Found problems: 85335
2007 Baltic Way, 1
For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$. In each subset $P_i$, let $p_i$ be the product of the two numbers in $P_i$. Prove that
\[\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1 \]
2007 Moldova National Olympiad, 11.4
The function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(\textrm{cot}x)=\sin2x+\cos2x$, for any $x\in(0,\pi)$. Find the minimum and maximum value of $g: [-1;1]\rightarrow\mathbb{R}$, $g(x)=f(x)\cdot f(1-x)$.
Kvant 2023, M2730
On each cell of a $3\times 6$ the board lies one coin. It is known that some two coins lying on adjacent cells are fake. They have the same weigh, but are lighter than the real ones. All the other coins are real. How can one find both counterfeit coins in three weightings on a double-pan balance, without using weights?
[i]Proposed by K. Knop[/i]
Champions Tournament Seniors - geometry, 2003.1
Consider the triangle $ABC$, in which $AB > AC$. Let $P$ and $Q$ be the feet of the perpendiculars dropped from the vertices $B$ and $C$ on the bisector of the angle $BAC$, respectively. On the line $BC$ note point $B$ such that $AD \perp AP.$ Prove that the lines $BQ, PC$ and $AD$ intersect at one point.
2013 Princeton University Math Competition, 2
How many ways are there to color the edges of a hexagon orange and black if we assume that two hexagons are indistinguishable if one can be rotated into the other? Note that we are saying the colorings OOBBOB and BOBBOO are distinct; we ignore flips.
1994 Baltic Way, 6
Prove that any irreducible fraction $p/q$, where $p$ and $q$ are positive integers and $q$ is odd, is equal to a fraction $\frac{n}{2^k-1}$ for some positive integers $n$ and $k$.
2008 Thailand Mathematical Olympiad, 9
Find the number of pairs of sets $(A, B)$ satisfying $A \subseteq B \subseteq \{1, 2, ...,10\}$
2023 Korea Summer Program Practice Test, P8
$n$ is a natural number larger than $3$ and denote all positive coprime numbers with $n$ as $1= b_1 < b_2 < \cdots b_k$. For a positive integer $m$ which is larger than $3$ and is coprime with $n$, let $A$ be the set of tuples $(a_1,a_2, \cdots a_k)$ satisfying the condition.
$$\textbf{Condition}: \text{For all integers } i, 0 \le a_i < m \text{ and } a_1b_1 + a_2b_2 + \cdots a_kb_k \text{ is a mutiple of } n$$
For elements of $A$, show that the difference of number of elements such that $a_1 = 1$ and the number of elements such that $a_2 = 2$ maximum $1$
2014 Benelux, 1
Find the smallest possible value of the expression \[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\]
in which $a,~ b,~ c$, and $d$ vary over the set of positive integers.
(Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)
2017 Sharygin Geometry Olympiad, P23
Let a line $m$ touch the incircle of triangle $ABC$. The lines passing through the incenter $I$ and perpendicular to $AI, BI, CI$ meet $m$ at points $A', B', C'$ respectively. Prove that $AA', BB'$ and $CC'$ concur.
1995 India National Olympiad, 2
Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.
2021 Indonesia TST, N
For a three-digit prime number $p$, the equation $x^3+y^3=p^2$ has an integer solution. Calculate $p$.
2003 Miklós Schweitzer, 4
Let $\{a_{n,1},\ldots, a_{n,n} \}_{n=1}^{\infty}$ integers such that $a_{n,i}\neq a_{n,j}$ for $1\le i<j\le n\, , n=2,3,\ldots$ and let $\left\langle y\right\rangle\in [0,1)$ denote the fractional part of the real number $y$. Show that there exists a real sequence $\{ x_n\}_{n=1}^{\infty}$ such that the numbers $\langle a_{n,1}x_n \rangle, \ldots, \langle a_{n,n}x_n \rangle$ are asymptotically uniformly distributed on the interval $[0,1]$.
(translated by L. Erdős)
1970 Miklós Schweitzer, 4
If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\]
J. Suranyi
1993 All-Russian Olympiad Regional Round, 11.6
Seven tetrahedra are placed on the table. For any three of them there exists a horizontal plane cutting them in triangles of equal areas. Show that there exists a plane cutting all seven tetrahedra in triangles of equal areas.
2020 Brazil National Olympiad, 6
Let $f (x) = 2x^2 + x - 1$, $f^0(x) = x$ and $f^{n + 1}(x) = f (f^n(x))$ for all real $x$ and $n \ge 0$ integer .
(a) Determine the number of real distinct solutions of the equation of $f^3(x) = x$.
(b) Determine, for each integer $n \ge 0$, the number of real distinct solutions of the equation $f^n(x) = 0$.
2018 Czech and Slovak Olympiad III A, 1
In a group of people, there are some mutually friendly pairs. For positive integer $k\ge3$ we say the group is $k$-great, if every (unordered) $k$-tuple of people from the group can be seated around a round table it the way that all pairs of neighbors are mutually friendly. [i](Since this was the 67th year of CZE/SVK MO,)[/i] show that if the group is 6-great, then it is 7-great as well.
[b]Bonus[/b] (not included in the competition): Determine all positive integers $k\ge3$ for which, if the group is $k$-great, then it is $(k+1)$-great as well.
2009 District Olympiad, 3
Let $a$ and $b$ be non-negative integers. Prove that the number $a^2 + b^2$ is the difference of two perfect squares if and only if $ab$ is even.
2022 Balkan MO Shortlist, A2
Let $k > 1{}$ be a real number, $n\geqslant 3$ be an integer, and $x_1 \geqslant x_2\geqslant\cdots\geqslant x_n$ be positive real numbers. Prove that \[\frac{x_1+kx_2}{x_2+x_3}+\frac{x_2+kx_3}{x_3+x_4}+\cdots+\frac{x_n+kx_1}{x_1+x_2}\geqslant\frac{n(k+1)}{2}.\][i]Ilija Jovcheski[/i]
2008 F = Ma, 25
Two satellites are launched at a distance $R$ from a planet of negligible radius. Both satellites are launched in the tangential direction. The first satellite launches correctly at a speed $v_\text{0}$ and enters a circular orbit. The second satellite, however, is launched at a speed $\frac{1}{2}v_\text{0}$. What is the minimum distance between the second satellite and the planet over the course of its orbit?
(a) $\frac{1}{\sqrt{2}}R$
(b) $\frac{1}{2}R$
(c) $\frac{1}{3}R$
(d) $\frac{1}{4}R$
(e) $\frac{1}{7}R$
1990 Baltic Way, 3
Given $a_0 > 0$ and $c > 0$, the sequence $(a_n)$ is defined by
\[a_{n+1}=\frac{a_n+c}{1-ca_n}\quad\text{for }n=1,2,\dots\]
Is it possible that $a_0, a_1, \dots , a_{1989}$ are all positive but $a_{1990}$ is negative?
1986 IMO Shortlist, 21
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that
\[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\]
where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.
2022 CCA Math Bonanza, L3.3
Determine the sum of all positive integers $n<100$ satisfying the following expression. \[\sum_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod \;{10^{k+1})}-n \;(\bmod \;{10^k)}\right)=\prod_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod\; 10^{k+1})-n \;(\bmod\; 10^k)\right).\] Here, $\textstyle\sum$ and $\textstyle\prod$ represent sum and product, respectively.
[i]2022 CCA Math Bonanza Lightning Round 3.3[/i]
2021 IMC, 3
We say that a positive real number $d$ is $good$ if there exists an infinite squence $a_1,a_2,a_3,...\in (0,d)$ such that for each $n$, the points $a_1,a_2,...,a_n$ partition the interval $[0,d]$ into segments of length at most $\frac{1}{n}$ each . Find
$\text{sup}\{d| d \text{is good}\}$.
1998 Iran MO (3rd Round), 1
A one-player game is played on a $m \times n$ table with $m \times n$ nuts. One of the nuts' sides is black, and the other side of them is white. In the beginning of the game, there is one nut in each cell of the table and all nuts have their white side upwards except one cell in one corner of the table which has the black side upwards. In each move, we should remove a nut which has its black side upwards from the table and reverse all nuts in adjacent cells (i.e. the cells which share a common side with the removed nut's cell). Find all pairs $(m,n)$ for which we can remove all nuts from the table.