Found problems: 85335
KoMaL A Problems 2017/2018, A. 727
For any finite sequence $(x_1,\ldots,x_n)$, denote by $N(x_1,\ldots,x_n)$ the number of ordered index pairs $(i,j)$ for which $1 \le i<j\le n$ and $x_i=x_j$. Let $p$ be an odd prime, $1 \le n<p$, and let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be arbitrary residue classes modulo $p$. Prove that there exists a permutation $\pi$ of the indices $1,2,\ldots,n$ for which
\[N(a_1+b_{\pi(1)},a_2+b_{\pi(2)},\ldots,a_n+b_{\pi(n)})\le \min(N(a_1,a_2,\ldots,a_n),N(b_1,b_2,\ldots,b_n)).\]
2009 Greece National Olympiad, 4
Consider pairwise distinct complex numbers $z_1,z_2,z_3,z_4,z_5,z_6$ whose images $A_1,A_2,A_3,A_4,A_5,A_6$ respectively are succesive points on the circle centered at $O(0,0)$ and having radius $r>0.$
If $w$ is a root of the equation $z^2+z+1=0$ and the next equalities hold \[z_1w^2+z_3w+z_5=0 \\ z_2w^2+z_4w+z_6=0\] prove that
[b]a)[/b] Triangle $A_1A_3A_5$ is equilateral
[b]b)[/b] \[|z_1-z_2|+|z_2-z_3|+|z_3-z_4|+|z_4-z_5|+z_5-z_6|+|z_6-z_1|=3|z_1-z_4|=3|z_2-z_5|=3|z_3-z_6|.\]
2017 Purple Comet Problems, 3
When Phil and Shelley stand on a scale together, the scale reads $151$ pounds. When Shelley and Ryan stand on the same scale together, the scale reads $132$ pounds. When Phil and Ryan stand on the same scale together, the scale reads $115$ pounds. Find the number of pounds Shelley weighs.
2017 AMC 10, 16
How many of the base-ten numerals for the positive integers less than or equal to 2017 contain the digit 0?
$\textbf{(A)} \text{ 469} \qquad \textbf{(B)} \text{ 471} \qquad \textbf{(C)} \text{ 475} \qquad \textbf{(D)} \text{ 478} \qquad \textbf{(E)} \text{ 481}$
2017 OMMock - Mexico National Olympiad Mock Exam, 4
Show that the equation
$$a^2b=2017(a+b)$$
has no solutions for positive integers $a$ and $b$.
[i]Proposed by Oriol Solé[/i]
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]