Found problems: 85335
1989 Tournament Of Towns, (218) 2
The point $M$ , inside $\vartriangle ABC$, satisfies the conditions that $\angle BMC = 90^o +\frac12 \angle BAC$ and that the line $AM$ contains the centre of the circumscribed circle of $\vartriangle BMC$. Prove that $M$ is the centre of the inscribed circle of $\vartriangle ABC$.
2018 Balkan MO Shortlist, G5
Let $ABC$ be an acute triangle with $AB<AC<BC$ and let $D$ be a point on it's extension of $BC$ towards $C$. Circle $c_1$, with center $A$ and radius $AD$, intersects lines $AC,AB$ and $CB$ at points $E,F$, and $G$ respectively. Circumscribed circle $c_2$ of triangle $AFG$ intersects again lines $FE,BC,GE$ and $DF$ at points $J,H,H' $ and $J'$ respectively. Circumscribed circle $c_3$ of triangle $ADE$ intersects again lines $FE,BC,GE$ and $DF$ at points $I,K,K' $ and $I' $ respectively. Prove that the quadrilaterals $HIJK$ and $H'I'J'K '$ are cyclic and the centers of their circumscribed circles coincide.
by Evangelos Psychas, Greece
2005 Thailand Mathematical Olympiad, 7
How many ways are there to express $2548$ as a sum of at least two positive integers, where two sums that differ in order are considered different?
2014 Contests, 1
Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that
$ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$
(Proposed by Gerhard Woeginger, Austria)
2021 AMC 10 Fall, 8
The largest prime factor of $16384$ is $2$, because $16384 = 2^{14}$. What is the sum of the digits of the largest prime factor of $16383$?
$\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }10\qquad\textbf{(D) }16\qquad\textbf{(E) }22$
1979 IMO Longlists, 23
Consider the set $E$ consisting of pairs of integers $(a, b)$, with $a \geq 1$ and $b \geq 1$, that satisfy in the decimal system the following properties:
[b](i)[/b] $b$ is written with three digits, as $\overline{\alpha_2\alpha_1\alpha_0}$, $\alpha_2 \neq 0$;
[b](ii)[/b] $a$ is written as $\overline{\beta_p \ldots \beta_1\beta_0}$ for some $p$;
[b](iii)[/b] $(a + b)^2$ is written as $\overline{\beta_p\ldots \beta_1 \beta_0 \alpha_2 \alpha_1 \alpha_0}.$
Find the elements of $E$.
2006 Romania National Olympiad, 4
Let $\displaystyle n \in \mathbb N$, $\displaystyle n \geq 2$. Determine $\displaystyle n$ sets $\displaystyle A_i$, $\displaystyle 1 \leq i \leq n$, from the plane, pairwise disjoint, such that:
(a) for every circle $\displaystyle \mathcal C$ from the plane and for every $\displaystyle i \in \left\{ 1,2,\ldots,n \right\}$ we have $\displaystyle A_i \cap \textrm{Int} \left( \mathcal C \right) \neq \phi$;
(b) for all lines $\displaystyle d$ from the plane and every $\displaystyle i \in \left\{ 1,2,\ldots,n \right\}$, the projection of $\displaystyle A_i$ on $\displaystyle d$ is not $\displaystyle d$.
2022 Latvia Baltic Way TST, P2
Prove that for positive real numbers $a,b,c$ satisfying $abc=1$ the following inequality holds:
$$ \frac{a}{b}+\frac{b}{c}+\frac{c}{a} \ge \frac{a^2+1}{2a}+\frac{b^2+1}{2b}+\frac{c^2+1}{2c}.$$
2014 BMT Spring, 20
A certain type of Bessel function has the form $I(x) = \frac{1}{\pi} \int_0^{\pi}e^{x \cos \theta} d\theta$ for all real $x$. Evaluate $\int_0^{\infty} x I(2x) e^{-x^2}dx$.
2024 CCA Math Bonanza, I4
Let $x$ and $y$ be positive integers that are at least $2$. Suppose Johnny hits $1$ out of every $10$ free throws, Abigail hits $1$ out of every $x$ free throws, and Demar hits $2$ out of every $y$ free throws. It turns out that the mean of Abigail's and Demar's individual free throw percentages are the same as Johnny's free throw percentage. Find the sum of all possible values of $x$.
[i]Individual #4[/i]
Swiss NMO - geometry, 2006.5
A circle $k_1$ lies within a second circle $k_2$ and touches it at point $A$. A line through $A$ intersects $k_1$ again in $B$ and $k_2$ in $C$. The tangent to $k_1$ through $B$ intersects $k_2$ at points $D$ and $E$. The tangents at $k_1$ passing through $C$ intersects $k_1$ in points $F$ and $G$. Prove that $D, E, F$ and $G$ lie on a circle.
2021 Oral Moscow Geometry Olympiad, 2
A trapezoid is given in which one base is twice as large as the other. Use one ruler (no divisions) to draw the midline of this trapezoid.
1958 February Putnam, A6
What is the smallest amount that may be invested at interest rate $i$, compounded annually, in order that one may withdraw $1$ dollar at the end of the first year, $4$ dollars at the end of the second year, $\ldots$ , $n^2$ dollars at the end of the $n$-th year, in perpetuity?
1973 AMC 12/AHSME, 7
The sum of all integers between 50 and 350 which end in 1 is
$ \textbf{(A)}\ 5880 \qquad
\textbf{(B)}\ 5539 \qquad
\textbf{(C)}\ 5208 \qquad
\textbf{(D)}\ 4877 \qquad
\textbf{(E)}\ 4566$
2018 China Team Selection Test, 5
Let $ABC$ be a triangle with $\angle BAC > 90 ^{\circ}$, and let $O$ be its circumcenter and $\omega$ be its circumcircle. The tangent line of $\omega$ at $A$ intersects the tangent line of $\omega$ at $B$ and $C$ respectively at point $P$ and $Q$. Let $D,E$ be the feet of the altitudes from $P,Q$ onto $BC$, respectively. $F,G$ are two points on $\overline{PQ}$ different from $A$, so that $A,F,B,E$ and $A,G,C,D$ are both concyclic. Let M be the midpoint of $\overline{DE}$. Prove that $DF,OM,EG$ are concurrent.
2010 AMC 10, 19
A circle with center $ O$ has area $ 156\pi$. Triangle $ ABC$ is equilateral, $ \overline{BC}$ is a chord on the circle, $ OA \equal{} 4\sqrt3$, and point $ O$ is outside $ \triangle ABC$. What is the side length of $ \triangle ABC$?
$ \textbf{(A)}\ 2\sqrt3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 4\sqrt3 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 18$
2009 Korea Junior Math Olympiad, 4
There are $n$ clubs composed of $4$ students out of all $9$ students. For two arbitrary clubs, there are no more than $2$ students who are a member of both clubs. Prove that $n\le 18$.
Translator’s Note. We can prove $n\le 12$, and we can prove that the bound is tight.
(Credits to rkm0959 for translation and document)
2001 Bulgaria National Olympiad, 1
Let $n \geq 2$ be a given integer. At any point $(i, j)$ with $i, j \in\mathbb{ Z}$ we write the remainder of $i+j$ modulo $n$. Find all pairs $(a, b)$ of positive integers such that the rectangle with vertices $(0, 0)$, $(a, 0)$, $(a, b)$, $(0, b)$ has the following properties:
[b](i)[/b] the remainders $0, 1, \ldots , n-1$ written at its interior points appear the same number of times;
[b](ii)[/b] the remainders $0, 1, \ldots , n -1$ written at its boundary points appear the same number of times.
2018 CCA Math Bonanza, L1.3
$ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\angle{ACE}$ is $90^{\circ}$. What is the average of the measures, in degrees, of $\angle{ABC}$ and $\angle{CDE}$?
[i]2018 CCA Math Bonanza Lightning Round #1.3[/i]
1988 Canada National Olympiad, 1
For what real values of $k$ do $1988x^2 + kx + 8891$ and $8891x^2 + kx + 1988$ have a common zero?
2022 Bangladesh Mathematical Olympiad, 4
Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n>m.$ Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n-2)$ and $(2m+2)$ respectively. Find minimum integer $x$ with proof such that if $n-m\geq x,$ then Pratyya's number will be larger than Payel's number everyday.
2006 Argentina National Olympiad, 1
Let $A$ be the set of positive real numbers less than $1$ that have a periodic decimal expansion with a period of ten different digits. Find a positive integer $n$ greater than $1$ and less than $10^{10}$ such that $na-a$ is a positive integer for all $a$. of set $A$.
1990 All Soviet Union Mathematical Olympiad, 525
A graph has $n$ points and $\frac{n(n-1)}{2}$ edges. Each edge is colored with one of $k$ colors so that there are no closed monochrome paths. What is the largest possible value of $n$ (given $k$)?
2020 CMIMC Geometry, 7
In triangle $ABC$, points $D$, $E$, and $F$ are on sides $BC$, $CA$, and $AB$ respectively, such that $BF = BD = CD = CE = 5$ and $AE - AF = 3$. Let $I$ be the incenter of $ABC$. The circumcircles of $BFI$ and $CEI$ intersect at $X \neq I$. Find the length of $DX$.
2022 Turkey Junior National Olympiad, 2
In a school with $101$ students, each student has at least one friend among the other students. Show that for every integer $1<n<101$, a group of $n$ students can be selected from this school in such a way that each selected student has at least one friend among the other selected students.