Define a function $f_n(x)\ (n=0,\ 1,\ 2,\ \cdots)$ by \[f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.\] (1) Let $f_n(x)=a_n\sin x+b_n\cos x.$ Express $a_{n+1},\ b_{n+1}$ in terms of $a_n,\ b_n.$ (2) Find $\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).$

Find all pairs of natural numbers $a,b$ , with $a\ne b$ , such that $a+b$ and $ab+1$ are powers of $2$.

In quadrilateral $ABCD$ , $AC$ is bisector of $\hat{A}$ and $\widehat{ADC}=\widehat{ACB}$. $X$ and $Y$ are feet of perpendicular from $A$ to $BC$ and $CD$,respectively.Prove that orthocenter of triangle $AXY$ is on $BD$.

In a convex pentagon every diagonal is parallel to one side. Show that the ratios between the lengths of diagonals and the sides parallel to them are equal and find their value.

In a group of $90$ children each has at least $30$ friends (friendship is mutual). Prove that they can be divided into three $30$-member groups so that each child has its own a group of at least one friend.

Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black) The figure can be rotated $90°, 180°$ or $270°$. Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.

Given that points $ D,E$ lie on the sidelines $ AB,BC$ of triangle $ ABC$, respectively, point $ P$ is in interior of triangle $ ABC$ such that $ PE \equal{} PC$ and $ \bigtriangleup DEP\sim \bigtriangleup PCA.$ Prove that $ BP$ is tangent of the circumcircle of triangle $ PAD.$

Let points $A_1$, $A_2$ and $A_3$ lie on the circle $\Gamma$ in a counter-clockwise order, and let $P$ be a point in the same plane. For $i \in \{1,2,3\}$, let $\tau_i$ denote the counter-clockwise rotation of the plane centred at $A_i$, where the angle of rotation is equial to the angle at vertex $A_i$ in $\triangle A_1A_2A_3$. Further, define $P_i$ to be the point $\tau_{i+2}(\tau_{i}(\tau_{i+1}(P)))$, where the indices are taken modulo $3$ (i.e., $\tau_4 = \tau_1$ and $\tau_5 = \tau_2$). Prove that the radius of the circumcircle of $\triangle P_1P_2P_3$ is at most the radius of $\Gamma$. [i]Proposed by Anant Mudgal[/i]

How many $ 7$ digit palindromes (numbers that read the same backward as forward) can be formed using the digits $ 2$, $ 2$, $ 3$, $ 3$, $ 5$, $ 5$, $ 5$? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 48$

Find the values of $p$ for which the equation $x^5 - px-1 = 0$ has two roots $r$ and $s$ which are the roots of equation $x^2-ax+b= 0$ for some integers $a,b$.

There exist real numbers $x$ and $y$ such that $x(a^3 + b^3 + c^3) + 3yabc \ge (x + y)(a^2b + b^2c + c^2a)$ holds for all positive real numbers $a, b$, and $c$. Determine the smallest possible value of $x/y$. .

A circle concentric with the inscribed circle of $ABC$ intersects the sides of the triangle at six points forming a convex hexagon $A_1A_2B_1B_2C_1C_2$ (points $C_1$ and $C_2$ on the $AB$ side, $A_1$ and $A_2$ on $BC$, $B_1$ and $B_2$ on $AC$). Prove that if line $A_1B_1$ is parallel to the bisector of angle $B$, then line $A_2C_2$ is parallel to the bisector of angle $C$.

Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet.

The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd? $\textbf{(A)} \frac{1}{21} \qquad \textbf{(B)} \frac{1}{19} \qquad \textbf{(C)} \frac{1}{18} \qquad \textbf{(D)} \frac{1}{2} \qquad \textbf{(E)} \frac{11}{21}$

What is the smallest square-free composite number that can divide a number of the form $4242\ldots 42\pm 1$?

How many zeroes appear at the end of $209$ factorial?

Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? $\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$

As shown in figure, given right $\vartriangle ABC$ with $\angle C=90^o$. $I$ is the incenter. The line $BI$ intersects segment $AC$ at the point $D$ . The line passing through $D$ parallel to $AI$ intersects $BC$ at point $E$. The line $EI$ intersects segment $AB$ at point $F$. Prove that $DF \perp AI$. [img]https://cdn.artofproblemsolving.com/attachments/2/4/6fc94adb4ce12c3bf07948b8c57170ca01b256.png[/img]

Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$. A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$, compute $100m+n$. [i]Proposed by Eugene Chen[/i]

Let $a_1>a_2>.....>a_r$ be positive real numbers . Compute $\lim_{n\to \infty} (a_1^n+a_2^n+.....+a_r^n)^{\frac{1}{n}}$

It is given $n$ a natural number and a circle with circumference $n$. On the circle, in clockwise direction, numbers $0,1,2,\dots n-1$ are written, in this order and in the same distance to each other. Every number is colored red or blue, and there exists a non-zero number of numbers of each color. It is known that there exists a set $S\subsetneq \{0,1,2,\dots n-1\}, |S|\geq 2$, for wich it holds: if $(x,y), x<y$ is a circle sector whose endpoints are of distinct colors, whose distance $y-x$ is in $S$, then $y$ is in $S$. Prove that there is a divisor $d$ of $n$ different from $1$ and $n$ for wich holds: if $(x,y),x<y$ are different points of distinct colors, such that their distance is divisible by $d$, then both $x,y$ are divisible by $d$.

Jae and Yoon are playing SunCraft. The probability that Jae wins the $n$-th game is $\frac{1}{n+2}.$ What is the probability that Yoon wins the first six games, assuming there are no ties?

Assume that $x, y \ge 0$. Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$.

Suppose that $m,n$ are integers greater than $1$ such that $m+n-1$ divides $m^2+n^2-1$. Prove that $m+n-1$ cannot be a prime number.

In an $8 \times 8$ grid of squares, each square was colored black or white so that no $2\times 2$ square has all its squares in the same color. A sequence of distinct squares $x_1,\dots, x_m$ is called a [b]snake of length $m$[/b] if for each $1\leq i <m$ the squares $x_i, x_{i+1}$ are adjacent and are of different colors. What is the maximum $m$ for which there must exist a snake of length $m$?