When the repeating decimal $ 0.363636\ldots$ is written in simplest fractional form, the sum of the numerator and denominator is: $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 114 \qquad \textbf{(D)}\ 135 \qquad \textbf{(E)}\ 150$

Consider the set of all integer points in $Z^3$. Sasha and Masha play such a game. At first, Masha marks an arbitrary point. After that, Sasha marks all the points on some a plane perpendicular to one of the coordinate axes and at no point, which Masha noted. Next, they continue to take turns (Masha can't to select previously marked points, Sasha cannot choose the planes on which there are points said Masha). Masha wants to mark $n$ consecutive points on some line that parallel to one of the coordinate axes, and Sasha seeks to interfere with it. Find all $n$, in which Masha can achieve the desired result.

Given a natural number $ n, $ prove that the set $ \{ -n+1,-n+2,\ldots ,-1,1,2,\ldots ,n-1,n\} $ can be partitioned into $ k $ subsets such that the sums of all elements of each of these subsets are equal, if and only if $ n $ is multiple of $ k. $

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?

Find all functions $f: R \rightarrow R$ such that $f(x^{2015} + (f(y))^{2015}) = (f(x))^{2015} + y^{2015}$ holds for all reals $x, y$

Do there exist distinct numbers $x,y,z$ from $[0,\dfrac{\pi}{2}]$, such that six number $\sin x$, $\sin y$,$\sin z$, $\cos x$, $\cos y$, $\cos z$ could be partitioned into 3 pairs with equal sums? [I]Proposed by A. Golovanov[/i]

Let $ n > 4$ be a non-negative integer. Given is the in a circle inscribed convex $ n$-gon $ A_0A_1A_2\dots A_{n \minus{} 1}A_n$ $ (A_n \equal{} A_0)$ where the side $ A_{i \minus{} 1}A_i \equal{} i$ (for $ 1 \le i \le n$). Moreover, let $ \phi_i$ be the angle between the line $ A_iA_{i \plus{} 1}$ and the tangent to the circle in the point $ A_i$ (where the angle $ \phi_i$ is less than or equal $ 90^o$, i.e. $ \phi_i$ is always the smaller angle of the two angles between the two lines). Determine the sum $ \Phi \equal{} \sum_{i \equal{} 0}^{n \minus{} 1} \phi_i$ of these $ n$ angles.

For positive numbers is true that $$ab+ac+bc=a+b+c$$ Prove $$a+b+c+1 \geq 4abc$$

In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ Determine the exponent $\beta$, as large as possible such that for some $C$ independent of $n$ \[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]

For which values of N is it possible to write numbers from 1 to N in some order so that for any group of two or more consecutive numbers, the arithmetic mean of these numbers is not whole?

Let $\mathcal{C}$ be the family of circumferences in $\mathbb{R}^2$ that satisfy the following properties: (i) if $C_n$ is the circumference with center $(n,1/2)$ and radius $1/2$, then $C_n\in \mathcal{C}$, for all $n\in \mathbb{Z}$. (ii) if $C$ and $C'$, both in $\mathcal{C}$, are externally tangent, then the circunference externally tangent to $C$ and $C'$ and tanget to $x$-axis also belongs to $\mathcal{C}$. (iii) $\mathcal{C}$ is the least family which these properties. Determine the set of the real numbers which are obtained as the first coordinate of the points of intersection between the elements of $\mathcal{C}$ and the $x$-axis.

For lunch, Lamy, Botan, Nene, and Polka each choose one of three options: a hot dog, a slice of pizza, or a hamburger. Lamy and Botan choose different items, and Nene and Polka choose the same item. In how many ways could they choose their items?

A circle $c$ with center $A$ passes through the vertices $B$ and $E$ of a regular pentagon $ABCDE$. The line $BC$ intersects the circle $c$ for second time at point $F$. Prove that the lines $DE$ and $EF$ are perpendicular.

Given positive numbers $a_1, \ldots , a_n$, $b_1, \ldots , b_n$, $c_1, \ldots , c_n$. Let $m_k$ be the maximum of the products $a_ib_jc_l$ over the sets $(i, j, l)$ for which $max(i, j, l) = k$. Prove that $$(a_1 + \ldots + a_n) (b_1 +\ldots + b_n) (c_1 +\ldots + c_n) \leq n^2 (m_1 + \ldots + m_n).$$

If the margin made on an article costing $ C$ dollars and selling for $ S$ dollars is $ M\equal{}\frac{1}{n}C$, then the margin is given by: $ \textbf{(A)}\ M\equal{}\frac{1}{n\minus{}1}S \qquad \textbf{(B)}\ M\equal{}\frac{1}{n}S \qquad \textbf{(C)}\ M\equal{}\frac{n}{n\plus{}1}S \\ \textbf{(D)}\ M\equal{}\frac{1}{n\plus{}1}S \qquad \textbf{(E)}\ M\equal{}\frac{n}{n\minus{}1}S$

A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial \[x^4 + 2ax^3 + (2a-2)x^2 + (-4a+3)x - 2\] are all real can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

We are given a balance with two bowls and a number of weights. (a) Give an example of four integer weights using which one can measure any weight of $1,2,...,40$ grams. (b) Are there four weights using which one can measure any weight of $1,2,...,50$ grams?

\[\int_{-\infty}^{0} \frac{1}{e^{-x}+2e^{x}+e^{3x}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Let triangle $\triangle{MNP}$ have side lengths $MN=13$, $NP=89$, and $PM=100$. Define points $S$, $R$, and $B$ as the midpoints of $\overline{MN}$, $\overline{NP}$, and $\overline{PM}$ respectively. A line $\ell$ cuts lines $\overline{MN}$, $\overline{NP}$, and $\overline{PM}$ at points $I$, $J$, and $A$ respectively. Find the minimum value of $(SI+RJ+BA)^2.$

Two teams take part in a competition. There are 7 members (numbered 1 to 7). Two member 1 start a competition first. The failer is sifted out, and the winner start a new competition with member 2 in the other team. ... When all members of a team are out, the competition ends. The number of possible situations is________.

Triangle $ABC$ has circumcenter $O$. $D$ is the foot from $A$ to $BC$, and $P$ is apoint on $AD$. The feet from $P$ to $CA, AB$ are $E, F$, respectively, and the foot from $D$ to $EF$ is $T$. $AO$ meets $(ABC)$ again at $A'$. $A'D$ meets $(ABC)$ again at $R$. If $Q$ is a point on $AO$ satisfying $\angle ABP = \angle QBC$, prove that $D, P, T, R$ lie on acircle and $DQ$ is tangent to it.

A given sequence $r_1, r_2, \dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more ``bubble passes''. A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the third term with the second term and exchanging them if and only if the third term is smaller, and so on in order, through comparing the last term, $r_n$, with its current predecessor and exchanging them if and only if the last term is smaller. The example below shows how the sequence 1, 9, 8, 7 is transformed into the sequence 1, 8, 7, 9 by one bubble pass. The numbers compared at each step are underlined. \[ \begin{array}{c} \underline{1 \quad 9} \quad 8 \quad 7 \\ 1 \quad \underline{9 \quad 8} \quad 7 \\ 1 \quad 8 \quad \underline{9 \quad 7} \\ 1 \quad 8 \quad 7 \quad 9 \end{array} \] Suppose that $n = 40$, and that the terms of the initial sequence $r_1, r_2, \dots, r_{40}$ are distinct from one another and are in random order. Let $p/q$, in lowest terms, be the probability that the number that begins as $r_{20}$ will end up, after one bubble pass, in the $30^{\text{th}}$ place. Find $p + q$.

Tessa the hyper-ant is at the origin of the four-dimensional Euclidean space $\mathbb R^4$. For each step she moves to another lattice point that is $2$ units away from the point she is currently on. How many ways can she return to the origin for the first time after exactly $6$ steps? [i]Proposed by Yannick Yao

Let $x_0$, $x_1$, $\ldots$, $x_{1368}$ be complex numbers. For an integer $m$, let $d(m)$, $r(m)$ be the unique integers satisfying $0\leq r(m) < 37$ and $m = 37d(m) + r(m)$. Define the $1369\times 1369$ matrix $A = \{a_{i,j}\}_{0\leq i, j\leq 1368}$ as follows: \[ a_{i,j} = \begin{cases} x_{37d(j)+d(i)} & r(i) = r(j),\ i\neq j\\ -x_{37r(i)+r(j)} & d(i) = d(j),\ i \neq j \\ x_{38d(i)} - x_{38r(i)} & i = j \\ 0 & \text{otherwise} \end{cases}. \]We say $A$ is $r$-\emph{murine} if there exists a $1369\times 1369$ matrix $M$ such that $r$ columns of $MA-I_{1369}$ are filled with zeroes, where $I_{1369}$ is the identity $1369\times 1369$ matrix. Let $\operatorname{rk}(A)$ be the maximum $r$ such that $A$ is $r$-murine. Let $S$ be the set of possible values of $\operatorname{rk}(A)$ as $\{x_i\}$ varies. Compute the sum of the $15$ smallest elements of $S$. [i]Proposed by Brandon Wang[/i]

Let $n$ be the answer to this problem. Find the number of distinct (i.e. non-congruent), non-degenerate triangles with integer side lengths and perimeter $n$.