Find the total number of solutions to the following system of equations: $ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37} \\ b(a + d)\equiv b \pmod{37} \\ c(a + d)\equiv c \pmod{37} \\ bc + d^2\equiv d \pmod{37} \\ ad - bc\equiv 1 \pmod{37} \end{array}$

Evaluate the following expression: $2013^2 + 2011^2 + … + 5^2 + 3^2 -2012^2 -2010^2-…-4^2-2^2$

Prove that for any three positive real numbers $ a,b,c, \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \ge \frac{9}{2} \cdot \frac{1}{a\plus{}b\plus{}c}$.

Lucky starts doodling on a $5\times 5$ Bingo board. He puts his pencil at the center of the upper-left square (marked by ‘·’) and draws a continuous doodle ending on the Free Space, never going off the board or through a corner of a square. (See Figure 1.) (a) Is it possible for Lucky’s doodle to visit all squares exactly once? (The starting and ending squares are considered visited.) (b) Is it possible for Lucky’s doodle to visit all squares exactly twice?

What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$ $\textbf{(A)}\ \pi+\sqrt{2} \qquad\textbf{(B)}\ \pi+2 \qquad\textbf{(C)}\ \pi+2\sqrt{2} \qquad\textbf{(D)}\ 2\pi+\sqrt{2} \qquad\textbf{(E)}\ 2\pi+2\sqrt{2}$

It is said that a positive integer is not GOOD, if there exists a permutation of the integers from 1 to n, $(a_1,a_2,...,a_n)$ such that $k + a_k$ is a perfect square for all $k$. For example $5$ is a GOOD number, since the permutation $(3,2,1,5,4)$ checks the condition: $1 + 3 = 2^2$, $2 + 2 = 2^2$, $3 + 1 = 2^2$; $4 + 5 = 3^2$ and $5 +4 = 3^2$. Find all GOOD numbers up to $12$.

In $\triangle ABC$, the incircle $\omega$ has center $I$ and is tangent to $\overline{CA}$ and $\overline{AB}$ at $E$ and $F$ respectively. The circumcircle of $\triangle{BIC}$ meets $\omega$ at $P$ and $Q$. Lines $AI$ and $BC$ meet at $D$, and the circumcircle of $\triangle PDQ$ meets $\overline{BC}$ again at $X$. Suppose that $EF = PQ = 16$ and $PX + QX = 17$. Then $BC^2$ can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$. [i]Proposed by Ankan Bhattacharya and Michael Ren

a) How many distinct ways are there are there of painting the faces of a cube six different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.) b)* How many distinct ways are there are there of painting the faces of a dodecahedron $12$ different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.)

Given are a positive integer $n$ and an infinite sequence of proper fractions $x_0 = \frac{a_0}{n}$, $\ldots$, $x_i=\frac{a_i}{n+i}$, with $a_i < n+i$. Prove that there exist a positive integer $k$ and integers $c_1$, $\ldots$, $c_k$ such that \[ c_1 x_1 + \ldots + c_k x_k = 1. \] [i]Proposed by M. Dubashinsky[/i]

Unit square $ABCD$ is divided into four rectangles by $EF$ and $GH$, with $BF = \frac14$. $EF$ is parallel to $AB$ and $GH$ parallel to $BC$. $EF$ and $GH$ meet at point $P$. Suppose $BF + DH = FH$, calculate the nearest integer to the degree of $\angle FAH$. [asy] size(100); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } // NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55. pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0]; draw(A--B--C--D--cycle); draw(F--A--H); draw(E--F); draw(G--H); label("$A$",D2(A),NW); label("$B$",D2(B),SW); label("$C$",D2(C),SE); label("$D$",D2(D),NE); label("$E$",D2(E),plain.N); label("$F$",D2(F),S); label("$G$",D2(G),W); label("$H$",D2(H),plain.E); label("$P$",D2(P),SE); [/asy]

In all tetrahedron $ABCD$ holds [list=1] [*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(h_a-r)^2}{(h_a^n-r^n)(h_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$ [*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(r_a-r)^2}{(r_a^n-r^n)(r_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$ [/list] for all $n\in \mathbb{N}^*$

Given three natural numbers greater than $100$, that are pairwise coprime and such that the square of the difference of any two of them is divisible by the third and any of them is less than the product of the other two. Prove that these numbers are squares of some natural numbers.

Palmer and James work at a dice factory, placing dots on dice. Palmer builds his dice correctly, placing the dots so that $1$, $2$, $3$, $4$, $5$, and $6$ dots are on separate faces. In a fit of mischief, James places his $21$ dots on a die in a peculiar order, putting some nonnegative integer number of dots on each face, but not necessarily in the correct configuration. Regardless of the configuration of dots, both dice are unweighted and have equal probability of showing each face after being rolled. Then Palmer and James play a game. Palmer rolls one of his normal dice and James rolls his peculiar die. If they tie, they roll again. Otherwise the person with the larger roll is the winner. What is the maximum probability that James wins? Give one example of a peculiar die that attains this maximum probability.

Let $ ABC$ be a triangle and $ O$ its circumcenter. Lines $ AB$ and $ AC$ meet the circumcircle of $ OBC$ again in $ B_1\neq B$ and $ C_1 \neq C$, respectively, lines $ BA$ and $ BC$ meet the circumcircle of $ OAC$ again in $ A_2\neq A$ and $ C_2\neq C$, respectively, and lines $ CA$ and $ CB$ meet the circumcircle of $ OAB$ in $ A_3\neq A$ and $ B_3\neq B$, respectively. Prove that lines $ A_2A_3$, $ B_1B_3$ and $ C_1C_2$ have a common point.

Let $g$ be the multiplicative function given by $$g(p^{\alpha}) = \alpha p^{\alpha-1},$$ for all $\alpha\in\mathbb Z^+$ and primes $p$. Prove that there exist infinitely many integers $n$ such that $$g(n+1) = g(n) + g(1).$$

There are $36$ participants at a BdMO event. Some of the participants shook hands with each other. But no two participants shook hands with each other more than once. Each participant recorded the number of handshakes they made. It was found that no two participants with the same number of handshakes made, had shaken hands with each other. Find the maximum possible number of handshakes at the party with proof. (When two participants shake hands with each other, this will be counted as one handshake.)

Maria and Luis play the following game: Maria throws three dice and Luis can select some of them (possibly none) and turn them changing their value for the value in the opposite face of each selected die. Prove that Luis can always play in such a way that the sum of the upper faces of the dice after the change is a multiple of $4$. Note: The game is played with normal dice, that is, the sum of opposite faces is $7$.

Let $ M \ge 3$ be an integer and let $ S \equal{} \{3,4,5,\ldots,m\}$. Find the smallest value of $ m$ such that for every partition of $ S$ into two subsets, at least one of the subsets contains integers $ a$, $ b$, and $ c$ (not necessarily distinct) such that $ ab \equal{} c$. [b]Note[/b]: a partition of $ S$ is a pair of sets $ A$, $ B$ such that $ A \cap B \equal{} \emptyset$, $ A \cup B \equal{} S$.

There are \(n^2 + n\) numbers, none of which appears more than \(\frac{n^2 + n}{2}\) times. Prove that they can be divided into \((n+1)\) groups of \(n\) numbers each in such a way that the sums of the numbers in these groups are pairwise distinct. [i]Proposed by Anton Trygub[/i]

Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$ where $d$ is the smallest divisor of $n$ which is greater than $1$.

$n \geq 5$ football teams participate in a round-robin tournament. For every game played, the winner receives 3 points, the loser receives 0 points, and in the event of a draw, both teams receive 1 point. The third-from-bottom team has fewer points than all the teams ranked before it, and more points than the last 2 teams; it won more games than all the teams before it, but fewer games than the 2 teams behind it. Find the smallest possible $n$.

Let $P(x)$ be a polynomial such that, when divided by $x - 2$, the remainder is $3$ and, when divided by $x - 3$, the remainder is $2$. If, when divided by $(x - 2)(x - 3)$, the remainder is $ax + b$, find $a^2 + b^2$.

Determine whether the equation $$x^4+y^3=z!+7$$ has an infinite number of solutions in positive integers.

Suppose that $k$ is a natural number. Prove that there exists a prime number in $\mathbb Z_{[i]}$ such that every other prime number in $\mathbb Z_{[i]}$ has a distance at least $k$ with it.

Evaluate $\int_0^{\frac 12} (x+1)\sqrt{1-2x^2}\ dx$. [i]2011 Kyoto University entrance exam/Science, Problem 1B[/i]