The inscribed circle of $\Delta ABC$ is tangent to $AC$ and $BC$ in points $M$ and $N$ respectively. Line $MN$ intersects line $AB$ in point $P$, so that $B$ is between $A$ and $P$. Determine $\angle ABC$, if $BP=CM$.

A function $f:\{1,2, \ldots, n\} \to \{1, \ldots, m\}$ is [i]multiplication-preserving[/i] if $f(i)f(j) = f(ij)$ for all $1 \le i \le j \le ij \le n$, and [i]injective[/i] if $f(i) = f(j)$ only when $i = j$. For $n = 9, m = 88$, the number of injective, multiplication-preserving functions is $N$. Find the sum of the prime factors of $N$, including multiplicity. (For example, if $N = 12$, the answer would be $2 + 2 + 3 = 7$.)

Given is a directed graph with $28$ vertices, such that there do not exist vertices $u, v$, such that $u \rightarrow v$ and $v \rightarrow u$. Every $16$ vertices form a directed cycle. Prove that among any $17$ vertices, we can choose $15$ which form a directed cycle.

Inside a regular triangle $ABC$, points $X$ and $Y$ are chosen such that $\angle{AXC} = 120^{\circ}$, $2\angle{XAC} + \angle{YBC} = 90^{\circ}$and $XY = YB = \frac{AC}{\sqrt{3}}$. Prove that point $Y$ lies on the incircle of triangle $ABC$.

Prove that there are infinite pairs of non-congruent triangles that have the same angles and two of their equal sides. Develop an algorithm or rule to obtain these pairs of triangles and indicate at least one pair that satisfies the asserted.

Given triangle $ ABC$. Point $ O$ is the center of the excircle touching the side $ BC$. Point $ O_1$ is the reflection of $ O$ in $ BC$. Determine angle $ A$ if $ O_1$ lies on the circumcircle of $ ABC$.

Prove the following inequality: $x_1 + 2x_2 + 3x_3 + ... + nx_n \leq \frac{n(n-1)}{2} + x_1 + x_2 ^2 + x_3 ^3 + ... + x_n ^n$ where $\forall _{x_i} x_i > 0$

Suppose $N$ is any positive integer. Add the digits of $N$ to obtain a smaller integer. Repeat this process of digit-addition till you get a single digit numbem. Find the number of positive integers $N \le 1000$, such that the final single-digit number $n$ is equal to $5$. Example: $N = 563\to (5 + 6 + 3) = 14 \to(1 + 4) = 5$ will be counted as one such integer.

In Oddland there are stamps with values of $1$ cent, $3$ cents, $5$ cents, etc., each for odd number there is exactly one stamp type. Oddland Post dictates: For two different values on a letter must be the number of stamps of the lower one value must be at least as large as the number of tokens of the higher value. In Squareland, on the other hand, there are stamps with values of $1$ cent, $4$ cents, $9$ cents, etc. there is exactly one stamp type for each square number. Brands can be found in Squareland can be combined as required without further regulations. Prove for every positive integer $n$: there are the same number in the two countries possibilities to send a letter with stamps worth a total of $n$ cents. It makes no difference if you have the same stamps on arrange a letter differently. (Stephan Wagner)

Let $ n>0$ be a natural number. Determine all the polynomials of degree $ 2n$ with real coefficients in the form $ P(X)\equal{}X^{2n}\plus{}(2n\minus{}10)X^{2n\minus{}1}\plus{}a_2X^{2n\minus{}2}\plus{}...\plus{}a_{2n\minus{}2}X^2\plus{}(2n\minus{}10)X\plus{}1$, if it is known that all the roots of them are positive reals. [i]Proposer[/i]: [b]Baltag Valeriu[/b]

Let $n\geq 2$ be integer. On a plane there are $n+2$ points $O,\ P_{0},\ P_{1},\ \cdots P_{n}$ which satisfy the following conditions as follows. [1] $\angle{P_{k-1}OP_{k}}=\frac{\pi}{n}\ (1\leq k\leq n),\ \angle{OP_{k-1}P_{k}}=\angle{OP_{0}P_{1}}\ (2\leq k\leq n).$ [2] $\overline{OP_{0}}=1,\ \overline{OP_{1}}=1+\frac{1}{n}.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}\overline{P_{k-1}P_{k}}.$

Consider the polynomial \[P(x)=x^3+3x^2+6x+10.\] Let its three roots be $a$, $b$, $c$. Define $Q(x)$ to be the monic cubic polynomial with roots $ab$, $bc$, $ca$. Compute $|Q(1)|$. [i]Proposed by Nathan Xiong[/i]

Let $x(t)$ be a solution to the differential equation \[\left(x+x^\prime\right)^2+x\cdot x^{\prime\prime}=\cos t\] with $x(0)=x^\prime(0)=\sqrt{\frac{2}{5}}$. Compute $x\left(\dfrac{\pi}{4}\right)$.

Solve the system of simultaneous equations \[\begin{cases}a^3+3ab^2+3ac^2-6abc=1\\ b^3+3ba^2+3bc^2-6abc=1\\c^3+3ca^2+3cb^2-6abc=1\end{cases}\] in real numbers.

Find the real solution(s) to the equation $ (x \plus{} y)^2 \equal{} (x \plus{} 1)(y \minus{} 1)$.

Given are $n$ pairwise intersecting convex $k$-gons on the plane. Any of them can be transferred to any other by a homothety with a positive coefficient. Prove that there is a point in a plane belonging to at least $1 +\frac{n-1}{2k}$ of these $k$-gons.

Convex equiangular hexagon $ABCDEF$ has $AB=CD=EF=1$ and $BC = DE = FA = 4$. Congruent and pairwise externally tangent circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ are drawn such that $\gamma_1$ is tangent to side $\overline{AB}$ and side $\overline{BC}$, $\gamma_2$ is tangent to side $\overline{CD}$ and side $\overline{DE}$, and $\gamma_3$ is tangent to side $\overline{EF}$ and side $\overline{FA}$. Then the area of $\gamma_1$ can be expressed as $\frac{m\pi}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Sean Li[/i]

Suppose $x_1,x_2,x_3,\dotsc$ is a strictly decreasing sequence of positive real numbers such that the series $x_1+x_2+x_3+\cdots$ diverges. Is it necessary true that the series $\sum_{n=2}^{\infty}{\min \left\{ x_n,\frac{1}{n\log (n)}\right\} }$ diverges?

Suppose that \[ \frac {2x}{3} \minus{} \frac {x}{6} \]is an integer. Which of the following statements must be true about $ x$? $ \textbf{(A)}\ \text{It is negative.} \qquad \textbf{(B)}\ \text{It is even, but not necessarily a multiple of }3\text{.}$ $ \textbf{(C)}\ \text{It is a multiple of }3\text{, but not necessarily even.}$ $ \textbf{(D)}\ \text{It is a multiple of }6\text{, but not necessarily a multiple of }12\text{.}$ $ \textbf{(E)}\ \text{It is a multiple of }12\text{.}$

For how many positive integers $n\le100$ is it true that $10n$ has exactly three times as many positive divisors as $n$ has?

There are $11$ integers (not necessarily distinct) written on the board. Can it turn out that the product of any five of them is greater than the product of the other six?

Cynthia loves Pokemon and she wants to catch them all. In Victory Road, there are a total of $80$ Pokemon. Cynthia wants to catch as many of them as possible. However, she cannot catch any two Pokemon that are enemies with each other. After exploring around for a while, she makes the following two observations: 1. Every Pokemon in Victory Road is enemies with exactly two other Pokemon. 2. Due to her inability to catch Pokemon that are enemies with one another, the maximum number of the Pokemon she can catch is equal to $n$. What is the sum of all possible values of $n$?

While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers $a$, $b$, and $c$, and recalled that their product is $24$, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than $25$ with fewer than $6$ divisors. Help James by computing $a+b+c$.

Let $n$ be a positive integer. Prove that at least one of the integers $[2^n \cdot \sqrt2]$, $[2^{n+1} \cdot \sqrt2]$, $...$, $[2^{2n} \cdot \sqrt2]$ is even, where $[a]$ denotes the integer part of $a$.

Integers are written in the cells of a table $2010 \times 2010$. Adding $1$ to all the numbers in a row or in a column is called a [i]move[/i]. We say that a table is [i]equilibrium[/i] if one can obtain after finitely many moves a table in which all the numbers are equal. [list=a] [*]Find the largest positive integer $n$, for which there exists an [i]equilibrium[/i] table containing the numbers $2^0, 2^1, \ldots , 2^n$. [*] For this $n$, find the maximal number that may be contained in such a table. [/list]