Found problems: 85335
2020 Mediterranean Mathematics Olympiad, 2
Let $S$ be a set of $n\ge2$ positive integers. Prove that there exist at least $n^2$ integers that can be written in the form $x+yz$ with $x,y,z\in S$.
[i]Proposed by Gerhard Woeginger, Austria[/i]
2002 Stanford Mathematics Tournament, 5
Solve for $a, b, c$ given that $a \le b \le c$, and
$a+b+c=-1$
$ab+bc+ac=-4$
$abc=-2$
1987 IMO Longlists, 66
At a party attended by $n$ married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques $C_1, C_2, \cdots, C_k$ with the following property: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to which both members belong. Prove that if $n \geq 4$, then $k \geq 2n$.
[i]Proposed by USA.[/i]
2007 IberoAmerican Olympiad For University Students, 2
Prove that for all positive integers $n$ and for all real numbers $x$ such that $0\le x\le1$, the following inequality holds:
$\left(1-x+\frac{x^2}{2}\right)^n-(1-x)^n\le\frac{x}{2}$.
2013 Gulf Math Olympiad, 4
Let $m,n$ be integers. It is known that there are integers $a,b$ such that $am+bn=1$ if, and only if, the greatest common divisor of $m,n$ is 1. [i]You are not required to prove this[/i].
Now suppose that $p,q$ are different odd primes. In each case determine if there are integers $a,b$ such that $ap+bq=1$ so that the given condition is satisfied:
[list]
a. $p$ divides $b$ and $q$ divides $a$;
b. $p$ divides $a$ and $q$ divides $b$;
c. $p$ does not divide $a$ and $q$ does not divide $b$.
[/list]
2011 IMO Shortlist, 6
Let $P(x)$ and $Q(x)$ be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both $P(x)$ and $Q(x).$ Suppose that for every positive integer $n$ the integers $P(n)$ and $Q(n)$ are positive, and $2^{Q(n)}-1$ divides $3^{P(n)}-1.$ Prove that $Q(x)$ is a constant polynomial.
[i]Proposed by Oleksiy Klurman, Ukraine[/i]
2018 Regional Olympiad of Mexico West, 3
A scalene acute triangle $ABC$ is drawn on the plane, in which $BC$ is the longest side. Points $P$ and $D$ are constructed, the first inside $ABC$ and the second outside, so that $\angle ABC = \angle CBD$, $\angle ACP = \angle BCD$ and that the area of triangle $ABC$ is equal to the area of quadrilateral $BPCD$. Prove that triangles $BCD$ and $ACP$ are similar.
2011 Tokyo Instutute Of Technology Entrance Examination, 1
Let $f_n\ (n=1,\ 2,\ \cdots)$ be a linear transformation expressed by a matrix $\left(
\begin{array}{cc}
1-n & 1 \\
-n(n+1) & n+2
\end{array}
\right)$ on the $xy$ plane. Answer the following questions:
(1) Prove that there exists 2 lines passing through the origin $O(0,\ 0)$ such that all points of the lines are mapped to the same lines, then find the equation of the lines.
(2) Find the area $S_n$ of the figure enclosed by the lines obtained in (1) and the curve $y=x^2$.
(3) Find $\sum_{n=1}^{\infty} \frac{1}{S_n-\frac 16}.$
[i]2011 Tokyo Institute of Technlogy entrance exam, Problem 1[/i]
PEN A Problems, 21
Let n be a positive integer. Show that the product of $ n$ consecutive positive integers is divisible by $ n!$
2022 Latvia Baltic Way TST, P16
Find all triples of positive integers $(a,b,p)$, where $p$ is a prime, such that both $a+b$ and $ab+1$ are some powers of $p$ (not necessarily the same).
2010 Junior Balkan Team Selection Tests - Romania, 1
We consider on a circle a finite number of real numbers with the sum strictly greater than $0$. Of all the sums that have as terms numbers on consecutive positions on the circle, let $S$ be the largest sum and $s$ the smallest sum. Show that $S + s> 0$.
2020 South Africa National Olympiad, 1
Find the smallest positive multiple of $20$ with exactly $20$ positive divisors.
Swiss NMO - geometry, 2011.2
Let $\triangle{ABC}$ be an acute-angled triangle and let $D$, $E$, $F$ be points on $BC$, $CA$, $AB$, respectively, such that \[\angle{AFE}=\angle{BFD}\mbox{,}\quad\angle{BDF}=\angle{CDE}\quad\mbox{and}\quad\angle{CED}=\angle{AEF}\mbox{.}\] Prove that $D$, $E$ and $F$ are the feet of the perpendiculars through $A$, $B$ and $C$ on $BC$, $CA$ and $AB$, respectively.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 2)[/i]
2005 Harvard-MIT Mathematics Tournament, 10
Find the sum of the absolute values of the roots of $x^4 - 4x^3 - 4x^2 + 16x - 8 = 0$.
2008 China Western Mathematical Olympiad, 2
In triangle $ ABC$, $ AB\equal{}AC$, the inscribed circle $ I$ touches $ BC, CA, AB$ at points $ D,E$ and $ F$ respectively. $ P$ is a point on arc $ EF$ opposite $ D$. Line $ BP$ intersects circle $ I$ at another point $ Q$, lines $ EP$, $ EQ$ meet line $ BC$ at $ M, N$ respectively. Prove that
(1) $ P, F, B, M$ concyclic
(2)$ \frac{EM}{EN} \equal{} \frac{BD}{BP}$
(P.S. Can anyone help me with using GeoGebra, the incircle function of the plugin doesn't work with my computer.)
2021 Harvard-MIT Mathematics Tournament., 7
In triangle $ABC$, let $M$ be the midpoint of $BC$ and $D$ be a point on segment $AM$. Distinct points $Y$ and $Z$ are chosen on rays $\overrightarrow{CA}$ and $\overrightarrow{BA}$ , respectively, such that $\angle DYC=\angle DCB$ and $\angle DBC=\angle DZB$. Prove that the circumcircle of $\Delta DYZ$ is tangent to the circumcircle of $\Delta DBC$.
2005 Miklós Schweitzer, 1
Let [n] be the set {1, 2,. . . , n}.
For any $a, b \in N$, denote $G (a, b)$ by a graph (not directed) defined by the following rule: the vertices have the form (i, f), where $i \in [a]$, and $f: [a] \to [b]$. A vertex (i, f) and a vertex (j, g) are connected if $i \neq j$, and $f (k) \neq g (k)$ holds exactly for k strictly between i and j. Prove that for any $c \in N$ there is $a, b \in N$ such that the vertices of G (a, b) cannot be well-colored with $c$ colors.
2017 Vietnam National Olympiad, 3
Given an acute, non isoceles triangle $ABC$ and $(O)$ be its circumcircle, $H$ its orthocenter and $E, F$ are the feet of the altitudes from $B$ and $C$, respectively. $AH$ intersects $(O)$ at $D$ ($D\ne A$).
a) Let $I$ be the midpoint of $AH$, $EI$ meets $BD$ at $M$ and $FI$ meets $CD$ at $N$. Prove that $MN\perp OH$.
b) The lines $DE$, $DF$ intersect $(O)$ at $P,Q$ respectively ($P\ne D,Q\ne D$). $(AEF)$ meets $(O)$ and $AO$ at $R,S$ respectively ($R\ne A, S\ne A$). Prove that $BP,CQ,RS$ are concurrent.
2013 Online Math Open Problems, 43
In a tennis tournament, each competitor plays against every other competitor, and there are no draws. Call a group of four tennis players ``ordered'' if there is a clear winner and a clear loser (i.e., one person who beat the other three, and one person who lost to the other three.) Find the smallest integer $n$ for which any tennis tournament with $n$ people has a group of four tennis players that is ordered.
[i]Ray Li[/i]
2014 Bulgaria JBMO TST, 6
If $a,b$ are real numbers such that $a^3 +12a^2 + 49a + 69 = 0$ and $ b^3 - 9b^2 + 28b - 31 = 0,$ find $a + b .$
1995 Mexico National Olympiad, 6
A $1$ or $0$ is placed on each square of a $4 \times 4$ board. One is allowed to change each symbol in a row, or change each symbol in a column, or change each symbol in a diagonal (there are $14$ diagonals of lengths $1$ to $4$). For which arrangements can one make changes which end up with all $0$s?
1987 Romania Team Selection Test, 8
Let $ABCD$ be a square and $a$ be the length of his edges. The segments $AE$ and $CF$ are perpendicular on the square's plane in the same half-space and they have the length $AE=a$, $CF=b$ where $a<b<a\sqrt 3$. If $K$ denoted the set of the interior points of the square $ABCD$ determine $\min_{M\in K} \left( \max ( EM, FM ) \right) $ and $\max_{M\in K} \left( \min (EM,FM) \right)$.
[i]Octavian Stanasila[/i]
2018 Belarusian National Olympiad, 11.3
For all pairs $(m, n)$ of positive integers that have the same number $k$ of divisors we define the operation $\circ$. Write all their divisors in an ascending order: $1=m_1<\ldots<m_k=m$, $1=n_1<\ldots<n_k=n$ and set
$$
m\circ n= m_1\cdot n_1+\ldots+m_k\cdot n_k.
$$
Find all pairs of numbers $(m, n)$, $m\geqslant n$, such that $m\circ n=497$.
2005 District Olympiad, 2
Find the functions $f:\mathbb{Z}\times \mathbb{Z}\to\mathbb{R}$ such that
a) $f(x,y)\cdot f(y,z) \cdot f(z,x) = 1$ for all integers $x,y,z$;
b) $f(x+1,x)=2$ for all integers $x$.
1998 India Regional Mathematical Olympiad, 6
Given the $7$-element set $A = \{ a ,b,c,d,e,f,g \}$, find a collection $T$ of $3$-element subsets of $A$ such that each pair of elements from $A$ occurs exactly once on one of the subsets of $T$.