Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where m and n are relatively prime positive integers. Find $m + n$.

Let $a,b,c$ be integers larger than $1$. Prove that \[a(a-1)+b(b-1)+c(c-1)\le (a+b+c-4)(a+b+c-5)+4.\]

Prove that if one of the numbers $25x+31y, 3x+7y$ (where $x,y \in Z$) is a multiple of $41$, then so is the other.

Let $[ {x} ]$ be the greatest integer less than or equal to $x$, and let $\{x\}=x-[x]$. Solve the equation: $[x] \cdot \{x\} = 2005x$

Three points are randomly placed on a circle. What is the probability that they lie on the same semicircle

a) Prove that it is possible to choose one number out of any 39 consecutive positive integers, having the sum of its digits divisible by 11; b) Find the first 38 consecutive positive integers none of which have the sum of its digits divisible by 11.

Let $a, b, c, d$ be positive reals strictly smaller than $1$, such that $a+b+c+d=2$. Prove that $$\sqrt{(1-a)(1-b)(1-c)(1-d)} \leq \frac{ac+bd}{2}. $$

Given are $n + 1$ points $P_1, P_2,..., P_n$ and $Q$ in the plane, no three collinear. For any two different points $P_i$ and $P_j$ , there is a point $P_k$ such that the point $Q$ lies inside the triangle $P_iP_jP_k$. Prove that $n$ is an odd number.

Let $x_1,x_2,\dots, x_n$ be positive real numbers such that $x_1+x_2+\cdots + x_n=1$. Prove that $$\sum_{i=1}^n \frac{\min\{x_{i-1},x_i\}\cdot \max\{x_i,x_{i+1}\}}{x_i}\leq 1,$$ where we denote $x_0=x_n$ and $x_{n+1}=x_1$.

Let $k$ be a positive integer. An arrangement of finitely many open intervals in $R$ is called [i]good [/i] if for any of the intervals the number of other intervals which intersect with it is a nonzero multiple of $k$. Find the maximum positive integer $n$ (as a function of $k$) for which there is no good arrangement with $n$ intervals

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties: (a). $x_1 < x_2 < \cdots < x_{n-1}$ ; (b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$; (c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.

Product of two integers is $1$ less than three times of their sum. Find those integers.

Prove that for any positive integer $n\ge $, $2 \cdot \sqrt3 \cdot \sqrt[3]{4} ...\sqrt[n-1]{n} > n$

Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$

For real numbers $ x$ and $ y$, define $ x\spadesuit y \equal{} (x \plus{} y)(x \minus{} y)$. What is $ 3\spadesuit(4\spadesuit 5)$? $ \textbf{(A) } \minus{} 72 \qquad \textbf{(B) } \minus{} 27 \qquad \textbf{(C) } \minus{} 24 \qquad \textbf{(D) } 24 \qquad \textbf{(E) } 72$

Andriyko has rectangle desk and a lot of stripes that lie parallel to sides of the desk. For every pair of stripes we can say that first of them is under second one. In desired configuration for every four stripes such that two of them are parallel to one side of the desk and two others are parallel to other side, one of them is under two other stripes that lie perpendicular to it. Prove that Andriyko can put stripes one by one such way that every next stripe lie upper than previous and get desired configuration. [i]Proposed by Denys Smirnov[/i]

Find all integers $m$ and $n$ such that $2m^2 +n^2 = 2mn+3n$

Determine all triples $(p, q, r)$ of prime numbers for which $p^q + p^r$ is a perfect square.

[b]p1.[/b] Let $S$ be the sum of the $99$ terms: $$(\sqrt1 + \sqrt2)^{-1},(\sqrt2 + \sqrt3)^{-1}, (\sqrt3 + \sqrt4)^{-1},..., (\sqrt{99} + \sqrt{100})^{-1}.$$ Prove that $S$ is an integer. [b]p2.[/b] Determine all pairs of positive integers $x$ and $y$ for which $N=x^4+4y^4$ is a prime. (Your work should indicate why no other solutions are possible.) [b]p3.[/b] Let $w,x,y,z$ be arbitrary positive real numbers. Prove each inequality: (a) $xy \le \left(\frac{x+y}{2}\right)^2$ (b) $wxyz \le \left(\frac{w+x+y+z}{4}\right)^4$ (c) $xyz \le \left(\frac{x+y+z}{3}\right)^3$ [b]p4.[/b] Twelve points $P_1$,$P_2$, $...$,$P_{12}$ are equally spaaed on a circle, as shown. Prove: that the chords $\overline{P_1P_9}$, $\overline{P_4P_{12}}$ and $\overline{P_2P_{11}}$ have a point in common. [img]https://cdn.artofproblemsolving.com/attachments/d/4/2eb343fd1f9238ebcc6137f7c84a5f621eb277.png[/img] [b]p5.[/b] Two very busy men, $A$ and $B$, who wish to confer, agree to appear at a designated place on a certain day, but no earlier than noon and no later than $12:15$ p.m. If necessary, $A$ will wait $6$ minutes for $B$ to arrive, while $B$ will wait $9$ minutes for $A$ to arrive but neither can stay past $12:15$ p.m. Express as a percent their chance of meeting. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Each coefficient of a polynomial is an integer with absolute value not exceeding $2015$. Prove that every positive root of this polynomial exceeds $\frac{1}{2016}$. [i]($6$ points)[/i]

$ABC$ is a triangle with $|AB|=6$, $|BC|=7$, and $|AC|=8$. Let the angle bisector of $\angle A$ intersect $BC$ at $D$. If $E$ is a point on $[AC]$ such that $|CE|=2$, what is $|DE|$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ \frac {17}5 \qquad\textbf{(C)}\ \frac 72 \qquad\textbf{(D)}\ 2\sqrt 3 \qquad\textbf{(E)}\ 3\sqrt 2 $

For $x, y$ positive integers, $x^2-4y+1$ is a multiple of $(x-2y)(1-2y)$. Prove that $|x-2y|$ is a square number.

Function ${{f\colon \mathbb[0, +\infty)}\to\mathbb[0, +\infty)}$ satisfies the condition $f(x)+f(y){\ge}2f(x+y)$ for all $x,y{\ge}0$. Prove that $f(x)+f(y)+f(z){\ge}3f(x+y+z)$ for all $x,y,z{\ge}0$. Mathematical induction? __________________________________ Azerbaijan Land of the Fire :lol:

Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length $3$ and the remaining four sides of length $2$. Give the answer in the form $r+s\sqrt{t}$ with $r,s, t$ positive integers.