Found problems: 85335
1960 AMC 12/AHSME, 16
In the numeration system with base $5$, counting is as follows: $1, 2, 3, 4, 10, 11, 12, 13, 14, 20, ...$ The number whose description in the decimal system is $69$, when described in the base $5$ system, is a number with:
$ \textbf{(A)}\ \text{two consecutive digits} \qquad\textbf{(B)}\ \text{two non-consecutive digits} \qquad$
$\textbf{(C)}\ \text{three consecutive digits} \qquad\textbf{(D)}\ \text{three non-consecutive digits} \qquad$
$\textbf{(E)}\ \text{four digits} $
2020 MBMT, 2
Daniel, Clarence, and Matthew split a \$20.20 dinner bill so that Daniel pays half of what Clarence pays. If Daniel pays \$6.06, what is the ratio of Clarence's pay to Matthew's pay?
[i]Proposed by Henry Ren[/i]
2010 Turkey MO (2nd round), 2
For integers $a$ and $b$ with $0 \leq a,b < {2010}^{18}$ let $S$ be the set of all polynomials in the form of $P(x)=ax^2+bx.$ For a polynomial $P$ in $S,$ if for all integers n with $0 \leq n <{2010}^{18}$ there exists a polynomial $Q$ in $S$ satisfying $Q(P(n)) \equiv n \pmod {2010^{18}},$ then we call $P$ as a [i]good polynomial.[/i]
Find the number of [i]good polynomials.[/i]
2001 Estonia National Olympiad, 1
The angles of a convex $n$-gon are $a,2a, ... ,na$. Find all possible values of $n$ and the corresponding values of $a$.
2012 CHMMC Spring, 5
Suppose $S$ is a subset of $\{1, 2, 3, 4, 5, 6, 7\}$. How many different possible values are there for the product of the elements in $S$?
2024 Saint Petersburg Mathematical Olympiad, 3
On the side $BC$ of acute triangle $ABC$ point $P$ was chosen. Point $E$ is symmetric to point $B$ onto line $AP$. Segment $PE$ meets circumcircle of triangle $ABP$ in point $D$. $M$ is midpoint of side $AC$. Prove that $DE+AC>2BM$.
2009 AMC 8, 6
Steve's empty swimming pool will hold $ 24,000$ gallons of water when full. It will be filled by $ 4$ hoses, each of which supplies $ 2.5$ gallons of water per minute. How many hours will it take to fill Steve's pool?
$ \textbf{(A)}\ 40 \qquad
\textbf{(B)}\ 42 \qquad
\textbf{(C)}\ 44 \qquad
\textbf{(D)}\ 46 \qquad
\textbf{(E)}\ 48$
1989 Iran MO (2nd round), 1
[b](a)[/b] Let $n$ be a positive integer, prove that
\[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}\]
[b](b)[/b] Find a positive integer $n$ for which
\[ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12\]
BIMO 2022, 5
Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ such that for all prime $p$ the following condition holds:
$$p \mid ab + bc + ca \iff p \mid f(a)f(b) + f(b)f(c) + f(c)f(a)$$
[i]Proposed by Anzo Teh Zhao Yang[/i]
2014 Baltic Way, 10
In a country there are $100$ airports. Super-Air operates direct flights between some pairs of airports (in both directions). The [i]traffic[/i] of an airport is the number of airports it has a direct Super-Air connection with. A new company, Concur-Air, establishes a direct flight between two airports if and only if the sum of their traffics is at least $100.$ It turns out that there exists a round-trip of Concur-Air flights that lands in every airport exactly once. Show that then there also exists a round-trip of Super-Air flights that lands in every airport exactly once.
2022 Federal Competition For Advanced Students, P1, 2
The points $A, B, C, D$ lie in this order on a circle with center $O$. Furthermore, the straight lines $AC$ and $BD$ should be perpendicular to each other. The base of the perpendicular from $O$ on $AB$ is $F$. Prove $CD = 2 OF$.
[i](Karl Czakler)[/i]
2018 Slovenia Team Selection Test, 4
Let $\mathcal{K}$ be a circle centered in $A$. Let $p$ be a line tangent to $\mathcal{K}$ in $B$ and let a line parallel to $p$ intersect $\mathcal{K}$ in $C$ and $D$. Let the line $AD$ intersect $p$ in $E$ and let $F$ be the intersection of the lines $CE$ and $AB$. Prove that the line through $D$, parallel to the tangent through $A$ to the circumcircle of $AFD$ intersects the line $CF$ on $\mathcal{K}$.
2010 National Chemistry Olympiad, 10
Magnesium chloride dissolves in water to form:
$ \textbf{(A)}\hspace{.05in}\text{hydrated MgCl}_2 \text{molecules}\qquad$
$\textbf{(B)}\hspace{.05in}\text{hydrated Mg}^{2+} \text{ions and hydrated Cl}^- \text{ions} \qquad$
$\textbf{(C)}\hspace{.05in}\text{hydrated Mg}^{2+} \text{ions and hydrated Cl}_2 ^{2-} \text{ions}\qquad$
$\textbf{(D)}\hspace{.05in}\text{hydrated Mg atoms and hydrated Cl}_2 \text{molecules}\qquad$
2015 Abels Math Contest (Norwegian MO) Final, 1a
Find all triples $(x, y, z) \in R^3$ satisfying the equations $\begin{cases} x^2 + 4y^2 = 4zx \\
y^2 + 4z^2 = 4xy \\
z^2 + 4x^2 = 4yz \end{cases}$
PEN A Problems, 40
Determine the greatest common divisor of the elements of the set \[\{n^{13}-n \; \vert \; n \in \mathbb{Z}\}.\]
2012 USAMO, 4
Find all functions $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ (where $\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m) - f(n)$ for all distinct positive integers $m, n$.
1984 Tournament Of Towns, (078) 3
We are given a regular decagon with all diagonals drawn. The number "$+ 1$ " is attached to each vertex and to each point where diagonals intersect (we consider only internal points of intersection). We can decide at any time to simultaneously change the sign of all such numbers along a given side or a given diagonal . Is it possible after a certain number of such operations to have changed all the signs to negative?
1985 Balkan MO, 1
In a given triangle $ABC$, $O$ is its circumcenter, $D$ is the midpoint of $AB$ and $E$ is the centroid of the triangle $ACD$. Show that the lines $CD$ and $OE$ are perpendicular if and only if $AB=AC$.
2008 All-Russian Olympiad, 1
Do there exist $ 14$ positive integers, upon increasing each of them by $ 1$,their product increases exactly $ 2008$ times?
2003 JBMO Shortlist, 3
Let $G$ be the centroid of triangle $ABC$, and $A'$ the symmetric of $A$ wrt $C$. Show that $G, B, C, A'$ are concyclic if and only if $GA \perp GC$.
LMT Speed Rounds, 2010.10
How many integers less than $2502$ are equal to the square of a prime number?
2011 Sharygin Geometry Olympiad, 17
a) Does there exist a triangle in which the shortest median is longer that the longest bisectrix?
b) Does there exist a triangle in which the shortest bisectrix is longer that the longest altitude?
2012 AIME Problems, 9
Let $x$ and $y$ be real numbers such that $\frac{\sin{x}}{\sin{y}} = 3$ and $\frac{\cos{x}}{\cos{y}} = \frac{1}{2}$. The value of $\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
1999 Mongolian Mathematical Olympiad, Problem 2
The rays $l_1,l_2,\ldots,l_{n-1}$ divide a given angle $ABC$ into $n$ equal parts. A line $l$ intersects $AB$ at $A_1$, $BC$ at $A_{n+1}$, and $l_i$ at $A_{i+1}$ for $i=1,\ldots,n-1$. Show that the quantity
$$\left(\frac1{BA_1}+\frac1{BA_{n+1}}\right)\left(\frac1{BA_1}+\frac1{BA_2}+\ldots+\frac1{BA_{n+1}}\right)^{-1}$$is independent of the line $l$, and compute its value if $\angle ABC=\phi$.
2013 Dutch IMO TST, 3
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.