Let $p$ be an odd prime number. For every integer $a,$ define the number $S_a = \sum^{p-1}_{j=1} \frac{a^j}{j}.$ Let $m,n \in \mathbb{Z},$ such that $S_3 + S_4 - 3S_2 = \frac{m}{n}.$ Prove that $p$ divides $m.$ [i]Proposed by Romeo Meštrović, Montenegro[/i]

In convex quadrilateral $ABCD, \angle BAC=\angle CAD.$ $E$ lies on segment $CD$, and $BE$ and $AC$ intersect at $F,$ $DF$ and $BC$ intersect at $G.$ Prove that $\angle GAC=\angle EAC.$

Let $ABCD$ be a cyclic quadrilateral with center $O$ such that $BD$ bisects $AC.$ Suppose that the angle bisector of $\angle ABC$ intersects the angle bisector of $\angle ADC$ at a single point $X$ different than $B$ and $D.$ Prove that the line passing through the circumcenters of triangles $XAC$ and $XBD$ bisects the segment $OX.$ [i]Proposed by Viktor Ahmeti and Leart Ajvazaj, Kosovo[/i]

Suppose that $ V$ is a locally compact topological space that admits no countable covering with compact sets. Let $ \textbf{C}$ denote the set of all compact subsets of the space $ V$ and $ \textbf{U}$ the set of open subsets that are not contained in any compact set. Let $ f$ be a function from $ \textbf{U}$ to $ \textbf{C}$ such that $ f(U)\subseteq U$ for all $ U \in \textbf{U}$. Prove that either (i) there exists a nonempty compact set $ C$ such that $ f(U)$ is not a proper subset of $ C$ whenever $ C \subseteq U \in \textbf{U}$, (ii) or for some compact set $ C$, the set \[ f^{-1}(C)= \bigcup \{U \in \textbf{U}\;: \ \;f(U)\subseteq C\ \}\] is an element of $ \textbf{U}$, that is, $ f^{-1}(C)$ is not contained in any compact set. [i]A. Mate[/i]

For a natural number $n,$ let $a_n$ be the sum of all products $xy$ over all integers $x$ and $y$ with $1 \leq x < y \leq n.$ For example, $a_3 = 1\cdot2 + 2\cdot3 + 1\cdot3 = 11.$ Determine the smallest $n \in \mathbb{N}$ such that $n > 1$ and $a_n$ is a multiple of $2020.$

There is $1$ integer in between $300$ and $400$ (base $10$) inclusive such that its last digit is $7$ when written in bases $8$, $10$, and $12$. Find this integer, in base $10$. [i]2015 CCA Math Bonanza Individual Round #9[/i]

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from left to right. Thus, $ a \times b \minus{} c$ in such languages means the same as $ a(b\minus{}c)$ in ordinary algebraic notation. If $ a \div b \minus{} c \plus{} d$ is evaluated in such a language, the result in ordinary algebraic notation would be $ \textbf{(A)}\ \frac{a}{b} \minus{} c \plus{} d \qquad \textbf{(B)}\ \frac{a}{b} \minus{} c \minus{} d \qquad \textbf{(C)}\ \frac{d \plus{} c \minus{} b}{a} \qquad \textbf{(D)}\ \frac{a}{b \minus{} c \plus{} d} \qquad \textbf{(E)}\ \frac{a}{b\minus{}c\minus{}d}$

Let $ABC$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M$, $N$, $P$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ and let $E$, $F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$, respectively. Let $U$, $V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $\widehat{BAC}$ of $\Gamma$. Given that $AB = 5$, $AC = 8$, and $\angle A = 60^{\circ}$, compute the area of triangle $XUV$.

Consider the set of grid points $(m,n)$ in the plane, $m,n$ integers. Let $\sigma$ be a finite subset and define \[S(\sigma)=\sum_{(m,n)\in\sigma}(100-|m|-|n|) \] Find the maximum of $S$, taken over the set of all such subsets $\sigma$.

Compute the smallest positive integer $x$ which satisfies $x^2-8x+1\equiv0\pmod{22}$ and $x^2-22x+1\equiv0\pmod{8}$.

Do there exist two sequences of real numbers $ \{a_i\}, \{b_i\},$ $ i \in \mathbb{N},$ satisfying the following conditions: \[ \frac{3 \cdot \pi}{2} \leq a_i \leq b_i\] and \[ \cos(a_i x) \minus{} \cos(b_i x) \geq \minus{} \frac{1}{i}\] $ \forall i \in \mathbb{N}$ and all $ x,$ with $ 0 < x < 1?$

Let $ f$ be a function given by $ f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$. a) Solve the equation $ f(x) = 0$. b) Find the number of the subsets of the set \[ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.\]

Find all non-decreasing functions $f:\mathbb R^+\cup\{0\}\rightarrow\mathbb R^+\cup\{0\}$ such that for each $x,y\in \mathbb R^+\cup\{0\}$ \[f\left(\frac{x+f(x)}2+y\right)=2x-f(x)+f(f(y)).\]

For each pair of real numbers $ a\not\equal{} b$, define the operation $ \star$ as \[(a \star b) \equal{} \frac{a \plus{} b}{a \minus{} b}.\] What is the value of $ ((1 \star 2) \star 3)$? $ \textbf{(A)}\ \minus{}\frac{2}{3}\qquad \textbf{(B)}\ \minus{}\frac{1}{5}\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ \frac{1}{2}\qquad \textbf{(E)}\ \text{This value is not defined.}$

Let $f: [0,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $$f(x) = \int_{x - 1}^xf(t)\mathrm{d}t$$ for all $x \geq 1$. Show that $f$ has bounded variation on $[1,\infty)$, i.e. $$\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.$$

The $2001$ towns in a country are connected by some roads, at least one road from each town, so that no town is connected by a road to every other city. We call a set $D$ of towns [i]dominant[/i] if every town not in $D$ is connected by a road to a town in $D$. Suppose that each dominant set consists of at least $k$ towns. Prove that the country can be partitioned into $2001-k$ republics in such a way that no two towns in the same republic are connected by a road.

Let $a,b,c>0$ so that $a+b+c=1$. Then prove that $$(a+2ab+2ac+bc)^a(b+2bc+2ba+ca)^b(c+2ca+2cb+ab)^c\le1.$$ [i]Proposed by Marius Drăgan[/i]

A circle of radius $10$ is tangent to two adjacent sides of a square and intersects its two remaining sides at the endpoints of a diameter of the circle. Find the side length of the square.

Find the two last digits of $2012^{2012}$.

Let $f$ be the quadratic function with leading coefficient $1$ whose graph is tangent to that of the lines $y=-5x+6$ and $y=x-1$. The sum of the coefficients of $f$ is $\tfrac pq$, where $p$ and $q$ are positive relatively prime integers. Find $100p + q$. [i]Proposed by David Altizio[/i]

Given a finite collection of lines in a plane $P$, show that it is possible to draw an arbitrarily large circle in $P$ which does not meet any of them. On the other hand, show that it is possible to arrange a countable infinite sequence of lines (first line, second line, third line, etc.) in $P$ so that every circle in $P$ meets at least one of the lines. (A point is not considered to be a circle.)

Determine all triples $\left( x,y,z\right)$ of positive integers for which the number \[ \sqrt{\frac{2005}{x+y}}+\sqrt{\frac{2005}{y+z}}+\sqrt{\frac{2005}{z+x}} \] is an integer .

Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$? [i]Proposed by Mahan Malihi[/i]

Let $ S $ be a finite [url=https://en.wikipedia.org/wiki/Cancellation_property]cancellative semigroup.[/url] [b]a)[/b] Prove that $ S $ contains an idempotent element. [b]b)[/b] Prove that $ S $ is a group. [b]c)[/b] Disprove subpoint [b]b)[/b] in the case that $ S $ would not be finite. [i]Vlad Mihaly[/i]