Found problems: 85335
2007 Princeton University Math Competition, 1
If $a, b, c$ are real numbers such that $a+b+c=6$ and $ab+bc+ca = 9$, find the sum of all possible values of the expression $\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor$.
1993 Tournament Of Towns, (387) 5
Let $S(n)$ denote the sum of digits of $n$ (in decimal representation). Do there exist three different natural numbers $n$, $p$ and $q$ such that
$$n +S(n) = p + S(p) = q + S(q)?$$
(M Gerver)
III Soros Olympiad 1996 - 97 (Russia), 10.9
For any positive $a$ and $b$, find positive solutions of the system
$$\begin{cases} \dfrac{a^2}{x^2}- \dfrac{b^2}{y^2}=8(y^4-x^4)
\\ ax-by=x^4-y^4
\end{cases}$$
2024 Argentina National Math Olympiad Level 3, 2
Consider a square $8 \times 8$ board with its $64$ cells initially white. Determine the minimum number of colors needed to color the cells (each one with only one color) in such a way that if four cells on the board can be covered by an $L$-shaped tile as shown in the figure, then the four cells are of different colors.
[asy]
size(3cm);
draw((1,0)--(1,1)--(2,1)--(2,0)--(1,0)--(0,0)--(0,1)--(0,2)--(1,2)--(1,1)--(0,1)--(1,1)--(2,1)--(3,1)--(3,0)--(2,0));
[/asy]
[b]Note:[/b] The $L$-shaped tile can be rotated or flipped.
1991 Arnold's Trivium, 67
What is the dimension of the space of solutions continuous on $x^2+y^2\ge1$ of the problem
\[\Delta u=0\text{ for }x^2+y^2>1\]
\[\partial u/\partial n=0\text{ for }x^2+y^2=1\]
2006 IberoAmerican Olympiad For University Students, 7
Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$.
Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$.
2007 Romania Team Selection Test, 4
Let $\mathcal O_{1}$ and $\mathcal O_{2}$ two exterior circles. Let $A$, $B$, $C$ be points on $\mathcal O_{1}$ and $D$, $E$, $F$ points on $\mathcal O_{1}$ such that $AD$ and $BE$ are the common exterior tangents to these two circles and $CF$ is one of the interior tangents to these two circles, and such that $C$, $F$ are in the interior of the quadrilateral $ABED$. If $CO_{1}\cap AB=\{M\}$ and $FO_{2}\cap DE=\{N\}$ then prove that $MN$ passes through the middle of $CF$.
2023 LMT Fall, 4
Fred chooses a positive two-digit number with distinct nonzero digits. Laura takes Fred’s number and swaps its digits. She notices that the sum of her number and Fred’s number is a perfect square and the positive difference between them is a perfect cube. Find the greater of the two numbers.
2015 Romania Team Selection Tests, 2
Given an integer $k \geq 2$, determine the largest number of divisors the binomial coefficient $\binom{n}{k}$ may have in the range $n-k+1, \ldots, n$ , as $n$ runs through the integers greater than or equal to $k$.
2021 Centroamerican and Caribbean Math Olympiad, 5
Let $n \geq 3$ be an integer and $a_1,a_2,...,a_n$ be positive real numbers such that $m$ is the smallest and $M$ is the largest of these numbers. It is known that for any distinct integers $1 \leq i,j,k \leq n$, if $a_i \leq a_j \leq a_k$ then $a_ia_k \leq a_j^2$. Show that
\[ a_1a_2 \cdots a_n \geq m^2M^{n-2} \]
and determine when equality holds