Found problems: 85335
1999 Estonia National Olympiad, 3
The incircle of the triangle $ABC$, with the center $I$ , touches the sides $AB, AC$ and $BC$ in the points $K, L$ and $M$ respectively. Points $P$ and $Q$ are taken on the sides $AC$ and $BC$ respectively, such that $|AP| = |CL|$ and $|BQ| = |CM|$. Prove that the difference of areas of the figures $APIQB$ and $CPIQ$ is equal to the area of the quadrangle $CLIM$
2010 VJIMC, Problem 1
a) Is it true that for every bijection $f:\mathbb N\to\mathbb N$ the series
$$\sum_{n=1}^\infty\frac1{nf(n)}$$is convergent?
b) Prove that there exists a bijection $f:\mathbb N\to\mathbb N$ such that the series
$$\sum_{n=1}^\infty\frac1{n+f(n)}$$is convergent.
($\mathbb N$ is the set of all positive integers.)
2000 German National Olympiad, 4
Find all nonnegative solutions $(x,y,z)$ to the system
$$\begin{cases} \sqrt{x+y}+\sqrt{z} = 7 \\
\sqrt{x+z}+\sqrt{y} = 7 \\
\sqrt{y+z}+\sqrt{x} = 5 \end{cases}$$
2014 AMC 12/AHSME, 1
What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$
${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$
MOAA Team Rounds, 2022.9
Emily has two cups $A$ and $B$, each of which can hold $400$ mL, A initially with $200$ mL of water and $B$ initially with $300$ mL of water. During a round, she chooses the cup with more water (randomly picking if they have the same amount), drinks half of the water in the chosen cup, then pours the remaining half into the other cup and refills the chosen cup to back to half full. If Emily goes for $20$ rounds, how much water does she drink, to the nearest integer?
2017 District Olympiad, 3
Let $ a $ be a positive real number. Prove that
$$ a^{\sin x}\cdot (a+1)^{\cos x}\ge a,\quad\forall x\in \left[ 0,\frac{\pi }{2} \right] . $$
2018 Indonesia MO, 8
Let $I, O$ be the incenter and circumcenter of the triangle $ABC$ respectively. Let the excircle $\omega_A$ of $ABC$ be tangent to the side $BC$ on $N$, and tangent to the extensions of the sides $AB, AC$ on $K, M$ respectively. If the midpoint of $KM$ lies on the circumcircle of $ABC$, prove that $O, I, N$ are collinear.
2007 Chile National Olympiad, 4
$31$ guests at a party sit in equally spaced chairs around a round table , but they have not noticed that there are cards with the names of the guests on the stalls.
(a) Assuming they have been so unlucky that no one is in the room which corresponds to him, show that it is possible to get at least two people to stay in their correct position, without anyone getting up from their seat, turning the table.
(b) Show a configuration where exactly one guest is in his assigned place and where in no way that the table is turned it is possible to achieve that at least two remain right.
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} $