Found problems: 85335
1995 Belarus National Olympiad, Problem 6
Let $p$ and $q$ be distinct positive integers. Prove that at least one of the equations $x^2+px+q=0$ and $x^2+qx+p=0$ has a real root.
2021 Brazil Team Selection Test, 2
There are $100$ books in a row, numbered from $1$ to $100$ in some order. An operation is choose three books and reorder in any order between them(the others $97$ books stay at the same place). Denote that a book is in [i]correct position[/i] if the book $i$ is in the position $i$. Determine the least integer $m$ such that, for any initial configuration, we can realize $m$ operations and all the books will be in the correct position.
2018 Greece Team Selection Test, 2
A triangle $ABC$ is inscribed in a circle $(C)$ .Let $G$ the centroid of $\triangle ABC$ .
We draw the altitudes $AD,BE,CF$ of the given triangle .Rays $AG$ and $GD$ meet (C) at $M$ and $N$.Prove that points $ F,E,M,N $ are concyclic.
2009 Greece Team Selection Test, 2
Given is a triangle $ABC$ with barycenter $G$ and circumcenter $O$.The perpendicular bisectors of $GA,GB,GC$ intersect at $A_1,B_1,C_1$.Show that $O$ is the barycenter of $\triangle{A_1B_1C_1}$.
2016 IMO Shortlist, A6
The equation
$$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$$
is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
1947 Putnam, A3
Given a triangle $ABC$ with an interior point $P$ and points $Q_1 , Q_2$ not lying on any of the segments $AB , AC ,BC,$ $AP ,BP ,CP,$ show that there does not exist a polygonal line $K$ joining $Q_1$ and $Q_2$ such that
i) $K$ crosses each segment exactly once,
ii) $K$ does not intersect itself
iii) $K$ does not pass through $A, B , C$ or $P.$
1974 All Soviet Union Mathematical Olympiad, 190
Among all the numbers representable as $36^k - 5^l$ ($k$ and $l$ are natural numbers) find the smallest. Prove that it is really the smallest.
2013 NZMOC Camp Selection Problems, 5
Consider functions $f$ from the whole numbers (non-negative integers) to the whole numbers that have the following properties:
$\bullet$ For all $x$ and $y$, $f(xy) = f(x)f(y)$,
$\bullet$ $f(30) = 1$, and
$\bullet$ for any $n$ whose last digit is $7$, $f(n) = 1$.
Obviously, the function whose value at $n$ is $ 1$ for all $n$ is one such function. Are there any others? If not, why not, and if so, what are they?
1948 Putnam, B1
Let $f(x)$ be a cubic polynomial with roots $x_1 , x_2$ and $x_3.$ Assume that $f(2x)$ is divisible by $f'(x)$ and compute the ratio $x_1 : x_2: x_3 .$
2016 China Team Selection Test, 6
Let $m,n$ be naturals satisfying $n \geq m \geq 2$ and let $S$ be a set consisting of $n$ naturals. Prove that $S$ has at least $2^{n-m+1}$ distinct subsets, each whose sum is divisible by $m$. (The zero set counts as a subset).
ICMC 2, 3
Show that if the faces of a tetrahedron have the same area, then they are congruent.
2003 Estonia Team Selection Test, 3
Let $N$ be the set of all non-negative integers and for each $n \in N$ denote $n'= n +1$. The function $A : N^3 \to N$ is defined as follows:
(i) $A(0, m, n) = m'$ for all $m, n \in N$
(ii) $A(k', 0, n) =\left\{ \begin{array}{ll}
n & if \, \, k = 0 \\
0 & if \, \,k = 1, \\
1 & if \, \, k > 1 \end{array} \right.$ for all $k, n \in N$
(iii) $A(k', m', n) = A(k, A(k',m,n), n)$ for all $k,m, n \in N$.
Compute $A(5, 3, 2)$.
(H. Nestra)
2013 Lusophon Mathematical Olympiad, 4
Find all the pairs $(x,y)$ of positive integers that satisfy the equation $x^2-xy+2x-3y=2013$.
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?
2004 USAMO, 4
Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number in that row is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from one to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players.
2023 Princeton University Math Competition, 6
6. Let $f(p)$ denote the number of ordered tuples $\left(x_{1}, x_{2}, \ldots, x_{p}\right)$ of nonnegative integers satisfying $\sum_{i=1}^{p} x_{i}=2022$, where $x_{i} \equiv i(\bmod p)$ for all $1 \leq i \leq p$. Find the remainder when $\sum_{p \in \mathcal{S}} f(p)$ is divided by 1000, where $\mathcal{S}$ denotes the set of all primes less than 2022 .
1967 Miklós Schweitzer, 3
Prove that if an infinite, noncommutative group $ G$ contains a proper normal subgroup with a commutative factor group, then $ G$ also contains an infinite proper normal subgroup.
[i]B. Csakany[/i]
1988 China Team Selection Test, 2
Let $ABCD$ be a trapezium $AB // CD,$ $M$ and $N$ are fixed points on $AB,$ $P$ is a variable point on $CD$. $E = DN \cap AP$, $F = DN \cap MC$, $G = MC \cap PB$, $DP = \lambda \cdot CD$. Find the value of $\lambda$ for which the area of quadrilateral $PEFG$ is maximum.
2018 District Olympiad, 2
Let $p$ be a natural number greater than or equal to $2$ and let $(M, \cdot)$ be a finite monoid such that $a^p \ne a$, for any $a\in M \backslash \{e\}$, where $e$ is the identity element of $M$. Show that $(M, \cdot)$ is a group.
2014 Math Prize for Girls Olympiad, 4
Let $n$ be a positive integer. A 4-by-$n$ rectangle is divided into $4n$ unit squares in the usual way. Each unit square is colored black or white. Suppose that every white unit square shares an edge with at least one black unit square. Prove that there are at least $n$ black unit squares.
2023 Chile National Olympiad, 2
In Cartesian space, let $\Omega = \{(a, b, c) : a, b, c$ are integers between $1$ and $30\}$.
A point of $\Omega$ is said to be [i]visible [/i] from the origin if the segment that joins said point with the origin does not contain any other elements of $\Omega$. Find the number of points of $\Omega$ that are [i]visible [/i] from the origin.
2019 District Olympiad, 4
Find all positive integers $p$ for which there exists a positive integer $n$ such that $p^n+3^n~|~p^{n+1}+3^{n+1}.$
2025 Sharygin Geometry Olympiad, 9
The line $l$ passing through the orthocenter $H$ of a triangle $ABC$ $(BC>AB)$ and parallel to $AC$ meets $AB$ and $BC$ at points $D$ and $E$ respectively. The line passing through the circumcenter of the triangle and parallel to the median $BM$ meets $l$ at point $F$. Prove that the length of segment $HF$ is three times greater than the difference of $FE$ and $DH$
Proposed by: A.Mardanov, K.Mardanova
2006 China Second Round Olympiad, 15
Suppose $f(x)=x^2+a$. Define $f^1(x)=f(x)$, $f^n(x)=f(f^{n-1}(x))$, $n=2, 3, \cdots$, and let $M=\{a\in\mathbb{R}| |f^n(0)|\le 2, \text{for any } n\in\mathbb{N}\} $. Prove that $M=[-2, \frac{1}{4}]$.
2019 AIME Problems, 12
Given $f(z) = z^2-19z$, there are complex numbers $z$ with the property that $z$, $f(z)$, and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$. There are positive integers $m$ and $n$ such that one such value of $z$ is $m+\sqrt{n}+11i$. Find $m+n$.