Found problems: 85335
2005 India IMO Training Camp, 3
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]
Kvant 2022, M2691
There are $N{}$ points marked on the plane. Any three of them form a triangle, the values of the angles of which in are expressed in natural numbers (in degrees). What is the maximum $N{}$ for which this is possible?
[i]Proposed by E. Bakaev[/i]
2001 Vietnam Team Selection Test, 3
Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition
\[0 < a_{n+1} - a_n \leq 2001\]
for all $n \in \mathbb{N}^{*}$
Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.
1997 Brazil Team Selection Test, Problem 4
Prove that it is impossible to arrange the numbers $1,2,\ldots,1997$ around a circle in such a way that, if $x$ and $y$ are any two neighboring numbers, then $499\le|x-y|\le997$.
2011 Canadian Open Math Challenge, 8
A group of n friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$, and four full-solution problems $F_1, F_2, F_3, F_4$. Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 table. Inside the square located in the $i$th row and $j$th column, we write down the number of people who correctly solved both problem $S_i$ and $F_j$. If the 32 entries in the table sum to 256, what is the value of n?
2011 Morocco National Olympiad, 3
Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing.
[i](Poland) Andrzej Komisarski and Marcin Kuczma[/i]
2012 Princeton University Math Competition, B2
Let $O_1, O_2, ..., O_{2012}$ be $2012$ circles in the plane such that no circle intersects or contains anyother circle and no two circles have the same radius. For each $1\le i < j \le 2012$, let $P_{i,j}$ denotethe point of intersection of the two external tangent lines to $O_i$ and $O_j$, and let $T$ be the set of all $P_{i,j}$ (so $|T|=\binom {2012}{2}= 2023066$). Suppose there exists a subset $S\subset T$ with $|S|= 2021056$ such that all points in $S$ lie on the same line. Prove that all points in $T$ lie on the same line.
2018 Belarusian National Olympiad, 10.5
Find all positive integers $n$ such that equation $$3a^2-b^2=2018^n$$ has a solution in integers $a$ and $b$.
2023 AMC 12/AHSME, 24
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$
2022 IFYM, Sozopol, 1
In a football tournament with $n\geq 2$ teams each two played a match. For a won match the victor gets 2 points and for a draw each one gets 1 point. In the final results there weren’t two teams with equal amount of points.
It turned out that because of a mistake each match that was written in the results as won was actually a draw and each one that was written as draw was actually won. In the new ranking there were also no two teams with the same amount of points.
Find all n for which it is possible for the two rankings to be opposite of each other, that is the first team in the first ranking is actually the last one, the second team is pre-last and so on.
2006 Federal Math Competition of S&M, Problem 3
For every natural number $a$, consider the set $S(a)=\{a^n+a+1|n=2,3,\ldots\}$. Does there exist an infinite set $A\subset\mathbb N$ with the property that for any two distinct
elements $x,y\in A$, $x$ and $y$ are coprime and $S(x)\cap S(y)=\emptyset$?
2004 India Regional Mathematical Olympiad, 7
Let $x$ and $y$ be positive real numbers such that $y^3 + y \leq x - x^3$. Prove that
(A) $y < x < 1$
(B) $x^2 + y^2 < 1$.
2017 NIMO Problems, 7
Let $ABC$ be a triangle with $AB=4$, $AC=5$, $BC=6$, and circumcircle $\Omega$. Points $E$ and $F$ lie on $AC$ and $AB$ respectively such that $\angle ABE=\angle CBE$ and $\angle ACF=\angle BCF$. The second intersection point of the circumcircle of $\triangle AEF$ with $\Omega$ (other than $A$) is $P$. Suppose $AP^2=\frac mn$ where $m$ and $n$ are positive relatively prime integers. Find $100m+n$.
[i]Proposed by David Altizio[/i]
1996 IMO Shortlist, 5
Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$:
\[ f(3mn \plus{} m \plus{} n) \equal{} 4f(m)f(n) \plus{} f(m) \plus{} f(n).
\]
2002 District Olympiad, 4
For any natural number $ n\ge 2, $ define $ m(n) $ to be the minimum number of elements of a set $ S $ that simultaneously satisfy:
$ \text{(i)}\quad \{ 1,n\} \subset S\subset \{ 1,2,\ldots ,n\} $
$ \text{(ii)}\quad $ any element of $ S, $ distinct from $ 1, $ is equal to the sum of two (not necessarily distinct) elements from $ S. $
[b]a)[/b] Prove that $ m(n)\ge 1+\left\lfloor \log_2 n \right\rfloor ,\quad\forall n\in\mathbb{N}_{\ge 2} . $
[b]b)[/b] Prove that there are infinitely many natural numbers $ n\ge 2 $ such that $ m(n)=m(n+1). $
$ \lfloor\rfloor $ denotes the usual integer part.
2022 District Olympiad, P1
Let $f:\mathbb{N}^*\rightarrow \mathbb{N}^*$ be a function such that $\frac{x^3+3x^2f(y)}{x+f(y)}+\frac{y^3+3y^2f(x)}{y+f(x)}=\frac{(x+y)^3}{f(x+y)},~(\forall)x,y\in\mathbb{N}^*.$
$a)$ Prove that $f(1)=1.$
$b)$ Find function $f.$
2024 Bangladesh Mathematical Olympiad, P9
Find all pairs of positive integers $(k, m)$ such that for any positive integer $n$, the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!$.
2008 USAMO, 6
At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).
2012 AMC 12/AHSME, 2
A circle of radius $5$ is inscribed in a rectangle as shown. The ratio of the the length of the rectangle to its width is $2\ :\ 1$. What is the area of the rectangle?
[asy]
draw((0,0)--(0,10)--(20,10)--(20,0)--cycle);
draw(circle((10,5),5));
[/asy]
$ \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 $
2001 Hungary-Israel Binational, 3
Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$.
If $e(G_{n}) \geq\frac{n\sqrt{n}}{2}+\frac{n}{4}$ ,prove that $G_{n}$ contains $C_{4}$ .
2017 Thailand Mathematical Olympiad, 9
Determine all functions $f$ on the set of positive rational numbers such that $f(xf(x) + f(y)) = f(x)^2 + y$ for all positive rational numbers $x, y$.
2024 LMT Fall, 14
Isabella assigns a distinct integer from $1$ to $6$ to each row and column of a $3\times 3$ grid. In each entry, she writes either the sum or the product of the values assigned to the corresponding row and column. Find the maximum possible value of the sum of all entries in the grid.
2018 ELMO Shortlist, 2
Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$
[i]Proposed by Carl Schildkraut[/i]
2014 Tuymaada Olympiad, 6
Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares.
[i](V. Dolnikov)[/i]
2005 Slovenia Team Selection Test, 5
Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$.
[i]Proposed by Hojoo Lee, Korea[/i]