In the expression $\left(\underline{\qquad}\times\underline{\qquad}\right)+\left(\underline{\qquad}\times\underline{\qquad}\right)$ each blank is to be filled in with one of the digits $1,2,3,$ or $4,$ with each digit being used once. How many different values can be obtained? $ \textbf{(A) }2 \qquad \textbf{(B) }3\qquad \textbf{(C) }4 \qquad \textbf{(D) }6 \qquad \textbf{(E) }24 \qquad $

Let $a_1 = 1, a_2, a_3,...$: be a sequence of positive integers such that $$a_k < 1 + a_1 + a_2 +... + a_{k-1}$$ for all $k > 1$. Prove that every positive integer can be expressed as a sum of $a_i$s.

Let $a$, $b$, $c$ be positive reals for which \begin{align*} (a+b)(a+c) &= bc + 2 \\ (b+c)(b+a) &= ca + 5 \\ (c+a)(c+b) &= ab + 9 \end{align*} If $abc = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Evan Chen[/i]

Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. What is the degree measure of $\angle ACB$? $ \textbf{(A) }105 \qquad \textbf{(B) }112\frac{1}{2} \qquad \textbf{(C) }120 \qquad \textbf{(D) }135 \qquad \textbf{(E) }150 \qquad $

Points $P,Q,R,S$ lie on a circle and $\angle PSR$ is right. $H,K$ are the projections of $Q$ on lines $PR,PS$. Prove that $HK$ bisects segment $ QS$.

Let $a$ and $b$ be positive integers such that\[a^2+(a+1)^2=b^4.\]Find the least possible value of $a+b$.

Let $\triangle ABC$ be a triangle with circumcenter $O$. The perpendicular bisectors of the segments $OA,OB$ and $OC$ intersect the lines $BC,CA$ and $AB$ at $D,E$ and $F$, respectively. Prove that $D,E,F$ are collinear.

Fuzzy draws a segment of positive length in a plane. How many locations can Fuzzy place another point in the same plane to form a non-degenerate isosceles right triangle with vertices consisting of his new point and the endpoints of the segment? [i]Proposed by Timothy Qian[/i]

Suppose $PAB$ and $PCD$ are two secants of circle $O$. Lines $AD \cap BC=Q$. Point $T$ lie on segment $BQ$ and point $K$ is intersection of segment $PT$ with circle $O$, $S=QK\cap PA$ Given that $ST \parallel PQ$, prove that $B,S,K,T$ lie on a circle.

[b]9.[/b] Lep $p$ be a connected non-closed broken line without self-intersection in the plane $\varphi $. Prove that if $v$ is a non-zero vector in $\varphi $ and $p$ has a commom point with the broken line $p+v$, then $p$ has a common point with the broken line $p+\alpha v$ too, where $\alpha =\frac{1}{n}$ and $n$ is a positive integer. Does a similar statemente hold for other positive values of $\alpha$? ($p+v$ denotes the broken line obtained from $p$ through displacemente by the vector $v$.) [b](G. 1)[/b]

Two straight lines intersecting at an angle of $46^o$ are the axes of symmetry of the figure $F$ on the plane. What is the smallest number of axes of symmetry this figure can have?

A spherical shell of mass $M$ and radius $R$ is completely filled with a frictionless fluid, also of mass M. It is released from rest, and then it rolls without slipping down an incline that makes an angle $\theta$ with the horizontal. What will be the acceleration of the shell down the incline just after it is released? Assume the acceleration of free fall is $g$. The moment of inertia of a thin shell of radius $r$ and mass $m$ about the center of mass is $I = \frac{2}{3}mr^2$; the momentof inertia of a solid sphere of radius r and mass m about the center of mass is $I = \frac{2}{5}mr^2$. $\textbf{(A) } g \sin \theta \\ \textbf{(B) } \frac{3}{4} g \sin \theta\\ \textbf{(C) } \frac{1}{2} g \sin \theta\\ \textbf{(D) } \frac{3}{8} g \sin \theta\\ \textbf{(E) } \frac{3}{5} g \sin \theta$

Alice is making a batch of cookies and needs $2 \frac{1}{2}$ cups of sugar. Unforunately, her measuring cup holds only $\frac{1}{4}$ cup of sugar. How many times must she fill that cup to get the correct amount of sugar? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 16 \qquad\textbf{(E)}\ 20$

Suppose $\mathcal{T}=A_0A_1A_2A_3$ is a tetrahedron with $\angle A_1A_0A_3 = \angle A_2A_0A_1 = \angle A_3A_0A_2 = 90^\circ$, $A_0A_1=5, A_0A_2=12$ and $A_0A_3=9$. A cube $A_0B_0C_0D_0E_0F_0G_0H_0$ with side length $s$ is inscribed inside $\mathcal{T}$ with $B_0\in \overline{A_0A_1}, D_0 \in \overline{A_0A_2}, E_0 \in \overline{A_0A_3}$, and $G_0\in \triangle A_1A_2A_3$; what is $s$?

Given an acute non-isosceles triangle $ABC$ with circumcircle $\Gamma$. $M$ is the midpoint of segment $BC$ and $N$ is the midpoint of arc $BC$ of $\Gamma$ (the one that doesn't contain $A$). $X$ and $Y$ are points on $\Gamma$ such that $BX\parallel CY\parallel AM$. Assume there exists point $Z$ on segment $BC$ such that circumcircle of triangle $XYZ$ is tangent to $BC$. Let $\omega$ be the circumcircle of triangle $ZMN$. Line $AM$ meets $\omega$ for the second time at $P$. Let $K$ be a point on $\omega$ such that $KN\parallel AM$, $\omega_b$ be a circle that passes through $B$, $X$ and tangents to $BC$ and $\omega_c$ be a circle that passes through $C$, $Y$ and tangents to $BC$. Prove that circle with center $K$ and radius $KP$ is tangent to 3 circles $\omega_b$, $\omega_c$ and $\Gamma$. [i]Proposed by Tran Quan - Vietnam[/i]

A5) Find the maximum value of $\int_{0}^{y}\sqrt{x^{4}+(y-y^{2})^{2}}dx$ for $0\leq y\leq 1$. I don't have a solution for this yet. I figure this may be useful: Let the integral be denoted $f(y)$, then according to the [url=http://mathworld.wolfram.com/LeibnizIntegralRule.html]Leibniz Integral Rule[/url] we have $\frac{df}{dy}=\int_{0}^{y}\frac{y(1-y)(1-2y)}{\sqrt{x^{4}+(y-y^{2})^{2}}}dx+\sqrt{y^{4}+(y-y^{2})^{2}}$ Now what?

Let $ABCD$ be an arbitrary quadrilateral. Squares with centers $M_1, M_2, M_3, M_4$ are constructed on $AB,BC,CD,DA$ respectively, all outwards or all inwards. Prove that $M_1 M_3=M_2 M_4$ and $M_1 M_3\perp M_2 M_4$.

A flame test was performed to confirm the identity of a metal ion in solution. The result was a green flame. Which of the following metal ions is indicated? ${ \textbf{(A)}\ \text{copper} \qquad\textbf{(B)}\ \text{sodium} \qquad\textbf{(C)}\ \text{strontium} \qquad\textbf{(D)}\ \text{zinc} } $

Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.

Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.

Points $X,Y$ lie on the side $CD$ of a convex pentagon $ABCDE$ with $X$ between $Y$ and $C$. Suppose that the triangles $\bigtriangleup XCB, \bigtriangleup ABX, \bigtriangleup AXY, \bigtriangleup AYE, \bigtriangleup YED$ are all similar (in this exact order). Prove that circumcircles of the triangles $\bigtriangleup ACD, \bigtriangleup AXY$ are tangent. [i]Pouria Mahmoudkhan Shirazi - Iran[/i]

Every square of a $3\times3$ board is assigned a sign $+$ or $-$. In every move, one square is selected and the signs are changed in the selected square and all the neighboring squares (two squares are neighboring if they have a common side). Is it true that, no matter how the signs were initially distributed, one can obtain a table in which all signs are $-$ after finitely many moves?

The angles of quadrilateral $ ABCD$ satisfy $ \angle A \equal{} 2 \angle B \equal{} 3 \angle C \equal{} 4 \angle D$. What is the degree measure of $ \angle A$, rounded to the nearest whole number? $ \textbf{(A)}\ 125 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 153 \qquad \textbf{(D)}\ 173 \qquad \textbf{(E)}\ 180$

In a basketball tournament every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$, and there are no draws. A single team wins the tournament with $26$ points and exactly two teams share the last position with $20$ points. How many teams participated in the tournament?

For a complex number $z=1+2\sqrt{6}i$ and natural number $n=1,\ 2,\ 3,\ \cdots$, express the complex number $z^n$ in using real numbers $a_n,\ b_n$ as $z^n=a_n+b_ni$. Answer the following questions. (1) Show that $a_n^2+b_n^2=5^{2n}\ (n=1,\ 2,\ 3,\ \cdots).$ (2) Find the constants $p,\ q$ such that $a_{n+2}=pa_{n+1}+qa_n$ holds for all $n$. (3) Show that $a_n$ is not a multiple of $5$ for any $n$. (4) Show that $z^n\ (n=1,\ 2,\ 3,\ \cdots)$ is not a real number.