Found problems: 85335
1985 Iran MO (2nd round), 3
Let $f: \mathbb R \to \mathbb R,g: \mathbb R \to \mathbb R$ and $\varphi: \mathbb R \to \mathbb R$ be three ascendant functions such that
\[f(x) \leq g(x) \leq \varphi(x) \qquad \forall x \in \mathbb R.\]
Prove that
\[f(f(x)) \leq g(g(x)) \leq \varphi(\varphi(x)) \qquad \forall x \in \mathbb R.\]
[i]Note. The function is $k(x)$ ascendant if for every $ x,y \in D_k, x \leq {y}$ we have $g(x)\leq{g(y)}$.[/i]
2023 Math Prize for Girls Problems, 20
Let $f_1(x) = 2\pi \sin (x)$. For $n > 1$, define $f_n(x)$ recursively by
\[
f_n(x) = 2\pi \sin(f_{n-1}(x)).
\]
How many intervals $[a, b]$ are there such that
$\quad \bullet \ $ $0 \le a < b \le 2\pi$,
$\quad \bullet \ $ $f_6(a) = -2\pi$,
$\quad \bullet \ $ $f_6(b)=2\pi$,
$\quad \bullet \ $ and $f_6$ is increasing on $[a, b]$?
2007 Harvard-MIT Mathematics Tournament, 17
During the regular season, Washington Redskins achieve a record of $10$ wins and $6$ losses. Compute the probability that their wins came in three streaks of consecutive wins, assuming that all possible arrangements of wins and losses are equally likely. (For example, the record $LLWWWWWLWWLWWWLL$ contains three winning streaks, while $WWWWWWWLLLLLLWWW$ has just two.)
2023 AIME, 1
Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2020 Miklós Schweitzer, 10
Let $f$ be a polynomial of degree $n$ with integer coefficients and $p$ a prime for which $f$, considered modulo $p$, is a degree-$k$ irreducible polynomial over $\mathbb{F}_p$. Show that $k$ divides the degree of the splitting field of $f$ over $\mathbb{Q}$.
1969 AMC 12/AHSME, 19
The number of distinct ordered pairs $(x,y)$, where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$, is:
$\textbf{(A) }0\qquad
\textbf{(B) }3\qquad
\textbf{(C) }4\qquad
\textbf{(D) }12\qquad
\textbf{(E) }\text{infinite}$
2019 BMT Spring, 10
A [i]3-4-5 point[/i] of a triangle $ ABC $ is a point $ P $ such that the ratio $ AP : BP : CP $ is equivalent to the ratio $ 3 : 4 : 5 $. If $ \triangle ABC $ is isosceles with base $ BC = 12 $ and $ \triangle ABC $ has exactly one $ 3-4-5 $ point, compute the area of $ \triangle ABC $.
2018 Latvia Baltic Way TST, P2
Find all ordered pairs $(x,y)$ of real numbers that satisfy the following system of equations:
$$\begin{cases}
y(x+y)^2=2\\
8y(x^3-y^3) = 13.
\end{cases}$$
2008 District Olympiad, 3
Let $ABCDA' B' C' D '$ be a cube , $M$ the foot of the perpendicular from $A$ on the plane $(A'CD)$, $N$ the foot of the perpendicular from $B$ on the diagonal $A'C$ and $P$ is symmetric of the point $D$ with respect to $C$. Show that the points $M, N, P$ are collinear.
1987 IberoAmerican, 3
Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be the points on the sides $AD$ and $BC$ respectively such that $\frac{AP}{PD}=\frac{BQ}{QC}=\frac{AB}{CD}$.
Prove that the line $PQ$ forms equal angles with the lines $AB$ and $CD$.
2023 Stanford Mathematics Tournament, 1
For all positive integers $n > 1$, let $f(n)$ denote the largest odd proper divisor of $n$ (a proper divisor of $n$ is a positive divisor of $n$ except for $n$ itself). Given that $N=20^{23}\cdot23^{20}$, compute
\[\frac{f(N)}{f(f(f(N)))}.\]
1971 IMO Longlists, 47
A sequence of real numbers $x_1,x_2,\ldots ,x_n$ is given such that $x_{i+1}=x_i+\frac{1}{30000}\sqrt{1-x_i^2},\ i=1,2,\ldots ,$ and $x_1=0$. Can $n$ be equal to $50000$ if $x_n<1$?
2023 AMC 12/AHSME, 9
A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
[asy]
size(200);
defaultpen(linewidth(0.6pt)+fontsize(10pt));
real y = sqrt(3);
pair A,B,C,D,E,F,G,H;
A = (0,0);
B = (0,y);
C = (y,y);
D = (y,0);
E = ((y + 1)/2,y);
F = (y, (y - 1)/2);
G = ((y - 1)/2, 0);
H = (0,(y + 1)/2);
fill(H--B--E--cycle, gray);
draw(A--B--C--D--cycle);
draw(E--F--G--H--cycle);
[/asy]
$\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1$
2012 Moldova Team Selection Test, 2
Positive integers $a,b$ are such that $137$ divides $a+139b$ and $139$ divides $a+137b$. Find the minimal posible value of $a+b$.
2011 AMC 12/AHSME, 21
Let $f_1(x)=\sqrt{1-x}$, and for integers $n \ge 2$, let $f_n(x)=f_{n-1}(\sqrt{n^2-x})$. If $N$ is the largest value of $n$ for which the domain of $f_n$ is nonempty, the domain of $f_N$ is ${c}$. What is $N+c$?
$ \textbf{(A)}\ -226 \qquad
\textbf{(B)}\ -144 \qquad
\textbf{(C)}\ -20 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 144$
2013 Balkan MO Shortlist, N6
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$
2013 National Chemistry Olympiad, 3
What mass of the compound $\ce{CrO3}$ $\text{(M = 100.0)}$ contains $4.5\times10^{23}$ oxygen atoms?
$ \textbf{(A) }\text{2.25 g}\qquad\textbf{(B) }\text{12.0 g}\qquad\textbf{(C) }\text{25.0 g}\qquad\textbf{(D) }\text{75.0 g}\qquad$
2011 Bosnia Herzegovina Team Selection Test, 3
In quadrilateral $ABCD$ sides $AD$ and $BC$ aren't parallel. Diagonals $AC$ and $BD$ intersect in $E$. $F$ and $G$ are points on sides $AB$ and $DC$ such $\frac{AF}{FB}=\frac{DG}{GC}=\frac{AD}{BC}$ Prove that if $E, F, G$ are collinear then $ABCD$ is cyclic.
2014 Purple Comet Problems, 9
Find $n$ such that\[\frac{1!\cdot2!\cdot3!\cdots10!}{(1!)^2(3!)^2(5!)^2(7!)^2(9!)^2}=15\cdot2^n.\]
1989 IMO Shortlist, 11
Define sequence $ (a_n)$ by $ \sum_{d|n} a_d \equal{} 2^n.$ Show that $ n|a_n.$
2023 Myanmar IMO Training, 4
Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$.
[i]Proposed by Angelo Di Pasquale, Australia[/i]
2009 VTRMC, Problem 1
A walker and a jogger travel along the same straight line in the same direction. The walker walks at one meter per second, while the jogger runs at two meters per second. The jogger starts one meter in front of the walker. A dog starts with the walker, and then runs back and forth between the walker and the jogger with constant speed of three meters per second. Let $f(n)$ meters denote the total distance travelled by the dog when it has returned to the walker for the nth time (so $f(0)=0$). Find a formula for $f(n)$.
2016 Argentina National Olympiad Level 2, 3
Nico wants to write the $100$ integers from $1$ to $100$ around a circle in some order and without repetition, such that they have the following property: when moving around the circle clockwise, the sum of the $100$ distances between each number and its next number is equal to $198$. Determine in how many ways the $100$ numbers can be ordered so that Nico achieves his goal.
[b]Note:[/b] The distance between two numbers $a$ and $b$ is equal to the absolute value of their difference: $|a - b|$.
2020 Harvard-MIT Mathematics Tournament, 6
Let $n > 1$ be a positive integer and $S$ be a collection of $\frac{1}{2}\binom{2n}{n}$ distinct $n$-element subsets of $\{1, 2, \dotsc, 2n\}$. Show that there exists $A, B\in S$ such that $|A\cap B|\leq 1$.
[i]Proposed by Michael Ren.[/i]
Ukrainian TYM Qualifying - geometry, 2019.9
On the base $BC$ of the isosceles triangle $ABC$ chose a point $D$ and in each of the triangles $ABD$ and $ACD$ inscribe a circle. Then everything was wiped, leaving only two circles. It is known from which side of their line of centers
the apex $A$ is located . Use a compass and ruler to restore the triangle $ABC$ , if we know that :
a) $AD$ is angle bisector,
b) $AD$ is median.