Prove that if $ a $, $ b $, $ c $, $ d $ are the sides of a quadrilateral in which a circle can be circumscribed and a circle can be inscribed in it, then the area $ S $ of the quadrilateral is given by $$S = \sqrt{abcd}.$$

A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle? [asy] unitsize(0.25cm); pair A, B, C, O; A = (-8, 0); B = (8, 0); C = (0, 15); O = (0, 0); draw(arc(O, 120/17, 0, 180)); draw(A--B--C--cycle); [/asy] $\textbf{(A) }4 \sqrt{3}\qquad\textbf{(B) } \dfrac{120}{17}\qquad\textbf{(C) }10\qquad\textbf{(D) }\dfrac{17\sqrt{2}}{2}\qquad \textbf{(E) }\dfrac{17\sqrt{3}}{2}$

Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition \[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 \] for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$ [i]Author: Nikolai Nikolov, Bulgaria[/i]

Define $H_n = 1+\frac{1}{2}+\cdots+\frac{1}{n}$. Let the sum of all $H_n$ that are terminating in base 10 be $S$. If $S = m/n$ where m and n are relatively prime positive integers, find $100m+n$. [i]Proposed by Lewis Chen[/i]

The orthocenter of triangle $ABC$ lies on its circumcircle. One of the angles of $ABC$ must equal: (The orthocenter of a triangle is the point where all three altitudes intersect.) $$ \mathrm a. ~ 30^\circ\qquad \mathrm b.~60^\circ\qquad \mathrm c. ~90^\circ \qquad \mathrm d. ~120^\circ \qquad \mathrm e. ~\text{It cannot be deduced from the given information.} $$

What is the area of the region determined by the points outside a triangle with perimeter length $\pi$ where none of these points has a distance greater than $1$ to any corner of the triangle? $ \textbf{(A)}\ 4\pi \qquad\textbf{(B)}\ 3\pi \qquad\textbf{(C)}\ \dfrac{5\pi}2 \qquad\textbf{(D)}\ 2\pi \qquad\textbf{(E)}\ \dfrac{3\pi}2 $

For all positive integer $ n \ge 2 $, prove that $ 2^n -1 $ can't be a divisor of $ 3^n -1 $.

Let $N = 11\cdots 122 \cdots 25$, where there are $n$ $1$s and $n+1$ $2$s. Show that $N$ is a perfect square.

Let $x$, $y$ be two positive integers such that $\displaystyle 3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect square.

Let $ ABC$ be a triangle with $ \angle A = 60^{\circ}$.Prove that if $ T$ is point of contact of Incircle And Nine-Point Circle, Then $ AT = r$, $ r$ being inradius.

For a positive integer $k,$ call an integer a $pure$ $k-th$ $power$ if it can be represented as $m^k$ for some integer $m.$ Show that for every positive integer $n,$ there exists $n$ distinct positive integers such that their sum is a pure $2009-$th power and their product is a pure $2010-$th power.

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$, $$\{ x,f(x),\cdots f^{p-1}(x) \} $$ is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$. [i]Proposed by IndoMathXdZ[/i]

Let $P(x)$ be a polynomial with degree $2017$ such that $P(k) =\frac{k}{k + 1}$, $\forall k = 0, 1, 2, ..., 2017$ . Calculate $P(2018)$.

Find the number of positive integers $n$ for which (i) $n \leq 1991$; (ii) 6 is a factor of $(n^2 + 3n +2)$.

Let the triangle $ABC$ be with $| AB | > | AC |$. Point M is the midpoint of the side $[BC]$, and point $I$ is the center of the circle inscribed in the triangle ABC such that the relation $| AI | = | MI |$. Prove that points $A, B, M, I$ are located on the same circle.

The figure below is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m}+\sqrt{n},$ where $m$ and $n$ are positive integers. What is $m+n?$ [asy] /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,S); label("D",D,S); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); [/asy] $\textbf{(A) }20 \qquad \textbf{(B) }21 \qquad \textbf{(C) }22\qquad \textbf{(D) }23 \qquad \textbf{(E) }24$ Proposed by [b]djmathman[/b]

Among all triangles with a given perimeter, find the one with the maximal radius of its incircle.

Prove that the product of four consecutive positive integers cannot be a perfect square or cube.

In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has two lanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane, and Sammy on the outer. They will race for one complete lap, measured by the inner track. What is the square of the distance between Willy and Sammy's starting positions so that they will both race the same distance? Assume that they are of point size and ride perfectly along their respective lanes

Find the number of ordered pairs $(a, b)$ of integers, where $1 \le a, b \le 2004$, such that $x^2 + ax + b = 167 y$ has integer solutions in $x$ and $y$. Justify your answer.

The three-digit number $2a3$ is added to the number $326$ to give the three-digit number $5b9$. If $5b9$ is divisible by 9, then $a+b$ equals $ \text{(A)}\ 2\qquad\text{(B)}\ 4\qquad\text{(C)}\ 6\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9$

Given are three circles $(O_1), (O_2), (O_3)$, pairwise intersecting each other, that is, every single circle meets the other two circles at two distinct points. Let $(X_1)$ be the circle externally tangent to $(O_1)$ and internally tangent to the circles $(O_2), (O_3),$ circles $(X_2), (X_3)$ are defined in the same manner. Let $(Y_1)$ be the circle internally tangent to $(O_1)$ and externally tangent to the circles $(O_2), (O_3)$, the circles $(Y_2), (Y_3)$ are defined in the same way. Let $(Z_1), (Z_2)$ be two circles internally tangent to all three circles $(O_1), (O_2), (O_3)$. Prove that the four lines $X_1Y_1, X_2Y_2, X_3Y_3, Z_1Z_2$ are concurrent. Nguyễn Văn Linh

Let $T_1$ be a triangle having $a, b, c$ as lengths of its sides and let $T_2$ be another triangle having $u, v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that \[16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).\] When does equality hold?

An L-shaped figure composed of $4$ unit squares (such as shown in the picture) we call L-dominoes. [img]https://cdn.artofproblemsolving.com/attachments/b/2/064b7c7de496f981cd937cbb7392efc1066420.png[/img] Determine the maximum number of L-dominoes that can be placed on a board of dimensions $n \times n$, where $n$ is natural number, so that no two dominoes overlap and it is possible get from the upper left to the lower right corner of the board by moving only across those squares that are not covered by dominoes. (By moving, we move from someone of the square on it the neighboring square, i.e. the square with which it shares the page). Note: L-Dominoes can be rotated as well as flipped, giving an symmetrical figure wrt axis compared to the one shown in the picture.

Points $A, B, C, D,$ and $F$ lie on a sphere with radius $\sqrt{10}$ such that lines $AD, BE,$ and $CF$ are concurrent at point $P$ inside the sphere and are pairwise perpendicular. If $PA=\sqrt{6}, PB=\sqrt{10},$ and $PC=\sqrt{15},$ what is the volume of tetrahedron $DEFP$?