Found problems: 85335
2005 Colombia Team Selection Test, 1
Let $a,b,c$ be integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$ prove that $abc$ is a perfect cube!
2016 Estonia Team Selection Test, 11
Find all positive integers $n$ such that $(n^2 + 11n - 4) \cdot n! + 33 \cdot 13^n + 4$ is a perfect square
2008 Bundeswettbewerb Mathematik, 2
Represent the number $ 2008$ as a sum of natural number such that the addition of the reciprocals of the summands yield 1.
2021 Romania National Olympiad, 1
Let $\mathcal C$ be a circle centered at $O$ and $A\ne O$ be a point in its interior. The perpendicular bisector of the segment $OA$ meets $\mathcal C$ at the points $B$ and $C$, and the lines $AB$ and $AC$ meet $\mathcal C$ again at $D$ and $E$, respectively. Show that the circles $(OBC)$ and $(ADE)$ have the same centre.
[i]Ion Pătrașcu, Ion Cotoi[/i]
1992 IMO Longlists, 43
Find the number of positive integers $n$ satisfying $\phi(n) | n$ such that
\[\sum_{m=1}^{\infty} \left( \left[ \frac nm \right] - \left[\frac{n-1}{m} \right] \right) = 1992\]
What is the largest number among them? As usual, $\phi(n)$ is the number of positive integers less than or equal to $n$ and relatively prime to $n.$
1994 Putnam, 3
Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f^{\prime}(x) > f(x)$ for all $x,$ there is some number $N$ such that $f(x) > e^{kx}$ for all $x > N.$
2007 iTest Tournament of Champions, 4
Find the smallest positive integer $k$ such that \[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] for some ordered triple of positive integers $(a,b,c)$.
2009 Math Prize For Girls Problems, 3
The [i]Fibonacci numbers[/i] are defined recursively by the equation
\[ F_n \equal{} F_{n \minus{} 1} \plus{} F_{n \minus{} 2}\]
for every integer $ n \ge 2$, with initial values $ F_0 \equal{} 0$ and $ F_1 \equal{} 1$. Let $ G_n \equal{} F_{3n}$ be every third Fibonacci number. There are constants $ a$ and $ b$ such that every integer $ n \ge 2$ satisfies
\[ G_n \equal{} a G_{n \minus{} 1} \plus{} b G_{n \minus{} 2}.\]
Compute the ordered pair $ (a, b)$.
2025 Serbia Team Selection Test for the IMO 2025, 4
For a permutation $\pi$ of the set $A = \{1, 2, \ldots, 2025\}$, define its [i]colorfulness [/i]as the greatest natural number $k$ such that:
- For all $1 \le i, j \le 2025$, $i \ne j$, if $|i - j| < k$, then $|\pi(i) - \pi(j)| \ge k$.
What is the maximum possible colorfulness of a permutation of the set $A$? Determine how many such permutations have maximal colorfulness.
[i]Proposed by Pavle Martinović[/i]
2014 ELMO Shortlist, 1
In a non-obtuse triangle $ABC$, prove that
\[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]
2007 Harvard-MIT Mathematics Tournament, 7
A student at Harvard named Kevin
Was counting his stones by $11$
He messed up $n$ times
And instead counted $9$s
And wound up at $2007$.
How many values of $n$ could make this limerick true?
2002 Croatia National Olympiad, Problem 3
If two triangles with side lengths $a,b,c$ and $a',b',c'$ and the corresponding angle $\alpha,\beta,\gamma$ and $\alpha',\beta',\gamma'$ satisfy $\alpha+\alpha'=\pi$ and $\beta=\beta'$, prove that $aa'=bb'+cc'$.
1995 Czech And Slovak Olympiad IIIA, 4
Do there exist $10000$ ten-digit numbers divisible by $7$, all of which can be obtained from one another by a reordering of their digits?
1953 Moscow Mathematical Olympiad, 250
Somebody wrote $1953$ digits on a circle. The $1953$-digit number obtained by reading these figures clockwise, beginning at a certain point, is divisible by $27$. Prove that if one begins reading the figures at any other place, one gets another $1953$-digit number also divisible by $27$.
1989 Putnam, B4
Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?
1988 Poland - Second Round, 5
Decide whether any rectangle that can be covered by 25 circles of radius 2 can also be covered by 100 circles of radius 1.
Russian TST 2019, P3
Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.
1992 Poland - Second Round, 5
Determine the upper limit of the volume of spheres contained in tetrahedra of all heights not longer than $ 1 $.
2019 District Olympiad, 3
Consider the rectangular parallelepiped $ABCDA'B'C'D' $ as such the measure of the dihedral angle formed by the planes $(A'BD)$ and $(C'BD)$ is $90^o$ and the measure of the dihedral angle formed by the planes $(AB'C)$ and $(D'B'C)$ is $60^o$. Determine and measure the dihedral angle formed by the planes $(BC'D)$ and $(A'C'D)$.
1957 AMC 12/AHSME, 7
The area of a circle inscribed in an equilateral triangle is $ 48\pi$. The perimeter of this triangle is:
$ \textbf{(A)}\ 72\sqrt{3} \qquad
\textbf{(B)}\ 48\sqrt{3}\qquad
\textbf{(C)}\ 36\qquad
\textbf{(D)}\ 24\qquad
\textbf{(E)}\ 72$
2010 Bundeswettbewerb Mathematik, 2
There are $9999$ rods with lengths $1, 2, ..., 9998, 9999$. The players Anja and Bernd alternately remove one of the sticks, with Anja starting. The game ends when there are only three bars left. If from those three bars, a not degenerate triangle can be constructed then Anja wins, otherwise Bernd.
Who has a winning strategy?
2004 District Olympiad, 1
Let $(x_n)_{n\ge 0}$ a sequence of real numbers defined by $x_0>0$ and $x_{n+1}=x_n+\frac{1}{\sqrt{x_n}}$. Compute $\lim_{n\to \infty}x_n$ and $\lim_{n\to \infty} \frac{x_n^3}{n^2}$.
1983 Bulgaria National Olympiad, Problem 1
Determine all natural numbers $n$ for which there exists a permutation $(a_1,a_2,\ldots,a_n)$ of the numbers $0,1,\ldots,n-1$ such that, if $b_i$ is the remainder of $a_1a_2\cdots a_i$ upon division by $n$ for $i=1,\ldots,n$, then $(b_1,b_2,\ldots,b_n)$ is also a permutation of $0,1,\ldots,n-1$.
2017 AMC 10, 5
The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$
2023 USAJMO Solutions by peace09, 5
A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.
After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.