Found problems: 85335
2015 Online Math Open Problems, 11
Let $S$ be a set. We say $S$ is $D^\ast$[i]-finite[/i] if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$, there exists a $y \in Y$ such that $f(y) \notin Y$. The function $f$ is called a [i]witness[/i] of $S$. How many witnesses does $\{0,1,\cdots,5\}$ have?
[i]Proposed by Evan Chen[/i]
2007 Putnam, 3
Let $ k$ be a positive integer. Suppose that the integers $ 1,2,3,\dots,3k \plus{} 1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $ 3$ ? Your answer should be in closed form, but may include factorials.
1999 VJIMC, Problem 4
Let $u_1,u_2,\ldots,u_n\in C([0,1]^n)$ be nonnegative and continuous functions, and let $u_j$ do not depend on the $j$-th variable for $j=1,\ldots,n$. Show that
$$\left(\int_{[0,1]^n}\prod_{j=1}^nu_j\right)^{n-1}\le\prod_{j=1}^n\int_{[0,1]^n}u_j^{n-1}.$$
2010 Tournament Of Towns, 3
Is it possible to cover the surface of a regular octahedron by several regular hexagons without gaps and overlaps? (A regular octahedron has $6$ vertices, each face is an equilateral triangle, each vertex belongs to $4$ faces.)
2017 Purple Comet Problems, 9
The diagram below shows $\vartriangle ABC$ with point $D$ on side $\overline{BC}$. Three lines parallel to side $\overline{BC}$ divide segment $\overline{AD}$ into four equal segments. In the triangle, the ratio of the area of the shaded region to the area of the unshaded region is $\frac{49}{33}$ and $\frac{BD}{CD} = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[img]https://cdn.artofproblemsolving.com/attachments/2/8/77239633b68f073f4193aa75cdfb9238461cae.png[/img]
2024 Azerbaijan IMO TST, 2
Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.
Prove that lines $AD, PM$, and $BC$ are concurrent.
2009 Czech-Polish-Slovak Match, 1
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^+\to\mathbb{R}^+$ that satisfy \[ \Big(1+yf(x)\Big)\Big(1-yf(x+y)\Big)=1\] for all $x,y\in\mathbb{R}^+$.
2023-24 IOQM India, 27
A quadruple $(a,b,c,d)$ of distinct integers is said to be $balanced$ if $a+c=b+d$. Let $\mathcal{S}$ be any set of quadruples $(a,b,c,d)$ where $1 \leqslant a<b<d<c \leqslant 20$ and where the cardinality of $\mathcal{S}$ is $4411$. Find the least number of balanced quadruples in $\mathcal{S}.$
1999 Yugoslav Team Selection Test, Problem 4
For a natural number $d$, $M_d$ denotes the set of natural numbers which are not representable as the sum of at least two consecutive terms of an arithmetic progression with the common difference d whose terms are integers. Prove that each $c\in M_3$ can be written in the form $c=ab$, where $a\in M_1$ and $b\in M_2\setminus\{2\}$.
2007 Middle European Mathematical Olympiad, 2
For a set $ P$ of five points in the plane, no three of them being collinear, let $ s(P)$ be the numbers of acute triangles formed by vertices in $ P$.
Find the maximum value of $ s(P)$ over all such sets $ P$.
2012 Bosnia and Herzegovina Junior BMO TST, 3
Internal angles of triangle are $(5x+3y)^{\circ}$, $(3x+20)^{\circ}$ and $(10y+30)^{\circ}$ where $x$ and $y$ are positive integers. Which values can $x+y$ get ?
I Soros Olympiad 1994-95 (Rus + Ukr), 10.1
The function $f: Z \to Z$ satisfies the following conditions:
1) $f(f(n))=n$ for all integers $n$
2) $f(f(n+2)+2) = n$ for all integers $n$
3) $f(0)=1$.
Find the value of $f(1995)$ and $f(-1994)$.
1941 Moscow Mathematical Olympiad, 075
Prove that $1$ plus the product of any four consecutive integers is a perfect square.
2009 Jozsef Wildt International Math Competition, W. 25
Let $ABCD$ be a quadrilateral in which $\widehat{A}=\widehat{C}=90^{\circ}$. Prove that $$\frac{1}{BD}(AB+BC+CD+DA)+BD^2\left (\frac{1}{AB\cdot AD}+\frac{1}{CB\cdot CD}\right )\geq 2\left (2+\sqrt{2}\right )$$
2007 Mexico National Olympiad, 1
The fraction $\frac1{10}$ can be expressed as the sum of two unit fraction in many ways, for example, $\frac1{30}+\frac1{15}$ and $\frac1{60}+\frac1{12}$.
Find the number of ways that $\frac1{2007}$ can be expressed as the sum of two distinct positive unit fractions.
1998 Irish Math Olympiad, 2
Prove that if $ a,b,c$ are positive real numbers, then:
$ \frac{9}{a\plus{}b\plus{}c} \le 2 \left( \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \right) \le \frac{1}{a}\plus{}\frac{1}{b}\plus{}\frac{1}{c}.$
2005 Cono Sur Olympiad, 3
On the cartesian plane we draw circunferences of radii 1/20 centred in each lattice point. Show that any circunference of radii 100 in the cartesian plane intersect at least one of the small circunferences.
2002 All-Russian Olympiad Regional Round, 11.5
Let $P(x)$ be a polynomial of odd degree. Prove that the equation $P(P(x)) = 0$ has at least as many different real roots as the equation $P(x) = 0$
[hide=original wording]Пусть P(x) — многочлен нечетной степени. Докажите, что уравнение P(P(x)) = 0 имеет не меньше различных действительных корней, чем уравнение P(x) = 0[/hide]
2018 China Team Selection Test, 4
Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.
2014 BMT Spring, 1
Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?
2001 Denmark MO - Mohr Contest, 4
Show that any number of the form
$$4444 ...44 88...8$$
where there are twice as many $4$s as $8$s is a square number.
1994 Turkey MO (2nd round), 2
Let $ABCD$ be a cyclic quadrilateral $\angle{BAD}< 90^\circ$ and $\angle BCA = \angle DCA$. Point $E$ is taken on segment $DA$ such that $BD=2DE$. The line through $E$ parallel to $CD$ intersects the diagonal $AC$ at $F$. Prove that \[ \frac{AC\cdot BD}{AB\cdot FC}=2.\]
2019 IMO, 2
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.
Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.
[i]Proposed by Anton Trygub, Ukraine[/i]
2023 Romania National Olympiad, 3
Let $n$ be a natural number $n \geq 2$ and matrices $A,B \in M_{n}(\mathbb{C}),$ with property $A^2 B = A.$
a) Prove that $(AB - BA)^2 = O_{n}.$
b) Show that for all natural number $k$, $k \leq \frac{n}{2}$ there exist matrices $A,B \in M_{n}(\mathbb{C})$ with property stated in the problem such that $rank(AB - BA) = k.$
2019 USAMTS Problems, 3
Circle $\omega$ is inscribed in unit square $PLUM$, and points $I$ and $E$ lie on $\omega$ such that $U,I,$ and $E$ are collinear. Find, with proof, the greatest possible area for $\triangle PIE$.