Found problems: 85335
2017 Online Math Open Problems, 3
The USAMO is a $6$ question test. For each question, you submit a positive integer number $p$ of pages on which your solution is written. On the $i$th page of this question, you write the fraction $i/p$ to denote that this is the $i$th page out of $p$ for this question. When you turned in your submissions for the $2017$ USAMO, the bored proctor computed the sum of the fractions for all of the pages which you turned in. Surprisingly, this number turned out to be $2017$. How many pages did you turn in?
[i]Proposed by Tristan Shin[/i]
2010 Purple Comet Problems, 2
Three boxes each contain four bags. Each bag contains five marbles. How many marbles are there altogether in the three boxes?
2013 Romania National Olympiad, 1
A series of numbers is called complete if it has non-zero natural terms and any nonzero integer has at least one among multiple series. Show that the arithmetic progression is a complete sequence if and only if it divides the first term relationship.
2020 Korean MO winter camp, #5
$\square ABCD$ is a quadrilateral with $\angle A=2\angle C <90^\circ$. $I$ is the incenter of $\triangle BAD$, and the line passing $I$ and perpendicular to $AI$ meets rays $CB$ and $CD$ at $E,F$ respectively. Denote $O$ as the circumcenter of $\triangle CEF$. The line passing $E$ and perpendicular to $OE$ meets ray $OF$ at $Q$, and the line passing $F$ and perpendicular to $OF$ meets ray $OE$ at $P$. Prove that the circle with diameter $PQ$ is tangent to the circumcircle of $\triangle BCD$.
II Soros Olympiad 1995 - 96 (Russia), 9.4
$100$ schoolchildren took part in the Mathematical Olympiad. $4$ tasks were proposed. The first problem was solved by $90$ people, the second by $80$, the third by $70$ and the fourth by $60$. However, no one solved all the problems. Students who solved both the third and fourth questions received an award. How many students were awarded?
2021 Israel National Olympiad, P5
Solve the following equation in positive numbers.
$$(2a+1)(2a^2+2a+1)(2a^4+4a^3+6a^2+4a+1)=828567056280801$$
2002 Moldova National Olympiad, 4
The circles $ C_1$ and $ C_2$ with centers $ O_1$ and $ O_2$ respectively are externally tangent. Their common tangent not intersecting the segment $ O_1O_2$ touches $ C_1$ at $ A$ and $ C_2$ at $ B$. Let $ C$ be the reflection of $ A$ in $ O_1O_2$ and $ P$ be the intersection of $ AC$ and $ O_1O_2$. Line $ BP$ meets $ C_2$ again at $ L$. Prove that line $ CL$ is tangent to the circle $ C_2$.
2008 Bosnia And Herzegovina - Regional Olympiad, 4
A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)?
1983 Czech and Slovak Olympiad III A, 2
Given a triangle $ABC$, prove that for every inner point $P$ of the side $AB$ the inequality $$PC\cdot AB<PA\cdot BC+PB\cdot AC$$ holds.
2019 Bulgaria EGMO TST, 1
Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$, $OG = 1$ and $OG \parallel BC$. (As usual $O$ is the circumcenter and $G$ is the centroid.)
2002 AMC 8, 5
Carlos Montado was born on Saturday, November 9, 2002. On what day of the week will Carlos be 706 days old?
$ \text{(A)}\ \text{Monday}\qquad\text{(B)}\ \text{Wednesday}\qquad\text{(C)}\ \text{Friday}\qquad\text{(D)}\ \text{Saturday}\qquad\text{(E)}\ \text{Sunday} $
2015 Princeton University Math Competition, A4/B7
What is the smallest positive integer $n$ such that $20 \equiv n^{15} \pmod{29}$?
LMT Guts Rounds, 32
Compute the infinite sum $\frac{1^3}{2^1}+\frac{2^3}{2^2}+\frac{3^3}{2^3}+\dots+\frac{n^3}{2^n}+\dots.$
1991 IMTS, 1
Use each of the digits 1,2,3,4,5,6,7,8,9 exactly twice to form distinct prime numbers whose sum is as small as possible. What must this minimal sum be? (Note: The five smallest primes are 2,3,5,7, and 11)
2022 CMIMC, 2.7
For polynomials $P(x) = a_nx^n + \cdots + a_0$, let $f(P) = a_n\cdots a_0$ be the product of the coefficients of $P$. The polynomials $P_1,P_2,P_3,Q$ satisfy $P_1(x) = (x-a)(x-b)$, $P_2(x) = (x-a)(x-c)$, $P_3(x) = (x-b)(x-c)$, $Q(x) = (x-a)(x-b)(x-c)$ for some complex numbers $a,b,c$. Given $f(Q) = 8$, $f(P_1) + f(P_2) + f(P_3) = 10$, and $abc>0$, find the value of $f(P_1)f(P_2)f(P_3)$.
[i]Proposed by Justin Hsieh[/i]
2016 Nigerian Senior MO Round 2, Problem 3
The integers $1, 2, \dots , 9$ are written on individual slips of paper and all are put into a bag. Ade chooses a slip at random, notes the integer on it, and replaces it in the bag. Bala then picks a slip at random and notes the integer written on it. Chioma then adds up Ade's and Bala's numbers. What is the probability that the unit's digit of this sum is less that $5$?
2018 Taiwan TST Round 3, 5
Find all functions $ f: \mathbb{N} \to \mathbb{N} $ such that $$ f\left(x+yf\left(x\right)\right) = x+f\left(y\right)f\left(x\right) $$ holds for all $ x,y \in \mathbb{N} $
2019 Jozsef Wildt International Math Competition, W. 9
Let $\alpha > 0$ be a real number. Compute the limit of the sequence $\{x_n\}_{n\geq 1}$ defined by $$x_n=\begin{cases} \sum \limits_{k=1}^n \sinh \left(\frac{k}{n^2}\right),& \text{when}\ n>\frac{1}{\alpha}\\ 0,& \text{when}\ n\leq \frac{1}{\alpha}\end{cases}$$
2011 Saint Petersburg Mathematical Olympiad, 6
$ABCD$ - convex quadrilateral. $M$ -midpoint $AC$ and $\angle MCB=\angle CMD =\angle MBA=\angle MBC-\angle MDC$.
Prove, that $AD=DC+AB$
1989 Greece National Olympiad, 3
If $a\ge 0$ prove that $a^4+ a^3-10 a^2+9 a+4>0$.
2001 Tournament Of Towns, 3
Kolya is told that two of his four coins are fake. He knows that all real coins have the same weight, all fake coins have the same weight, and the weight of a real coin is greater than that of a fake coin. Can Kolya decide whether he indeed has exactly two fake coins by using a balance twice?
2014 Postal Coaching, 4
Denote by $F_n$ the $n^{\text{th}}$ Fibonacci number $(F_1=F_2=1)$.Prove that if $a,b,c$ are positive integers such that $a| F_b,b|F_c,c|F_a$,then either $5$ divides each of $a,b,c$ or $12$ divides each of $a,b,c$.
2014 Cezar Ivănescu, 3
Find the real numbers $ \lambda $ that have the property that there is a nonconstant, continuous function $ u: [0,1]\longrightarrow\mathbb{R} $ satisfying
$$ u(x)=\lambda\int_0^1 (x-3y)u(y)dy , $$
for any $ x $ in the interval $ [0,1]. $
Estonia Open Junior - geometry, 2007.1.2
The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.
2019 China Team Selection Test, 5
Let $M$ be the midpoint of $BC$ of triangle $ABC$. The circle with diameter $BC$, $\omega$, meets $AB,AC$ at $D,E$ respectively. $P$ lies inside $\triangle ABC$ such that $\angle PBA=\angle PAC, \angle PCA=\angle PAB$, and $2PM\cdot DE=BC^2$. Point $X$ lies outside $\omega$ such that $XM\parallel AP$, and $\frac{XB}{XC}=\frac{AB}{AC}$. Prove that $\angle BXC +\angle BAC=90^{\circ}$.