Found problems: 85335
2020 AIME Problems, 1
Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$.
2014 Stanford Mathematics Tournament, 6
Let $E$ be an ellipse with major axis length $4$ and minor axis length $2$. Inscribe an equilateral triangle $ABC$ in $E$ such that $A$ lies on the minor axis and $BC$ is parallel to the major axis. Compute the area of $\vartriangle ABC$.
2015 Harvard-MIT Mathematics Tournament, 4
Let $ABCD$ be a cyclic quadrilateral with $AB=3$, $BC=2$, $CD=2$, $DA=4$. Let lines perpendicular to $\overline{BC}$ from $B$ and $C$ meet $\overline{AD}$ at $B'$ and $C'$, respectively. Let lines perpendicular to $\overline{BC}$ from $A$ and $D$ meet $\overline{AD}$ at $A'$ and $D'$, respectively. Compute the ratio $\frac{[BCC'B']}{[DAA'D']}$, where $[\overline{\omega}]$ denotes the area of figure $\overline{\omega}$.
2024 International Zhautykov Olympiad, 6
Let $G$ be the centroid of triangle $ABC$. Find the biggest $\alpha$ such that there exists a triangle for which there are at least three angles among $\angle GAB, \angle GAC, \angle GBA, \angle GBC, \angle GCA, \angle GCB$ which are $\geq \alpha$.
2014 IPhOO, 6
A square plate has side length $L$ and negligible thickness. It is laid down horizontally on a table and is then rotating about the axis $\overline{MN}$ where $M$ and $N$ are the midpoints of two adjacent sides of the square. The moment of inertia of the plate about this axis is $kmL^2$, where $m$ is the mass of the plate and $k$ is a real constant. Find $k$.
[color=red]Diagram will be added to this post very soon. If you want to look at it temporarily, see the PDF.[/color]
[i]Problem proposed by Ahaan Rungta[/i]
1979 IMO Longlists, 77
By $h(n)$, where $n$ is an integer greater than $1$, let us denote the greatest prime divisor of the number $n$. Are there infinitely many numbers $n$ for which $h(n) < h(n+1)< h(n+2)$ holds?
2012 AMC 8, 11
The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, $x$ are all equal. What is the value of $x$?
$\textbf{(A)}\hspace{.05in}5 \qquad \textbf{(B)}\hspace{.05in}6 \qquad \textbf{(C)}\hspace{.05in}7 \qquad \textbf{(D)}\hspace{.05in}11 \qquad \textbf{(E)}\hspace{.05in}12 $
2013 BMT Spring, 7
If $x,y$ are positive real numbers satisfying $x^3-xy+1=y^3$, find the minimum possible value of $y$.
2021 Nordic, 2
Find all functions $f:R->R$ satisfying that for every $x$ (real number):
$f(x)(1+|f(x)|)\geq x \geq f(x(1+|x|))$
MOAA Gunga Bowls, 2021.6
Determine the number of triangles, of any size and shape, in the following figure:
[asy]
size(4cm);
draw(2*dir(0)--dir(120)--dir(240)--cycle);
draw(dir(60)--2*dir(180)--dir(300)--cycle);
[/asy]
[i]Proposed by William Yue[/i]
2015 Costa Rica - Final Round, A3
Knowing that $ b$ is a real constant such that $b\ge 1$, determine the sum of the real solutions of the equation $$x =\sqrt{b-\sqrt{b+x}}$$
2010 Irish Math Olympiad, 5
Find all polynomials $f(x)=x^3+bx^2+cx+d$, where $b,c,d,$ are real numbers, such that $f(x^2-2)=-f(-x)f(x)$.
2015 Danube Mathematical Competition, 3
Solve in N $a^2 = 2^b3^c + 1$.
2020 MBMT, 26
Let $\triangle MBT$ be a triangle with $\overline{MB} = 4$ and $\overline{MT} = 7$. Furthermore, let circle $\omega$ be a circle with center $O$ which is tangent to $\overline{MB}$ at $B$ and $\overline{MT}$ at some point on segment $\overline{MT}$. Given $\overline{OM} = 6$ and $\omega$ intersects $ \overline{BT}$ at $I \neq B$, find the length of $\overline{TI}$.
[i]Proposed by Chad Yu[/i]
2016 IFYM, Sozopol, 5
Find all pairs of integers $(x,y)$ for which $x^z+z^x=(x+z)!$.
2011 Morocco National Olympiad, 2
Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle.
$(a)$ Prove that
\[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\]
$(b)$ When do we have equality?
1964 Spain Mathematical Olympiad, 2
The RTP tax is a function $f(x)$, where $x$ is the total of the annual profits (in pesetas). Knowing that:
a) $f(x)$ is a continuous function
b) The derivative $\frac{df(x)}{dx}$ on the interval $0 \leq 6000$ is constant and equals zero; in the interval $6000< x < P$ is constant and equals $1$; and when $x>P$ is constant and equal 0.14.
c) $f(0)=0$ and $f(140000)=14000$.
Determine the value of the amount $P$ (in pesetas) and represent graphically the function $y=f(x)$.
2020 USA TSTST, 4
Find all pairs of positive integers $(a,b)$ satisfying the following conditions:
[list]
[*] $a$ divides $b^4+1$,
[*] $b$ divides $a^4+1$,
[*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$.
[/list]
[i]Yang Liu[/i]
2000 All-Russian Olympiad Regional Round, 11.8
There are $2000$ cities in the country, some pairs of cities are connected by roads. It is known that no more than $N$ different non-self-intersecting cyclic routes of odd length. Prove that the country can be divided into $N + 2$ republics so that no two cities from the same republic are connected by a road.
2018 Balkan MO Shortlist, N5
Let $x,y$ be positive integers. If for each positive integer $n$ we have that $$(ny)^2+1\mid x^{\varphi(n)}-1.$$
Prove that $x=1$.
[i](Silouanos Brazitikos, Greece)[/i]
2011 China Western Mathematical Olympiad, 3
Let $n \geq 2$ be a given integer
$a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$
$b)$ Determine all possible values of the sum $\sum \limits_{i = 1}^{2^n} (-1)^{i}S(A_{i})$ where $S(A_{i})$ denotes the sum of all elements in $A_{i}$ and $S(\emptyset) = 0$, for any subset sequence $A_{1},A_{2},\cdots ,A_{2^n}$ satisfying the condition in $a)$
2016 Taiwan TST Round 3, 4
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
2016 PUMaC Number Theory B, 4
For a positive integer $n$, let $P(n)$ be the product of the factors of $n$ (including $n$ itself).
A positive integer $n$ is called [i]deplorable [/i] if $n > 1$ and $\log_n P(n)$ is an odd integer.
How many factors of $2016$ are [i]deplorable[/i]?
1998 Tuymaada Olympiad, 7
All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.
1968 Miklós Schweitzer, 4
Let $ f$ be a complex-valued, completely multiplicative,arithmetical function. Assume that there exists an infinite increasing sequence $ N_k$ of natural numbers such that \[ f(n)\equal{}A_k \not\equal{} 0 \;\textrm{provided}\ \; N_k \leq n \leq N_k\plus{}4 \sqrt{N_k}\
.\] Prove that $ f$ is identically $ 1$.
[i]I. Katai[/i]