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 \[3x_1-x_2-x_3-x_5 = 0,\]\[-x_1+3x_2-x_4-x_6= 0,\]\[-x_1 + 3x_3 - x_4 - x_7 = 0,\]\[-x_2 - x_3 + 3x_4 - x_8 = 0,\]\[-x_1 + 3x_5 - x_6 - x_7 = 0,\]\[-x_2 - x_5 + 3x_6 - x_8 = 0,\]\[-x_3 - x_5 + 3x_7 - x_8 = 0,\]\[-x_4 - x_6 - x_7 + 3x_8 = 0.\]

Eve and Martti have a whole number of euros. Martti said to Eve: ''If you give give me three euros, so I have $n$ times the money compared to you. '' Eve in turn said to Martti: ''If you give me $n$ euros then I have triple the amount of money compared to you'' . Suppose, that both claims are valid. What values can a positive integer $n$ get?

Let $ABC$ be a triangle and $X$ be a point on its circumcircle. $Q,P$ lie on a line $BC$ such that $XQ\perp AC , XP\perp AB$. Let $Y$ be the circumcenter of $\triangle XQP$. Prove that $ABC$ is equilateral triangle if and if only $Y$ moves on a circle when $X$ varies on the circumcircle of $ABC$.

Alice and Bob live on the edges and vertices of the unit cube. Alice begins at point $(0, 0, 0)$ and Bob begins at $(1, 1, 1)$. Every second, each of them chooses one of the three adjacent corners and walks at a constant rate of $1$ unit per second along the edge until they reach the corner, after which they repeat the process. What is the expected amount of time in seconds before Alice and Bob meet?

Find the roots of $6x^4+17x^3+7x^2-8x-4$

Given $n > 2$ nonnegative distinct integers $a_1,...,a_n$, find all nonnegative integers $y$ and $x_1,...,x_n$ satisfying $gcd(x_1,...,x_n) = 1$ and $$\begin{cases} a_1x_1 + a_2x_2 +...+ a_nx_n = yx_1 \\ a_2x_1 + a_3x_2 +...+ a_1x_n = yx_2 \\ ... \\ a_nx_1 + a_1x_2 +...+ a_{n-1}x_n = yx_n \end{cases}$$

In the figure on the right, $O$ is the center of the circle, $OK$ and $OA$ are perpendicular to one another, $M$ is the midpoint of $OK$, $BN$ is parallel to $OK$, and $\angle AMN=\angle NMO$. Determine the measure of $\angle A B N$ in degrees. [asy] defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, A=dir(90), K=dir(180), M=0.5*dir(180), N=2/5*dir(90), B=dir(degrees((2/5, sqrt(21/25)))+90); draw(K--O--A--M--N--B--A^^Circle(origin,1)); label("$A$", A, dir(O--A)); label("$K$", K, dir(O--K)); label("$B$", B, dir(O--B)); label("$N$", N, E); label("$M$", M, S); label("$O$", O, SE);[/asy]

A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were $ 100$ tourists on the ferry boat, and that on each successive trip, the number of tourists was $ 1$ fewer than on the previous trip. How many tourists did the ferry take to the island that day? $ \textbf{(A)}\ 585\qquad \textbf{(B)}\ 594\qquad \textbf{(C)}\ 672\qquad \textbf{(D)}\ 679\qquad \textbf{(E)}\ 694$

For real positive numbers $a, b, c$, the equality $abc = 1$ holds. Prove that $$\frac{a^{2014}}{1 + 2 bc}+\frac{b^{2014}}{1 + 2ac}+\frac{c^{2014}}{1 + 2ab} \ge \frac{3}{ab+bc+ca}.$$

Let $ S(n) $ be the sum of the squares of the positive integers less than and coprime to $ n $. For example, $ S(5) = 1^2 + 2^2 + 3^2 + 4^2 $, but $ S(4) = 1^2 + 3^2 $. Let $ p = 2^7 - 1 = 127 $ and $ q = 2^5 - 1 = 31 $ be primes. The quantity $ S(pq) $ can be written in the form $$ \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right) $$ where $ a $, $ b $, and $ c $ are positive integers, with $ b $ and $ c $ coprime and $ b < c $. Find $ a $.

We have $4$ sticks. It is known, that for every $3$ sticks we can build a triangle with the same area. Is it true, that sticks have the same length?