Found problems: 85335
2022 HMNT, 1
Emily’s broken clock runs backwards at five times the speed of a regular clock. Right now, it is displaying the wrong time. How many times will it display the correct time in the next 24 hours? It is an analog clock (i.e. a clock with hands), so it only displays the numerical time, not AM or PM. Emily’s clock also does not tick, but rather updates continuously.
2013 Hanoi Open Mathematics Competitions, 13
Solve the system of equations $\begin{cases} \frac{1}{x}+\frac{1}{y}=\frac{1}{6} \\
\frac{3}{x}+\frac{2}{y}=\frac{5}{6} \end{cases}$
2010 AMC 12/AHSME, 20
Arithmetic sequences $ (a_n)$ and $ (b_n)$ have integer terms with $ a_1 \equal{} b_1 \equal{} 1 < a_2 \le b_2$ and $ a_nb_n \equal{} 2010$ for some $ n$. What is the largest possible value of $ n$?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 8 \qquad
\textbf{(D)}\ 288 \qquad
\textbf{(E)}\ 2009$
2012 Mathcenter Contest + Longlist, 6 sl14
For a real number $a,b,c>0$ where $bc-ca-ab=1$ find the maximum value of $$P=\frac{4024}{1+a^2}-\frac{4024}{1+b^2}-\frac{2555}{1+c^2}$$ and find out when that holds .
[i](PP-nine)[/i]
1966 Swedish Mathematical Competition, 4
Let $f(x) = 1 + \frac{2}{x}$. Put $f_1(x) = f(x)$, $f_2(x) = f(f_1(x))$, $f_3(x) = f(f_2(x))$, $... $. Find the solutions to $x = f_n(x)$ for $n > 0$.
2013 Junior Balkan Team Selection Tests - Romania, 5
a) Prove that for every positive integer n, there exist $a, b \in R - Z$ such that
the set $A_n = \{a - b, a^2 - b^2, a^3 - b^3,...,a^n - b^n\}$ contains only positive integers.
b) Let $a$ and $b$ be two real numbers such that the set $A = \{a^k - b^k | k \in N*\}$ contains only positive integers.
Prove that $a$ and $b$ are integers.
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 $p$