Found problems: 85335
2009 AMC 10, 9
Positive integers $ a$, $ b$, and $ 2009$, with $ a<b<2009$, form a geometric sequence with an integer ratio. What is $ a$?
$ \textbf{(A)}\ 7 \qquad
\textbf{(B)}\ 41 \qquad
\textbf{(C)}\ 49 \qquad
\textbf{(D)}\ 289 \qquad
\textbf{(E)}\ 2009$
2019 Moroccan TST, 5
Let $n$ be a nonzero even integer. We fill up all the cells of an $n\times n$ grid with $+$ and $-$ signs ensuring that the number of $+$ signs equals the number of $-$ signs.
Show that there exists two rows with the same number of $+$ signs or two collumns with the same number of $+$ signs.
2004 Germany Team Selection Test, 1
A function $f$ satisfies the equation
\[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\]
for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.
2015 Czech-Polish-Slovak Junior Match, 6
The vertices of the cube are assigned $1, 2, 3..., 8$ and then each edge we assign the product of the numbers assigned to its two extreme points. Determine the greatest possible the value of the sum of the numbers assigned to all twelve edges of the cube.
LMT Speed Rounds, 2
Eddie has a study block that lasts $1$ hour. It takes Eddie $25$ minutes to do his homework and $5$ minutes to play a game of Clash Royale. He can’t do both at the same time. How many games can he play in this study block while still completing his homework?
[i]Proposed by Edwin Zhao[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{7}$
Study block lasts 60 minutes, thus he has 35 minutes to play Clash Royale, during which he can play $\frac{35}{5}=\boxed{7}$ games.
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1997 Irish Math Olympiad, 3
Let $ A$ be a subset of $ \{ 0,1,2,...,1997 \}$ containing more than $ 1000$ elements. Prove that either $ A$ contains a power of $ 2$ (that is, a number of the form $ 2^k$ with $ k\equal{}0,1,2,...)$ or there exist two distinct elements $ a,b \in A$ such that $ a\plus{}b$ is a power of $ 2$.
2013 AMC 10, 20
The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $
2021 Baltic Way, 19
Find all polynomials $p$ with integer coefficients such that the number $p(a) - p(b)$ is divisible by $a + b$ for all integers $a, b$, provided that $a + b \neq 0$.
1996 National High School Mathematics League, 3
$\odot O_1$ and $\odot O_2$ are escribed circles of $\triangle ABC$ ($\odot O_1$ is in $\angle ACB$, $\odot O_2$ is in $\angle ABC$). $\odot O_1$ touches $CB,CA$ at $E,G$; $\odot O_1$ touches $BC,BA$ at $F,H$. $EG\cap FG=P$, prove that $AP\perp BC$.
2003 District Olympiad, 2
Let $M \subset R$ be a finite set containing at least two elements. We say that the function $f$ has property $P$ if $f : M \to M$ and there are $a \in R^*$ and $b \in R$ such that $f(x) = ax + b$.
(a) Show that there is at least a function having property $P$.
(b) Show that there are at most two functions having property $P$.
(c) If $M$ has $2003$ elements with sum $0$ and if there are two functions with property $P$, prove that $0 \in M$.
2013 European Mathematical Cup, 4
Given a triangle $ABC$ let $D$, $E$, $F$ be orthogonal projections from $A$, $B$, $C$ to the opposite sides respectively. Let $X$, $Y$, $Z$ denote midpoints of $AD$, $BE$, $CF$ respectively. Prove that perpendiculars from $D$ to $YZ$, from $E$ to $XZ$ and from $F$ to $XY$ are concurrent.
2020 South East Mathematical Olympiad, 7
Given any prime $p \ge 3$. Show that for all sufficient large positive integer $x$, at least one of $x+1,x+2,\cdots,x+\frac{p+3}{2}$ has a prime divisor greater than $p$.
2024 Harvard-MIT Mathematics Tournament, 8
Let $P$ be a point in the interior of quadrilateral $ABCD$ such that the circumcircles of triangles $PDA, PAB,$ and $PBC$ are pairwise distinct but congruent. Let the lines $AD$ and $BC$ meet at $X$. If $O$ is the circumcenter of triangle $XCD$, prove that $OP \perp AB$.
2015 Online Math Open Problems, 20
Amandine and Brennon play a turn-based game, with Amadine starting.
On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers.
For example, if $3, 5$ have been selected so far, only $1, 2, 4, 7$ are available to be picked;
if only $3$ has been selected so far, all numbers not divisible by three are eligible.
A player loses immediately if they select the integer $1$.
Call a number $n$ [i]feminist[/i] if $\gcd(n, 6) = 1$ and if Amandine wins if she starts with $n$. Compute the sum of the [i]feminist[/i] numbers less than $40$.
[i]Proposed by Ashwin Sah[/i]
2006 All-Russian Olympiad, 6
Let $K$ and $L$ be two points on the arcs $AB$ and $BC$ of the circumcircle of a triangle $ABC$, respectively, such that $KL\parallel AC$. Show that the incenters of triangles $ABK$ and $CBL$ are equidistant from the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$.
2013 Switzerland - Final Round, 3
Let $ABCD$ be a cyclic quadrilateral with $\angle ADC = \angle DBA$. Furthermore, let $E$ be the projection of $A$ on $BD$. Show that $BC = DE - BE$ .
2012 Irish Math Olympiad, 1
Let $$C=\{1,22,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}$$ and let
$$S=\{4,5,9,14,23,37\}$$ Find two sets $A$ and $B$ with the properties
(a) $A\cap B=\emptyset$.
(b) $A\cup B=C$.
(c) The sum of two distinct elements of $A$ is not in $S$.
(d) The sum of two distinct elements of $B$ is not in $S$.
ICMC 6, 5
Let $[0, 1]$ be the set $\{x \in \mathbb{R} : 0 \leq x \leq 1\}$. Does there exist a continuous function $g : [0, 1] \to [0, 1]$ such that no line intersects the graph of $g$ infinitely many times, but for any positive integer $n$ there is a line intersecting $g$ more than $n$ times?
[i]Proposed by Ethan Tan[/i]
1996 IberoAmerican, 3
We have a grid of $k^2-k+1$ rows and $k^2-k+1$ columns, where $k=p+1$ and $p$ is prime. For each prime $p$, give a method to put the numbers 0 and 1, one number for each square in the grid, such that on each row there are exactly $k$ 0's, on each column there are exactly $k$ 0's, and there is no rectangle with sides parallel to the sides of the grid with 0s on each four vertices.
2020 AMC 10, 10
Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
$\textbf{(A) } 644 \qquad \textbf{(B) } 658 \qquad \textbf{(C) } 664 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 749$
2014 Silk Road, 1
What is the maximum number of coins can be arranged in cells of the table $n \times n$ (each cell is not more of the one coin) so that any coin was not simultaneously below and to the right than any other?
2011 AIME Problems, 5
The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits $1$ through $9$ in such a way that the sum of the numbers on every three consecutive vertices is a multiple of $3$. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.
2000 Taiwan National Olympiad, 1
Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.
2002 Paraguay Mathematical Olympiad, 2
In the rectangular parallelepiped in the figure, the lengths of the segments $EH$, $HG$, and $EG$ are consecutive integers. The height of the parallelepiped is $12$. Find the volume of the parallelepiped.
[img]https://cdn.artofproblemsolving.com/attachments/6/4/f74e7fed38c815bff5539613f76b0c4ca9171b.png[/img]
2016 India Regional Mathematical Olympiad, 8
At some integer points a polynomial with integer coefficients take values $1, 2$ and $3$. Prove that there exist not more than one integer at which the polynomial is equal to $5$.