Found problems: 85335
Bangladesh Mathematical Olympiad 2020 Final, #7
Tiham is trying to find [b]6[/b] digit positive integers$ PQRSTU$ (where $PQRSTU $are not necessarily distinct). But he only wants the numbers where the sum of the [b]3[/b] digit number$ PQR$, and the [b]3[/b] digit number $STU$ is divisible by [b]37[/b]. How many such numbers Tiham can find?
2010 Purple Comet Problems, 7
Find the sum of the digits in the decimal representation of the number $5^{2010} \cdot 16^{502}.$
1960 IMO Shortlist, 2
For what values of the variable $x$ does the following inequality hold: \[ \dfrac{4x^2}{(1-\sqrt{2x+1})^2}<2x+9 \ ? \]
1995 Poland - First Round, 4
A line tangent to the incircle of the equilateral triangle ABC intersects the sides AB and BC at points D and E respectively. Prove that
$\frac{AD}{DB}+\frac{AE}{EC} = 1$.
2007 iTest Tournament of Champions, 2
The area of triangle $ABC$ is $2007$. One of its sides has length $18$, and the tangent of the angle opposite that side is $2007/24832$. When the altitude is dropped to the side of length $18$, it cuts that side into two segments. Find the sum of the squares of those two segments.
2024 Argentina National Olympiad Level 2, 3
[b]a)[/b] Find an example of an infinite list of numbers of the form $a + n \cdot d$, with $n \geqslant 0$, where $a$ and $d$ are positive integers, such that no number in the list is equal to the $k$-th power of an integer, for all $k = 2, 3, 4, \dots$
[b]b)[/b] Find an example of an infinite list of numbers of the form $a + n \cdot d$, with $n \geqslant 0$, where $a$ and $d$ are positive integers, such that no number in the list is equal to the square of an integer, but the list contains infinitely many numbers that are equal to the cubes of positive integers.
2011 Stars Of Mathematics, 3
The checkered plane is painted black and white, after a chessboard fashion. A polygon $\Pi$ of area $S$ and perimeter $P$ consists of some of these unit squares (i.e., its sides go along the borders of the squares).
Prove the polygon $\Pi$ contains not more than $\dfrac {S} {2} + \dfrac {P} {8}$, and not less than $\dfrac {S} {2} - \dfrac {P} {8}$ squares of a same color.
(Alexander Magazinov)
1988 Spain Mathematical Olympiad, 2
We choose $n > 3$ points on a circle and number them $1$ to $ n$ in some order. We say that two non-adjacent points $A$ and $B$ are related if, in one of the arcs $AB$, all the points are marked with numbers less than those at $A,B$. Show that the number of pairs of related points is exactly $n-3$.
2017 Olympic Revenge, 4
Let $f:\mathbb{R}_{+}^{*}$$\rightarrow$$\mathbb{R}_{+}^{*}$ such that $f'''(x)>0$ for all $x$ $\in$ $\mathbb{R}_{+}^{*}$. Prove that:
$f(a^{2}+b^{2}+c^{2})+2f(ab+bc+ac)$ $\geq$ $f(a^{2}+2bc)+f(b^{2}+2ca)+f(c^{2}+2ab)$, for all $a,b,c$ $\in$ $\mathbb{R}_{+}^{*}$.
2018 Iran Team Selection Test, 3
Let $a_1,a_2,a_3,\cdots $ be an infinite sequence of distinct integers. Prove that there are infinitely many primes $p$ that distinct positive integers $i,j,k$ can be found such that $p\mid a_ia_ja_k-1$.
[i]Proposed by Mohsen Jamali[/i]
2017 AMC 12/AHSME, 17
There are 24 different complex numbers $z$ such that $z^{24} = 1$. For how many of these is $z^6$ a real number?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }12\qquad\textbf{(E) }24$
2019 Taiwan TST Round 2, 2
Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.
2024 Indonesia TST, C
Let $A$ be a set with $1000$ members and $\mathcal F =${$A_1,A_2,\ldots,A_n$} a family of subsets of A such that
(a) Each element in $\mathcal F$ consists of 3 members
(b) For every five elements in $\mathcal F$, the union of them all will have at least $12$ members
Find the largest value of $n$
2009 Polish MO Finals, 6
Let $ n$ be a natural number equal or greater than 3 . A sequence of non-negative numbers $ (c_0,c_1,\ldots,c_n)$ satisfies the condition: $ c_{p}c_{s}\plus{}c_{r}c_{t}\equal{} c_{p\plus{}r}c_{r\plus{}s}$ for all non-negative $ p,q,r,s$ such that $ p\plus{}q\plus{}r\plus{}s\equal{}n$. Determine all possible values of $ c_2$ when $ c_1\equal{}1$.
2005 MOP Homework, 4
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(x^3)-f(y^3)=(x^2+xy+y^2)(f(x)-f(y))$.
2012 Indonesia TST, 4
Determine all integer $n > 1$ such that
\[\gcd \left( n, \dfrac{n-m}{\gcd(n,m)} \right) = 1\]
for all integer $1 \le m < n$.
2013 Estonia Team Selection Test, 6
A class consists of $7$ boys and $13$ girls. During the first three months of the school year, each boy has communicated with each girl at least once. Prove that there exist two boys and two girls such that both boys communicated with both girls first time in the same month.
2007 USA Team Selection Test, 4
Determine whether or not there exist positive integers $ a$ and $ b$ such that $ a$ does not divide $ b^n \minus{} n$ for all positive integers $ n$.
2016 Harvard-MIT Mathematics Tournament, 5
Find all prime numbers $p$ such that $y^2 = x^3+4x$ has exactly $p$ solutions in integers modulo $p$.
In other words, determine all prime numbers $p$ with the following property:
there exist exactly $p$ ordered pairs of integers $(x,y)$ such that $x,y \in \{0,1,\dots,p-1\}$
and \[ p \text{ divides } y^2 - x^3 - 4x. \]
2023 Mongolian Mathematical Olympiad, 1
Let $u, v$ be arbitrary positive real numbers. Prove that \[\min{(u, \frac{100}{v}, v+\frac{2023}{u})} \leq \sqrt{2123}.\]
2023 BMT, 2
Consider an equilateral triangle with side length $9$. Each side is divided into $3$ equal segments by $2$ points, for a total of $6$ points. Compute the area of the circle passing through these$ 6$ points.
[img]https://cdn.artofproblemsolving.com/attachments/7/b/1860a3ff86a0e4b93a4891861300dcb09adad4.png[/img]
1982 Vietnam National Olympiad, 2
Let $p$ be a positive integer and $q, z$ be real numbers with $0\le q\le 1$ and $q^{p+1}\le z\le 1$. Prove that
\[\prod_{k=1}^p \left|\frac{z - q^k}{z + q^k}\right| \le\prod_{k=1}^p \left|\frac{1 - q^k}{1 + q^k}\right|.\]
2018 Taiwan TST Round 3, 1
Suppose that $x,y$ are distinct positive reals, and $n>1$ is a positive integer. If
\[x^n-y^n=x^{n+1}-y^{n+1},\]
then show that
\[1<x+y<\frac{2n}{n+1}.\]
2016 CCA Math Bonanza, T1
It takes $3$ rabbits $5$ hours to dig $9$ holes. It takes $5$ beavers $36$ minutes to build $2$ dams. At this rate, how many more minutes does it take $1$ rabbit to dig $1$ hole than it takes $1$ beaver to build $1$ dam?
[i]2016 CCA Math Bonanza Team #1[/i]
1993 Hungary-Israel Binational, 3
In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group.
Show that every element of $S_{n}$ is a product of $2$-cycles.