Found problems: 85335
2019 AMC 10, 14
The base-ten representation for $19!$ is $121,6T5,100,40M,832,H00$, where $T$, $M$, and $H$ denote digits that are not given. What is $T+M+H$?
$\textbf{(A) }3
\qquad\textbf{(B) }8
\qquad\textbf{(C) }12
\qquad\textbf{(D) }14
\qquad\textbf{(E) } 17 $
2023 Federal Competition For Advanced Students, P1, 3
Given a positive integer $n$, find the proportion of the subsets of $\{1,2, \ldots, 2n\}$ such that their smallest element is odd.
1996 Estonia National Olympiad, 1
Find all pairs of integers $(x, y)$ such that ths sum of the fractions $\frac{19}{x}$ and $\frac{96}{y}$ would be equal to their product.
2015 Princeton University Math Competition, 11
Given a rational number $r$ that, when expressed in base-$10$, is a repeating, non-terminating decimal, we define $f(r)$ to be the number of digits in the decimal representation of $r$ that are after the decimal point but before the repeating part of $r$. For example, $f(1.2\overline{7}) = 0$ and $f(0.35\overline{2}) = 2$. What is the smallest positive integer $n$ such that $\tfrac{1}{n}, \tfrac{2}{n}$, and $\tfrac{4}{n}$ are non-terminating decimals, where $f\left( \tfrac{1}{n} \right) = 3, f\left( \tfrac{2}{n} \right) = 3$, and $f\left( \tfrac{4}{n} \right) = 2$?.
1998 Moldova Team Selection Test, 11
Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.
2005 IMC, 3
What is the maximal dimension of a linear subspace $ V$ of the vector space of real $ n \times n$ matrices such that for all $ A$ in $ B$ in $ V$, we have $ \text{trace}\left(AB\right) \equal{} 0$ ?
2013 Tournament of Towns, 5
Do there exist two integer-valued functions $f$ and $g$ such that for every integer $x$ we have
(a) $f(f(x)) = x, g(g(x)) = x, f(g(x)) > x, g(f(x)) > x$ ?
(b) $f(f(x)) < x, g(g(x)) < x, f(g(x)) > x, g(f(x)) > x$ ?
2009 Rioplatense Mathematical Olympiad, Level 3, 2
Find all pairs $(a,b)$ of integers with $a>1$ and $b>1$ such that $a$ divides $b+1$ and $b$ divides $a^3-1$.
1986 Iran MO (2nd round), 1
$O$ is a point in the plane. Let $O'$ be an arbitrary point on the axis $Ox$ of the plane and let $M$ be an arbitrary point. Rotate $M$, $90^\circ$ clockwise around $O$ to get the point $M'$ and rotate $M$, $90^\circ$ anticlockwise around $O'$ to get the point $M''.$ Prove that the midpoint of the segment $MM''$ is a fixed point.
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 $$x =\sqrt{b-\sqrt{b+x}}$$