Point $P$ is inside triangle $\triangle{ABC}$ such that $\angle{ABP}=\angle{ACP}.$ Given that $AB=6, AC=8, BC=7,$ and $\tfrac{BP}{PC}=\tfrac{1}{2},$ compute $\tfrac{[BPC]}{[ABC]}.$ (Here, $[XYZ]$ denotes the area of $\triangle{XYZ}.$)

A $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ where $h_i\left(x_1,x_2, \cdots , x_n\right)$ are $n$ variable polynomials with real coefficients is called [i]good[/i] if the following condition holds: For any $n$ functions $f_1,f_2, \cdots ,f_n : \mathbb R \to \mathbb R$ if for all $1 \le i \le n+1$, $P_i(x)=h_i \left(f_1(x),f_2(x), \cdots, f_n(x) \right)$ is a polynomial with variable $x$, then $f_1(x),f_2(x), \cdots, f_n(x)$ are polynomials. $a)$ Prove that for all positive integers $n$, there exists a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that the degree of all $h_i$ is more than $1$. $b)$ Prove that there doesn't exist any integer $n>1$ that for which there is a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that all $h_i$ are symmetric polynomials. [i]Proposed by Alireza Shavali[/i]

Suppose we choose two numbers $x,y\in[0,1]$ uniformly at random. If the probability that the circle with center $(x,y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1]\times [0,1]$ is written as $\tfrac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?

Let $ABC$ be a triangle with $AB>AC$. Let $D$ be a point on side $AB$ such that $BD=AC$. Consider the circle $\gamma$ passing through point $D$ and tangent to side $AC$ at point $A$. Consider the circumscribed circle $\omega$ of the triangle $ABC$ that interesects the circle $\gamma$ at points $A$ and $E$. Prove that point $E$ is the intersection point of the perpendicular bisectors of line segments $BC$ and $AD$.

Find all positive integer solutions $(x,y,z,n)$ of equation $x^{2n+1}-y^{2n+1}=xyz+2^{2n+1}$, where $n\ge 2$ and $z \le 5\times 2^{2n}$.

[b]Problem.[/b] Let $f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ be a function defined as $$f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+.$$ Prove that for any positive integer $m,$ the sequence $$m,f(m),f(f(m)),f(f(f(m))),\ldots$$ contains a perfect square.

There are two math exams called A and B. 2014 students took the A exam and/or the B exam. Each student took one or both exams, so the total number of exam papers was between 2014 and 4028, inclusive. The score for each exam is an integer from 0 through 40. The average score of all the exam papers was 20. The grade for a student is the best score from one or both exams that she took. The average grade of all 2014 students was 14. Let $G$ be the [i]greatest[/i] possible number of students who took both exams. Let $L$ be the [i]least[/i] possible number of students who took both exams. Compute $G - L$.

Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$. [i]Oleksii Masalitin, Ukraine[/i]

Show that the sequence \[\binom{2002}{2002},\binom{2003}{2002},\binom{2004}{2002},\ldots \] considred modulo $2002$, is periodic.

Ted flips seven fair coins. there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that Ted flips at least two heads given that he flips at least three tails. Find $m+n$.

Does there exist a continuously differentiable function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for every $x \in \mathbb{R}$ we have $f(x) > 0$ and $f'(x) = f(f(x))$?

Prove that the equation $ \frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c}\plus{}\frac{1}{abc} \equal{} \frac {12}{a \plus{} b \plus{} c}$ has infinitely many solutions $ (a,b,c)$ in natural numbers.

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

Let \(\mathbb{R}^2\) denote the set of points in the Euclidean plane. For points \(A,P\in\mathbb{R}^2\) and a real number \(k\), define the [i]dilation[/i] of \(A\) about \(P\) by a factor of \(k\) as the point \(P+k(A-P)\). Call a sequence of point \(A_0, A_1, A_2,\ldots\in\mathbb{R}^2\) [i]unbounded[/i] if the sequence of lengths \(\left|A_0-A_0\right|,\left|A_1-A_0\right|,\left|A_2-A_0\right|,\ldots\) has no upper bound. Now consider \(n\) distinct points \(P_0,P_1,\ldots,P_{n-1}\in\mathbb{R}^2\), and fix a real number \(r\). Given a starting point \(A_0\in\mathbb{R}^2\), iteratively define \(A_{i+1}\) by dilating \(A_i\) about \(P_j\) by a factor of \(r\), where \(j\) is the remainder of \(i\) when divided by \(n\). Prove that if \(\left|r\right|\geq 1\), then for any starting point \(A_0\in\mathbb{R}^2\), the sequence \(A_0,A_1,A_2,\ldots\) is either periodic or unbounded. [i]Proposed by the ICMC Problem Committee[/i]

Find $ \lim_{n\to\infty} \int_0^1 x^2|\sin n\pi x|\ dx\ (n\equal{}1,\ 2,\cdots)$.

For which integers $n \geq 2$ is it possible to draw $n$ straight lines in the plane in such a way that there are at least $n - 2$ points where exactly three of the lines meet?

Neel and Roshan are going to the Newark Liberty International Airport to catch separate flights. Neel plans to arrive at some random time between 5:30 am and 6:30 am, while Roshan plans to arrive at some random time between 5:40 am and 6:40 am. The two want to meet, however briefly, before going through airport security. As such, they agree that each will wait for $n$ minutes once he arrives at the airport before going through security. What is the smallest $n$ they can select such that they meet with at least 50% probability? The answer will be of the form $a + b\sqrt{c}$ for integers $a$, $b$, and $c$, where $c$ has no perfect square factor other than $1$. Report $a + b + c.$

Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$

Let $a = (\sqrt2 +\sqrt3 +\sqrt6)(\sqrt2 +\sqrt3 -\sqrt6)(\sqrt3 +\sqrt6 -\sqrt2)(\sqrt6 +\sqrt2 -\sqrt3)$ $b = (\sqrt2 +\sqrt3 +\sqrt5)(\sqrt2 +\sqrt3 -\sqrt5)(\sqrt3 +\sqrt5 -\sqrt2)(\sqrt5 +\sqrt2 -\sqrt3)$ The difference $a - b$ belongs to the set: A. $(-\infty,-4)$ B. $[-4,0)$ C.$\{0\}$ D. $(0,4]$ E. $(4,\infty)$

$ABCDEF$ is a regular hexagon. A frog sarts jumping at $A$, each time it can jump to one of the two adjacent points. If the frog jump to $D$ in no more than five times, it stops. After five jumpings, if the frog hasn't jumped to $D$ yet, it will stop as well. Then the number of different ways to jump is________.

A positive integer $n$ is said to possess Property ($A$) if there exists a positive integer $N$ such that $N^2$ can be written as the sum of the squares of $n$ consecutive positive integers. Is it true that there are infinitely many positive integers which possess Property ($A$)? Justify your answer. (As an example, the number $n = 2$ possesses Property ($A$) since $5^2 = 3^2 + 4^2$).

Find all real solutions to the system of equations $$\begin{cases} x^2 +y^2 = 6z \\ y^2 +z^2 = 6x \\ z^2 +x^2 = 6y \end{cases}$$

Two cirlces $ C_1$ and $ C_2$, with center $ O_1$ and $ O_2$ respectively, intersect at $ A$ and $ B$. Let $ O_1$ lies on $ C_2$. A line $ l$ passes through $ O_1$ but does not pass through $ O_2$. Let $ P$ and $ Q$ be the projection of $ A$ and $ B$ respectively on the line $ l$ and let $ M$ be the midpoint of $ \overline{AB}$. Prove that $ MPQ$ is an isoceles triangle.

The points $A, B, C$ and $D$ lie in this order on the circle $k$. Let $t$ be the tangent at $k$ through $C$ and $s$ the reflection of $AB$ at $AC$. Let $G$ be the intersection of the straight line $AC$ and $BD$ and $H$ the intersection of the straight lines $s$ and $CD$. Show that $GH$ is parallel to $t$.

Let $a, b, c$ be the lengths of the sides of a triangle and $A, B, C$, the opposite angles. Prove that $$Aa + Bb + Cc \ge \frac{Ab + Ac + Ba + Bc + Ca + Cb}{2}$$