Found problems: 85335
2008 AMC 8, 7
If $\frac{3}{5}=\frac{M}{45}=\frac{60}{N}$, what is $M+N$?
$\textbf{(A)}\ 27\qquad
\textbf{(B)}\ 29 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 105\qquad
\textbf{(E)}\ 127$
2022 Ecuador NMO (OMEC), 2
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y$
\[f(x + y)=f(f(x)) + y + 2022\]
2013 NIMO Problems, 8
The number $\frac{1}{2}$ is written on a blackboard. For a real number $c$ with $0 < c < 1$, a [i]$c$-splay[/i] is an operation in which every number $x$ on the board is erased and replaced by the two numbers $cx$ and $1-c(1-x)$. A [i]splay-sequence[/i] $C = (c_1,c_2,c_3,c_4)$ is an application of a $c_i$-splay for $i=1,2,3,4$ in that order, and its [i]power[/i] is defined by $P(C) = c_1c_2c_3c_4$.
Let $S$ be the set of splay-sequences which yield the numbers $\frac{1}{17}, \frac{2}{17}, \dots, \frac{16}{17}$ on the blackboard in some order. If $\sum_{C \in S} P(C) = \tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Lewis Chen[/i]
1999 Croatia National Olympiad, Problem 3
For each $a$, $1<a<2$, the graphs of functions $y=1-|x-1|$ and $y=|2x-a|$ determine a figure. Prove that the area of this figure is less than $\frac13$.
1980 IMO, 7
Prove that $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x,y$ with $x\le 1980$.
[i] Variant: [/i] Prove the equation $4x^3-3x+1=2y^2$ has infinitely many solutions in positive integers x,y.
2009 Hanoi Open Mathematics Competitions, 6
Suppose that $4$ real numbers $a, b,c,d$ satisfy the conditions $\begin{cases} a^2 + b^2 = 4\\
c^2 + d^2 = 4 \\
ac + bd = 2 \end{cases}$
Find the set of all possible values the number $M = ab + cd$ can take.
Swiss NMO - geometry, 2022.1
Let $k$ be a circle with centre $M$ and let $AB$ be a diameter of $k$. Furthermore, let $C$ be a point on $k$ such that $AC = AM$. Let $D$ be the point on the line $AC$ such that $CD = AB$ and $C$ lies between $A$ and $D$. Let $E$ be the second intersection of the circumcircle of $BCD$ with line $AB$ and $F$ be the intersection of the lines $ED$ and $BC$. The line $AF$ cuts the segment $BD$ in $X$. Determine the ratio $BX/XD$.
1992 Flanders Math Olympiad, 3
a conic with apotheme 1 slides (varying height and radius, with $r < \frac12$) so that the conic's area is $9$ times that of its inscribed sphere. What's the height of that conic?
2020 Iran MO (3rd Round), 1
Find all positive integers $n$ such that the following holds.
$$\tau(n)|2^{\sigma(n)}-1$$
2022 Stanford Mathematics Tournament, 8
Given that
\[A=\sum_{n=1}^\infty\frac{\sin(n)}{n},\]
determine $\lfloor100A\rfloor$.
2019 Romania Team Selection Test, 3
Let be three positive integers $ a,b,c $ and a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ defined as
$$ f(n)=\left\{ \begin{matrix} n-a, & n>c\\ f\left( f(n+b) \right) ,& n\le c \end{matrix} \right. . $$
Determine the number of fixed points this function has.
2010 Malaysia National Olympiad, 6
Find the number of different pairs of positive integers $(a,b)$ for which $a+b\le100$ and \[\dfrac{a+\frac{1}{b}}{\frac{1}{a}+b}=10\]
1969 IMO Longlists, 60
$(SWE 3)$ Find the natural number $n$ with the following properties:
$(1)$ Let $S = \{P_1, P_2, \cdots\}$ be an arbitrary finite set of points in the plane, and $r_j$ the distance from $P_j$ to the origin $O.$ We assign to each $P_j$ the closed disk $D_j$ with center $P_j$ and radius $r_j$. Then some $n$ of these disks contain all points of $S.$
$(2)$ $n$ is the smallest integer with the above property.
2010 Postal Coaching, 4
How many ordered triples $(a, b, c)$ of positive integers are there such that none of $a, b, c$ exceeds $2010$ and each of $a, b, c$ divides $a + b + c$?
2014 JHMMC 7 Contest, 25
If a triangle has three altitudes of lengths $6, 6, \text{and} 6,$ what is its perimeter?
1972 IMO Longlists, 1
Find all integer solutions of the equation
\[1 + x + x^2 + x^3 + x^4 = y^4.\]
2020 BMT Fall, 23
Circle $\Gamma$ has radius $10$, center $O$, and diameter $AB$. Point $C$ lies on $\Gamma$ such that $AC = 12$. Let $P$ be the circumcenter of $\vartriangle AOC$. Line $AP$ intersects $\Gamma$ at $Q$, where $Q$ is different from $A$. Then the value of $\frac{AP}{AQ}$ can be expressed in the form $\frac{m}{n}$, where m and n are relatively prime positive integers. Compute $m + n$.
2007 Moldova National Olympiad, 11.2
Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$.
Find $\lim_{n\rightarrow\infty}a_{n}$.
1999 IberoAmerican, 2
An acute triangle $\triangle{ABC}$ is inscribed in a circle with centre $O$. The altitudes of the triangle are $AD,BE$ and $CF$. The line $EF$ cut the circumference on $P$ and $Q$.
a) Show that $OA$ is perpendicular to $PQ$.
b) If $M$ is the midpoint of $BC$, show that $AP^2=2AD\cdot{OM}$.
2005 AMC 10, 11
A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$?
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 7$
2005 China Team Selection Test, 1
Prove that for any $n$ ($n \geq 2$) pairwise distinct fractions in the interval $(0,1)$, the sum of their denominators is no less than $\frac{1}{3} n^{\frac{3}{2}}$.
2016 Serbia Additional Team Selection Test, 1
Let $P_0(x)=x^3-4x$. Sequence of polynomials is defined as following:\\
$P_{n+1}=P_n(1+x)P_n(1-x)-1$.\\
Prove that $x^{2016}|P_{2016}(x)$.
2022 CMIMC, 12
Let $ABCD$ be a cyclic quadrilateral with $AB=3, BC=2, CD=6, DA=8,$ and circumcircle $\Gamma.$ The tangents to $\Gamma$ at $A$ and $C$ intersect at $P$ and the tangents to $\Gamma$ at $B$ and $D$ intersect at $Q.$ Suppose lines $PB$ and $PD$ intersect $\Gamma$ at points $W \neq B$ and $X \neq D,$ respectively. Similarly, suppose lines $QA$ and $QC$ intersect $\Gamma$ at points $Y \neq A$ and $Z \neq C,$ respectively. What is the value of $\frac{{WX}^2}{{YZ}^2}?$
[i]Proposed by Kyle Lee[/i]
2011 AIME Problems, 1
Gary purchased a large beverage, but drank only $m/n$ of this beverage, where $m$ and $n$ are relatively prime positive integers. If Gary had purchased only half as much and drunk twice as much, he would have wasted only $\frac{2}{9}$ as much beverage. Find $m+n$.
2004 AMC 10, 3
Alicia earns $ \$20$ per hour, of which $ 1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
$ \textbf{(A)}\ 0.0029\qquad
\textbf{(B)}\ 0.029\qquad
\textbf{(C)}\ 0.29\qquad
\textbf{(D)}\ 2.9\qquad
\textbf{(E)}\ 29$