For every positive integer $n$, let $s(n)$ be the sum of the exponents of $71$ and $97$ in the prime factorization of $n$; for example, $s(2021) = s(43 \cdot 47) = 0$ and $s(488977) = s(71^2 \cdot 97) = 3$. If we define $f(n)=(-1)^{s(n)}$, prove that the limit \[ \lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n} \] exists and determine its value.

Let $p$ be a prime number greater than 10. Prove that there exist positive integers $m$ and $n$ such that $m+n < p$ and $5^m 7^n-1$ is divisible by $p$.

$ABC$ is an equilateral triangle with side length $1$. Point $D$ lies on $\overline{AB}$, point $E$ lies on $\overline{AC}$, and points $G$ and $F$ lie on $\overline{BC}$, such that $DEFG$ is a square. What is the area of $DEFG$?

Find both extrema of the function $ x\to\frac{\sin x-3}{\cos x +2} .$

Let $ ABCD$ be a quadrilateral in which $ AB$ is parallel to $ CD$ and perpendicular to $ AD; AB \equal{} 3CD;$ and the area of the quadrilateral is $ 4$. if a circle can be drawn touching all the four sides of the quadrilateral, find its radius.

Let $n$ be some natural number. One boss writes $n$ letters a day numerated from 1 to $n$ consecutively. When he writes a letter he piles it up (on top) in a box. When his secretary is free, she gets the letter on the top of the pile and prints it. Sometimes the secretary isn’t able to print the letter before her boss puts another one or more on the pile in the box. Though she is always able to print all of the letters at the end of the day. A permutation is called [i]“printable”[/i] if it is possible for the letters to be printed in this order. Find a formula for the number of [i]“printable”[/i] permutations.

When writing the product of two three-digit numbers, the multiplication sign was omitted, forming a six-digit number. It turns out that the six-digit number is equal to three times the product. Find the six-digit number.

Let $f$ and $g$ be functions from the set $A$ to the same set $A$. We define $f$ to be a functional $n$-th root of $g$ ($n$ is a positive integer) if $f^n(x) = g(x)$, where $f^n(x) = f^{n-1}(f(x)).$ (a) Prove that the function $g : \mathbb R \to \mathbb R, g(x) = 1/x$ has an infinite number of $n$-th functional roots for each positive integer $n.$ (b) Prove that there is a bijection from $\mathbb R$ onto $\mathbb R$ that has no nth functional root for each positive integer $n.$

In triangle ABC, consider the sizes $\tan \angle A, \tan \angle B$, and $\tan \angle C$ into another such as the numbers $1, 2$ and $3$. Find the ratio of the sidelenghts $AC$ and $AB$ of the triangle.

Let $a, b, c$ be the lengths of the sides of a triangle, and let $S$ be it's area. Prove that $$S \le \frac{a^2+b^2+c^2}{4\sqrt3}$$ and the equality is achieved only for an equilateral triangle.

Given parallelogram $ABCD$, construct point $F$ so that $CF\perp BC$, as shown. Also $F$ is placed so that $\angle DFC = 120^o$. If $DF = 4$ and $BC =CF = 2$, what is the area of the parallelogram? [img]https://cdn.artofproblemsolving.com/attachments/7/4/0cdb0752760686acb891580da55f55212098fb.jpg[/img]

Let $n$ be the number of ways that $10$ dollars can be changed into dimes and quarters, with at least one of each coin being used. Then $n$ equals: $\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 38 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 19 $

Prove that the number $\left(\frac{76}{\frac{1}{\sqrt[3]{77}-\sqrt[3]{75}}-\sqrt[3]{5775}}+\frac{1}{\frac{76}{\sqrt[3]{77}+\sqrt[3]{75}}+\sqrt[3]{5775}}\right)^3$ is an integer.

Find $2^{2006}$ positive integers satisfying the following conditions. (i) Each positive integer has $2^{2005}$ digits. (ii) Each positive integer only has 7 or 8 in its digits. (iii) Among any two chosen integers, at most half of their corresponding digits are the same.

The figure below depicts two congruent triangles with angle measures $40^\circ$, $50^\circ$, and $90^\circ$. What is the measure of the obtuse angle $\alpha$ formed by the hypotenuses of these two triangles? [asy] import olympiad; size(80); defaultpen(linewidth(0.8)); draw((0,0)--(3,0)--(0,4.25)--(0,0)^^(0,3)--(4.25,0)--(3,0)^^rightanglemark((0,3),(0,0),(3,0),10)); pair P = intersectionpoint((3,0)--(0,4.25),(0,3)--(4.25,0)); draw(anglemark((4.25,0),P,(0,4.25),10)); label("$\alpha$",P,2 * NE); [/asy]

Let the set $A=(a_{1},a_{2},a_{3},a_{4})$ . If the sum of elements in every 3-element subset of $A$ makes up the set $B=(-1,5,3,8)$ , then find the set $A$.

For positive integer $n$ let $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example, $a_3 = 999,488$ and $a_7 = 9,999,999,488$. Find the value of $n$ so that an is divisible by the highest power of $2$.

Three gamblers play against each other for money. They each start by placing a pile of one-krone coins on the table, and from this point on the total number of coins on the table does not change. The ratio between the number of coins they start with is $6 : 5 : 4$. At the end of the game, the ratio of the number of coins they have is $7 : 6 : 5$ in some order. At the end of the game, one of the gamblers has three coins more than at the beginning. How many coins does this gambler have at the end?

Find all triplets $(x, y, z)$ of real numbers that satisfy the equation $$2^{x^2-3y+z}+2^{y^2-3z+x}+2^{z^2-3x+y}=1,5.$$

Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$ [asy] size(160); defaultpen(linewidth(1.1)); path square = (1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle; filldraw(square,white); filldraw(rotate(30)*square,white); filldraw(rotate(60)*square,white); dot((0,0),linewidth(7)); [/asy] $\textbf{(A)}\: 75\qquad\textbf{(B)} \: 93\qquad\textbf{(C)} \: 96\qquad\textbf{(D)} \: 129\qquad\textbf{(E)} \: 147$

Find the smallest positive integer $n$ which can not be expressed as $n =\frac{2^a - 2^b}{2^c - 2^d}$ for some positive integers $a, b, c, d$

A square is triangulated in such way that no three vertices are collinear. For every vertex (including vertices of the square) the number of sides issuing from it is counted. Can it happen that all these numbers are even?

Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$

Let $N\ge2$ be a fixed positive integer. There are $2N$ people, numbered $1,2,...,2N$, participating in a tennis tournament. For any two positive integers $i,j$ with $1\le i<j\le 2N$, player $i$ has a higher skill level than player $j$. Prior to the first round, the players are paired arbitrarily and each pair is assigned a unique court among $N$ courts, numbered $1,2,...,N$. During a round, each player plays against the other person assigned to his court (so that exactly one match takes place per court), and the player with higher skill wins the match (in other words, there are no upsets). Afterwards, for $i=2,3,...,N$, the winner of court $i$ moves to court $i-1$ and the loser of court $i$ stays on court $i$; however, the winner of court 1 stays on court 1 and the loser of court 1 moves to court $N$. Find all positive integers $M$ such that, regardless of the initial pairing, the players $2, 3, \ldots, N+1$ all change courts immediately after the $M$th round. [i]Proposed by Ray Li[/i]

For a three-digit prime number $p$, the equation $x^3+y^3=p^2$ has an integer solution. Calculate $p$.