Found problems: 85335
2017 Brazil Team Selection Test, 4
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$
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?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$