Found problems: 85335
2017 HMNT, 7
There are $ 12$ students in a classroom; $6$ of them are Democrats and 6 of them are Republicans. Every hour the students are randomly separated into four groups of three for political debates. If a group contains students from both parties, the minority in the group will change his/her political alignment to that of the majority at the end of the debate. What is the expected amount of time needed for all $ 12$ students to have the same political alignment, in hours?
2007 IMO Shortlist, 5
Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$.
[i]Author: Christopher Bradley, United Kingdom [/i]
2001 Moldova National Olympiad, Problem 6
Prove that for any integer $n>1$ there are distinct integers $a,b,c$ between $n^2$ and $(n+1)^2$ such that $c$ divides $a^2+b^2$.
Oliforum Contest I 2008, 2
Find all non-negative integers $ x,y,z$ such that $ 5^x \plus{} 7^y \equal{} 2^z$.
:lol:
([i]Daniel Kohen, University of Buenos Aires - Buenos Aires,Argentina[/i])
2021 USAMTS Problems, 4
Let ABC be a triangle whose vertices are inside a circle $\Omega$. Prove that we can choose two of the vertices of ABC such that there are infinitely many circles $\omega$ that satisfy the following properties:
1. $\omega$ is inside of $\Omega$,
2. $\omega$ passes through the two chosen vertices, and
3. the third vertex is in the interior of $\omega$ .
2017 ASDAN Math Tournament, 3
What is the remainder when $2^{1023}$ is divided by $1023$?
2006 China Team Selection Test, 2
The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that
\[ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots \]
Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.
2007 Junior Balkan Team Selection Tests - Moldova, 2
The real numbers $a_1, a_2, a_3$ are greater than $1$ and have the sum equal to $S$.
If for any $i = 1, 2, 3$, holds the inequality $\frac{a_i^2}{a_i-1}>S$ , prove the inequality
$$\frac{1}{a_1+ a_2}+\frac{1}{a_2+ a_3}+\frac{1}{a_3+ a_1}>1$$
2018 CCA Math Bonanza, T9
$21$ Savage has a $12$ car garage, with a row of spaces numbered $1,2,3,\ldots,12$. How many ways can he choose $6$ of them to park his $6$ identical cars in, if no $3$ spaces with consecutive numbers may be all occupied?
[i]2018 CCA Math Bonanza Team Round #9[/i]
1922 Eotvos Mathematical Competition, 1
Given four points $A,B,C,D$ in space, find a plane, $S$, equidistant from all four points and having $A$ and $C$ on one side, $B$ and $D$ on the other.
2002 Chile National Olympiad, 2
Determine all natural numbers $n$ for which it is possible to construct a rectangle of sides $15$ and $n$, with pieces congruent to:
[asy]
unitsize(0.6 cm);
draw((0,0)--(3,0));
draw((0,1)--(3,1));
draw((0,2)--(1,2));
draw((2,2)--(3,2));
draw((0,0)--(0,2));
draw((1,0)--(1,2));
draw((2,0)--(2,2));
draw((3,0)--(3,2));
draw((5,-0.5)--(6,-0.5));
draw((4,0.5)--(7,0.5));
draw((4,1.5)--(7,1.5));
draw((5,2.5)--(6,2.5));
draw((4,0.5)--(4,1.5));
draw((5,-0.5)--(5,2.5));
draw((6,-0.5)--(6,2.5));
draw((7,0.5)--(7,1.5));
[/asy]
The squares of the pieces have side $1$ and the pieces cannot overlap or leave free spaces
2021 Abels Math Contest (Norwegian MO) Final, 4a
A tetrahedron $ABCD$ satisfies $\angle BAC=\angle CAD=\angle DAB=90^o$. Show that the areas of its faces satisfy the equation $area(BAC)^2 + area(CAD)^2 + area(DAB)^2 = area(BCD)^2$.
.
2004 National Olympiad First Round, 12
What is the least value of $(x-1)(x-2)(x-3)(x-4)$ where $x$ is a real number?
$
\textbf{(A)}\ -\dfrac 14
\qquad\textbf{(B)}\ - \dfrac 13
\qquad\textbf{(C)}\ -\dfrac 12
\qquad\textbf{(D)}\ -1
\qquad\textbf{(E)}\ -2
$
2006 Hanoi Open Mathematics Competitions, 5
Suppose $ n$ is a positive integer and 3 arbitrary numbers numbers are chosen from the set $ 1,2,3,...,3n+1$ with their sum equal to $ 3n+1$. What is the largest possible product of those 3 numbers?
2019 BAMO, A
Let $a$ and $b$ be positive whole numbers such that $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$.
Find the fraction $\frac{a}{b}$ for which the sum $a+b$ is as small as possible.
Justify your answer
2020 Greece Team Selection Test, 3
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
2008 Tournament Of Towns, 4
Five distinct positive integers form an arithmetic progression. Can their product be equal to $a^{2008}$ for some positive integer $a$ ?
Ukrainian From Tasks to Tasks - geometry, 2012.13
The sides of a triangle are consecutive natural numbers, and the radius of the inscribed circle is $4$. Find the radius of the circumscribed circle.
2016 AMC 10, 11
Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?
$\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$
2015 Dutch IMO TST, 4
Let $\Gamma_1$ and $\Gamma_2$ be circles - with respective centres $O_1$ and $O_2$ - that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\Gamma_2$ in $A$ and $C$ and the line $O_2A$ intersects $\Gamma_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\Gamma_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.
2018 India National Olympiad, 6
Let $\mathbb N$ denote set of all natural numbers and let $f:\mathbb{N}\to\mathbb{N}$ be a function such that
$\text{(a)} f(mn)=f(m).f(n)$ for all $m,n \in\mathbb{N}$;
$\text{(b)} m+n$ divides $f(m)+f(n)$ for all $m,n\in \mathbb N$.
Prove that, there exists an odd natural number $k$ such that $f(n)= n^k$ for all $n$ in $\mathbb{N}$.
2022 Bulgaria JBMO TST, 2
Let $a$, $b$ and $c$ be positive real numbers with $abc = 1$. Determine the minimum possible value of
$$ \left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \cdot \left(\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}\right) $$
as well as all triples $(a,b,c)$ which attain the minimum.
2014 NIMO Summer Contest, 5
We have a five-digit positive integer $N$. We select every pair of digits of $N$ (and keep them in order) to obtain the $\tbinom52 = 10$ numbers $33$, $37$, $37$, $37$, $38$, $73$, $77$, $78$, $83$, $87$. Find $N$.
[i]Proposed by Lewis Chen[/i]
2022 Assam Mathematical Olympiad, 7
In how many ways can $10$ balls of same size be distributed among $4$ children in the following cases:
(a) all the balls are of the same colour?
(b) each ball is of a different colour?
1979 Chisinau City MO, 175
Prove that if the sum of positive numbers $a, b, c$ is equal to $1$, then $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 9.$