Found problems: 85335
2004 Thailand Mathematical Olympiad, 3
$18$ students with pairwise distinct heights line up. Ideally, the teacher wants the students to be ordered by height so that the tallest student is in the back of the line. However, it turns out that this is not the case, so when the teacher sees two consecutive students where the taller of the two is in front, the two students are swapped. It turns out that $150$ swaps must be made before the students are lined up in the correct order. How many possible starting orders are there?
2024 Pan-African, 4
Consider $m$ segments on the real line. Each segment has its two endpoints in the set of integers $\{1, 2, \ldots, 2024\}$, and no two segments have the same length. No segment is entirely contained in another segment, but two segments may partially overlap each other.
What is the maximum value of $m$?
2011 Iran MO (2nd Round), 1
find the smallest natural number $n$ such that there exists $n$ real numbers in the interval $(-1,1)$ such that their sum equals zero and the sum of their squares equals $20$.
2005 National High School Mathematics League, 5
Which kind of curve does the equation $\frac{x^2}{\sin\sqrt2-\sin\sqrt3}+\frac{y^2}{\cos\sqrt2-\cos\sqrt3}=1$ refer to?
$\text{(A)}$ An ellipse, whose focal points are on $x$-axis.
$\text{(B)}$ A hyperbola, whose focal points are on $x$-axis.
$\text{(C)}$ An ellipse, whose focal points are on $y$-axis.
$\text{(D)}$ A hyperbola, whose focal points are on $y$-axis.
2006 Stanford Mathematics Tournament, 15
Let $c_i$ denote the $i$th composite integer so that $\{c_i\}=4,6,8,9,...$ Compute
\[\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}\]
(Hint: $\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$)
1952 Miklós Schweitzer, 4
Let $ K$ be a finite field of $ p$ elements, where $ p$ is a prime. For every polynomial
$ f(x)\equal{}\sum_{i\equal{}0}^na_ix^i$ ($ \in K[x]$)
put
$ \overline{f(x)}\equal{}\sum_{i\equal{}0}^n a_ix^{p^i}$.
Prove that for any pair of polynomials $ f(x),g(x)\in K[x]$, $ \overline{f(x)}|\overline{g(x)}$ if and only if $ f(x)|g(x)$.
2010 Indonesia MO, 8
Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram.
[i]Raja Oktovin, Pekanbaru[/i]
2006 Romania National Olympiad, 1
We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.
2008 IMAR Test, 2
A point $ P$ of integer coordinates in the Cartesian plane is said [i]visible[/i] if the segment $ OP$ does not contain any other points with integer coordinates (except its ends). Prove that for any $ n\in\mathbb{N}^*$ there exists a visible point $ P_{n}$, at distance larger than $ n$ from any other visible point.
[b]Dan Schwarz[/b]
2000 Greece Junior Math Olympiad, 3
On a past Mathematical Olympiad the maximum possible score on a problem was 5. The average score of boys was 4, the average score of girls was 3.25, and the overall average score was 3.60. Find the total number of participants, knowing that it was in the range from 31 to 50.
2021 Sharygin Geometry Olympiad, 24
A truncated trigonal pyramid is circumscribed around a sphere touching its bases at points $T_1, T_2$. Let $h$ be the altitude of the pyramid, $R_1, R_2$ be the circumradii of its bases, and $O_1, O_2$ be the circumcenters of the bases. Prove that $$R_1R_2h^2 = (R_1^2-O_1T_1^2)(R_2^2-O_2T_2^2).$$
2015 Junior Balkan Team Selection Tests - Romania, 1
Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?
2011 Iran MO (2nd Round), 2
In triangle $ABC$, we have $\angle ABC=60$. The line through $B$ perpendicular to side $AB$ intersects angle bisector of $\angle BAC$ in $D$ and the line through $C$ perpendicular $BC$ intersects angle bisector of $\angle ABC$ in $E$. prove that $\angle BED\le 30$.
2014-2015 SDML (High School), 8
A penny is placed in the coordinate plane $\left(0,0\right)$. The penny can be moved $1$ unit to the right, $1$ unit up, or diagonally $1$ unit to the right and $1$ unit up. How many different ways are there for the penny to get to the point $\left(5,5\right)$?
$\text{(A) }8\qquad\text{(B) }25\qquad\text{(C) }99\qquad\text{(D) }260\qquad\text{(E) }351$
V Soros Olympiad 1998 - 99 (Russia), 11.1
Solve the equation
$$x^5 + (x + 1)^5 + (x + 2)^5 + ... + (x + 1998)^5 = 0.$$
2023 Caucasus Mathematical Olympiad, 1
Let $n{}$ and $m$ be positive integers, $n>m>1$. Let $n{}$ divided by $m$ have partial quotient $q$ and remainder $r$ (so that $n = qm + r$, where $r\in\{0,1,...,m-1\}$). Let $n-1$ divided by $m$ have partial quotient $q^{'}$ and remainder $r^{'}$.
a) It appears that $q+q^{'} =r +r^{'} = 99$. Find all possible values of $n{}$.
b) Prove that if $q+q^{'} =r +r^{'}$, then $2n$ is a perfect square.
2016 Spain Mathematical Olympiad, 5
From all possible permutations from $(a_1,a_2,...,a_n)$ from the set $\{1,2,..,n\}$, $n\geq 1$, consider the sets that satisfies the $2(a_1+a_2+...+a_m)$ is divisible by $m$, for every $m=1,2,...,n$. Find the total number of permutations.
2024 Mexican University Math Olympiad, 4
Given \( b > 0 \), consider the following matrix:
\[
B = \begin{pmatrix} b & b^2 \\ b^2 & b^3 \end{pmatrix}
\]
Denote by \( e_i \) the top left entry of \( B^i \). Prove that the following limit exists and calculate its value:
\[
\lim_{i \to \infty} \sqrt[i]{e_i}.
\]
2021 Romania National Olympiad, 1
In the cuboid $ABCDA'B'C'D'$ with $AB=a$, $AD=b$ and $AA'=c$ such that $a>b>c>0$, the points $E$ and $F$ are the orthogonal projections of $A$ on the lines $A'D$ and $A'B$, respectively, and the points $M$ and $N$ are the orthogonal projections of $C$ on the lines $C'D$ and $C'B$, respectively. Let $DF\cap BE=\{G\}$ and $DN\cap BM=\{P\}$.
[list=a]
[*] Show that $(A'AG)\parallel (C'CP)$ and determine the distance between these two planes;
[*] Show that $GP\parallel (ABC)$ and determine the distance between the line $GP$ and the plane $(ABC)$.
[/list]
[i]Petre Simion, Nicolae Victor Ioan[/i]
1989 Romania Team Selection Test, 1
Let $M$ denote the set of $m\times n$ matrices with entries in the set $\{0,1,2,3,4\}$ such that in each row and each column the sum of elements is divisible by $5$. Find the cardinality of set $M$.
2021 Macedonian Mathematical Olympiad, Problem 3
Let $ABCD$ be a trapezoid with $AD \parallel BC$ and $\angle BCD < \angle ABC < 90^\circ$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$. The circumcircle $\omega$ of $\triangle BEC$ intersects the segment $CD$ at $X$. The lines $AX$ and $BC$ intersect at $Y$, while the lines $BX$ and $AD$ intersect at $Z$. Prove that the line $EZ$ is tangent to $\omega$ iff the line $BE$ is tangent to the circumcircle of $\triangle BXY$.
2006 Harvard-MIT Mathematics Tournament, 6
A circle of radius $t$ is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius $r=1+\sin \frac{\pi}{8}$. Find $rt$.
2024 Malaysia IMONST 2, 1
Suppose $a, b, c, d$ are positive reals such that $a \geq b \geq c \geq d$ and $ab^2c^3d^4 = 1$.
Help Janson prove that $a+b+c+d \geq 4$.
2023 Costa Rica - Final Round, 3.5
Let $t$ be a positive real number such that $t^4 + t^{-4} = 2023$. Determine the value of $t^3 + t^{-3}$ in the form of $a\sqrt b$, where $a$ and $b$ are positive integers.
2021 China Team Selection Test, 1
Let $ n(\ge2) $ be a positive integer. Find the minimum $ m $, so that there exists $x_{ij}(1\le i ,j\le n)$ satisfying:
(1)For every $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} $ or $ x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$
(2)For every $1\le i \le n$, there are at most $m$ indices $k$ with $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$
(3)For every $1\le j \le n$, there are at most $m$ indices $k$ with $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$