The precent ages in years of two brothers $A$ and $B$,and their father $C$ are three distinct positive integers $a ,b$ and $c$ respectively .Suppose $\frac{b-1}{a-1}$ and $\frac{b+1}{a+1}$ are two consecutive integers , and $\frac{c-1}{b-1}$ and $\frac{c+1}{b+1}$ are two consecutive integers . If $a+b+c\le 150$ , determine $a,b$ and $c$.

Let $N$ be the number of consecutive 0's at the right end of the decimal representation of the product $1!\times2!\times3!\times4!\cdots99!\times100!.$ Find the remainder when $N$ is divided by 1000.

Xavier takes a permutation of the numbers $1$ through $2011$ at random, where each permutation has an equal probability of being selected. He then cuts the permutation into increasing contiguous subsequences, such that each subsequence is as long as possible. Compute the expected number of such subsequences. [i]Author: Alex Zhu[/i] [hide="Clarification"]An increasing contiguous subsequence is an increasing subsequence all of whose terms are adjacent in the original sequence. For example, 1,3,4,5,2 has two maximal increasing contiguous subsequences: (1,3,4,5) and (2). [/hide]

Find all positive integers n such that $ (n ^{ 2} + 11n - 4) n! + 33.13 ^ n + 4 $ is the perfect square

Let $ABC$ be a triangle, and $M_a, M_b, M_c$ be the midpoints of $BC, CA, AB$, respectively. Extend $M_bM_c$ so that it intersects $\odot (ABC)$ at $P$. Let $AP$ and $BC$ intersect at $Q$. Prove that the tangent at $A$ to $\odot(ABC)$ and the tangent at $P$ to $\odot (P QM_a)$ intersect on line $BC$. (Li4)

The sequence $a_1, a_2, \ldots$ is geometric with $a_1=a$ and common ratio $r$, where $a$ and $r$ are positive integers. Given that $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r)$.

Two different prime numbers $p$ and $q$ differ in less than $2$ times. Prove that exists two consecutive natural numbers, such that largest prime divisor of first number is $p$, and largest prime divisor of second number is $q$.

Find the least value of $x>0$ such that for all positive real numbers $a,b,c,d$ satisfying $abcd=1$, the inequality $a^x+b^x+c^x+d^x\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$ is true.

Suppose that $p$ is a prime number. Find all natural numbers $n$ such that $p|\varphi(n)$ and for all $a$ such that $(a,n)=1$ we have \[ n|a^{\frac{\varphi(n)}{p}}-1 \]

Eve randomly chooses two $\textbf{distinct}$ points on the coordinate plane from the set of all $11^2$ lattice points $(x, y)$ with $0 \le x \le 10$, $0 \le y \le 10$. Then, Anne the ant walks from the point $(0,0)$ to the point $(10, 10)$ using a sequence of one-unit right steps and one-unit up steps. Let $P$ be the number of paths Anne could take that pass through both of the points that Eve chose. The expected value of $P$ is $\dbinom{20}{10} \cdot \dfrac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$. [i]Proposed by Michael Tang[/i]

A woman has $\$1.58$ in pennies, nickels, dimes, quarters, half-dollars and silver dollars. If she has a different number of coins of each denomination, how many coins does she have?

In each line and column of a table $ (2n \plus{} 1)\times (2n \plus{} 1)$ are written arbitrarly the numbers $ 1,2,...,2n \plus{} 1$. It was constated that the repartition of the numbers is symmetric to the main diagonal of this table. Prove that all the numbers on the main diagonal are distinct.

Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$. Prove that there exists an integer $a$, $1<a<\frac{n}{k}+1$, such that $n \mid a^2-a$.

Will stands at a point $P$ on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of $n^o$ and $(n + 1)^o$ with the tangent at $P$, where $n$ is a positive integer less than $90$. The lasers reflect off of the walls, illuminating the points they hit on the walls, until they reach $P$ again. ($P$ is also illuminated at the end.) What is the minimum possible number of illuminated points on the walls of the room? [img]https://cdn.artofproblemsolving.com/attachments/a/9/5548d7b34551369d1b69eae682855bcc406f9e.jpg[/img]

Let $M$ be a set with $n$ elements. How many pairs $(A, B)$ of subsets of $M$ are there such that $A$ is a subset of $B?$

A positive integer $b$ is called good if there exist positive integers $1=a_1, a_2, \ldots, a_{2023}=b$ such that $|a_{i+1}-a_i|=2^i$. Find the number of the good integers.

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \]. Compute $2^A$.

Determine all values of $n$ satisfied the following condition: there's exist a cyclic $(a_1,a_2,a_3,...,a_n)$ of $(1,2,3,...,n)$ such that $\left\{ {{a_1},{a_1}{a_2},{a_1}{a_2}{a_3},...,{a_1}{a_2}...{a_n}} \right\}$ is a complete residue systems modulo $n$.

A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle. [asy] size(150); defaultpen(linewidth(0.65)+fontsize(11)); real r=10; pair O=(0,0),A=r*dir(45),B=(A.x,A.y-r),C; path P=circle(O,r); C=intersectionpoint(B--(B.x+r,B.y),P); draw(Arc(O, r, 45, 360-17.0312)); draw(A--B--C);dot(A); dot(B); dot(C); label("$A$",A,NE); label("$B$",B,SW); label("$C$",C,SE); [/asy]

Let $n$ be a positive integer. Find all real numbers $x_1,x_2 ,..., x_n$ such that $$\prod_{k=1}^{n}(x_k^2+ (k + 2)x_k + k^2 + k + 1) =\left(\frac{3}{4}\right)^n (n!)^2$$

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$ [i]Proposed by Liam Baker, South Africa[/i]

Consider a $m\times n$ rectangular grid with $m$ rows and $n$ columns. There are water fountains on some of the squares. A water fountain can spray water onto any of it's adjacent squares, or a square in the same column such that there is exactly one square between them. Find the minimum number of fountains such that each square can be sprayed in the case that a) $m=4$; b) $m=3$.

There are three number-transforming machines. We input the pair $(a_1, a_2)$, and the machine returns $(b_1, b_2)$. We denote this transformation as $(a_1, a_2) \to (b_1, b_2)$. (a) The first machine can perform two transformations: - $(a, b) \to (a - 1, b - 1)$ - $(a, b) \to (a + 13, b + 5)$ If the input pair is $(25,32)$, is it possible to obtain the pair $(82,98)$ after a series of transformations? (b) The second machine can perform two transformations: - $(a, b) \to (a - 1, b - 1)$ - $(a, b) \to (2a, 2b)$ If the input pair is $(34,60)$, is it possible to obtain the pair $(2000, 2008)$ after a series of transformations? (c) The third machine can perform two transformations: - $(a, b) \to (a - 2, b + 2)$ - $(a, b) \to (2a - b + 1, 2b - 1 - a)$ If the input pair is $(145,220)$, is it possible to obtain the pair $(363,498)$ after a series of transformations?

[b]a)[/b] Prove that $ 4040100 $ divides $ 2009\cdot 2011^{2011}+1. $ [i]Gabriel Iorgulescu[/i] [b]b)[/b] Let be three natural numbers $ x,y,z $ with the property that $ (1+\sqrt 2)^x=y^2+2z^2+2yz\sqrt 2. $ Show that $ x $ is even. [i]Marius Cavachi[/i]