Let $ABC$ be a triangle and $L$ its circumscribed circle. The internal bisector of angle $A$ meets $BC$ at point $P$. Let $L_1$ be the circle tangent to $AP,BP$ and $L$. Similarly, let $L_2$ be the circle tangent to $AP,CP$ and $L$. Prove that the tangency points of $L_1$ and $L_2$ with $AP$ coincide.

A class of $50$ students took an exam with $4$ questions. At least $1$ of any $40$ students gave exactly $3$, at least $2$ of any $40$ gave exactly $2$, and at least $3$ of any $40$ gave exactly $1$ correct answers. At least $4$ of any $40$ students gave exactly $4$ wrong answers. What is the least number of students who gave an odd number of correct answers? $ \textbf{(A)}\ 18 \qquad\textbf{(B)}\ 24 \qquad\textbf{(C)}\ 26 \qquad\textbf{(D)}\ 28 \qquad\textbf{(E)}\ \text{None of the above} $

Let $p,q$ be positive integers. Consider a rectangle $ABCD$ with lengths of sides $p$ and $q$ that consists of $pq$ unital squares. How many of these squares are crossed by diagonal $AC$?

Find all pairs of positive integers $m$ and $n$ such that $$m! + n! = m^n.$$ .

Find the minimum value of $\int_0^{\frac{\pi}{2}} |a\sin 2x-\cos ^ 2 x|dx\ (a>0).$ 2009 Shimane University entrance exam/Medicine

SEEMOUS 2016 COMPETITION PROBLEMS

Triangle $LMT$ has $\overline{MA}$ as an altitude. Given that $MA = 16$, $MT = 20$, and $LT = 25$, find the length of the altitude from $L$ to $\overline{MT}$. [i]Proposed by Kevin Zhao[/i]

Find all polynomials $ p$ of one variable with integer coefficients such that if $ a$ and $ b$ are natural numbers such that $ a \plus{} b$ is a perfect square, then $ p\left(a\right) \plus{} p\left(b\right)$ is also a perfect square.

Consider the four circles tangent to all three lines containing the sides of a triangle $ABC$; let $k$ and $k_c$ be those tangent to side $AB$ between $A$ and $B$. Prove that the geometric mean of the radii of k and $k_c$, does not exceed half the length of $AB$.

Let $f(n)=\sum_{k=0}^{n-1}x^ky^{n-1-k}$ with, $x$, $y$ real numbers. If $f(n)$, $f(n+1)$, $f(n+2)$, $f(n+3)$, are integers for some $n$, prove $f(n)$ is integer for all $n$.

A sequence starts at some rational number $x_1>1$, and is subsequently defined using the recurrence relation \[x_{n+1}=\frac{x_n\cdot n}{\lfloor x_n\cdot n\rfloor }\] Show that $k>0$ exists with $x_k=1$.

Suppose $S = \{1, 2, 3, x\}$ is a set with four distinct real numbers for which the difference between the largest and smallest values of $S$ is equal to the sum of elements of $S.$ What is the value of $x?$ $$ \mathrm a. ~ {-1}\qquad \mathrm b.~{-3/2}\qquad \mathrm c. ~{-2} \qquad \mathrm d. ~{-2/3} \qquad \mathrm e. ~{-3} $$

Geodude wants to assign one of the integers $1,2,3,\ldots,11$ to each lattice point $(x,y,z)$ in a 3D Cartesian coordinate system. In how many ways can Geodude do this if for every lattice parallelogram $ABCD$, the positive difference between the sum of the numbers assigned to $A$ and $C$ and the sum of the numbers assigned to $B$ and $D$ must be a multiple of $11$? (A [i]lattice point[/i] is a point with all integer coordinates. A [i]lattice parallelogram[/i] is a parallelogram with all four vertices lying on lattice points. Here, we say four not necessarily distinct points $A,B,C,D$ form a [i]parallelogram[/i] $ABCD$ if and only if the midpoint of segment $AC$ coincides with the midpoint of segment $BD$.) [hide="Clarifications"] [list] [*] The ``positive difference'' between two real numbers $x$ and $y$ is the quantity $|x-y|$. Note that this may be zero. [*] The last sentence was added to remove confusion about ``degenerate parallelograms.''[/list][/hide] [i]Victor Wang[/i]

Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k \plus{} 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta \equal{} \pi/6$.

$A = \{a, b, c\}$ is a set containing three positive integers. Prove that we can find a set $B \subset A$, $B = \{x, y\}$ such that for all odd positive integers $m, n$ we have $$10\mid x^my^n-x^ny^m.$$ [i]Tomi Dimovski[/i]