are such that $x/y +y/x = 7$ and $y/z +z/y = 7$. There are two possible values for $z/x + x/z;$ find the greater value. [b]p1B[/b] Real values $x$ and $y$ are such that $x+y = 2$ and $x^3+y^3 = 3$. Find $x^2+y^2$. [b]p2[/b] Set $A = \{5, 6, 8, 13, 20, 22, 33, 42\}$. Let $\sum S$ denote the sum of the members of $S$; then $\sum A = 149$. Find the number of (not necessarily proper) subsets $B$ of $A$ for which $\sum B \ge 75$. [b]p3[/b] $99$ dots are evenly spaced around a circle. Call two of these dots ”close” if they have $0$, $1$, or $2$ dots between them on the circle. We wish to color all $99$ dots so that any two dots which are close are colored differently. How many such colorings are possible using no more than $4$ different colors? [b]p4[/b] Given a $9 \times 9$ grid of points, count the number of nondegenerate squares that can be drawn whose vertices are in the grid and whose center is the middle point of the grid. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

Medians $AD$ and and $CE$ of triangle $ABC$ intersect in $M$. The midpoint of $AE$ is $N$. Let the area of triangle $MNE$ be $k$ times the area of triangle $ABC$. Then $k$ equals: ${{ \textbf{(A)}\ \frac{1}{6} \qquad\textbf{(B)}\ \frac{1}{8} \qquad\textbf{(C)}\ \frac{1}{9} \qquad\textbf{(D)}\ \frac{1}{12} }\qquad\textbf{(E)}\ \frac{1}{16} } $

Given a positive integer $n$, a collection $\mathcal{S}$ of $n-2$ unordered triples of integers in $\{1,2,\ldots,n\}$ is [i]$n$-admissible[/i] if for each $1 \leq k \leq n - 2$ and each choice of $k$ distinct $A_1, A_2, \ldots, A_k \in \mathcal{S}$ we have $$ \left|A_1 \cup A_2 \cup \cdots A_k \right| \geq k+2.$$ Is it true that for all $n > 3$ and for each $n$-admissible collection $\mathcal{S}$, there exist pairwise distinct points $P_1, \ldots , P_n$ in the plane such that the angles of the triangle $P_iP_jP_k$ are all less than $61^{\circ}$ for any triple $\{i, j, k\}$ in $\mathcal{S}$? [i]Ivan Frolov, Russia[/i]

Let $f(t)=\frac{t}{1-t}$, $t \not= 1$. If $y=f(x)$, then $x$ can be expressed as $\textbf{(A)}\ f\left(\frac{1}{y}\right)\qquad \textbf{(B)}\ -f(y)\qquad \textbf{(C)}\ -f(-y)\qquad \textbf{(D)}\ f(-y)\qquad \textbf{(E)}\ f(y)$

[i]Triangular numbers[/i] are numbers of the form $1 + 2 + . . . + n$ with positive integer $n$, that is $1, 3, 6, 10$, . . . . Find the largest non-triangular positive integer number that cannot be represented as the sum of distinct triangular numbers. [i]Proposed by A. Golovanov[/i]

Prove the following assertion: The four altitudes of a tetrahedron $ABCD$ intersect in a point if and only if \[AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2.\]

Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow? [i]Proposed by Gerhard Woeginger, Netherlands[/i]

Let $n$ be a positive integer and $a_1\le a_2\le\ldots\le a_{n+1}$ and $b_1\le b_2\le\ldots\le b_n$ be real numbers such that for all $k\le n$, \[\binom nk\sum_{\substack{1\le i_1<i_2<\ldots<i_k\le n+1,\\i_1,i_2,\ldots,i_k\in\mathbb N}}a_{i_1}a_{i_2}\ldots a_{i_k} = \binom{n+1}k\sum_{\substack{1\le j_1<j_2<\ldots<j_k\le n,\\j_1,j_2,\ldots,j_k\in\mathbb N}}b_{j_1}b_{j_2}\ldots b_{j_k}.\] Show that \[a_1\le b_1\le a_2\le b_2\le \ldots \le a_n\le b_n\le a_{n+1}.\] [i](Proposed by Ivan Chan Guan Yu)[/i]

We have four charged batteries, four uncharged batteries and a radio which needs two charged batteries to work. Suppose we don't know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the radio will work. An attempt consists in putting two batteries in the radio and check if the radio works or not.

At a volleyball tournament, each team plays exactly once against each other team. Each game has a winning team, which gets $1$ point. The losing team gets $0$ points. Draws do not occur. In the nal ranking, only one team turns out to have the least number of points (so there is no shared last place). Moreover, each team, except for the team having the least number of points, lost exactly one game against a team that got less points in the final ranking. a) Prove that the number of teams cannot be equal to $6$. b) Show, by providing an example, that the number of teams could be equal to $7$.