Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied. (Walther Janous)

Prove that for every prime number $p\ge5$, (a) $p^3$ divides $\binom{2p}p-2$; (b) $p^3$ divides $\binom{kp}p-k$ for every natural number $k$.

Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$ holds. [i](Walther Janous)[/i]

Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is $44$ years, then Ann's age is $\textbf{(A) }22\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad \textbf{(E) }28$

If $ p \geq 5$ is a prime number, then $ 24$ divides $ p^2 \minus{} 1$ without remainder $ \textbf{(A)}\ \text{never} \qquad \textbf{(B)}\ \text{sometimes only} \qquad \textbf{(C)}\ \text{always} \qquad$ $ \textbf{(D)}\ \text{only if } p \equal{}5 \qquad \textbf{(E)}\ \text{none of these}$

Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its sides.

Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$, determine $ f(9) .$

In a triangle $ABC$, $M$ is the midpoint of the $AB$. A point $B_1$ is marked on $AC$ such that $CB=CB_1$. Circle $\omega$ and $\omega_1$, the circumcircles of triangles $ABC$ and $BMB_1$, respectively, intersect again at $K$. Let $Q$ be the midpoint of the arc $ACB$ on $\omega$. Let $B_1Q$ and $BC$ intersect at $E$. Prove that $KC$ bisects $B_1E$. [i]M. Kungozhin[/i]

Find the largest possible value of $a+b$ less than or equal to $2007$, for which $a$ and $b$ are relatively prime, and such that there is some positive integer $n$ for which \[\frac{2^3-1}{2^3+1}\cdot\frac{3^3-1}{3^3+1}\cdot\frac{4^3-1}{4^3+1}\cdots\frac{n^3-1}{n^3+1} = \frac ab.\]

In how many different ways can the set $\{1,2,\dots, 2006\}$ be divided into three non-empty sets such that no set contains two successive numbers? $ \textbf{(A)}\ 3^{2006}-3\cdot 2^{2006}+1 \qquad\textbf{(B)}\ 2^{2005}-2 \qquad\textbf{(C)}\ 3^{2004} \qquad\textbf{(D)}\ 3^{2005}-1 \qquad\textbf{(E)}\ \text{None of above} $

Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[ n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality. [i]Proposed by Finbarr Holland, Ireland[/i]

Show that from a set of $11$ square integers one can select six numbers $a^2,b^2,c^2,d^2,e^2,f^2$ such that $a^2+b^2+c^2 \equiv d^2+e^2+f^2\pmod{12}$.

In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b \geq 2$. A Martian student writes down \begin{align*}3 \log(\sqrt{x}\log x) &= 56\\\log_{\log (x)}(x) &= 54 \end{align*} and finds that this system of equations has a single real number solution $x > 1$. Find $b$.

The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime? [asy]unitsize(30); draw(unitcircle); draw((0,0)--(0,-1)); draw((0,0)--(cos(pi/6),sin(pi/6))); draw((0,0)--(-cos(pi/6),sin(pi/6))); label("$1$",(0,.5)); label("$3$",((cos(pi/6))/2,(-sin(pi/6))/2)); label("$5$",(-(cos(pi/6))/2,(-sin(pi/6))/2));[/asy] [asy]unitsize(30); draw(unitcircle); draw((0,0)--(0,-1)); draw((0,0)--(cos(pi/6),sin(pi/6))); draw((0,0)--(-cos(pi/6),sin(pi/6))); label("$2$",(0,.5)); label("$4$",((cos(pi/6))/2,(-sin(pi/6))/2)); label("$6$",(-(cos(pi/6))/2,(-sin(pi/6))/2));[/asy] $ \textbf{(A)}\ \frac {1}{2} \qquad \textbf{(B)}\ \frac {2}{3} \qquad \textbf{(C)}\ \frac {3}{4} \qquad \textbf{(D)}\ \frac {7}{9} \qquad \textbf{(E)}\ \frac {5}{6}$

Let $P = \{P_1, P_2, ..., P_{1997}\}$ be a set of $1997$ points in the interior of a circle of radius 1, where $P_1$ is the center of the circle. For each $k=1.\ldots,1997$, let $x_k$ be the distance of $P_k$ to the point of $P$ closer to $P_k$, but different from it. Show that $(x_1)^2 + (x_2)^2 + ... + (x_{1997})^2 \le 9.$

At a table tennis competition, every pair of players played each other exactly once. Every boy beat thrice as many boys as girls, and every girl was beaten by twice as many girls as boys. How many competitors were there, if we know that there were $10$ more boys than girls? There are no draws in table tennis, every match was won by one of the two players.

The diagram below shows a sequence of equally spaced parallel lines with a triangle whose vertices lie on these lines. The segment $\overline{CD}$ is $6$ units longer than the segment $\overline{AB}$. Find the length of segment $\overline{EF}$. [img]https://cdn.artofproblemsolving.com/attachments/8/0/abac87d63d366bf4c4e913fdb1022798379a73.png[/img]

Let $x_1, x_2,\cdots, x_n$ be $n$ integers. Let $n = p + q$, where $p$ and $q$ are positive integers. For $i = 1, 2, \cdots, n$, put \[S_i = x_i + x_{i+1} +\cdots + x_{i+p-1} \text{ and } T_i = x_{i+p} + x_{i+p+1} +\cdots + x_{i+n-1}\] (it is assumed that $x_{i+n }= x_i$ for all $i$). Next, let $m(a, b)$ be the number of indices $i$ for which $S_i$ leaves the remainder $a$ and $T_i$ leaves the remainder $b$ on division by $3$, where $a, b \in \{0, 1, 2\}$. Show that $m(1, 2)$ and $m(2, 1)$ leave the same remainder when divided by $3.$

What fraction of the large $12$ by $18$ rectangular region is shaded? [asy] draw((0,0)--(18,0)--(18,12)--(0,12)--cycle); draw((0,6)--(18,6)); for(int a=6; a<12; ++a) { draw((1.5a,0)--(1.5a,6)); } fill((15,0)--(18,0)--(18,6)--(15,6)--cycle,black); label("0",(0,0),W); label("9",(9,0),S); label("18",(18,0),S); label("6",(0,6),W); label("12",(0,12),W); [/asy] $\text{(A)}\ \frac{1}{108} \qquad \text{(B)}\ \frac{1}{18} \qquad \text{(C)}\ \frac{1}{12} \qquad \text{(D)}\ \frac29 \qquad \text{(E)}\ \frac13$

The point $K{}$ is the middle of the arc $BAC$ of the circumcircle of the triangle $ABC$. The point $I{}$ is the center of its inscribed circle $\omega$. The line $KI$ intersects the circumcircle of the triangle $ABC$ at $T{}$ for the second time. Prove that the circle passing through the midpoints of the segments $BC, BT$ and $CT$ is tangent to the circle which is symmetric to $\omega$ with respect to $BC$.

Calculate the Gauss integral \[\oint\frac{(d\overrightarrow{A},d\overrightarrow{B},\overrightarrow{A}-\overrightarrow{B})}{|\overrightarrow{A}-\overrightarrow{B}|^3}\] where $\overrightarrow{A}$ runs along the curve $x=\cos\alpha$, $y=\sin\alpha$, $z=0$, and $\overrightarrow{B}$ along the curve $x=2\cos^2\beta$, $y=\frac12\sin\beta$, $z=\sin2\beta$. Note: that $\oint$ was supposed to be oiint (i.e. $\iint$ with a circle) but the command does not work on AoPS.

Let $ABC$ be a triangle with $AB=209$, $AC=243$, and $\angle BAC = 60^\circ$, and denote by $N$ the midpoint of the major arc $\widehat{BAC}$ of circle $\odot(ABC)$. Suppose the parallel to $AB$ through $N$ intersects $\overline{BC}$ at a point $X$. Compute the ratio $\tfrac{BX}{XC}$.

Given a sequence of integers $A_1,A_2,\cdots A_{99}$ such that for every sub-sequence that contains $m$ consecutive elements, there exist not more than $max\{ \frac{m}{3} ,1\}$ odd integers. Let $S=\{ (i,j) \ | i<j \}$ such that $A_i$ is even and $A_j$ is odd. Find $max\{ |S|\}$.

The least value of the function $ ax^2\plus{}bx\plus{}c$ with $a>0$ is: $\textbf{(A)}\ -\dfrac{b}{a} \qquad \textbf{(B)}\ -\dfrac{b}{2a} \qquad \textbf{(C)}\ b^2-4ac \qquad \textbf{(D)}\ \dfrac{4ac-b^2}{4a}\qquad \textbf{(E)}\ \text{None of these}$

Find all (real) solutions of the system \[3x_1-x_2-x_3-x_5 = 0,\]\[-x_1+3x_2-x_4-x_6= 0,\]\[-x_1 + 3x_3 - x_4 - x_7 = 0,\]\[-x_2 - x_3 + 3x_4 - x_8 = 0,\]\[-x_1 + 3x_5 - x_6 - x_7 = 0,\]\[-x_2 - x_5 + 3x_6 - x_8 = 0,\]\[-x_3 - x_5 + 3x_7 - x_8 = 0,\]\[-x_4 - x_6 - x_7 + 3x_8 = 0.\]