Found problems: 85335
2001 Regional Competition For Advanced Students, 2
Find all real solutions to the equation
$$(x+1)^{2001}+(x+1)^{2000}(x-2)+(x+1)^{1999}(x-2)^2+...+(x+1)^2(x-2)^{1999}+(x+1)^{2000}(x-2)+(x+1)^{2001}=0$$
2017 Morocco TST-, 4
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.
2001 Bulgaria National Olympiad, 3
Let $p$ be a prime number congruent to $3$ modulo $4$, and consider the equation $(p+2)x^{2}-(p+1)y^{2}+px+(p+2)y=1$.
Prove that this equation has infinitely many solutions in positive integers, and show that if $(x,y) = (x_{0}, y_{0})$ is a solution of the equation in positive integers, then $p | x_{0}$.
2024 IFYM, Sozopol, 6
The positive integers \( a \), \( b \), \( c \), \( d \) are such that \( (a+c)(b+d) = (ab-cd)^2 \). Prove that \( 4ad + 1 \) and \( 4bc + 1 \) are perfect squares of natural numbers.
1985 AMC 8, 10
The fraction halfway between $ \frac{1}{5}$ and $ \frac{1}{3}$ (on the number line) is
\[ \textbf{(A)}\ \frac{1}{4} \qquad
\textbf{(B)}\ \frac{2}{15} \qquad
\textbf{(C)}\ \frac{4}{15} \qquad
\textbf{(D)}\ \frac{53}{200} \qquad
\textbf{(E)}\ \frac{8}{15}
\]
2003 Canada National Olympiad, 4
Prove that when three circles share the same chord $AB$, every line through $A$ different from $AB$ determines the same ratio $X Y : Y Z$, where $X$ is an arbitrary point different from $B$ on the first circle while $Y$ and $Z$ are the points where AX intersects the other two circles (labeled so that $Y$ is between $X$ and $Z$).
2010 Iran MO (2nd Round), 1
Let $a,b$ be two positive integers and $a>b$.We know that $\gcd(a-b,ab+1)=1$ and $\gcd(a+b,ab-1)=1$. Prove that $(a-b)^2+(ab+1)^2$ is not a perfect square.
2007 IberoAmerican Olympiad For University Students, 1
For each pair of integers $(i,k)$ such that $1\le i\le k$, the linear transformation $P_{i,k}:\mathbb{R}^k\to\mathbb{R}^k$ is defined as:
$P_{i,k}(a_1,\cdots,a_{i-1},a_i,a_{i+1},\cdots,a_k)=(a_1,\cdots,a_{i-1},0,a_{i+1},\cdots,a_k)$
Prove that for all $n\ge2$ and for every set of $n-1$ linearly independent vectors $v_1,\cdots,v_{n-1}$ in $\mathbb{R}^n$, there is an integer $k$ such that $1\le k\le n$ and such that the vectors $P_{k,n}(v_1),\cdots,P_{k,n}(v_{n-1})$ are linearly independent.
1983 IMO Longlists, 38
Let $\{u_n \}$ be the sequence defined by its first two terms $u_0, u_1$ and the recursion formula
\[u_{n+2 }= u_n - u_{n+1}.\]
[b](a)[/b] Show that $u_n$ can be written in the form $u_n = \alpha a^n + \beta b^n$, where $a, b, \alpha, \beta$ are constants independent of $n$ that have to be determined.
[b](b)[/b] If $S_n = u_0 + u_1 + \cdots + u_n$, prove that $S_n + u_{n-1}$ is a constant independent of $n.$ Determine this constant.
2013 Tournament of Towns, 4
Integers $1, 2,...,100$ are written on a circle, not necessarily in that order. Can it be that the absolute value of the di erence between any two adjacent integers is at least $30$ and at most $50$?
2018 Purple Comet Problems, 11
Find the number of positive integers less than $2018$ that are divisible by $6$ but are not divisible by at least one of the numbers $4$ or $9$.
2022 Princeton University Math Competition, 10
Let $\alpha, \beta, \gamma \in C$ be the roots of the polynomial $x^3 - 3x2 + 3x + 7$. For any complex number $z$, let $f(z)$ be defined as follows:
$$f(z) = |z -\alpha | + |z - \beta|+ |z-\gamma | - 2 \underbrace{\max}_{w \in \{\alpha, \beta, \gamma}\} |z - w|.$$
Let $A$ be the area of the region bounded by the locus of all $z \in C$ at which $f(z)$ attains its global minimum. Find $\lfloor A \rfloor$.
2016 Israel Team Selection Test, 4
A regular 60-gon is given. What is the maximum size of a subset of its vertices containing no isosceles triangles?
1977 IMO Longlists, 51
Several segments, which we shall call white, are given, and the sum of their lengths is $1$. Several other segments, which we shall call black, are given, and the sum of their lengths is $1$. Prove that every such system of segments can be distributed on the segment that is $1.51$ long in the following way: Segments of the same colour are disjoint, and segments of different colours are either disjoint or one is inside the other. Prove that there exists a system that cannot be distributed in that way on the segment that is $1.49$ long.
2004 Moldova Team Selection Test, 11
Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$.
[i]Proposed by Hojoo Lee[/i]
2006 Singapore Team Selection Test, 2
Let n be an integer greater than 1 and let $x_1, x_2, . . . , x_n$ be real numbers such that
$|x_1| + |x_2| + ... + |x_n| = 1$ and $x_1 + x_2 + ... + x_n = 0$
Prove that
$\left| \frac{x_1}{1}+\frac{x_2}{2}+\cdots+\frac{x_n}{n} \right| \leq \frac{1}{2} \left(1-\frac{1}{n}\right)$
2023 Math Prize for Girls Problems, 5
Acute triangle $ABC$ has area 870. The triangle whose vertices are the feet of the altitudes of $\triangle ABC$ has area 48. Determine
\[
\sin^2 A + \sin^2 B + \sin^2 C .
\]
2016 CMIMC, 5
Let $\mathcal{S}$ be a regular 18-gon, and for two vertices in $\mathcal{S}$ define the $\textit{distance}$ between them to be the length of the shortest path along the edges of $\mathcal{S}$ between them (e.g. adjacent vertices have distance 1). Find the number of ways to choose three distinct vertices from $\mathcal{S}$ such that no two of them have distance 1, 8, or 9.
1979 Canada National Olympiad, 1
Given: (i) $a$, $b > 0$; (ii) $a$, $A_1$, $A_2$, $b$ is an arithmetic progression; (iii) $a$, $G_1$, $G_2$, $b$ is a geometric progression. Show that
\[A_1 A_2 \ge G_1 G_2.\]
2021 Malaysia IMONST 2, 2
The five numbers $a, b, c, d,$ and $e$ satisfy the inequalities
$$a+b < c+d < e+a < b+c < d+e.$$
Among the five numbers, which one is the smallest, and which one is the largest?
2019 Brazil Team Selection Test, 1
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
2015 Indonesia MO Shortlist, A5
Let $a,b,c$ be positive real numbers. Prove that
$\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$
2011 IberoAmerican, 3
Let $k$ and $n$ be positive integers, with $k \geq 2$. In a straight line there are $kn$ stones of $k$ colours, such that there are $n$ stones of each colour. A [i]step[/i] consists of exchanging the position of two adjacent stones. Find the smallest positive integer $m$ such that it is always possible to achieve, with at most $m$ steps, that the $n$ stones are together, if:
a) $n$ is even.
b) $n$ is odd and $k=3$
2013 Stanford Mathematics Tournament, 5
An unfair coin lands heads with probability $\tfrac1{17}$ and tails with probability $\tfrac{16}{17}$. Matt flips the coin repeatedly until he flips at least one head and at least one tail. What is the expected number of times that Matt flips the coin?
2023 Belarus - Iran Friendly Competition, 5
Define $M_n = \{ 1, 2, \ldots , n \} $ for all positive integers $n$. A collection of $3$-element subsets
of $M_n$ is said to be fine if for any colouring of elements of $M_n$ in two colours there is a subset of the
collection all three elements of which are of the same colour. For each $n \geq 5$ find the minimal
possible number of the $3$-element subsets of a fine collection