Compute the number of positive integer divisors of $100000$ which do not contain the digit $0.$

Determine all bounded functions $f: R \to R$, such that $f (f (x) + y) = f (x) + f (y)$, for all real $x, y$.

Consider a convex pentagon $ABCDE$ and a variable point $X$ on its side $CD$. Suppose that points $K, L$ lie on the segment $AX$ such that $AB = BK$ and $AE = EL$ and that the circumcircles of triangles $CXK$ and $DXL$ intersect for the second time at $Y$ . As $X$ varies, prove that all such lines $XY$ pass through a fixed point, or they are all parallel. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

Determine the maximum number of bishops that we can place in a $8 \times 8$ chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop. Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals the $2$ main diagonals and the ones parallel to those ones.

[b]p1.[/b] We say that integers $a$ and $b$ are [i]friends [/i] if their product is a perfect square. Prove that if $a$ is a friend of $b$, then $a$ is a friend of $gcd (a, b)$. [b]p2.[/b] On the island of knights and liars, a traveler visited his friend, a knight, and saw him sitting at a round table with five guests. "I wonder how many knights are among you?" he asked. " Ask everyone a question and find out yourself" advised him one of the guests. "Okay. Tell me one: Who are your neighbors?" asked the traveler. This question was answered the same way by all the guests. "This information is not enough!" said the traveler. "But today is my birthday, do not forget it!" said one of the guests. "Yes, today is his birthday!" said his neighbor. Now the traveler was able to find out how many knights were at the table. Indeed, how many of them were there if [i]knights always tell the truth and liars always lie[/i]? [b]p3.[/b] A rope is folded in half, then in half again, then in half yet again. Then all the layers of the rope were cut in the same place. What is the length of the rope if you know that one of the pieces obtained has length of $9$ meters and another has length $4$ meters? [b]p4.[/b] The floor plan of the palace of the Shah is a square of dimensions $6 \times 6$, divided into rooms of dimensions $1 \times 1$. In the middle of each wall between rooms is a door. The Shah orders his architect to eliminate some of the walls so that all rooms have dimensions $2 \times 1$, no new doors are created, and a path between any two rooms has no more than $N$ doors. What is the smallest value of $N$ such that the order could be executed? [b]p5.[/b] There are $10$ consecutive positive integers written on a blackboard. One number is erased. The sum of remaining nine integers is $2011$. Which number was erased? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the function $f : R^+ \to R^+$ and satisfying $$f(x + 2y + f(x + y)) = f(2x) + f(3y), \,\, \forall \,\, x, y > 0.$$ 1. Find all functions $f(x)$ that satisfy the given condition. 2. Suppose that $f(4\sin^4x)f(4\cos^4x) \ge f^2(1)$ for all $x \in \left(0\frac{\pi}{2}\right) $. Find the minimum value of $f(2022)$.

On the plane, $n$ points are given ($n>2$). No three of them are collinear. Through each two of them the line is drawn, and among the other given points, the one nearest to this line is marked (in each case this point occurred to be unique). What is the maximal possible number of marked points for each given $n$? [i]Proposed by Boris Frenkin (Russia)[/i]

Natural numbers $ x, y, z $ whose greatest common divisor is equal to 1 satisfy the equation $$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$$ Prove that $ x + y $ is the square of a natural number.

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

We say that a function $f: \mathbb{Z}_{\ge 0} \times \mathbb{Z}_{\ge 0} \to \mathbb{Z}$ is [i]great[/i] if for any nonnegative integers $m$ and $n$, \[f(m + 1, n + 1) f(m, n) - f(m + 1, n) f(m, n + 1) = 1.\] If $A = (a_0, a_1, \dots)$ and $B = (b_0, b_1, \dots)$ are two sequences of integers, we write $A \sim B$ if there exists a great function $f$ satisfying $f(n, 0) = a_n$ and $f(0, n) = b_n$ for every nonnegative integer $n$ (in particular, $a_0 = b_0$). Prove that if $A$, $B$, $C$, and $D$ are four sequences of integers satisfying $A \sim B$, $B \sim C$, and $C \sim D$, then $D \sim A$. [i]Ankan Bhattacharya[/i]

Consider $\vartriangle ABC$, with $AD$ bisecting $\angle BAC$, $D$ on segment $BC$. Let $E$ be a point on $BC$, such that $BD = EC$. Through $E$ we draw the line $\ell$ parallel to $AD$ and consider a point $ P$ on it and inside the $\vartriangle ABC$. Let $G$ be the point where line $BP$ cuts side $AC$ and let F be the point where line $CP$ to side $AB$. Show that $BF = CG$.

[color=darkred]Prove that if $n\ge 2$ is a natural number and $x_1,x_2,\ldots,x_n$ are positive real numbers, then: \[4\left(\frac {x_1^3-x_2^3}{x_1+x_2}+\frac {x_2^3-x_3^3}{x_2+x_3}+\ldots+\frac {x_{n-1}^3-x_n^3}{x_{n-1}+x_n}+\frac {x_n^3-x_1^3}{x_n+x_1}\right)\le \\ \\ \le(x_1-x_2)^2+(x_2-x_3)^2+\ldots+(x_{n-1}-x_n)^2+(x_n-x_1)^2\, .\][/color]

Let $a,b,c$ be positive real numbers. Prove that \[(a+b)(b+c)(c+a)\ge 8(a+b-c)(b+c-a)(c+a-b)\] and determine when equality occurs.

Billiam is distributing his ample supply of balls among an ample supply of boxes. He distributes the balls as follows: he places a ball in the first empty box, and then for the greatest positive integer n such that all $n$ boxes from box $1$ to box $n$ have at least one ball, he takes all of the balls in those $n$ boxes and puts them into box $n +1$. He then repeats this process indefinitely. Find the number of repetitions of this process it takes for one box to have at least $2022$ balls.

Find all polynomials $f$ that satisfy the equation \[\frac{f(3x)}{f(x)} = \frac{729 (x-3)}{x-243}\] for infinitely many real values of $x$.

Harry the knight is positioned at the origin of the Cartesian plane. In a "knight hop", Harry can move from the point $(i,j)$ to a point with integer coordinates that is a distance of $\sqrt{5}$ away from $(i,j)$. What is the number of ways that Harry can return to the origin after 6 knight hops? [i]Proposed by Harry Kim[/i]

A fair coin is repeatedly tossed. We receive one marker for every ”head” and two markers for every ”tail”. We win the game if, at some moment, we possess exactly $100$ markers. Is the probability of winning the game greater than, equal to, or less than $2/3$?

For two base-10 positive integers $a$ and $b$, we say $a \sim b$ if we can rearrange the digits of $a$ in some way to obtain $b$, where the leading digit of both $a$ and $b$ is nonzero. For instance, $463 \sim 463$ and $634 \sim 463$. Find the number of $11$-digit positive integers $K$ such that $K$ is divisible by $2$, $3$, and $5$, and there is some positive integer $K'$ such that $K' \sim K$ and $K'$ is divisible by $7$, $11$, $13$, $17$, $101$, and $9901$.

Given a permutation $\sigma = (a_1,a_2,a_3,...a_n)$ of $(1,2,3,...n)$ , an ordered pair $(a_j,a_k)$ is called an inversion of $\sigma$ if $a \leq j < k \leq n$ and $a_j > a_k$. Let $m(\sigma)$ denote the no. of inversions of the permutation $\sigma$. Find the average of $m(\sigma)$ as $\sigma$ varies over all permutations.

There are $ 100$ players in a singles tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest $ 28$ players are given a bye, and the remaining $ 72$ players are paired off to play. After each round, the remaining players play in the next round. The match continues until only one player remains unbeaten. The total number of matches played is $ \textbf{(A)}\ \text{a prime number} \qquad \textbf{(B)}\ \text{divisible by 2} \qquad \textbf{(C)}\ \text{divisible by 5}$ $ \textbf{(D)}\ \text{divisible by 7} \qquad \textbf{(E)}\ \text{divisible by 11}$

Let $f(x) = x^n + a_{n-2} x^{n-2} + a_{n-3}x^{n-3} + \dots + a_1x + a_0$ be a polynomial with real coefficients $(n \ge 2)$. Suppose all roots of $f$ are real. Prove that the absolute value of each root is at most $\sqrt{\frac{2(1-n)}n a_{n-2}}$.

Triangle $ABC$ and a function $f:\mathbb{R}^+\to\mathbb{R}$ have the following property: for every line segment $DE$ from the interior of the triangle with midpoint $M$, the inequality $f(d(D))+f(d(E))\le 2f(d(M))$, where $d(X)$ is the distance from point $X$ to the nearest side of the triangle ($X$ is in the interior of $\triangle ABC$). Prove that for each line segment $PQ$ and each point interior point $N$ the inequality $|QN|f(d(P))+|PN|f(d(Q))\le |PQ|f(d(N))$ holds.

Define the [i]bigness [/i]of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and [i]bigness [/i]$N$ and another one with integer side lengths and [i]bigness [/i]$N + 1$.

Let $ABC$ be a triangle whose incircle $(I)$ touches $BC, CA, AB$ at $D, E, F$, respectively. The line passing through $A$ and parallel to $BC$ cuts $DE, DF$ at $M, N$, respectively. The circumcircle of triangle $DMN$ cuts $(I)$ again at $L$. a) Let $K$ be the intersection of $N E$ and $M F$. Prove that $K$ is the orthocenter of the triangle $DMN$. b) Prove that $A, K, L$ are collinear.

A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle $ABC$ of side length $3$. The perimeter of the remaining quadrilateral is [asy] draw((0,0)--(2,0)--(2.5,.87)--(1.5,2.6)--cycle, linewidth(1)); draw((2,0)--(3,0)--(2.5,.87)); label("3", (0.75,1.3), NW); label("1", (2.5, 0), S); label("1", (2.75,.44), NE); label("A", (1.5,2.6), N); label("B", (3,0), S); label("C", (0,0), W); label("D", (2.5,.87), NE); label("E", (2,0), S);[/asy] $\text{(A)} \ 6 \qquad \text{(B)} \ 6\frac12 \qquad \text{(C)} \ 7 \qquad \text{(D)} \ 7\frac12 \qquad \text{(E)} \ 8$