Found problems: 85335
2015 Middle European Mathematical Olympiad, 4
Find all pairs of positive integers $(m,n)$ for which there exist relatively prime integers $a$ and $b$ greater than $1$ such that
$$\frac{a^m+b^m}{a^n+b^n}$$
is an integer.
2018 Belarus Team Selection Test, 1.1
Let $A=2^7(7^{14}+1)+2^6\cdot 7^{11}\cdot 10^2+2^6\cdot 7^7\cdot 10^{4}+2^4\cdot 7^3\cdot 10^6$. Prove that the number $A$ ends with $14$ zeros.
[i](I. Gorodnin)[/i]
2004 IMO Shortlist, 3
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
2010 Kazakhstan National Olympiad, 3
Positive real $A$ is given. Find maximum value of $M$ for which inequality
$ \frac{1}{x}+\frac{1}{y}+\frac{A}{x+y} \geq \frac{M}{\sqrt{xy}} $
holds for all $x, y>0$
1916 Eotvos Mathematical Competition, 2
Let the bisector of the angle at $C$ of triangle $ABC$ intersect side $AB$ in point $D$. Show that the segment $CD$ is shorter than the geometric mean of the sides $CA$ and $CB$.
(The geometric mean of two positive numbers is the square root of their product; the geometric mean of $n$ numbers is the $n$-th root of their product.
2010 Princeton University Math Competition, 6
In regular hexagon $ABCDEF$, $AC$, $CE$ are two diagonals. Points $M$, $N$ are on $AC$, $CE$ respectively and satisfy $AC: AM = CE: CN = r$. Suppose $B, M, N$ are collinear, find $100r^2$.
[asy]
size(120); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
pair A=dir(0), B=dir(60), C=dir(120), D=dir(180), E=dir(240), F=dir(300), N=(4*E+C)/5,M=intersectionpoints(A--C,B--N)[0];
draw(A--B--C--D--E--F--cycle); draw(A--C--E); draw(B--N);
label("$A$",D2(A),plain.E);
label("$B$",D2(B),NE);
label("$C$",D2(C),NW);
label("$D$",D2(D),W);
label("$E$",D2(E),SW);
label("$F$",D2(F),SE);
label("$M$",D2(M),(0,-1.5));
label("$N$",D2(N),SE);
[/asy]
2012 Miklós Schweitzer, 1
Is there any real number $\alpha$ for which there exist two functions $f,g: \mathbb{N} \to \mathbb{N}$ such that
$$\alpha=\lim_{n \to \infty} \frac{f(n)}{g(n)},$$
but the function which associates to $n$ the $n$-th decimal digit of $\alpha$ is not recursive?
1998 National Olympiad First Round, 8
$ a_{1} \equal{}1$, $ a_{n\plus{}1} \equal{}\frac{a_{n} }{\sqrt{1\plus{}4a_{n}^{2} } }$ for $ n\ge 1$. What is the least $ k$ such that $ a_{k} <10^{\minus{}2}$ ?
$\textbf{(A)}\ 2501 \qquad\textbf{(B)}\ 251 \qquad\textbf{(C)}\ 2499 \qquad\textbf{(D)}\ 249 \qquad\textbf{(E)}\ \text{None}$
2023 Harvard-MIT Mathematics Tournament, 32
Let $ABC$ be a triangle with $\angle{BAC}>90^\circ.$ Let $D$ be the foot of the perpendicular from $A$ to side $BC.$ Let $M$ and $N$ be the midpoints of segments $BC$ and $BD,$ respectively. Suppose that $AC=2, \angle{BAN}=\angle{MAC},$ and $AB \cdot BC = AM.$ Compute the distance from $B$ to line $AM.$
1998 North Macedonia National Olympiad, 3
A triangle $ABC$ is given. For every positive numbers $p,q,r$, let $A',B',C'$ be the points such that $\overrightarrow{BA'} = p\overrightarrow{AB}, \overrightarrow{CB'}=q\overrightarrow{BC} $, and $\overrightarrow{AC'}=r\overrightarrow{CA}$. Define $f(p,q,r)$ as the ratio of the area of $\vartriangle A'B'C'$ to that of $\vartriangle ABC$. Prove that for all positive numbers $x,y,z$ and every positive integer $n$, $\sum_{k=0}^{n-1}f(x+k,y+k,z+k) = n^3f\left(\frac{x}{n},\frac{y}{n},\frac{z}{n}\right)$.
2012 Czech-Polish-Slovak Junior Match, 1
Point $P$ lies inside the triangle $ABC$. Points $K, L, M$ are symmetrics of point $P$ wrt the midpoints of the sides $BC, CA, AB$. Prove that the straight $AK, BL, CM$ intersect at one point.
2021 CHMMC Winter (2021-22), 6
There is a unique degree-$10$ monic polynomial with integer coefficients $f(x)$ such that
$$f \left( \sum^9_{j=0}\sqrt[10]{2021^j}\right)= 0.$$
Find the remainder when $f(1)$ is divided by $1000$.
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?
2019 HMNT, 5
Let $a, b, c$ be positive real numbers such that $a\le b \le c \le 2a$. Find the maximum possible value of $$\frac{b}{a}
+\frac{c}{b}
+\frac{a}{c}.$$
2001 Singapore Senior Math Olympiad, 1
Let $n$ be a positive integer. Suppose that the following simultaneous equations
$$\begin{cases} \sin x_1 + \sin x_2+ ...+ \sin x_n = 0 \\
\sin x_1 + 2\sin x_2+ ...+ n \sin x_n = 100 \end{cases}$$
has a solution, where $x_1 x_2,.., x_n$ are the unknowns. Find the smallest possible positive integer $n$. Justify your answer.
2016 ITAMO, 4
Determine all pairs of positive integers $(a,n)$ with $a\ge n\ge 2$ for which $(a+1)^n+a-1$ is a power of $2$.
2003 All-Russian Olympiad Regional Round, 8.6
For some natural numbers $a, b, c$ and $d$ the following equations holds: $$\frac{a}{c}= \frac{b}{d}= \frac{ab + 1}{cd + 1} .$$ Prove that $a = c$ and $b = d$.
1970 All Soviet Union Mathematical Olympiad, 129
Given a circle, its diameter $[AB]$ and a point $C$ on it. Construct (with the help of compasses and ruler) two points $X$ and $Y$, that are symmetric with respect to $(AB)$ line, such that $(YC)$ is orthogonal to $(XA)$.
2014 Junior Balkan MO, 4
For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?
2006 Austrian-Polish Competition, 5
Prove that for all positive integers $n$ and all positive reals $a,b,c$ the following inequality holds: \[\frac{a^{n+1}}{a^{n}+a^{n-1}b+\ldots+b^{n}}+\frac{b^{n+1}}{b^{n}+b^{n-1}c+\ldots+c^{n}}+\frac{c^{n+1}}{c^{n}+c^{n-1}a+\ldots+a^{n}}\\ \ge \frac{a+b+c}{n+1}\]
2000 Chile National Olympiad, 4
Let $ AD $ be the bisector of a triangle $ ABC $ $ (D \in BC) $ such that $ AB + AD = CD $ and $ AC + AD = BC $. Determine the measure of the angles of $ \vartriangle ABC $
2000 IMO, 5
Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n \plus{} 1$?
1997 APMO, 3
Let $ABC$ be a triangle inscribed in a circle and let
\[ l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ , \]
where $m_a$,$m_b$, $m_c$ are the lengths of the angle bisectors (internal to the triangle) and $M_a$, $M_b$, $M_c$ are the lengths of the angle bisectors extended until they meet the circle. Prove that
\[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3 \]
and that equality holds iff $ABC$ is an equilateral triangle.
2014 Purple Comet Problems, 12
The first number in the following sequence is $1$. It is followed by two $1$'s and two $2$'s. This is followed by three $1$'s, three $2$'s, and three $3$'s. The sequence continues in this fashion.
\[1,1,1,2,2,1,1,1,2,2,2,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,\dots.\]
Find the $2014$th number in this sequence.
2020 Vietnam National Olympiad, 3
Let a sequence $(a_n)$ satisfy: $a_1=5,a_2=13$ and $a_{n+1}=5a_n-6a_{n-1},\forall n\ge2$
a) Prove that $(a_n, a_{n+1})=1,\forall n\ge1$
b) Prove that: $2^{k+1}|p-1\forall k\in\mathbb{N}$, if p is a prime factor of $a_{2^k}$