Found problems: 85335
2022 Thailand TST, 3
Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$
[i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]
2018 HMNT, 5
Compute the smallest positive integer $n$ for which $$\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}}$$ is an integer.
1992 Czech And Slovak Olympiad IIIA, 4
Solve the equation $\cos 12x = 5\sin 3x+9\ tan ^2x+\ cot ^2x$
2012 All-Russian Olympiad, 1
$101$ wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions.
2009 Tuymaada Olympiad, 2
$ M$ is the midpoint of base $ BC$ in a trapezoid $ ABCD$. A point $ P$ is chosen on the base $ AD$. The line $ PM$ meets the line $ CD$ at a point $ Q$ such that $ C$ lies between $ Q$ and $ D$. The perpendicular to the bases drawn through $ P$ meets the line $ BQ$ at $ K$. Prove that $ \angle QBC \equal{} \angle KDA$.
[i]Proposed by S. Berlov[/i]
Indonesia MO Shortlist - geometry, g8
Prove that there is only one triangle whose sides are consecutive natural numbers and one of the angles is twice the other angle.
2005 Thailand Mathematical Olympiad, 6
Find the number of positive integer solutions to the equation $(x_1+x_2+x_3)^2(y_1+y_2) = 2548$.
2019 Romanian Master of Mathematics Shortlist, original P6
Let $P(x)$ be a nonconstant complex coefficient polynomial and let $Q(x,y)=P(x)-P(y).$ Suppose that polynomial $Q(x,y)$ has exactly $k$ linear factors unproportional two by tow (without counting repetitons). Let $R(x,y)$ be factor of $Q(x,y)$ of degree strictly smaller than $k$. Prove that $R(x,y)$ is a product of linear polynomials.
[b]Note: [/b] The [i]degree[/i] of nontrivial polynomial $\sum_{m}\sum_{n}c_{m,n}x^{m}y^{n}$ is the maximum of $m+n$ along all nonzero coefficients $c_{m,n}.$ Two polynomials are [i]proportional[/i] if one of them is the other times a complex constant.
[i]Proposed by Navid Safaie[/i]
2024 ITAMO, 2
We are given a unit square in the plane. A point $M$ in the plane is called [i]median [/i]if there exists points $P$ and $Q$ on the boundary of the square such that $PQ$ has length one and $M$ is the midpoint of $PQ$.
Determine the geometric locus of all median points.
2010 Contests, 3
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$\boxed{1} \ f(1) = 1$
$\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$
Ukrainian TYM Qualifying - geometry, 2014.8
In the triangle $ABC$ on the ray $BA$ mark the point $K$ so that $\angle BCA= \angle KCA$ , and on the median $BM$ mark the point $T$ so that $\angle CTK=90^o$ . Prove that $\angle MTC=\angle MCB$ .
2009 All-Russian Olympiad, 3
How many times changes the sign of the function \[ f(x)\equal{}\cos x\cos\frac{x}{2}\cos\frac{x}{3}\cdots\cos\frac{x}{2009}\] at the interval $ \left[0, \frac{2009\pi}{2}\right]$?
2014 India Regional Mathematical Olympiad, 6
Let $D,E,F$ be the points of contact of the incircle of an acute-angled triangle $ABC$ with $BC,CA,AB$ respectively. Let $I_1,I_2,I_3$ be the incentres of the triangles $AFE, BDF, CED$, respectively. Prove that the lines $I_1D, I_2E, I_3F$ are concurrent.
2015 HMNT, 7
Consider a $7 \times 7$ grid of squares. Let $f:\{1,2,3,4,5,6,7\} \rightarrow \{1,2,3,4,5,6,7\}$ be a function; in other words, $f(1), f(2), \dots, f(7)$ are each (not necessarily distinct) integers from 1 to 7. In the top row of the grid, the numbers from 1 to 7 are written in order; in every other square, $f(x)$ is written where $x$ is the number above the square. How many functions have the property that the bottom row is identical to the top row, and no other row is identical to the top row?
2015 Caucasus Mathematical Olympiad, 2
There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$. What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?
2024 Kyiv City MO Round 2, Problem 1
Prove that for any real numbers $x, y, z$ at least one of numbers $x^2 + y + \frac{1}{4}, y^2 + z + \frac{1}{4}, z^2 + x + \frac{1}{4}$ is nonnegative.
[i]Proposed by Oleksii Masalitin[/i]
2013 Peru IMO TST, 2
Let $a \geq 3$ be a real number, and $P$ a polynomial of degree $n$ and having real coefficients. Prove that at least one of the following numbers is greater than or equal to $1:$ $$|a^0- P(0)|, \ |a^1-P(1)| , \ |a^2-P(2)|, \cdots, |a^{n + 1}-P(n + 1)|.$$
2018 239 Open Mathematical Olympiad, 10-11.5
Given a trapezoid $ABCD$, with $AB\parallel CD$. Lines $AC$ and $BD$ intersect at point $E$, and lines $AD$ and $BC$ intersect at point $F$. It turns out that the circle with diameter $EF$ is tangent to the midline of the trapezoid. Prove that there exists a square such that there is a mutual correspondence between all six lines containing pairs of its vertices, and points $A$, $B$, $C$, $D$, $E$, and $F$: each line corresponds to a point lying on it.
[i]Proposed by V. Mokin[/i]
2020 Brazil National Olympiad, 3
Consider an inifinte sequence $x_1, x_2,\dots$ of positive integers such that, for every integer $n\geq 1$:
[list] [*]If $x_n$ is even, $x_{n+1}=\dfrac{x_n}{2}$;
[*]If $x_n$ is odd, $x_{n+1}=\dfrac{x_n-1}{2}+2^{k-1}$, where $2^{k-1}\leq x_n<2^k$.[/list]
Determine the smaller possible value of $x_1$ for which $2020$ is in the sequence.
LMT Guts Rounds, 3
A circle has circumference $8\pi.$ Determine its radius.
2016 China Western Mathematical Olympiad, 4
For an $n$-tuple of integers, define a transformation to be:
$$(a_1,a_2,\cdots,a_{n-1},a_n)\rightarrow (a_1+a_2, a_2+a_3, \cdots, a_{n-1}+a_n, a_n+a_1)$$
Find all ordered pairs of integers $(n,k)$ with $n,k\geq 2$, such that for any $n$-tuple of integers $(a_1,a_2,\cdots,a_{n-1},a_n)$, after a finite number of transformations, every element in the of the $n$-tuple is a multiple of $k$.
2021 Taiwan TST Round 1, N
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
2009 Greece Team Selection Test, 1
Suppose that $a$ is an even positive integer and $A=a^{n}+a^{n-1}+\ldots +a+1,n\in \mathbb{N^{*}}$ is a perfect square.Prove that $8\mid a$.
2009 Purple Comet Problems, 1
The pentagon below has three right angles. Find its area.
[asy]
size(150);
defaultpen(linewidth(1));
draw((4,10)--(0,10)--origin--(10,0)--(10,2)--cycle);
label("4",(2,10),N);
label("10",(0,5),W);
label("10",(5,0),S);
label("2",(10,1),E);
label("10",(7,6),NE);
[/asy]
2019 AMC 10, 15
A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
$$a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}$$for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive inegers. What is $p+q ?$
$\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$