Found problems: 85335
1987 AMC 12/AHSME, 26
The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $.357$, or into $\sqrt{3}$ and $2.5-\sqrt{3}.$ Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is the probability that the two integers sum to $3$?
$ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{2}{5} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{3}{4} $
2020 Jozsef Wildt International Math Competition, W20
Let $p\in(0,1)$ and $a>0$ be real numbers. Determine the asymptotic behavior of the sequence $\{a_n\}_{n=1}^\infty$ defined recursively by
$$a_1=a,a_{n+1}=\frac{a_n}{1+a_n^p},n\in\mathbb N$$
[i]Proposed by Arkady Alt[/i]
JOM 2015 Shortlist, C1
Baron and Peter are playing a game. They are given a simple finite graph $G$ with $n\ge 3$ vertex and $k$ edges that connects the vertices. First Peter labels two vertices A and B, and places a counter at A. Baron starts first. A move for Baron is move the counter along an edge. Peter's move is to remove an edge from the graph. Baron wins if he reaches $B$, otherwise Peter wins.
Given the value of $n$, what is the largest $k$ so that Peter can always win?
2019 Auckland Mathematical Olympiad, 4
Find the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$, where $a$, $ b$, $c$, $d$ are non-negative integers.
2015 Saudi Arabia JBMO TST, 2
Let $a,b,c$ be positive real numbers. Prove that
$$\frac{a}{\sqrt{(2a+b)(2a+c)}} +\frac{b}{\sqrt{(2b+c)(2b+a)}} +\frac{c}{\sqrt{(2c+a)(2c+b)}} \le 1 $$
Bangladesh Mathematical Olympiad 2020 Final, #9
You have 2020 piles of coins in front Of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins and so on. So, the 2020th pile contains 2020 coins. Guess a positive integer[b] k[/b], in which piles contain at least[b] k [/b]coins, take away exact[b] k[/b] coins from these piles. Find the [b]minimum number of turns[/b] you need to take way all of these coins?
2011 Today's Calculation Of Integral, 676
Let $f(x)=\cos ^ 4 x+3\sin ^ 4 x$.
Evaluate $\int_0^{\frac{\pi}{2}} |f'(x)|dx$.
[i]2011 Tokyo University of Science entrance exam/Management[/i]
1991 AMC 12/AHSME, 1
If for any three distinct numbers $a$, $b$ and $c$ we define \[\boxed{a,b,c} = \frac{c + a}{c - b},\] then $\boxed{1,-2,-3}=$
$ \textbf{(A)}\ -2\qquad\textbf{(B)}\ -\frac{2}{5}\qquad\textbf{(C)}\ -\frac{1}{4}\qquad\textbf{(D)}\ \frac{2}{5}\qquad\textbf{(E)}\ 2 $
2021 Brazil Team Selection Test, 3
Let $P$ be a point on the circumcircle of acute triangle $ABC$. Let $D,E,F$ be the reflections of $P$ in the $A$-midline, $B$-midline, and $C$-midline. Let $\omega$ be the circumcircle of the triangle formed by the perpendicular bisectors of $AD, BE, CF$.
Show that the circumcircles of $\triangle ADP, \triangle BEP, \triangle CFP,$ and $\omega$ share a common point.
CIME I 2018, 15
A positive integer $n$ is said to be $m$-free if $n \leq m!$ and $\gcd(i,n)=1$ for each $i=1,2,...,m$. Let $\mathcal{S}_k$ denote the sum of the squares of all the $k$-free integers. Find the remainder when $\mathcal{S}_7-\mathcal{S}_6$ is divided by $1000$.
[i]Proposed by [b]FedeX333X[/b][/i]
2011 Miklós Schweitzer, 10
Let $X_0, \xi_{i, j}, \epsilon_k$ (i, j, k ∈ N) be independent, non-negative integer random variables. Suppose that $\xi_{i, j}$ (i, j ∈ N) have the same distribution, $\epsilon_k$ (k ∈ N) also have the same distribution.
$\mathbb{E}(\xi_{1,1})=1$ , $\mathbb{E}(X_0^l)<\infty$ , $\mathbb{E}(\xi_{1,1}^l)<\infty$ , $\mathbb{E}(\epsilon_1^l)<\infty$ for some $l\in\mathbb{N}$
Consider the random variable $X_n := \epsilon_n + \sum_{j=1}^{X_{n-1}} \xi_{n,j}$ (n ∈ N) , where $\sum_{j=1}^0 \xi_{n,j} :=0$
Introduce the sequence $M_n := X_n-X_{n-1}-\mathbb{E}(\epsilon_n)$ (n ∈ N)
Prove that there is a polynomial P of degree $\leq l/2$ such that $\mathbb{E}(M_n^l) = P_l(n)$ (n ∈ N).
2005 Alexandru Myller, 1
Let $ x,y,z $ be numbers distinct from $ -1 $ that verify the equation
$$ \frac{1}{1+a} +\frac{1}{1+b} +\frac{1}{1+c} =\frac{3}{2} . $$
Prove that if $ abc=1, $ then $ a $ or $ b $ or $ c $ is equal to $ 1. $
1989 All Soviet Union Mathematical Olympiad, 492
$ABC$ is a triangle. $A' , B' , C'$ are points on the segments $BC, CA, AB$ respectively. $\angle B' A' C' = \angle A$ , $\frac{AC'}{C'B} = \frac{BA' }{A' C} = \frac{CB'}{B'A}$. Show that $ABC$ and $A'B'C'$ are similar.
2022 Iran Team Selection Test, 7
Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$-axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than vertices.
Proposed by Mohammad Ahmadi
2015 JHMT, 1
Clyde is making a Pacman sticker to put on his laptop. A Pacman sticker is a circular sticker of radius $3$ inches with a sector of $120^o$ cut out. What is the perimeter of the Pacman sticker in inches?
2009 Today's Calculation Of Integral, 492
Find the volume formed by the revolution of the region satisfying $ 0\leq y\leq (x \minus{} p)(q \minus{} x)\ (0 < p < q)$ in the coordinate plane about the $ y$ -axis.
You are not allowed to use the formula: $ V \equal{} \boxed{\int_a^b 2\pi x|f(x)|\ dx\ (a < b)}$ here.
2016 EGMO, 1
Let $n$ be an odd positive integer, and let $x_1,x_2,\cdots ,x_n$ be non-negative real numbers. Show that \[ \min_{i=1,\ldots,n} (x_i^2+x_{i+1}^2) \leq \max_{j=1,\ldots,n} (2x_jx_{j+1}) \]where $x_{n+1}=x_1$.
2024 Pan-American Girls’ Mathematical Olympiad, 6
Let $ABC$ be a triangle, and let $a$, $b$, and $c$ be the lengths of the sides opposite vertices $A$, $B$, and $C$, respectively. Let $R$ be its circumradius and $r$ its inradius. Suppose that $b + c = 2a$ and $R = 3r$.
The excircle relative to vertex $A$ intersects the circumcircle of $ABC$ at points $P$ and $Q$. Let $U$ be the midpoint of side $BC$, and let $I$ be the incenter of $ABC$.
Prove that $U$ is the centroid of triangle $QIP$.
2011 Middle European Mathematical Olympiad, 3
In a plane the circles $\mathcal K_1$ and $\mathcal K_2$ with centers $I_1$ and $I_2$, respectively, intersect in two points $A$ and $B$. Assume that $\angle I_1AI_2$ is obtuse. The tangent to $\mathcal K_1$ in $A$ intersects $\mathcal K_2$ again in $C$ and the tangent to $\mathcal K_2$ in $A$ intersects $\mathcal K_1$ again in $D$. Let $\mathcal K_3$ be the circumcircle of the triangle $BCD$. Let $E$ be the midpoint of that arc $CD$ of $\mathcal K_3$ that contains $B$. The lines $AC$ and $AD$ intersect $\mathcal K_3$ again in $K$ and $L$, respectively. Prove that the line $AE$ is perpendicular to $KL$.
2009 Argentina Team Selection Test, 4
Find all positive integers $ n$ such that $ 20^n \minus{} 13^n \minus{} 7^n$ is divisible by $ 309$.
1981 All Soviet Union Mathematical Olympiad, 309
Three equilateral triangles $ABC, CDE, EHK$ (the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors $\overrightarrow{AD}$ and $\overrightarrow{DK}$ are equal. Prove that the triangle $BHD$ is also equilateral
MBMT Guts Rounds, 2015.8
The school store is running out of supplies, but it still has five items: one pencil (costing $\$1$), one pen (costing $\$1$), one folder (costing $\$2$), one pack of paper (costing $\$3$), and one binder (costing $\$4$). If you have $\$10$, in how many ways can you spend your money? (You don't have to spend all of your money, or any of it.)
2014 Contests, 4
Let $m\ge 3$ and $n$ be positive integers such that $n>m(m-2)$. Find the largest positive integer $d$ such that $d\mid n!$ and $k\nmid d$ for all $k\in\{m,m+1,\ldots,n\}$.
2006 Romania National Olympiad, 3
We have in the plane the system of points $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$, which have different centers of mass. Prove that there is a point $P$ such that \[ PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n . \]
2021 New Zealand MO, 4
Find all triples $(x, p, n)$ of non-negative integers such that $p$ is prime and $2x(x + 5) = p^n + 3(x - 1)$.