Found problems: 85335
MathLinks Contest 5th, 6.1
Let $ABC$ be a triangle and let $C$ be a circle that intersects the sides $BC, CA$ and $AB$ in the points $A_1, A_2, B_1, B_2$ and $C_1, C_2$ respectively. Prove that if $AA_1, BB_1$ and $CC_1$ are concurrent lines then $AA_2, BB_2$ and $CC_2$ are also concurrent lines.
2014 Online Math Open Problems, 3
Suppose that $m$ and $n$ are relatively prime positive integers with $A = \tfrac mn$, where
\[ A = \frac{2+4+6+\dots+2014}{1+3+5+\dots+2013} - \frac{1+3+5+\dots+2013}{2+4+6+\dots+2014}. \] Find $m$. In other words, find the numerator of $A$ when $A$ is written as a fraction in simplest form.
[i]Proposed by Evan Chen[/i]
1997 VJIMC, Problem 2
Let $\alpha\in(0,1]$ be a given real number and let a real sequence $\{a_n\}^\infty_{n=1}$ satisfy the inequality
$$a_{n+1}\le\alpha a_n+(1-\alpha)a_{n-1}\qquad\text{for }n=2,3,\ldots$$Prove that if $\{a_n\}$ is bounded, then it must be convergent.
2024 Miklos Schweitzer, 8
Prove that for any finite bipartite planar graph, a circle can be assigned to each vertex so that all circles are coplanar, the circles assigned to any two adjacent vertices are tangent to one another, while the circles assigned to any two distinct, non-adjacent vertices intersect in two points.
2016 Romanian Master of Mathematics Shortlist, N2
Given a prime $p$, prove that the sum $\sum_{k=1}^{\lfloor \frac{q}{p} \rfloor}{k^{p-1}}$ is not divisible by $q$ for all but finitely many primes $q$.
2023 BMT, Tie 4
Let $N = 2^{18} \cdot 3^{19} \cdot5^{20} \cdot7^{21} \cdot 11^{22}$. Compute the number of positive integer divisors of $N$ whose units digit is $7$.
2013 Federal Competition For Advanced Students, Part 2, 4
For a positive integer $n$, let $a_1, a_2, \ldots a_n$ be nonnegative real numbers such that for all real numbers $x_1>x_2>\ldots>x_n>0$ with $x_1+x_2+\ldots+x_n<1$, the inequality $\sum_{k=1}^na_kx_k^3<1$ holds. Show that \[na_1+(n-1)a_2+\ldots+(n-j+1)a_j+\ldots+a_n\leqslant\frac{n^2(n+1)^2}{4}.\]
2015 BAMO, 5
We are given $n$ identical cubes, each of size $1\times 1\times 1$. We arrange all of these $n$ cubes to produce one or more congruent rectangular solids, and let $B(n)$ be the number of ways to do this.
For example, if $n=12$, then one arrangement is twelve $1\times1\times1$ cubes, another is one $3\times 2\times2$ solid, another is three $2\times 2\times1$ solids, another is three $4\times1\times1$ solids, etc. We do not consider, say, $2\times2\times1$ and $1\times2\times2$ to be different; these solids are congruent. You may wish to verify, for example, that $B(12) =11$.
Find, with proof, the integer $m$ such that $10^m<B(2015^{100})<10^{m+1}$.
2011 ELMO Shortlist, 6
Do there exist positive integers $k$ and $n$ such that for any finite graph $G$ with diameter $k+1$ there exists a set $S$ of at most $n$ vertices such that for any $v\in V(G)\setminus S$, there exists a vertex $u\in S$ of distance at most $k$ from $v$?
[i]David Yang.[/i]
1990 IMO Longlists, 96
Suppose that points $X, Y,Z$ are located on sides $BC, CA$, and $AB$, respectively, of triangle $ABC$ in such a way that triangle $XY Z$ is similar to triangle $ABC$. Prove that the orthocenter of triangle $XY Z$ is the circumcenter of triangle $ABC.$
2004 China Girls Math Olympiad, 1
We say a positive integer $ n$ is [i]good[/i] if there exists a permutation $ a_1, a_2, \ldots, a_n$ of $ 1, 2, \ldots, n$ such that $ k \plus{} a_k$ is perfect square for all $ 1\le k\le n$. Determine all the good numbers in the set $ \{11, 13, 15, 17, 19\}$.
2021 China Second Round A1, 3
Let $\{a_n\}$, $\{b_n\}$ be sequences of positive real numbers satisfying $$a_n=\sqrt{\frac{1}{100} \sum\limits_{j=1}^{100} b_{n-j}^2}$$ and $$b_n=\sqrt{\frac{1}{100} \sum\limits_{j=1}^{100} a_{n-j}^2}$$
For all $n\ge 101$. Prove that there exists $m\in \mathbb{N}$ such that $|a_m-b_m|<0.001$
[url=https://zhuanlan.zhihu.com/p/417529866] Link [/url]
1991 Flanders Math Olympiad, 3
Given $\Delta ABC$ equilateral, with $X\in[A,B]$. Then we define unique points Y,Z so that $Y\in[B,C]$, $Z\in[A,C]$, $\Delta XYZ$ equilateral.
If $Area\left(\Delta ABC\right) = 2 \cdot Area\left(\Delta XYZ\right)$, find the ratio of $\frac{AX}{XB},\frac{BY}{YC},\frac{CZ}{ZA}$.
2000 Harvard-MIT Mathematics Tournament, 44
A function $f:\mathbb{Z}\implies\mathbb{Z}$ satisfies
$f(x+4)-f(x)=8x+20$
$f(x^2-1)=(f(x)-x)^2+x^2-2$
Find $f(0)$ and $f(1)$.
2018 Sharygin Geometry Olympiad, 17
Let each of circles $\alpha, \beta, \gamma$ touches two remaining circles externally, and all of them touche a circle $\Omega$ internally at points $A_1, B_1, C_1$ respectively. The common internal tangent to $\alpha$ and $\beta$ meets the arc $A_1B_1$ not containing $C_1$ at point $C_2$. Points $A_2$, $B_2$ are defined similarly. Prove that the lines $A_1A_2, B_1B_2, C_1C_2$ concur.
2006 Cezar Ivănescu, 2
[b]a)[/b] Prove that $ \{ a \} +\{ 1/a \} <3/2, $ for any positive real number $ a. $
[b]b)[/b] Give an example of a number $ b $ satisfying $ \{ b \} +\{ 1/b \} =1. $
[i]{} means fractional part[/i]
2020-2021 OMMC, 10
An [i]indivisible tiling [/i]is a tiling of an $m \times n$ rectangular grid using only rectangles with a width and/or length of 1, such that nowhere in the tiling is a smaller complete tiling of a rectangle with more than 1 tile. Find the smallest integer $a$ such that an indivisible tiling of an $a \times a$ square may contain exactly $2021$ $1 \times 1$ tiles.
2012 IFYM, Sozopol, 5
We denote with $p_n(k)$ the number of permutations of the numbers $1,2,...,n$ that have exactly $k$ fixed points.
a) Prove that $\sum_{k=0}^n kp_n (k)=n!$.
b) If $s$ is an arbitrary natural number, then:
$\sum_{k=0}^n k^s p_n (k)=n!\sum_{i=1}^m R(s,i)$,
where with $R(s,i)$ we denote the number of partitions of the set $\{1,2,...,s\}$ into $i$ non-empty non-intersecting subsets and $m=min(s,n)$.
2004 Manhattan Mathematical Olympiad, 1
Is there a whole number, so that if we multiply its digits we get $528$?
2017 Princeton University Math Competition, A6/B8
Together, Kenneth and Ellen pick a real number $a$. Kenneth subtracts $a$ from every thousandth root of unity (that is, the thousand complex numbers $\omega$ for which $\omega^{1000}=1$) then inverts each, then sums the results. Ellen inverts every thousandth root of unity, then subtracts $a$ from each, and then sums the results. They are surprised to find that they actually got the same answer! How many possible values of $a$ are there?
Novosibirsk Oral Geo Oly VIII, 2021.5
On the legs $AC$ and $BC$ of an isosceles right-angled triangle with a right angle $C$, points $D$ and $E$ are taken, respectively, so that $CD = CE$. Perpendiculars on line $AE$ from points $C$ and $D$ intersect segment $AB$ at points $P$ and $Q$, respectively. Prove that $BP = PQ$.
2004 India IMO Training Camp, 1
A set $A_1 , A_2 , A_3 , A_4$ of 4 points in the plane is said to be [i]Athenian[/i] set if there is a point $P$ of the plane satsifying
(*) $P$ does not lie on any of the lines $A_i A_j$ for $1 \leq i < j \leq 4$;
(**) the line joining $P$ to the mid-point of the line $A_i A_j$ is perpendicular to the line joining $P$ to the mid-point of $A_k A_l$, $i,j,k,l$ being distinct.
(a) Find all [i]Athenian[/i] sets in the plane.
(b) For a given [i]Athenian[/i] set, find the set of all points $P$ in the plane satisfying (*) and (**)
2012 Online Math Open Problems, 22
Let $c_1,c_2,\ldots,c_{6030}$ be 6030 real numbers. Suppose that for any 6030 real numbers $a_1,a_2,\ldots,a_{6030}$, there exist 6030 real numbers $\{b_1,b_2,\ldots,b_{6030}\}$ such that \[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\] and \[b_n = \sum_{d\mid n} c_d a_{n/d}\] for $n=1,2,\ldots,6030$. Find $c_{6030}$.
[i]Victor Wang.[/i]
2005 Estonia National Olympiad, 1
The height drawn on the hypotenuse of a right triangle divides the hypotenuse into two sections with a length ratio of $9: 1$ and two triangles of the starting triangle with a difference of areas of $48$ cm$^2$. Find the original triangle sidelengths.
2017 All-Russian Olympiad, 1
There are $n>3$ different natural numbers, less than $(n-1)!$ For every pair of numbers Ivan divides bigest on lowest and write integer quotient (for example, $100$ divides $7$ $= 14$) and write result on the paper. Prove, that not all numbers on paper are different.