Found problems: 85335
2020 Azerbaijan IZHO TST, 4
Consider an odd prime number $p$ and $p$ consecutive positive integers $m_1,m_2,…,m_p$.
Choose a permutation $\sigma$ of $1,2,…,p$ .
Show that there exist two different numbers $k,l\in{(1,2,…,p)}$ such that
$p\mid{m_k.m_{\sigma(k)}-m_l.m_{\sigma(l)}}$
2001 Putnam, 5
Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.
2015 AMC 12/AHSME, 20
Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Which of the following numbers is closest to $b$?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
2016 Romanian Master of Mathematics, 4
Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $x^{2016}+y> 1-\frac{1}{100}$
2018 Pan-African Shortlist, N6
Prove that there are infinitely many integers $n$ such that both the arithmetic mean of its divisors and the geometric mean of its divisors are integers.
(Recall that for $k$ positive real numbers, $a_1, a_2, \dotsc, a_k$, the arithmetic mean is $\frac{a_1 +a_2 +\dotsb +a_k}{k}$, and the geometric mean is $\sqrt[k]{a_1 a_2\dotsb a_k}$.)
2023 Macedonian Mathematical Olympiad, Problem 2
Let $p$ and $q$ be odd prime numbers and $a$ a positive integer so that $p|a^q+1$ and $q|a^p+1$. Show that $p|a+1$ or $q|a+1$.
[i]Authored by Nikola Velov[/i]
2011 Kosovo National Mathematical Olympiad, 4
Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove:
\[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3}
\]
In what case does equality hold?
2017 Bulgaria JBMO TST, 1
Find all positive integers $ a, b, c, d $ so that
$ a^2+b^2+c^2+d^2=13 \cdot 4^n $
2021 LMT Fall, 8
Three distinct positive integers are chosen at random from $\{1,2,3...,12\}$. The probability that no two elements of the set have an absolute difference less than or equal to $2$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$.
2019 LIMIT Category A, Problem 1
Let $p(x)$ be a polynomial of degree $4$ with leading coefficient $1$. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$ and $p(4)=4$. Then $p(5)=$?
$\textbf{(A)}~5$
$\textbf{(B)}~\frac{25}6$
$\textbf{(C)}~29$
$\textbf{(D)}~35$
2021 Durer Math Competition (First Round), 3
Let $k_1$ and $k_2$ be two circles that are externally tangent at point $C$. We have a point $A$ on $k_1$ and a point $B$ on $k_2$ such that $C$ is an interior point of segment $AB$. Let $k_3$ be a circle that passes through points $A$ and $B$ and intersects circles $k_1$ and $k_2$ another time at points $M$ and $N$ respectively. Let $k_4$ be the circumscribed circle of triangle $CMN$. Prove that the centres of circles $k_1, k_2, k_3$ and $k_4$ all lie on the same circle.
1993 Tournament Of Towns, (396) 4
A convex $1993$-gon is divided into convex $7$-gons. Prove that there are $3$ neighbouring sides of the $1993$-gon belonging to one such $7$-gon. (A vertex of a $7$-gon may not be positioned on the interior of a side of the $1993$-gon, and two $7$-gons either have no common points, exactly one common vertex or a complete common side.)
(A Kanel-Belov)
2004 National Olympiad First Round, 31
For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exacly $2004$ of other lines?
$
\textbf{(A)}\ 12
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 9
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 1
$
2016 Rioplatense Mathematical Olympiad, Level 3, 6
When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits $123456789101112 ....$ Denote $A_k$ the number formed by the first $k$ digits of this sequence . Prove that for all positive integer $n$ there is a positive integer $m$ which simultaneously verifies the following three conditions:
(i) $n$ divides $A_m$,
(ii) $n$ divides $m$,
(iii) $n$ divides the sum of the digits of $A_m$.
2021 Taiwan TST Round 3, A
Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has
\[ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. \]
2016 Iran MO (3rd Round), 1
Let $ABC$ be an arbitrary triangle,$P$ is the intersection point of the altitude from $C$ and the tangent line from $A$ to the circumcircle. The bisector of angle $A$ intersects $BC$ at $D$ . $PD$ intersects $AB$ at $K$, if $H$ is the orthocenter then prove : $HK\perp AD$
2009 China Team Selection Test, 3
Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1.$ Prove that
$ max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2.$
Where for all integers $ i, a_{i \plus{} 4} \equal{} a_{i}$ holds.
2024 OMpD, 2
Let \( ABCDE \) be a convex pentagon whose vertices lie on a circle \( \Gamma \). The tangents to \( \Gamma \) at \( C \) and \( E \) intersect at \( X \), and the segments \( CE \) and \( AD \) intersect at \( Y \). Given that \( CE \) is perpendicular to \( BD \), that \( XY \) is parallel to \( BD \), that \( AY = BD \), and that \( \angle BAD = 30^\circ \), what is the measure of the angle \( \angle BDA \)?
Proposed by João Gilberti Alves Tavares
2003 Indonesia MO, 6
The hall in a castle is a regular hexagon where its sides' length is 6 meters. The floor of the hall is to be tiled with equilateral triangular tiles where its sides' length is 50 centimeters. Each tile is divided into three congruent triangles by their altitudes up to its orthocenter (see below). Each of these small triangles are colored such that each tile has different colors and no two tiles have identical colorings. How many colors at least are required?
A tile's pattern is:
[asy]
draw((0,0.000)--(2,0.000));
draw((2,0.000)--(1,1.732));
draw((1,1.732)--(0,0.000));
draw((1,0.577)--(0,0.000));
draw((1,0.577)--(2,0.000));
draw((1,0.577)--(1,1.732));
[/asy]
2017 Macedonia National Olympiad, Problem 4
Let $O$ be the circumcenter of the acute triangle $ABC$ ($AB < AC$). Let $A_1$ and $P$ be the feet of the perpendicular lines drawn from $A$ and $O$ to $BC$, respectively. The lines $BO$ and $CO$ intersect $AA_1$ in $D$ and $E$, respectively. Let $F$ be the second intersection point of $\odot ABD$ and $\odot ACE$. Prove that the angle bisector od $\angle FAP$ passes through the incenter of $\triangle ABC$.
2011 Peru MO (ONEM), 2
If $\alpha, \beta, \gamma$ are angles whose measures in radians belong to the interval $\left[0, \frac{\pi}{2}\right]$ such that: $$\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1$$ calculate the minimum possible value of $\cos \alpha + \cos \beta + \cos \gamma$.
2016 Philippine MO, 3
Let \(n\) be any positive integer. Prove that \[\sum^n_{i=1} \frac{1}{(i^2+i)^{3/4}} > 2-\frac{2}{\sqrt{n+1}}\].
2009 Sharygin Geometry Olympiad, 12
Let $ CL$ be a bisector of triangle $ ABC$. Points $ A_1$ and $ B_1$ are the reflections of $ A$ and $ B$ in $ CL$, points $ A_2$ and $ B_2$ are the reflections of $ A$ and $ B$ in $ L$. Let $ O_1$ and $ O_2$ be the circumcenters of triangles $ AB_1B_2$ and $ BA_1A_2$ respectively. Prove that angles $ O_1CA$ and $ O_2CB$ are equal.
2018 ASDAN Math Tournament, 4
What is the remainder when $13^{16} + 17^{12}$ is divided by $221$?
1982 IMO Shortlist, 18
Let $O$ be a point of three-dimensional space and let $l_1, l_2, l_3$ be mutually perpendicular straight lines passing through $O$. Let $S$ denote the sphere with center $O$ and radius $R$, and for every point $M$ of $S$, let $S_M$ denote the sphere with center $M$ and radius $R$. We denote by $P_1, P_2, P_3$ the intersection of $S_M$ with the straight lines $l_1, l_2, l_3$, respectively, where we put $P_i \neq O$ if $l_i$ meets $S_M$ at two distinct points and $P_i = O$ otherwise ($i = 1, 2, 3$). What is the set of centers of gravity of the (possibly degenerate) triangles $P_1P_2P_3$ as $M$ runs through the points of $S$?