Let $ D \subset \mathbb {R} ^ {2} $ be a finite Lebesgue measure of a connected open set and $ u: D \rightarrow \mathbb {R} $ a harmonic function. Show that it is either a constant $ u $ or for almost every $ p \in D $ $$ f ^ {\prime} (t) = (\operatorname {grad} u) (f (t)), \quad f (0) = p $$has no initial value problem(differentiable everywhere) solution to $ f:[0,\infty) \rightarrow D $.

The bottom, side, and front areas of a rectangular box are known. The product of these areas is equal to: $ \textbf{(A)}\ \text{the volume of the box} \qquad\textbf{(B)}\ \text{the square root of the volume} \qquad\textbf{(C)}\ \text{twice the volume}$ $ \textbf{(D)}\ \text{the square of the volume} \qquad\textbf{(E)}\ \text{the cube of the volume}$

Let $N$ be the number of ordered pairs $(x,y)$ of integers such that $x^2+xy+y^2 \le 2007$. Remember, integers may be positive, negative, or zero! (a) Prove that $N$ is odd. (b) Prove that $N$ is not divisible by $3$.

A triangle $ABC$ is given. Let $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ be circles centered at points $X$, $Y$, $Z$, $T$ respectively such that each of lines $BC$, $CA$, $AB$ cuts off on them four equal chords. Prove that the centroid of $ABC$ divides the segment joining $X$ and the radical center of $\omega_2$, $\omega_3$, $\omega_4$ in the ratio $2:1$ from $X$.

A rubber band is wrapped around two pipes as shown. One has radius $3$ inches and the other has radius $9$ inches. The length of the band can be expressed as $a\pi+b\sqrt{c}$ where $a$, $b$, $c$ are integers and $c$ is square free. What is $a+b+c$? [asy] size(4cm); draw(circle((0,0),3)); draw(circle((12,0),9)); draw(3*dir(120)--(12,0)+9*dir(120)); draw(3*dir(240)--(12,0)+9*dir(240)); [/asy]

For any permutation $ f : \{ 1, 2, \cdots , n \} \to \{1, 2, \cdots , n \} $, and define \[ A = \{ i | i > f(i) \} \] \[ B = \{ (i, j) | i<j \le f(j) < f(i) \ or \ f(j) < f(i) < i < j \} \] \[ C = \{ (i, j) | i<j \le f(i) < f(j) \ or \ f(i) < f(j) < i < j \} \] \[ D = \{ (i, j) | i< j \ and \ f(i) > f(j)\} \] Prove that $ |A| + 2|B| + |C| = |D| $.

find all pairs of non-negative integer pairs $(m,n)$,satisfies $107^{56}(m^{2}-1)+2m+3=\binom{113^{114}}{n}$

A young baseball player thinks he has hit a home run and gets excited, but, instead, he has just hit it to an outfielder who is just able to catch the ball, and does so at ground level. The ball was hit at a height of 1.5 meters from the ground at an angle $\phi$ above the horizontal axis. The catch was taken at a horizontal distance 30 meters from home plate, which was where the batter hit the ball. The ball left the bat at a speed of 21 m/s. Find all possible values $0<\phi<90^\circ$, in degrees, rounded to the nearest integer. You may use WolframAlpha, Mathematica, or a graphing aid to compute $\phi$ after you derive an expression to solve for it. [i](Proposed by Ahaan Rungta)[/i]

Show that: $$ 2015\in\left\{ x_1+2x_2+3x_3\cdots +2015x_{2015}\big| x_1,x_2,\ldots ,x_{2015}\in \{ -2,3\}\right\}\not\ni 2016. $$

Around the convex quadrilateral $ABCD$, three rectangles are circumscribed . It is known that two of these rectangles are squares. Is it true that the third one is necessarily a square? (A rectangle is circumscribed around the quadrilateral $ABCD$ if there is one vertex $ABCD$ on each side of the rectangle).

Find all real pairs $x,y$ such that \begin{align*} x-|y+1|&=1, \\ x^2+y&=10. \end{align*}

In triangle \( ABC \), the incircle is tangent to side \( BC \) at point \( D \), the excircle opposite vertex \( B \) is tangent to line \( AB \) at point \( X \), and the excircle opposite vertex \( C \) is tangent to line \( AC \) at point \( Y \). If \( T \) is the midpoint of segment \( [AD] \) and \( U \) is the circumcenter of triangle \( AXY \), show that \( UT \perp BC \).

Set $M=\{u|u=12m+8n+4l,m,n,l\in\mathbb{Z}\},N=\{u|u=20p+16q+12x,p,q,x\in\mathbb{Z}\}$. Then $\text{(A)}M=N\qquad\text{(B)}M\not\subset N,N\not\subset M\qquad\text{(C)}M\subset N\qquad\text{(D)}N\subset M$

Determine all tuples of integers $(a,b,c)$ such that: $$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$$

Prove that every real function, defined on all of $\mathbb R$, can be represented as a sum of two functions whose graphs both have an axis of symmetry. [i]D. Tereshin[/i]

Let $n$ be a positive integer and $f(x)=a_mx^m+\ldots + a_1X+a_0$, with $m\ge 2$, a polynomial with integer coefficients such that: a) $a_2,a_3\ldots a_m$ are divisible by all prime factors of $n$, b) $a_1$ and $n$ are relatively prime. Prove that for any positive integer $k$, there exists a positive integer $c$, such that $f(c)$ is divisible by $n^k$.

Let $AB$ be a chord, not a diameter, of a circle with center $O$. The smallest arc $AB$ is divided into three congruent arcs $AC$, $CD$, $DB$. The chord $AB$ is also divided into three equal segments $AC'$, $C'D'$, $D'B$. Let $P$ be the intersection point of between the lines $CC'$ and $DD'$. Prove that $\angle APB = \frac13 \angle AOB$.

Let $p \geq 5$ be a prime number. Prove that there exists a positive integer $a < p-1$ such that neither of $a^{p-1}-1$ and $(a+1)^{p-1}-1$ is divisible by $p^{2}$ .

Let $A$, $B$, $C$, $D$ be four points in space not all lying on the same plane. The segments $AB$, $BC$, $CD$, and $DA$ are tangent to the same sphere. Prove that their four points of tangency are coplanar.

Consider the set of natural numbers $ U = \{1,2,3, ..., 6024 \} $ Prove that for any partition of the $ U $ in three subsets with $ 2008 $ elements each, we can choose a number in each subset so that one of the numbers is the sum of the other two numbers.

A subset $ B$ of the set of integers from $ 1$ to $ 100$, inclusive, has the property that no two elements of $ B$ sum to $ 125$. What is the maximum possible number of elements in $ B$? $ \textbf{(A)}\ 50\qquad \textbf{(B)}\ 51\qquad \textbf{(C)}\ 62\qquad \textbf{(D)}\ 65\qquad \textbf{(E)}\ 68$

Given a triangle $A, B, C, X$ is on side $AB$, $Y$ is on side $AC$, and $P$ and $Q$ are on side $BC$ such that $AX = AY , BX = BP$ and $CY = CQ$. Let $XP$ and $YQ$ intersect at $T$. Prove that $AT$ passes through the midpoint of $PQ$.

Let $\Gamma$ be a circle and $P$ a point outside of $\Gamma$ . A tangent from $P$ to the circle intersects it in $A$. Another line through $P$ intersects $\Gamma$ at the points $B$ and $C$. The bisector of $\angle APB$ intersects $AB$ at $D$ and $AC$ at $E$. Prove that the triangle $ADE$ is isosceles.

Let $ \left( a_n \right)_{n\ge 1} $ be an arithmetic progression with $ a_1=1 $ and natural ratio. [b]a)[/b] Prove that $$ a_n^{1/a_k} <1+\sqrt{\frac{2\left( a_n-1 \right)}{a_k\left( a_k -1 \right)}} , $$ for any natural numbers $ 2\le k\le n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{1}{a_n}\sum_{k=1}^n a_n^{1/a_k} . $ [i]Nicolae Bourbăcuț[/i]

A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. What is the area of the border, in square inches? $\textbf{(A)}\hspace{.05in}36 \qquad \textbf{(B)}\hspace{.05in}40 \qquad \textbf{(C)}\hspace{.05in}64 \qquad \textbf{(D)}\hspace{.05in}72 \qquad \textbf{(E)}\hspace{.05in}88 $