Found problems: 85335
1988 China Team Selection Test, 1
Let $f(x) = 3x + 2.$ Prove that there exists $m \in \mathbb{N}$ such that $f^{100}(m)$ is divisible by $1988$.
2000 All-Russian Olympiad Regional Round, 9.6
Among $2000$ outwardly indistinguishable balls, wines - aluminum weighing 1$0$ g, and the rest - duralumin weighing $9.9$ g. It is required to select two piles of balls so that the masses of the piles are different, and the number of balls in them - the same. What is the smallest number of weighings on a cup scale without weights that can be done?
2023 Lusophon Mathematical Olympiad, 5
Let $ABCDEF$ be a regular hexagon with side 1. Point $X, Y$ are on sides $CD$ and $DE$ respectively, such that the perimeter of $DXY$ is $2$. Determine $\angle XAY$.
2011 International Zhautykov Olympiad, 2
Let $n$ be integer, $n>1.$ An element of the set $M=\{ 1,2,3,\ldots,n^2-1\}$ is called [i]good[/i] if there exists some element $b$ of $M$ such that $ab-b$ is divisible by $n^2.$ Furthermore, an element $a$ is called [i]very good[/i] if $a^2-a$ is divisible by $n^2.$ Let $g$ denote the number of [i]good[/i] elements in $M$ and $v$ denote the number of [i]very good[/i] elements in $M.$ Prove that
\[v^2+v \leq g \leq n^2-n.\]
2002 HKIMO Preliminary Selection Contest, 18
Let $A_1A_2\cdots A_{2002}$ be a regular 2002 sided polygon. Each vertex $A_i$ is associated with a positive integer $a_i$ such that the following condition is satisfied: If $j_1,j_2,\cdots, j_k$ are positive integers such that $k<500$ and $A_{j_1}, A_{j_2}, \cdots A_{j_k}$ is a regular $k$ sided polygon, then the values of $a_{j_1},A_{j_2}, \cdots A_{j_k}$ are all different. Find the smallest possible value of $a_1+a_2+\cdots a_{2002}$
2019 India IMO Training Camp, P2
Let $ABC$ be a triangle with $\angle A=\angle C=30^{\circ}.$ Points $D,E,F$ are chosen on the sides $AB,BC,CA$ respectively so that $\angle BFD=\angle BFE=60^{\circ}.$ Let $p$ and $p_1$ be the perimeters of the triangles $ABC$ and $DEF$, respectively. Prove that $p\le 2p_1.$
2006 Bulgaria Team Selection Test, 2
a) Let $\{a_n\}_{n=1}^\infty$ is sequence of integers bigger than 1. Proove that if $x>0$ is irrational, then $\ds x_n>\frac{1}{a_{n+1}}$ for infinitely many $n$, where $x_n$ is fractional part of $a_na_{n-1}\dots a_1x$.
b)Find all sequences $\{a_n\}_{n=1}^\infty$ of positive integers, for which exist infinitely many $x\in(0,1)$ such that $\ds x_n>\frac{1}{a_{n+1}}$ for all $n$.
[i]Nikolai Nikolov, Emil Kolev[/i]
2009 239 Open Mathematical Olympiad, 1
In a sequence of natural numbers, the first number is $a$, and each subsequent number is the smallest number coprime to all the previous ones and greater than all of them. Prove that in this sequence from some place all numbers will be primes.
2018 Putnam, B5
Let $f = (f_1, f_2)$ be a function from $\mathbb{R}^2$ to $\mathbb{R}^2$ with continuous partial derivatives $\tfrac{\partial f_i}{\partial x_j}$ that are positive everywhere. Suppose that
\[\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2} - \frac{1}{4} \left(\frac{\partial f_1}{\partial x_2} + \frac{\partial f_2}{\partial x_1} \right)^2 > 0\]
everywhere. Prove that $f$ is one-to-one.
2004 China Team Selection Test, 2
Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$.
Prove that $ M$, $ N$, $ O$ are collinear.
2012 China Second Round Olympiad, 3
Suppose that $x,y,z\in [0,1]$. Find the maximal value of the expression
\[\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}.\]
2001 Balkan MO, 2
A convex pentagon $ABCDE$ has rational sides and equal angles. Show that it is regular.
2010 CHMMC Winter, Mixer
[b]p1.[/b] Compute $x$ such that $2009^{2010} \equiv x$ (mod $2011$) and $0 \le x < 2011$.
[b]p2.[/b] Compute the number of "words" that can be formed by rearranging the letters of the word "syzygy" so that the y's are evenly spaced. (The $y$'s are evenly spaced if the number of letters (possibly zero) between the first $y$ and the second $y$ is the same as the number of letters between the second $y$ and the third $y$.)
[b]p3.[/b] Let $A$ and $B$ be subsets of the integers, and let $A + B$ be the set containing all sums of the form $a + b$, where $a$ is an element of $A$, and $b$ is an element of $B$. For example, if $A = \{0, 4, 5\}$ and $B =\{-3,-1, 2, 6\}$, then $A + B = \{-3,-1, 1, 2, 3, 4, 6, 7, 10, 11\}$. If $A$ has $1955$ elements and $B$ has $1891$ elements, compute the smallest possible number of elements in $A + B$.
[b]p4.[/b] Compute the sum of all integers of the form $p^n$ where $p$ is a prime, $n \ge 3$, and $p^n \le 1000$.
[b]p5.[/b] In a season of interhouse athletics at Caltech, each of the eight houses plays each other house in a particular sport. Suppose one of the houses has a $1/3$ chance of beating each other house. If the results of the games are independent, compute the probability that they win at least three games in a row.
[b]p6.[/b] A positive integer $n$ is special if there are exactly $2010$ positive integers smaller than $n$ and relatively prime to $n$. Compute the sum of all special numbers.
[b]p7.[/b] Eight friends are playing informal games of ultimate frisbee. For each game, they split themselves up into two teams of four. They want to arrange the teams so that, at the end of the day, each pair of players has played at least one game on the same team. Determine the smallest number of games they need to play in order to achieve this.
[b]p8.[/b] Compute the number of ways to choose five nonnegative integers $a, b, c, d$, and $e$, such that $a + b + c + d + e = 20$.
[b]p9.[/b] Is $23$ a square mod $41$? Is $15$ a square mod $41$?
[b]p10.[/b] Let $\phi (n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Compute $ \sum_{d|15015} \phi (d)$.
[b]p11.[/b] Compute the largest possible volume of an regular tetrahedron contained in a cube with volume $1$.
[b]p12.[/b] Compute the number of ways to cover a $4 \times 4$ grid with dominoes.
[b]p13.[/b] A collection of points is called mutually equidistant if the distance between any two of them is the same. For example, three mutually equidistant points form an equilateral triangle in the plane, and four mutually equidistant points form a regular tetrahedron in three-dimensional space. Let $A$, $B$, $C$, $D$, and $E$ be five mutually equidistant points in four-dimensional space. Let $P$ be a point such that $AP = BP = CP = DP = EP = 1$. Compute the side length $AB$.
[b]p14. [/b]Ten turtles live in a pond shaped like a $10$-gon. Because it's a sunny day, all the turtles are sitting in the sun, one at each vertex of the pond. David decides he wants to scare all the turtles back into the pond. When he startles a turtle, it dives into the pond. Moreover, any turtles on the two neighbouring vertices also dive into the pond. However, if the vertex opposite the startled turtle is empty, then a turtle crawls out of the pond and sits at that vertex. Compute the minimum number of times David needs to startle a turtle so that, by the end, all but one of the turtles are in the pond.
[b]p15.[/b] The game hexapawn is played on a $3 \times 3$ chessboard. Each player starts with three pawns on the row nearest him or her. The players take turns moving their pawns. Like in chess, on a player's turn he or she can either
$\bullet$ move a pawn forward one space if that square is empty, or
$\bullet$ capture an opponent's pawn by moving his or her own pawn diagonally forward one space into the opponent's pawn's square.
A player wins when either
$\bullet$ he or she moves a pawn into the last row, or
$\bullet$ his or her opponent has no legal moves.
Eve and Fred are going to play hexapawn. However, they're not very good at it. Each turn, they will pick a legal move at random with equal probability, with one exception: If some move will immediately win the game (by either of the two winning conditions), then he or she will make that move, even if other moves are available. If Eve moves first, compute the probability that she will win.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2020 CMIMC Algebra & Number Theory, 4
For all real numbers $x$, let $P(x)=16x^3 - 21x$. What is the sum of all possible values of $\tan^2\theta$, given that $\theta$ is an angle satisfying \[P(\sin\theta) = P(\cos\theta)?\]
2018 CCA Math Bonanza, I12
For how many integers $n\neq1$ does $\left(n-1\right)^3$ divide $n^{2018\left(n-1\right)}-1$?
[i]2018 CCA Math Bonanza Individual Round #12[/i]
1993 Vietnam Team Selection Test, 2
A sequence $\{a_n\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}}$ for $n = 1, 2, 3, \ldots$. Find all real numbers $q$ such that the sequence $\{u_n\}$ defined by $u_n = a_n^q$, $n = 1, 2, 3, \ldots$ has nonzero finite limit when $n$ goes to infinity.
THERE MIGHT BE A TYPO!
1995 Denmark MO - Mohr Contest, 4
Solve the equation
$$(2^x-4)^3 +(4^x-2)^3=(4^x+2^x-6)^3$$
where $x$ is a real number.
2024 CMIMC Integration Bee, 15
\[\int_0^\infty 1+\cos\left(\tfrac 1{\sqrt x}\right)-2\cos\left(\tfrac 1{\sqrt {2x}}\right)\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2017 CMIMC Algebra, 8
Suppose $a_1$, $a_2$, $\ldots$, $a_{10}$ are nonnegative integers such that \[\sum_{k=1}^{10}a_k=15\qquad\text{and}\qquad \sum_{k=1}^{10}ka_k = 80.\] Let $M$ and $m$ denote the maximum and minimum respectively of $\sum_{k=1}^{10}k^2a_k$. Compute $M-m$.
2010 239 Open Mathematical Olympiad, 5
Among $33$ balls, there are $2$ radioactive ones. You can put several balls in the detector and it will show if the both radioactive balls are among the balls. What is the smallest number that we have to use the detector so that one can certainly find at least one of the radioactive balls?
2012 Bosnia And Herzegovina - Regional Olympiad, 3
Quadrilateral $ABCD$ is cyclic. Line through point $D$ parallel with line $BC$ intersects $CA$ in point $P$, line $AB$ in point $Q$ and circumcircle of $ABCD$ in point $R$. Line through point $D$ parallel with line $AB$ intersects $AC$ in point $S$, line $BC$ in point $T$ and circumcircle of $ABCD$ in point $U$. If $PQ=QR$, prove that $ST=TU$
1974 IMO, 5
The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression \[ S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} \] takes?
2014 Contests, 3
Find all nonnegative integer numbers such that $7^x- 2 \cdot 5^y = -1$
2023 VN Math Olympiad For High School Students, Problem 9
Given a quadrilateral $ABCD$ inscribed in $(O)$. Let $L, J$ be the [i]Lemoine[/i] point of $\triangle ABC$ and $\triangle ACD$.
Prove that: $AC, BD, LJ$ are concurrent.
2008 District Olympiad, 1
Let $ z \in \mathbb{C}$ such that for all $ k \in \overline{1, 3}$, $ |z^k \plus{} 1| \le 1$. Prove that $ z \equal{} 0$.