Let $ABC$ be a triangle in which the length of side $AB$ is $4$ units, and that of $BC$ is $2$ units. Let $D$ be the point on $AB$ at distance $3$ units from $A$. Prove that the line perpendicular to $AB$ through $D$, the angle bisector of $\angle ABC$, and the perpendicular bisector of $BC$ all meet at a single point.

The diameters of two circles are $ 8$ inches and $ 12$ inches respectively. The ratio of the area of the smaller to the area of the larger circle is: $ \textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{4}{9} \qquad\textbf{(C)}\ \frac{9}{4} \qquad\textbf{(D)}\ \frac{1}{2} \qquad\textbf{(E)}\ \text{none of these}$

A cube is made by soldering twelve $ 3$-inch lengths of wire properly at the vertices of the cube. If a fly alights at one of the vertices and then walks along the edges, the greatest distance it could travel before coming to any vertex a second time, without retracing any distance, is: $ \textbf{(A)}\ 24\text{ in.}\qquad \textbf{(B)}\ 12\text{ in.}\qquad \textbf{(C)}\ 30\text{ in.}\qquad \textbf{(D)}\ 18\text{ in.}\qquad \textbf{(E)}\ 36\text{ in.}$

Let $ABC$ be a non-degenerate triangle with incenter $I$ and circumcircle $\Gamma$. Denote by $M_a$ the midpoint of the arc $\widehat{BC}$ of $\Gamma$ not containing $A$, and define $M_b$, $M_c$ similarly. Suppose $\triangle ABC$ has inradius $4$ and circumradius $9$. Compute the maximum possible value of \[IM_a^2+IM_b^2+IM_c^2.\][i]Proposed by David Altizio[/i]

Let $n \geq 2$ be an integer. Find the smallest positive integer $m$ for which the following holds: given $n$ points in the plane, no three on a line, there are $m$ lines such that no line passes through any of the given points, and for all points $X \neq Y$ there is a line with respect to which $X$ and $Y$ lie on opposite sides

Into how many regions do $n$ circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?

Let $\Omega =\{ (x,y,z)\in \mathbb{Z}^3:y+1\geqslant x\geqslant y\geqslant z\geqslant 0\}$. A frog moves along the points of $\Omega$ by jumps of length $1$. For every positive integer $n$, determine the number of paths the frog can take to reach $(n,n,n)$ starting from $(0,0,0)$ in exactly $3n$ jumps. [i]Proposed by Fedor Petrov and Anatoly Vershik, St. Petersburg State University[/i]

Suppose $ M(y)$ is a polynomial of degree $ n$ such that $ M(y) \equal{} 2^y$ for $ y \equal{} 1, 2, \ldots, n \plus{} 1$. Compute $ M(n \plus{} 2)$.

A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.

Compute $99(99^2+3) + 3\cdot99^2$. [i]Proposed by Evan Chen[/i]

For the triple $(a,b,c)$ of positive integers we say it is interesting if $c^2+1\mid (a^2+1)(b^2+1)$ but none of the $a^2+1, b^2+1$ are divisible by $c^2+1$. Let $(a,b,c)$ be an interesting triple, prove that there are positive integers $u,v$ such that $(u,v,c)$ is interesting and $uv<c^3$.

Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$.

Let $f$ be a function $R \to R$ that satisfies the following equation: $$f (x)^2 + f (y)^2 = f (x^2 + y^2)+ f (0)$$ If there are $n$ possibilities for the function, find the sum of all values of $n \cdot f (12)$

Let $k > 2$ be an integer. A positive integer $l$ is said to be $k-pable$ if the numbers $1, 3, 5, . . . , 2k - 1$ can be partitioned into two subsets $A$ and $B$ in such a way that the sum of the elements of $A$ is exactly $l$ times as large as the sum of the elements of $B$. Show that the smallest $k-pable$ integer is coprime to $k$.

Denote by $M$ and $N$ feets of perpendiculars from $A$ to angle bisectors of exterior angles at $B$ and $C,$ in triangle $\triangle ABC.$ Prove that the length of segment $MN$ is equal to semiperimeter of triangle $\triangle ABC.$

We call a positive integer $n$ [i]delightful[/i] if there exists an integer $k$, $1 < k < n$, such that \[1+2+\cdots+(k-1)=(k+1)+(k+2)+\cdots+n\] Does there exist a delightful number $N$ satisfying the inequalities \[2013^{2013}<\dfrac{N}{2013^{2013}}<2013^{2013}+4 ?\]

Let $ABC$ be a triangle inscribed in circle $\omega$ with center $O$, let $\omega_A$ be its $A$-mixtilinear incircle, $\omega_B$ be its $B$-mixtilinear incircle, $\omega_C$ be its $C$-mixtilinear incircle, and $X$ be the radical center of $\omega_A$, $\omega_B$, $\omega_C$. Let $A'$, $B'$, $C'$ be the points at which $\omega_A$, $\omega_B$, $\omega_C$ are tangent to $\omega$. Prove that $AA'$, $BB'$, $CC'$ and $OX$ are concurrent. [i]Proposed by Robin Park[/i]

The product $55\cdot60\cdot65$ is written as a product of 5 distinct numbers. Find the least possible value of the largest number, among these 5 numbers.

As shown in the figure, two circles $\Gamma_1$ and $\Gamma_2$ on the plane intersect at two points $A$ and $B$. The two rays passing through $A$, $\ell_1$ and $\ell_2$ intersect $\Gamma_1$ at points $D$ and $E$ respectively, and $\Gamma_2$ at points $F$ and $C$ respectively (where $E$ and $F$ lie on line segments $AC$ and $AD$ respectively, and neither of them coincides with the endpoints). It is known that the three lines $AB$, $CF$ and $DE$ have a common point, the circumscribed circle of $\vartriangle AEF$ intersects $AB$ at point $G$, the straight line $EG$ intersects the circle $\Gamma_1$ at point $P$, the straight line $FG$ intersects the circle $\Gamma_2$ at point $Q$. Let the symmetric points of $C$ and $D$ wrt the straight line $AB$ be $C'$ and $D'$ respectively. If $PD'$ and $QC'$ intersect at point$ J$, prove that $J$ lies on the straight line $AB$. [img]https://cdn.artofproblemsolving.com/attachments/3/7/eb3acdbad52750a6879b4b6955dfdb7de19ed3.png[/img]

On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set. [i]Proposed by Tonći Kokan, Croatia[/i]

In triangle $ABC$, interior point $D$ is chosen such that triangle $BCD$ is equilateral. Let $E$ be the isogonal conjugate of point $D$ with respect to triangle $ABC$. Define point $P$ on the ray $AB$ such that $AP=BE$. Similarly, define point $Q$ on the ray $AC$ such that $AQ=CE$. Prove that line $AD$ bisects segment $PQ$. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

A positive integer $a > 1$ is given (in decimal notation). We copy it twice and obtain a number $b = \overline{aa}$ which happens to be a multiple of $a^2$. Find all possible values of $b/a^2$.

For three positive real numbers $ a,b,c $ satisfying the condition $ \frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca} =1, $ prove that $$ 3/2\le \frac{ab-1}{ab+1} +\frac{bc-1}{bc+1} +\frac{ca-1}{ca+1} <2. $$ [i]Mircea Becheanu[/i]

On a line $\ell$ there exists $3$ points $A, B$, and $C$ where $B$ is located between $A$ and $C$. Let $\Gamma_1, \Gamma_2, \Gamma_3$ be circles with $AC, AB$, and $BC$ as diameter respectively; $BD$ is a segment, perpendicular to $\ell$ with $D$ on $\Gamma_1$. Circles $\Gamma_4, \Gamma_5, \Gamma_6$ and $\Gamma_7$ satisfies the following conditions: $\bullet$ $\Gamma_4$ touches $\Gamma_1, \Gamma_2$, and$ BD$. $\bullet$ $\Gamma_5$ touches $\Gamma_1, \Gamma_3$, and $BD$. $\bullet$ $\Gamma_6$ touches $\Gamma_1$ internally, and touches $\Gamma_2$ and $\Gamma_3$ externally. $\bullet$ $\Gamma_7$ passes through $B$ and the tangent points of $\Gamma_2$ with $\Gamma_6$, and $\Gamma_3$ with $\Gamma_6$. Show that the circles $\Gamma_4, \Gamma_5$, and $\Gamma_7$ are congruent.

DeAndre Jordan shoots free throws that are worth $1$ point each. He makes $40\%$ of his shots. If he takes two shots find the probability that he scores at least $1$ point.