Found problems: 85335
2008 Tournament Of Towns, 4
Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.
2020-21 IOQM India, 16
The sides $x$ and $y$ of a scalene triangle satisfy $x + \frac{2\Delta }{x}=y+ \frac{2\Delta }{y}$ , where $\Delta$ is the area of the triangle. If $x = 60, y = 63$, what is the length of the largest side of the triangle?
2011 ISI B.Stat Entrance Exam, 2
Consider three positive real numbers $a,b$ and $c$. Show that there cannot exist two distinct positive integers $m$ and $n$ such that both $a^m+b^m=c^m$ and $a^n+b^n=c^n$ hold.
1985 Vietnam National Olympiad, 1
Let $ a$, $ b$ and $ m$ be positive integers. Prove that there exists a positive integer $ n$ such that $ (a^n \minus{} 1)b$ is divisible by $ m$ if and only if $ \gcd (ab, m) \equal{} \gcd (b, m)$.
2013 Harvard-MIT Mathematics Tournament, 22
Sherry and Val are playing a game. Sherry has a deck containing $2011$ red cards and $2012$ black cards, shuffled randomly. Sherry flips these cards over one at a time, and before she flips each card over, Val guesses whether it is red or black. If Val guesses correctly, she wins $1$ dollar; otherwise, she loses $1$ dollar. In addition, Val must guess red exactly $2011$ times. If Val plays optimally, what is her expected profit from this game?
2022 IMO, 1
The Bank of Oslo issues two types of coin: aluminum (denoted A) and bronze (denoted B). Marianne has $n$ aluminum coins and $n$ bronze coins arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k \leq 2n$, Gilberty repeatedly performs the following operation: he identifies the longest chain containing the $k^{th}$ coin from the left and moves all coins in that chain to the left end of the row. For example, if $n=4$ and $k=4$, the process starting from the ordering $AABBBABA$ would be $AABBBABA \to BBBAAABA \to AAABBBBA \to BBBBAAAA \to ...$
Find all pairs $(n,k)$ with $1 \leq k \leq 2n$ such that for every initial ordering, at some moment during the process, the leftmost $n$ coins will all be of the same type.
2001 Saint Petersburg Mathematical Olympiad, 11.4
For any two positive integers $n>m$ prove the following inequality:
$$[m,n]+[m+1,n+1]\geq \dfrac{2nm}{\sqrt{m-n}}$$
As always, $[x,y]$ means the least common multiply of $x,y$.
[I]Proposed by A. Golovanov[/i]
2025 Israel National Olympiad (Gillis), P6
Let $a$, $b$ and $c$ be non-negative numbers such that $ab+ac+bc+abc=4.$ . Prove that:
$$\sqrt{\frac{ab+ac+1}{a+2}}+\sqrt{\frac{ab+bc+1}{b+2}}+\sqrt{\frac{ac+bc+1}{c+2}}\leq3.$$
[hide="PS"]Dedicated to dear KhuongTrang :-D [/hide]
1995 Tuymaada Olympiad, 4
It is known that the merchant’s $n$ clients live in locations laid along the ring road. Of these, $k$ customers have debts to the merchant for $a_1,a_2,...,a_k$ rubles, and the merchant owes the remaining $n-k$ clients, whose debts are $b_1,b_2,...,b_{n-k}$ rubles, moreover, $a_1+a_2+...+a_k=b_1+b_2+...+b_{n-k}$. Prove that a merchant who has no money can pay all his debts and have paid all the customer debts, by starting a customer walk along the road from one of points and not missing any of their customers.
2020-21 IOQM India, 30
Find the number of pairs $(a,b)$ of natural nunbers such that $b$ is a 3-digit number, $a+1$ divides $b-1$ and $b$ divides $a^{2} + a + 2$.
2003 Swedish Mathematical Competition, 5
Given two positive numbers $a, b$, how many non-congruent plane quadrilaterals are there such that $AB = a$, $BC = CD = DA = b$ and $\angle B = 90^o$ ?
1998 Korea - Final Round, 1
Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.
2021 China Team Selection Test, 3
Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following:
There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements,
$$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$
where $S(n)$ denotes sum of digits of decimal representation of $n$.
1992 Tournament Of Towns, (345) 3
Do there exist two polynomials $P(x)$ and $Q(x)$ with integer coefficients such that
$$(P-Q)(x), \,\,\,\, P(x) \,\,\,\, and \,\,\,\,(P+Q)(x)$$
are squares of polynomials (and $Q$ is not equal to $cP$, where $c$ is a real number)?
(V Prasolov)
1973 Polish MO Finals, 6
Prove that for every centrally symmetric polygon there is at most one ellipse containing the polygon and having the minimal area.
2009 Today's Calculation Of Integral, 449
Evaluate $ \sum_{k\equal{}1}^n \int_0^{\pi} (\sin x\minus{}\cos kx)^2dx.$
ICMC 4, 4
Does there exist a set $\mathcal{R}$ of positive rational numbers such that every positive rational number is the sum of the elements of a unique finite subset of $\mathcal{R}$?
[i]Proposed by Tony Wang[/i]
2016 Kosovo National Mathematical Olympiad, 5
If $a,b,c$ are sides of right triangle with $c$ hypothenuse then show that for every positive integer $n>2$ we have $c^n>a^n+b^n$ .
2008 ITest, 10
Tony has an old sticky toy spider that very slowly "crawls" down a wall after being stuck to the wall. In fact, left untouched, the toy spider crawls down at a rate of one inch for every two hours it's left stuck to the wall. One morning, at around $9$ o' clock, Tony sticks the spider to the wall in the living room three feet above the floor. Over the next few mornings, Tony moves the spider up three feet from the point where he finds it. If the wall in the living room is $18$ feet high, after how many days (days after the first day Tony places the spider on the wall) will Tony run out of room to place the spider three feet higher?
2025 Turkey EGMO TST, 2
Does there exist a sequence of positive real numbers $\{a_i\}_{i=1}^{\infty}$ satisfying:
\[
\sum_{i=1}^{n} a_i \geq n^2 \quad \text{and} \quad \sum_{i=1}^{n} a_i^2 \leq n^3 + 2025n
\]
for all positive integers $n$.
2018 PUMaC Combinatorics B, 8
Frankie the Frog starts his morning at the origin in $\mathbb{R}^2$. He decides to go on a leisurely stroll, consisting of $3^1+3^{10}+3^{11}+3^{100}+3^{111}+3^{1000}$ moves, starting with the first move. On the $n$th move, he hops a distance of
$$\max\{k\in\mathbb{Z}:3^k|n\}+1,$$
then turns $90^{\circ}$ counterclockwise. What is the square of the distance from his final position to the origin?
2002 Mexico National Olympiad, 1
The numbers $1$ to $1024$ are written one per square on a $32 \times 32$ board, so that the first row is $1, 2, ... , 32$, the second row is $33, 34, ... , 64$ and so on. Then the board is divided into four $16 \times 16$ boards and the position of these boards is moved round clockwise, so that
$AB$ goes to $DA$
$DC \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \, CB$
then each of the $16 \times 16 $ boards is divided into four equal $8 \times 8$ parts and each of these is moved around in the same way (within the $ 16 \times 16$ board). Then each of the $8 \times 8$ boards is divided into four $4 \times 4$ parts and these are moved around, then each $4 \times 4$ board is divided into $2 \times 2$ parts which are moved around, and finally the squares of each $2 \times 2$ part are moved around. What numbers end up on the main diagonal (from the top left to bottom right)?
1990 ITAMO, 3
Let $a,b,c$ be distinct real numbers and $P(x)$ a polynomial with real coefficients. Suppose that the remainders of $P(x)$ upon division by $(x-a), (x-b)$ and $(x-c)$ are $a,b$ and $c$, respectively. Find the polynomial that is obtained as the remainder of $P(x)$ upon division by $(x-a)(x-b)(x-c)$.
II Soros Olympiad 1995 - 96 (Russia), 11.5
The space is filled in the usual way with unit cubes. (Each cube is adjacent to $6$ others that have a common face with it.) On three edges of one of the cubes emerging from one vertex, points are marked at a distance of $1/19$, $1/9$ and $1/7$ from it, respectively. A plane is drawn through these points. Let's consider the many different polygons formed when this plane intersects with the cubes filling the space. How many different (unequal) polygons are there in this set?
Russian TST 2015, P2
Given an acute triangle $ABC, H$ is the foot of the altitude drawn from the point $A$ on the line $BC, P$ and $K \ne H$ are arbitrary points on the segments $AH$ and$ BC$ respectively. Segments $AC$ and $BP$ intersect at point $B_1$, lines $AB$ and $CP$ at point $C_1$. Let $X$ and $Y$ be the projections of point $H$ on the lines $KB_1$ and $KC_1$, respectively. Prove that points $A, P, X$ and $Y$ lie on one circle.