Found problems: 85335
2019 Balkan MO, 1
Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that:
$$f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q$$
holds for all $p,q\in\mathbb{P}$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
1897 Eotvos Mathematical Competition, 3
Let $ABCD$ be a rectangle and let $M, N$ and $P, Q$ be the points of intersections of some line $e$ with the sides $AB, CD$ and $AD, BC$, respectively (or their extensions). Given the points $M, N, P, Q$ and the length $p$ of side $AB$, construct the rectangle. Under what conditions can this problem be solved, and how many solutions does it have?
1961 Leningrad Math Olympiad, grade 7
[b]7.1. / 6.5[/b] Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers.
[b]7.2[/b] Given a circle $O$ and a square $K$, as well as a line $L$. Construct a segment of given length parallel to $L$ and such that its ends lie on $O$ and $K$ respectively
[b]7.3[/b] The three-digit number $\overline{abc}$ is divisible by $37$. Prove that the sum of the numbers $\overline{bca}$ and $\overline{cab}$ is also divisible by $37$.[b] (typo corrected)[/b]
[b]7.4.[/b] Point $C$ is the midpoint of segment $AB$. On an arbitrary ray drawn from point $C$ and not lying on line $AB$, three consecutive points $P$, $M$ and $Q$ so that $PM=MQ$. Prove that $AP+BQ>2CM$.
[img]https://cdn.artofproblemsolving.com/attachments/f/3/a8031007f5afc31a8b5cef98dd025474ac0351.png[/img]
[b]7.5.[/b] Given $2n+1$ different objects. Prove that you can choose an odd number of objects from them in as many ways as an even number.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].
2005 AMC 10, 22
For how many positive integers $ n$ less than or equal to $ 24$ is $ n!$ evenly divisible by $ 1 \plus{} 2 \plus{} \dots \plus{} n$?
$ \textbf{(A)}\ 8\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ 17\qquad
\textbf{(E)}\ 21$
2019 Turkey EGMO TST, 6
There are $k$ piles and there are $2019$ stones totally. In every move we split a pile into two or remove one pile. Using finite moves we can reach conclusion that there are $k$ piles left and all of them contain different number of stonws. Find the maximum of $k$.
1964 All Russian Mathematical Olympiad, 045
a) Given a convex hexagon $ABCDEF$ with all the equal angles. Prove that $$|AB|-|DE| = |EF|-|BC| = |CD|-|FA|$$
b) The opposite problem:
Prove that it is possible to construct a convex hexagon with equal angles of six segments $a_1,a_2,...,a_6$, whose lengths satisfy the condition $$a_1-a_4 = a_5-a_2 = a_3-a_6$$
2018 BMT Spring, 1
A circle with radius $5$ is inscribed in a right triangle with hypotenuse $34$ as shown below. What is the area of the triangle? Note that the diagram is not to scale.
2023-24 IOQM India, 10
The Sequence $\{a_{n}\}_{n \geqslant 0}$ is defined by $a_{0}=1, a_{1}=-4$ and $a_{n+2}=-4a_{n+1}-7a_{n}$ , for $n \geqslant 0$. Find the number of positive integer divisors of $a^2_{50}-a_{49}a_{51}$.
1948 Putnam, A5
If $\xi_1,\ldots,\xi_n$ denote the $n$-th roots of unity, evaluate
$$\prod_{1\leq i<j \leq n} (\xi_{i}-\xi_j )^2 .$$
2012 Brazil Team Selection Test, 4
Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle.
[i]Proposed by Canada[/i]
Math Hour Olympiad, Grades 5-7, 2013.67
[u]Round 1[/u]
[b]p1.[/b] Goldilocks enters the home of the three bears – Papa Bear, Mama Bear, and Baby Bear. Each bear is wearing a different-colored shirt – red, green, or blue. All the bears look the same to Goldilocks, so she cannot otherwise tell them apart.
The bears in the red and blue shirts each make one true statement and one false statement.
The bear in the red shirt says: “I'm Blue's dad. I'm Green's daughter.”
The bear in the blue shirt says: “Red and Green are of opposite gender. Red and Green are my parents.”
Help Goldilocks find out which bear is wearing which shirt.
[b]p2.[/b] The University of Washington is holding a talent competition. The competition has five contests: math, physics, chemistry, biology, and ballroom dancing. Any student can enter into any number of the contests but only once for each one. For example, a student may participate in math, biology, and ballroom.
It turned out that each student participated in an odd number of contests. Also, each contest had an odd number of participants. Was the total number of contestants odd or even?
[b]p3.[/b] The $99$ greatest scientists of Mars and Venus are seated evenly around a circular table. If any scientist sees two colleagues from her own planet sitting an equal number of seats to her left and right, she waves to them. For example, if you are from Mars and the scientists sitting two seats to your left and right are also from Mars, you will wave to them. Prove that at least one of the $99$ scientists will be waving, no matter how they are seated around the table.
[b]p4.[/b] One hundred boys participated in a tennis tournament in which every player played each other player exactly once and there were no ties. Prove that after the tournament, it is possible for the boys to line up for pizza so that each boy defeated the boy standing right behind him in line.
[b]p5.[/b] To celebrate space exploration, the Science Fiction Museum is going to read Star Wars and Star Trek stories for $24$ hours straight. A different story will be read each hour for a total of $12$ Star Wars stories and $12$ Star Trek stories. George and Gene want to listen to exactly $6$ Star Wars and $6$ Star Trek stories. Show that no matter how the readings are scheduled, the friends can find a block of $12$ consecutive hours to listen to the stories together.
[u]Round 2[/u]
[b]p6.[/b] $2013$ people attended Cinderella's ball. Some of the guests were friends with each other. At midnight, the guests started turning into mice. After the first minute, everyone who had no friends at the ball turned into a mouse. After the second minute, everyone who had exactly one friend among the remaining people turned into a mouse. After the third minute, everyone who had two human friends left in the room turned into a mouse, and so on. What is the maximal number of people that could have been left at the ball after $2013$ minutes?
[b]p7.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2018 Romanian Master of Mathematics Shortlist, G1
Let $ABC$ be a triangle and let $H$ be the orthogonal projection of $A$ on the line $BC$. Let $K$ be a point on the segment $AH$ such that $AH = 3 KH$. Let $O$ be the circumcenter of triangle $ABC$ and let $M$ and $N$ be the midpoints of sides $AC$ and $AB$ respectively. The lines $KO$ and $MN$ meet at a point $Z$ and the perpendicular at $Z$ to $OK$ meets lines $AB, AC$ at $X$ and $Y$ respectively. Show that $\angle XKY = \angle CKB$.
[i]Italy[/i]
1982 IMO Longlists, 50
Let $O$ be the midpoint of the axis of a right circular cylinder. Let $A$ and $B$ be diametrically opposite points of one base, and $C$ a point of the other base circle that does not belong to the plane $OAB$. Prove that the sum of dihedral angles of the trihedral $OABC$ is equal to $2\pi$.
2022 Turkey EGMO TST, 2
We are given some three element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ [i]nice [/i] if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more element to every 29-element [i]nice [/i] subset while keeping it nice, find the minimum value of $n$.
2020 LMT Fall, A20
Let $ABCD$ be a cyclic quadrilateral with center $O$ with $AB > CD$ and $BC > AD$. Let $M$ and $N$ be the midpoint of sides $AD$ and $BC$, respectively, and let $X$ and $Y$ be on $AB$ and $CD$, respectively, such that $AX \cdot CY = BX \cdot DY = 20000$, and $AX \le CY$. Let lines $AD$ and $BC$ hit at $P$, and let lines $AB$ and $CD$ hit at $Q$. The circumcircles of $\triangle MNP$ and $\triangle XYQ$ hit at a point $R$ that is on the opposite side of $CD$ as $O$. Let $R_1$ be the midpoint of $PQ$ and $B$, $D$, and $R$ be collinear. Let $O_1$ be the circumcenter of $\triangle BPQ$. Let the lines $BO_1$ and $DR_1$ intersect at a point $I$. If $BP \cdot BQ = 823875$, $AB=429$, and $BC=495$, then $IR=\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of a prime, and $\gcd(a,c) = 1$. Find the value of $a+b+c$.
[i]Proposed by Kevin Zhao[/i]
2011 Rioplatense Mathematical Olympiad, Level 3, 6
Let $d(n)$ be the sum of positive integers divisors of number $n$ and $\phi(n)$ the quantity of integers in the interval $[0,n]$ such that these integers are coprime with $n$. For instance $d(6)=12$ and $\phi(7)=6$.
Determine if the set of the integers $n$ such that, $d(n)\cdot \phi (n)$ is a perfect square, is finite or infinite set.
2020 USA EGMO Team Selection Test, 2
Let $ABC$ be a triangle and let $P$ be a point not lying on any of the three lines $AB$, $BC$, or $CA$. Distinct points $D$, $E$, and $F$ lie on lines $BC$, $AC$, and $AB$, respectively, such that $\overline{DE}\parallel \overline{CP}$ and $\overline{DF}\parallel \overline{BP}$. Show that there exists a point $Q$ on the circumcircle of $\triangle AEF$ such that $\triangle BAQ$ is similar to $\triangle PAC$.
[i]Andrew Gu[/i]
2019 AMC 8, 22
A store increased the original price of a shirt by a certain percent and then decreased the new price by the same percent. Given that the resulting price was 84% of the original price, by what percent was the price increased and decreased?
$\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40$
1988 Balkan MO, 1
Let $ABC$ be a triangle and let $M,N,P$ be points on the line $BC$ such that $AM,AN,AP$ are the altitude, the angle bisector and the median of the triangle, respectively. It is known that
$\frac{[AMP]}{[ABC]}=\frac{1}{4}$ and $\frac{[ANP]}{[ABC]}=1-\frac{\sqrt{3}}{2}$.
Find the angles of triangle $ABC$.
2015 Vietnam Team selection test, Problem 6
Find the smallest positive interger number $n$ such that there exists $n$ real numbers $a_1,a_2,\ldots,a_n$ satisfied three conditions as follow:
a. $a_1+a_2+\cdots+a_n>0$;
b. $a_1^3+a_2^3+\cdots+a_n^3<0$;
c. $a_1^5+a_2^5+\cdots+a_n^5>0$.
1995 Chile National Olympiad, 2
In a circle of radius $1$, six arcs of radius $1$ are drawn, which cut the circle as in the figure. Determine the black area.
[img]https://cdn.artofproblemsolving.com/attachments/8/9/0323935be8406ea0c452b3c8417a8148c977e3.jpg[/img]
2018 All-Russian Olympiad, 7
In a card game, each card is associated with a numerical value from 1 to 100, with each card beating less, with one exception: 1 beats 100. The player knows that 100 cards with different values lie in front of him. The dealer who knows the order of these cards can tell the player which card beats the other for any pair of cards he draws. Prove that the dealer can make one hundred such messages, so that after that the player can accurately determine the value of each card.
2005 International Zhautykov Olympiad, 1
For the positive real numbers $ a,b,c$ prove the inequality
\[ \frac {c}{a \plus{} 2b} \plus{} \frac {a}{b \plus{} 2c} \plus{} \frac {b}{c \plus{} 2a}\ge1.
\]
2023 Assara - South Russian Girl's MO, 6
Aunt Raya has $14$ wheels of cheese. She found out that out of any $6$ wheels, she could choose $4$ and put them on the scales so that the scales came into balance. Aunt Raya wants to give Daud Kazbekovich two of these $14$ wheels , and divide the rest equally (by weight) between Pavel and Kirill. Prove that she can make her wish come true.
2024 AMC 10, 23
The Fibonacci numbers are defined by $F_1=1,$ $F_2=1,$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3.$ What is $$\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?$$
$\textbf{(A) }318 \qquad\textbf{(B) }319\qquad\textbf{(C) }320\qquad\textbf{(D) }321\qquad\textbf{(E) }322$