Found problems: 85335
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 $p+q ?$
$\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$
2022 MIG, 19
Cozi makes a two-way table on chalkboard describing the right or left hand usage of students and teachers in her school. However, when she returns to the chalkboard from lunch, she is dismayed to find that most of the numbers on her table have been erased, leaving behind:
\begin{tabular}{c c c c}
5 & ? & ? & Total \\
? & ? & 6 & Total \\
? & 11 & ? & \\
Total & Total & & \\
\end{tabular}
Fortunately, Cozi remembers that the difference between two of the missing numbers is equal to $12.$ Which of the following could be the total number of students and teachers on the table?
$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
2022 Austrian MO Beginners' Competition, 3
A semicircle is erected over the segment $AB$ with center $M$. Let $P$ be one point different from $A$ and $B$ on the semicircle and $Q$ the midpoint of the arc of the circle $AP$. The point of intersection of the straight line $BP$ with the parallel to $P Q$ through $M$ is $S$. Prove that $PM = PS$ holds.
[i](Karl Czakler)[/i]
2011 ISI B.Math Entrance Exam, 5
Consider a sequence denoted by $F_n$ of non-square numbers . $F_1=2$,$F_2=3$,$F_3=5$ and so on . Now , if $m^2\leq F_n<(m+1)^2$ . Then prove that $m$ is the integer closest to $\sqrt{n}$.
2007 Bundeswettbewerb Mathematik, 1
For which numbers $ n$ is there a positive integer $ k$ with the following property: The sum of digits for $ k$ is $ n$ and the number $ k^2$ has sum of digits $ n^2.$
2022 China Girls Math Olympiad, 3
In triangle $ABC,AB>AC,I$ is the incenter, $AM$ is the midline. The line crosses $I$ and is perpendicular to $BC $ intersect $AM$ at point $L$, and the symmetry of $I$ with respect to point $A$ is $J$
Prove that: $\angle ABJ= \angle LBI$.
2024 Baltic Way, 2
Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that
\[
\frac{f(a)}{1+a+ca}+\frac{f(b)}{1+b+ab}+\frac{f(c)}{1+c+bc} = 1
\]
for all $a,b,c \in \mathbb{R}^+$ that satisfy $abc=1$.
2000 VJIMC, Problem 2
Let $f:\mathbb N\to\mathbb R$ be given by
$$f(n)=n^{\frac12\tau(n)}$$for $n\in\mathbb N=\{1,2,\ldots\}$ where $\tau(n)$ is the number of divisors of $n$. Show that $f$ is an injection.
2014 NIMO Problems, 1
Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$.
[i]Proposed by Evan Chen[/i]
2018 Sharygin Geometry Olympiad, 22
Six circles of unit radius lie in the plane so that the distance between the centers of any two of them is greater than $d$. What is the least value of $d$ such that there always exists a straight line which does not intersect any of the circles and separates the circles into two groups of three?
2017 ELMO Shortlist, 1
Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$
[i]Proposed by Daniel Liu[/i]
2008 IMO Shortlist, 3
Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic.
[i]Proposed by John Cuya, Peru[/i]
2015 Mathematical Talent Reward Programme, SAQ: P 4
Find all real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying,$$\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\cdots+n \sqrt{x_{n}-n^{2}}=\frac{1}{2}\left(x_{1}+x_{2}+\cdots+x_{n}\right)$$
1995 Miklós Schweitzer, 9
A serpentine is a sequence of points $P_1 , ..., P_m$ in a plane, not necessarily all different, such that the distance between $P_i$ and $P_{i+1}$ is at least 1, and the segments $P_i P_{i +1}$ are alternately horizontal and vertical. Construct a compact set in which there is a sequence of serpentines with arbitrary long lengths but there is no closed serpentine ($P_m = P_i$ for some i < m).
2016 Regional Olympiad of Mexico Southeast, 4
The diagonals of a convex quadrilateral $ABCD$ intersect in $E$. Let $S_1, S_2, S_3$ and $S_4$ the areas of the triangles $AEB, BEC, CED, DEA$ respectively. Prove that, if exists real numbers $w, x, y$ and $z$ such that
$$S_1=x+y+xy, S_2=y+z+yz, S_3=w+z+wz, S_4=w+x+wx,$$
then $E$ is the midpoint of $AC$ or $E$ is the midpoint of $BD$.
Kvant 2023, M2744
A regular $100$-gon was cut into several parallelograms and two triangles. Prove that these triangles are congruent.
2020 Moldova Team Selection Test, 1
All members of geometrical progression $(b_n)_{n\geq1}$ are members of some arithmetical progression. It is known that $b_1$ is an integer. Prove that all members of this geometrical progression are integers. (progression is infinite)
2025 CMIMC Algebra/NT, 9
Find the largest prime factor of $45^5-1.$
2010 Baltic Way, 1
Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations
\[\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}\]
1964 IMO, 4
Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.
2011 Pre-Preparation Course Examination, 4
A star $K_{1,3}$ is called a paw. suppose that $G$ is a graph without any induced paws. prove that $\chi(G)\le(\omega(G))^2$. (15 points)
2012 AIME Problems, 14
In a group of nine people each person shakes hands with exactly two of the other people from the group. Let N be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when N is divided by 1000.
2017 All-Russian Olympiad, 1
In country some cities are connected by oneway flights( There are no more then one flight between two cities). City $A$ called "available" for city $B$, if there is flight from $B$ to $A$, maybe with some transfers. It is known, that for every 2 cities $P$ and $Q$ exist city $R$, such that $P$ and $Q$ are available from $R$. Prove, that exist city $A$, such that every city is available for $A$.
2011 AIME Problems, 12
Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.