Found problems: 85335
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 $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 9.$
Indonesia MO Shortlist - geometry, g9
It is known that $ABCD$ is a parallelogram. The point $E$ is taken so that $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line that passes through $A$, intersects the segment $DC$ at point $F$ and intersects the extension of the line $BC$ at $G$. Given $EF = EG = EC$. Prove that $\ell$ is the bisector of the angle $\angle BAD$.
1961 AMC 12/AHSME, 38
Triangle $ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:
$ \textbf{(A)}\ s^2\le8r^2$
$\qquad\textbf{(B)}\ s^2=8r^2$
$\qquad\textbf{(C)}\ s^2 \ge 8r^2$
${\qquad\textbf{(D)}\ s^2\le4r^2 }$
${\qquad\textbf{(E)}\ x^2=4r^2 } $
1947 Moscow Mathematical Olympiad, 139
In the numerical triangle
$................1..............$
$...........1 ...1 ...1.........$
$......1... 2... 3 ... 2 ... 1....$
$.1...3...6...7...6...3...1$
$...............................$
each number is equal to the sum of the three nearest to it numbers from the row above it; if the number is at the beginning or at the end of a row then it is equal to the sum of its two nearest numbers or just to the nearest number above it (the lacking numbers above the given one are assumed to be zeros). Prove that each row, starting with the third one, contains an even number.
2015 Princeton University Math Competition, A1/B1
A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$. How many distinct $3$-letter words can be made from a subset of the letters $c, o, m, b, o$, where each letter in the list is used no more than the number of times it appears?
1978 Canada National Olympiad, 2
Find all pairs of $a$, $b$ of positive integers satisfying the equation $2a^2 = 3b^3$.
2019 Centers of Excellency of Suceava, 1
Prove that if a prime is the sum of four perfect squares then the product of two of these is equal to the product of the other two.
[i]Gherghe Stoica[/i]
1993 Iran MO (2nd round), 1
Suppose that $p$ is a prime number and is greater than $3$. Prove that $7^{p}-6^{p}-1$ is divisible by $43$.
Kvant 2021, M2644
Petya and Vasya are playing on an $100\times 100$ board. Initially, all the cells of the board are white. With each of his moves, Petya paints one or more white cells standing on the same diagonal in black. With each of his moves, Vasya paints one or more white cells standing on the same column in black. Petya makes the first move. The one who can't make a move loses. Who has a winning strategy?
[i]Proposed by M. Didin[/i]
2018 Austria Beginners' Competition, 3
For a given integer $n \ge 4$ we examine whether there exists a table with three rows and $n$ columns which can be filled by the numbers $1, 2,...,, 3n$ such that
$\bullet$ each row totals to the same sum $z$ and
$\bullet$ each column totals to the same sum $s$.
Prove:
(a) If $n$ is even, such a table does not exist.
(b) If $n = 5$, such a table does exist.
(Gerhard J. Woeginger)
Mid-Michigan MO, Grades 10-12, 2002
[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$.
[b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral.
[b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms?
[b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].