Found problems: 85335
2024 China National Olympiad, 1
Find the smallest $\lambda \in \mathbb{R}$ such that for all $n \in \mathbb{N}_+$, there exists $x_1, x_2, \ldots, x_n$ satisfying $n = x_1 x_2 \ldots x_{2023}$, where $x_i$ is either a prime or a positive integer not exceeding $n^\lambda$ for all $i \in \left\{ 1,2, \ldots, 2023 \right\}$.
[i]Proposed by Yinghua Ai[/i]
2009 Princeton University Math Competition, 1
If $\phi$ is the Golden Ratio, we know that $\frac1\phi = \phi - 1$. Define a new positive real number, called $\phi_d$, where $\frac1{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$, $a, b, c$ positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $a + b + c$.
1999 All-Russian Olympiad, 4
Initially numbers from 1 to 1000000 are all colored black. A move consists of picking one number, then change the color (black to white or white to black) of itself and all other numbers NOT coprime with the chosen number. Can all numbers become white after finite numbers of moves?
Edited by pbornsztein
2018 Dutch IMO TST, 3
Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is de fined as follows:
we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer.
Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.
2021 AMC 12/AHSME Spring, 8
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34.$ What is the distance between two adjacent parallel lines?
$\textbf{(A)}\ 5\frac{1}{2} \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 6\frac{1}{2} \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 7\frac{1}{2}$
2011 IMO Shortlist, 7
Let $p$ be an odd prime number. For every integer $a,$ define the number $S_a = \sum^{p-1}_{j=1} \frac{a^j}{j}.$ Let $m,n \in \mathbb{Z},$ such that $S_3 + S_4 - 3S_2 = \frac{m}{n}.$ Prove that $p$ divides $m.$
[i]Proposed by Romeo Meštrović, Montenegro[/i]
2010 AMC 12/AHSME, 14
Nondegenerate $ \triangle ABC$ has integer side lengths, $ BD$ is an angle bisector, $ AD \equal{} 3$, and $ DC \equal{} 8$. What is the smallest possible value of the perimeter?
$ \textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 33 \qquad
\textbf{(C)}\ 35 \qquad
\textbf{(D)}\ 36 \qquad
\textbf{(E)}\ 37$
1997 Tournament Of Towns, (532) 4
$AC' BA'C B'$ is a convex hexagon such that $AB' = AC'$, $BC' = BA'$, $CA' = CB'$ and $\angle A +\angle B + \angle C = \angle A' + \angle B' + \angle C'$. Prove that the area of the triangle $ABC$ is half the area of the hexagon.
(V Proizvolov)
2014 Math Prize For Girls Problems, 3
Four different positive integers less than 10 are chosen randomly. What is the probability that their sum is odd?
2009 Korea - Final Round, 5
There is a $m \times (m-1)$ board. (i.e. there are $m+1$ horizontal lines and $m$ vertical lines) A stone is put on an intersection of the lowest horizontal line. Now two players move this stone with the following rules.
(i) Each players move the stone to a neighboring intersection along a segment, by turns.
(ii) A segment, which is already passed by the stone, cannot be used more.
(iii) One who cannot move the stone anymore loses.
Prove that there is a winning strategy for the former player.
2022/2023 Tournament of Towns, P1
There are two letter sequences $A$ and $B$, both with length $100$ letters. In one move you can insert in any place of sequence ( possibly to start or to end) any number of same letters or remove any number of consecutive same letters.
Prove that it is possible to make second sequence from first sequence using not more than $100$ moves.
2000 Irish Math Olympiad, 2
In a cyclic quadrilateral $ ABCD, a,b,c,d$ are its side lengths, $ Q$ its area, and $ R$ its circumradius. Prove that:
$ R^2\equal{}\frac{(ab\plus{}cd)(ac\plus{}bd)(ad\plus{}bc)}{16Q^2}$.
Deduce that $ R \ge \frac{(abcd)^{\frac{3}{4}}}{Q\sqrt{2}}$ with equality if and only if $ ABCD$ is a square.
2012 Kazakhstan National Olympiad, 3
The cell of a $(2m +1) \times (2n +1)$ board are painted in two colors - white and black. The unit cell of a row (column) is called [i]dominant[/i] on the row (the column) if more than half of the cells that row (column) have the same color as this cell. Prove that at least $m + n-1$ cells on the board are dominant in both their row and column.
2019 HMNT, 8
In $\vartriangle ABC$, the external angle bisector of $\angle BAC$ intersects line $BC$ at $D$. $E$ is a point on ray $\overrightarrow{AC}$ such that $\angle BDE = 2\angle ADB$. If $AB = 10$, $AC = 12$, and $CE = 33$, compute $\frac{DB}{DE}$ .
1980 Vietnam National Olympiad, 3
Let $P$ be a point inside a triangle $A_1A_2A_3$. For $i = 1, 2, 3$, line $PA_i$ intersects the side opposite to $A_i$ at $B_i$. Let $C_i$ and $D_i$ be the midpoints of $A_iB_i$ and $PB_i$, respectively. Prove that the areas of the triangles $C_1C_2C_3$ and $D_1D_2D_3$ are equal.
1970 IMO, 1
Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.
2011 Belarus Team Selection Test, 3
Find all functions $f:R\to R$ such that for all real $x,y$ with $y\ne 0$ $$f(x-f(x/y))=xf(1-f(1/y))$$ and
a) $f(1-f(1))\ne 0$
b) $ f(1-f(1))= 0$
S. Kuzmich, I.Voronovich
2009 Princeton University Math Competition, 1
You have an unlimited supply of monominos, dominos, and L-trominos. How many ways, in terms of $n$, can you cover a $2 \times n$ grid with these shapes? Please note that you do [i]NOT[/i] have to use all the shapes. Also, you are allowed to [i]rotate[/i] any of the pieces, so they do not have to be aligned exactly as they are in the diagram below.
[asy]
pen db = rgb(0,0,0.5); real r = 0.08; pair s1 = (3,0), s2 = 2*s1;
fill(unitsquare, db); fill(shift(s1)*unitsquare, db); fill(shift(s1-(0,1+r))*unitsquare, db); fill(shift(s2)*unitsquare, db); fill(shift(s2-(0,1+r))*unitsquare, db); fill(shift(s2+(1+r,-1-r))*unitsquare, db);
[/asy]
2015 NIMO Problems, 7
Find the number of ways a series of $+$ and $-$ signs can be inserted between the numbers $0,1,2,\cdots, 12$ such that the value of the resulting expression is divisible by 5.
[i]Proposed by Matthew Lerner-Brecher[/i]
2001 Irish Math Olympiad, 4
Find all nonnegative real numbers $ x$ for which $ \sqrt[3]{13\plus{}\sqrt{x}}\plus{}\sqrt[3]{13\minus{}\sqrt{x}}$ is an integer.
2012 Argentina National Olympiad, 2
Determine all natural numbers $n$ for which there are $2n$ distinct positive integers $x_1,…,x_n,y_1,…,y_n$
such that the product $$(11x^2_1+12y^2_1)(11x^2_2+12y^2_2)…(11x^2_n+12y^2_n)$$ is a perfect square.
2021 Malaysia IMONST 2, 2
The five numbers $a, b, c, d,$ and $e$ satisfy the inequalities
$$a+b < c+d < e+a < b+c < d+e.$$
Among the five numbers, which one is the smallest, and which one is the largest?
2008 Mongolia Team Selection Test, 3
Given positive integers $ m,n > 1$. Prove that the equation
$ (x \plus{} 1)^n \plus{} (x \plus{} 2)^n \plus{} ... \plus{} (x \plus{} m)^n \equal{} (y \plus{} 1)^{2n} \plus{} (y \plus{} 2)^{2n} \plus{} ... \plus{} (y \plus{} m)^{2n}$ has finitely number of solutions $ x,y \in N$
2016 India IMO Training Camp, 2
Given that $n$ is a natural number such that the leftmost digits in the decimal representations of $2^n$ and $3^n$ are the same, find all possible values of the leftmost digit.
2020 Durer Math Competition Finals, 14
How many ways are there to fill in the $ 8$ spots in the picture with letters $A, B, C$ and $D$, using two copies of each letter, such that the spots with identical letters can be connected with a continuous line that stays within the box, without these four lines crossing each other or going through other spots?
The lines do not have to be straight.
[img]https://cdn.artofproblemsolving.com/attachments/f/f/66c30eaf6fa3b42c5197d0e3a3d59e9160bb8e.png[/img]