Consider the sequence $ \left( I_n \right)_{n\ge 1} , $ where $ I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, $ for any natural number $ n. $ [b]a)[/b] Find a relation between any two consecutive terms of $ I_n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } nI_n. $ [i]c)[/i] Show that $ \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx. $ [i]Cătălin Țigăeru[/i]

If $ \angle\text{CBD} $ is a right angle, then this protractor indicates that the measure of $ \angle\text{ABC} $ is approximately [asy] unitsize(36); pair A,B,C,D; A=3*dir(160); B=origin; C=3*dir(110); D=3*dir(20); draw((1.5,0)..(0,1.5)..(-1.5,0)); draw((2.5,0)..(0,2.5)..(-2.5,0)--cycle); draw(A--B); draw(C--B); draw(D--B); label("O",(-2.5,0),W); label("A",A,W); label("B",B,S); label("C",C,W); label("D",D,E); label("0",(-1.8,0),W); label("20",(-1.7,.5),NW); label("160",(1.6,.5),NE); label("180",(1.7,0),E);[/asy] $ \text{(A)}\ 20^\circ\qquad\text{(B)}\ 40^\circ\qquad\text{(C)}\ 50^\circ\qquad\text{(D)}\ 70^\circ\qquad\text{(E)}\ 120^\circ $

There are integers $a$ and $b$, such that $a>b>1$ and $b$ is the largest divisor of $a$ different from $a$. Prove that the number $a+b$ is not a power of $2$ with integer exponent.

For two positive integers a and b, which are relatively prime, find all integer that can be the great common divisor of $a+b$ and $\frac{a^{2005}+b^{2005}}{a+b}$.

$M$ is an integer set with a finite number of elements. Among any three elements of this set, it is always possible to choose two such that the sum of these two numbers is an element of $M.$ How many elements can $M$ have at most?

On an infinite “squared” sheet six squares are shaded as in the diagram. On some squares there are pieces. It is possible to transform the positions of the pieces according to the following rule: if the neighbour squares to the right and above a given piece are free, it is possible to remove this piece and put pieces on these free squares. The goal is to have all the shaded squares free of pieces. Is it possible to reach this goal if (a) In the initial position there are $6$ pieces and they are placed on the $6$ shaded squares? (b) In the initial position there is only one piece, located in the bottom left shaded square? [img]https://cdn.artofproblemsolving.com/attachments/2/d/0d5cbc159125e2a84edd6ac6aca5206bf8d83b.png[/img] (M Kontsevich, Moscow)

Does there exist a function $f: \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}$ such that \[ f(ab) = f(a)b + af(b) \] for all $a,b \in \mathbb{Z}_{\geq 0}$ and $f(p) > p^p$ for every prime number $p$? [i] (Here, $\mathbb{Z}_{\geq 0}$ denotes the set of non-negative integers.)[/i]

Prove that no matter what digits are placed in the four empty boxes, the eight-digit number \[ \textbf{9999}\Box\Box\Box\Box \] is not a perfect square. (A $\textit{perfect square}$ is a whole number times itself. For example, $25$ is a perfect square because $25 = 5 \times 5$.)

Find all real numbers $a$ such that $4 < a < 5$ and $a(a-3\{a\})$ is an integer. ({x} represents the fractional part of x)

2. Find the minimum area of quadrilateral satisfy two condition as follows, (a) At least two interior angles are right angles. (b) A circle radius of $1$ inscribed.

Let $ABC$ be an acute triangle and $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. Call $Y$ and $Z$ the feet of the perpendicular lines from $B$ and $C$ to $FD$ and $DE$, respectively. Let $F_1$ be the symmetric of $F$ with respect to $E$ and $E_1$ be the symmetric of $E$ with respect to $F$. If $3EF=FD+DE$, prove that $\angle BZF_1=\angle CYE_1$.

In right triangle $ABC$, let $D$ be the midpoint of hypotenuse $[AB]$, circumradius be $\dfrac 52$ and $|BC|=3$. What is the distance between circumcenter of $\triangle ACD$ and incenter of $\triangle BCD$? $ \textbf{(A)}\ \dfrac {29}{2} \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ \dfrac 52 \qquad\textbf{(D)}\ \dfrac{5\sqrt{34}}{12} \qquad\textbf{(E)}\ 2\sqrt 2 $

Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy $a \le b \le c$ and $abc = 2(a + b + c)$.

Let $n$ be a positive integer. A [i]Nordic[/i] square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a [i]valley[/i]. An [i]uphill path[/i] is a sequence of one or more cells such that: (i) the first cell in the sequence is a valley, (ii) each subsequent cell in the sequence is adjacent to the previous cell, and (iii) the numbers written in the cells in the sequence are in increasing order. Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square. Author: Nikola Petrović

Find all nonnegative integers $x,y,z$ such that $(x+1)^{y+1} + 1= (x+2)^{z+1}$.

Let $r,n$ be positive integers. For a set $A$, let ${A \choose r}$ denote the family of all $r$-element subsets of $A$. Prove that if $A$ is infinite and $f : {A \choose r} \to {1,2,...,n}$ is any function, then there exists an infinite subset $B$ of $A$ such that $f(X) = f(Y)$ for all $X,Y \in {B \choose r}$.

Convex quadrilateral $ABCD$ has $AB = 3, BC = 4, CD = 13, AD = 12,$ and $\angle ABC = 90^\circ,$ as shown. What is the area of the quadrilateral? [asy] unitsize(.4cm); defaultpen(linewidth(.8pt)+fontsize(14pt)); dotfactor=2; pair A,B,C,D; C = (0,0); B = (0,4); A = (3,4); D = (12.8,-2.8); draw(C--B--A--D--cycle); draw(rightanglemark(C,B,A,20)); dot("$A$",A,N); dot("$B$",B,NW); dot("$C$",C,SW); dot("$D$",D,E); [/asy] $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 58.5 $

Find all pairs of naturals $a,b$ with $a <b$, such that the sum of the naturals greater than $a$ and less than $ b$ equals $1998$.

Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.

Integers $a_1, a_2, . . . , a_n$ satisfy $|a_k| = 1$ and \[ \sum_{k=1}^{n} a_ka_{k+1}a_{k+2}a_{k+3} = 2,\] where $a_{n+j} = a_j$. Prove that $n \neq 1992.$

Find three-digit numbers such that any its positive integer power ends with the same three digits and in the same order.

Find all positive integers $k$ for which there exists a nonlinear function $f:\mathbb{Z} \rightarrow\mathbb{Z}$ such that the equation $$f(a)+f(b)+f(c)=\frac{f(a-b)+f(b-c)+f(c-a)}{k}$$ holds for any integers $a,b,c$ satisfying $a+b+c=0$ (not necessarily distinct). [i]Evan Chen[/i]

Two segments $A_1B_1$ and $A_2B_2$ are given on the plane, with $\frac{A_2B_2}{A_1B_1} = k < 1$. On segment $A_1A_2$, point $A_3$ is taken, and on the extension of this segment beyond point $A_2$, point $A_4$ is taken, so $\frac{A_3A_2}{A_3A_1} =\frac{A_4A_2}{A_4A_1}= k$. Similarly, point $B_3$ is taken on segment $B_1B_2$ , and on the extension of this the segment beyond point $B_2$ is point $B_4$, so $\frac{B_3B_2}{B_3B_1} =\frac{B_4B_2}{B_4B_1}= k$. Find the angle between lines $A_3B_3$ and $A_4B_4$. (Netherlands)

The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments. [asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy] How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself? [list] [*] some rotation around a point of line $\ell$ [*] some translation in the direction parallel to line $\ell$ [*] the reflection across line $\ell$ [*] some reflection across a line perpendicular to line $\ell$ [/list] $\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.