Found problems: 85335
2021 Romania National Olympiad, 3
Given is an positive integer $a>2$
a) Prove that there exists positive integer $n$ different from $1$, which is not a prime, such that $a^n=1(mod n)$
b) Prove that if $p$ is the smallest positive integer, different from $1$, such that $a^p=1(mod p)$, then $p$ is a prime.
c) There does not exist positive integer $n$, different from $1$, such that $2^n=1(mod n)$
2016 ASDAN Math Tournament, 7
Let $x$, $y$, and $z$ be real numbers satisfying the equations
\begin{align*}
4x+2yz-6z+9xz^2&=4\\
xyz&=1.
\end{align*}
Find all possible values of $x+y+z$.
2021 HMNT, 4
The sum of the digits of the time $19$ minutes ago is two less than the sum of the digits of the time right now. Find the sum of the digits of the time in $19$ minutes. (Here, we use a standard $12$-hour clock of the form $hh:mm$.)
1963 Polish MO Finals, 3
From a given triangle, cut out the rectangle with the largest area.
2007 Tournament Of Towns, 1
(from The Good Soldier Svejk) Senior military doctor Bautze exposed $abccc$ malingerers among $aabbb$ draftees who claimed not to be fit for the military service. He managed to expose all but one draftees. (He would for sure expose this one too, if the lucky guy was not taken by a stroke at the very moment when the doctor yelled at him "Turn around !. . . ") How many malingerers were exposed by the vigilant doctor?
Each digit substitutes a letter. The same digits substitute the same letters, while distinct digits substitute distinct letters.
[i](1 point)[/i]
2016 Postal Coaching, 2
Determine all functions $f : \mathbb R \to \mathbb R$ such that $$f(f(x)- f(y)) = f(f(x)) - 2x^2f(y) + f\left(y^2\right),$$ for all reals $x, y$.
2017 India PRMO, 21
Attached below:
2016 Singapore MO Open, 5
A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$.
1984 AMC 12/AHSME, 2
If $x,y$ and $y - \frac{1}{x}$ are not 0, then \[\frac{x - \frac{1}{y}}{y - \frac{1}{x}}\] equals
$\textbf{(A) }1\qquad\textbf{(B) } \frac{x}{y}\qquad\textbf{(C) }\frac{y}{x}\qquad\textbf{(D) }\frac{x}{y} - \frac{y}{x}\qquad\textbf{(E) } xy - \frac{1}{xy}$
2024 Putnam, A3
Let $S$ be the set of bijections
\[
T\colon\{1,\,2,\,3\}\times\{1,\,2,\,\ldots,\,2024\}\to\{1,\,2,\,\ldots,\,6072\}
\]
such that $T(1,\,j)<T(2,\,j)<T(3,\,j)$ for all $j\in\{1,\,2,\,\ldots,\,2024\}$ and $T(i,\,j)<T(i,\,j+1)$ for all $i\in\{1,\,2,\,3\}$ and $j\in\{1,\,2,\,\ldots,\,2023\}$. Do there exist $a$ and $c$ in $\{1,\,2,\,3\}$ and $b$ and $d$ in $\{1,\,2,\,\ldots,\,2024\}$ such that the fraction of elements $T$ in $S$ for which $T(a,\,b)<T(c,\,d)$ is at least $1/3$ and at most $2/3$.
2002 AIME Problems, 13
In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
2005 Miklós Schweitzer, 9
prove that if $r_n$ is a rational function whose numerator and denominator have at most degrees $n$, then $$||r_n||_{1/2}+\left\|\frac{1}{r_n}\right\|_2\geq\frac{1}{2^{n-1}}$$ where $||\cdot||_a$ denotes the supremum over a circle of radius $a$ around the origin.
2024 Belarusian National Olympiad, 10.4
A parallelogram $ABCD$ is given. The incircle of triangle $ABC$ with center $I$ touches $AB,BC,CA$ at $R,P,Q$. Ray $DI$ intersects segment $AB$ at $S$. It turned out that $\angle DPR=90$
Prove that the circle with diameter $AS$ is tangent to the circumcircle of triangle $DPQ$
[i]M. Zorka[/i]
2014 AMC 12/AHSME, 25
What is the sum of all positive real solutions $x$ to the equation \[2\cos(2x)\left(\cos(2x)-\cos\left(\frac{2014\pi^2}{x}\right)\right)=\cos(4x)-1?\]
$\textbf{(A) }\pi\qquad
\textbf{(B) }810\pi\qquad
\textbf{(C) }1008\pi\qquad
\textbf{(D) }1080\pi\qquad
\textbf{(E) }1800\pi\qquad$
1982 IMO Longlists, 43
[b](a)[/b] What is the maximal number of acute angles in a convex polygon?
[b](b)[/b] Consider $m$ points in the interior of a convex $n$-gon. The $n$-gon is partitioned into triangles whose vertices are among the $n + m$ given points (the vertices of the $n$-gon and the given points). Each of the $m$ points in the interior is a vertex of at least one triangle. Find the number of triangles obtained.
2021 Peru MO (ONEM), 2
The numbers $1$ to $25$ will be written in a table $5 \times 5$. First, Ana chooses $k$ of these numbers($1$ to $25$), and write in some cells. Then, Enrique writes the remaining numbers with the following goal: The product of the numbers in some column/row is a perfect square.
[b]a)[/b] Prove that if $k=5$, Ana can [b]avoid[/b] Enrique to reach his goal.
[b]b)[/b] Prove that if $k=4$, Enrique can reach his goal.
1982 All Soviet Union Mathematical Olympiad, 342
What minimal number of numbers from the set $\{1,2,...,1982\}$ should be deleted to provide the property:
[i]none of the remained numbers equals to the product of two other remained numbers[/i]?
1998 All-Russian Olympiad Regional Round, 8.1
Are there $n$-digit numbers M and N such that all digits $M$ are even, all $N$ digits are odd, every digit from $0$ to $9$ occurs in decimal notation M or N at least once, and $M$ is divisible by $N$?
2021 Iran Team Selection Test, 6
Point $D$ is chosen on the Euler line of triangle $ABC$ and it is inside of the triangle. Points $E,F$ are were the line $BD,CD$ intersect with $AC,AB$ respectively. Point $X$ is on the line $AD$ such that $\angle EXF =180 - \angle A$, also $A,X$ are on the same side of $EF$. If $P$ is the second intersection of circumcircles of $CXF,BXE$ then prove the lines $XP,EF$ meet on the altitude of $A$
Proposed by [i]Alireza Danaie[/i]
1999 Romania Team Selection Test, 11
Let $a,n$ be integer numbers, $p$ a prime number such that $p>|a|+1$. Prove that the polynomial $f(x)=x^n+ax+p$ cannot be represented as a product of two integer polynomials.
[i]Laurentiu Panaitopol[/i]
2023 Rioplatense Mathematical Olympiad, 5
A positive integer $N$ is [i]rioplatense[/i] if it satifies the following conditions:
1 -There exist $34$ consecutive integers such that its product is divisible by $N$, but none of them is divisible by $N$.
2 - There [b]not[/b] exist $30$ consecutive integers such that its product is divisible by $N$, but none of them is divisible by $N$.
Determine all rioplatense numbers.
2005 iTest, 4
If the product of $(\sqrt2 +\sqrt3+\sqrt5) (\sqrt2 +\sqrt3-\sqrt5) (\sqrt2 -\sqrt3+\sqrt5) (-\sqrt2 +\sqrt3+\sqrt5)$ is $12\sqrt6+ 6\sqrt{x}$ , find $x$.
([i]0 points[/i] - [b]THROWN OUT[/b])
1984 Bulgaria National Olympiad, Problem 5
Let $0<x_i<1$ and $x_i+y_i=1$ for $i=1,2,\ldots,n$. Prove that
$$(1-x_1x_2\cdots x_n)^m+(1-y_1^m)(1-y_2^m)\cdots(1-y_n^m)>1$$for any natural numbers $m$ and $n$.
2021 USMCA, 28
How many functions $f : \mathbb{Z} \rightarrow \{0, 1, 2, \cdots, 2020 \}$ are there such that $f(n) = f(n+2021)$ and $2021 \mid f(2n) - f(n) - f(n-1)$ for all integers $n$?
2018 Harvard-MIT Mathematics Tournament, 3
There are two prime numbers $p$ so that $5p$ can be expressed in the form $\left\lfloor \dfrac{n^2}{5}\right\rfloor$ for some positive integer $n.$ What is the sum of these two prime numbers?