Found problems: 85335
2005 National Olympiad First Round, 18
How many integers $0\leq x < 121$ are there such that $x^5+5x^2 + x + 1 \equiv 0 \pmod{121}$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
2020 AMC 8 -, 13
Jamal has a drawer containing $6$ green socks, $18$ purple socks, and $12$ orange socks. After adding more purple socks, Jamal noticed that there is now a $60\%$ chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
$\textbf{(A)}\ 6\qquad~~\textbf{(B)}\ 9\qquad~~\textbf{(C)}\ 12\qquad~~\textbf{(D)}\ 18\qquad~~\textbf{(E)}\ 24$
2008 IMO Shortlist, 1
Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$.
Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic.
[i]Author: Andrey Gavrilyuk, Russia[/i]
2020 Tournament Of Towns, 4
For some integer n the equation $x^2 + y^2 + z^2 -xy -yz - zx = n$ has an integer solution $x, y, z$. Prove that the equation$ x^2 + y^2 - xy = n$ also has an integer solution $x, y$.
Alexandr Yuran
2011 Cono Sur Olympiad, 1
Find all triplets of positive integers $(x,y,z)$ such that $x^{2}+y^{2}+z^{2}=2011$.
1983 AMC 12/AHSME, 17
The diagram to the right shows several numbers in the complex plane. The circle is the unit circle centered at the origin. One of these numbers is the reciprocal of $F$. Which one?
$\text{(A)} \ A \qquad \text{(B)} \ B \qquad \text{(C)} \ C \qquad \text{(D)} \ D \qquad \text{(E)} \ E$
2000 Vietnam Team Selection Test, 3
A collection of $2000$ congruent circles is given on the plane such that no
two circles are tangent and each circle meets at least two other circles.
Let $N$ be the number of points that belong to at least two of the circles.
Find the smallest possible value of $N$.
2000 Stanford Mathematics Tournament, 14
The author of this question was born on April 24, 1977. What day of the week was that?
2005 AMC 10, 6
The average (mean) of $ 20$ numbers is $ 30$, and the average of $ 30$ other numbers is $ 20$. What is the average of all $ 50$ numbers?
$ \textbf{(A)}\ 23 \qquad
\textbf{(B)}\ 24 \qquad
\textbf{(C)}\ 25 \qquad
\textbf{(D)}\ 26 \qquad
\textbf{(E)}\ 27$
1932 Eotvos Mathematical Competition, 1
Let $a, b$ and $n$ be positive integers such that $ b$ is divisible by $a^n$. Prove that $(a+1)^b-1$ is divisible by $a^{n+1}$.
1999 Czech and Slovak Match, 3
Find all natural numbers $k$ for which there exists a set $M$ of ten real numbers such that there are exactly $k$ pairwise non-congruent triangles whose side lengths are three (not necessarily distinct) elements of $M$.
LMT Team Rounds 2021+, B7
Given that $x$ and $y$ are positive real numbers such that $\frac{5}{x}=\frac{y}{13}=\frac{x}{y}$, find the value of $x^3 + y^3$.
Proposed by Ephram Chun
2024 Sharygin Geometry Olympiad, 20
Lines $a_1, b_1, c_1$ pass through the vertices $A, B, C$ respectively of a triange $ABC$; $a_2, b_2, c_2$ are the reflections of $a_1, b_1, c_1$ about the corresponding bisectors of $ABC$; $A_1 = b_1 \cap c_1, B_1 = a_1 \cap c_1, C_1 = a_1 \cap b_1$, and $A_2, B_2, C_2$ are defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ have the same ratios of the area and circumradius (i.e. $\frac{S_1}{R_1} = \frac{S_2}{R_2}$, where $S_i = S(\triangle A_iB_iC_i)$, $R_i = R(\triangle A_iB_iC_i)$)
1988 IMO Longlists, 51
The positive integer $n$ has the property that, in any set of $n$ integers, chosen from the integers $1,2, \ldots, 1988,$ twenty-nine of them form an arithmetic progression. Prove that $n > 1788.$
1982 Brazil National Olympiad, 1
The angles of the triangle $ABC$ satisfy $\angle A / \angle C = \angle B / \angle A = 2$. The incenter is $O. K, L$ are the excenters of the excircles opposite $B$ and $A$ respectively. Show that triangles $ABC$ and $OKL$ are similar.
2011 IMO Shortlist, 4
For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences
\[t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)\] are divisible by 4.
[i]Proposed by Gerhard Wöginger, Austria[/i]
1979 Austrian-Polish Competition, 9
Find the greatest power of $2$ that divides $a_n = [(3+\sqrt{11} )^{2n+1}]$, where $n$ is a given positive integer.
2010 Contests, 4
Let $a_1,a_2,..,a_n,b_1,b_2,...,b_n$ be non-negative numbers satisfying the following conditions simultaneously:
(1) $\displaystyle\sum_{i=1}^{n} (a_i + b_i) = 1$;
(2) $\displaystyle\sum_{i=1}^{n} i(a_i - b_i) = 0$;
(3) $\displaystyle\sum_{i=1}^{n} i^2(a_i + b_i) = 10$.
Prove that $\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}$ for all $1 \le k \le n$.
1996 National High School Mathematics League, 3
For a prime number $p$, there exists $n\in\mathbb{Z}_+$, $\sqrt{p+n}+\sqrt{n}$ is an integer, then
$\text{(A)}$ there is no such $p$
$\text{(B)}$ there in only one such $p$
$\text{(C)}$ there is more than one such $p$, but finitely many
$\text{(D)}$ there are infinitely many such $p$
2019 Federal Competition For Advanced Students, P2, 6
Find the smallest possible positive integer n with the following property:
For all positive integers $x, y$ and $z$ with $x | y^3$ and $y | z^3$ and $z | x^3$ always to be true that $xyz| (x + y + z) ^n$.
(Gerhard J. Woeginger)
2017 Balkan MO Shortlist, N3
Prove that for all positive integer $n$, there is a positive integer $m$ that $7^n | 3^m +5^m -1$.
2023 Thailand TST, 2
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.
2025 Kosovo National Mathematical Olympiad`, P2
Find all real numbers $a$ and $b$ that satisfy the system of equations:
$$\begin{cases}
a &= \frac{2}{a+b} \\
\\
b &= \frac{2}{3a-b} \\
\end{cases}$$
MOAA Individual Speed General Rounds, 2023.2
In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$.
[i]Proposed by Andy Xu[/i]
2021-IMOC, A4
Find all functions f : R-->R such that
f (f (x) + y^2) = x −1 + (y + 1)f (y)
holds for all real numbers x, y