Found problems: 85335
2008 China Team Selection Test, 3
Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).$
2014 BMT Spring, 9
Two different functions $f, g$ of $x$ are selected from the set of real-valued functions $$\left \{sin x, e^{-x}, x \ln x, \arctan x, \sqrt{x^2 + x} -\sqrt{x^2 + x} -x, \frac{1}{x} \right \}$$ to create a product function $f(x)g(x)$. For how many such products is $\lim_{x\to infty} f(x)g(x)$ finite?
2009 QEDMO 6th, 2
Let there be a finite number of straight lines in the plane, none of which are three in one point to cut. Show that the intersections of these straight lines can be colored with $3$ colors so that that no two points of the same color are adjacent on any of the straight lines. (Two points of intersection are called [i]adjacent [/i] if they both lie on one of the finitely many straight lines and there is no other such intersection on their connecting line.)
VI Soros Olympiad 1999 - 2000 (Russia), 9.6
The sequence of integers $a_1,a_2,a_3 ,.. $such that $a_1 = 1$, $a_2 = 2$ and for every natural $n \ge 1$
$$a_{n+2}=\begin{cases} 2001a_{n+1} - 1999a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,even\,\,number} /\\
a_{n+1}-a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,odd\,\,number} \end{cases}$$
Is there such a natural $m$ that $a_m= 2000$?
2022 Kazakhstan National Olympiad, 3
Given $m\in\mathbb{N}$. Find all functions $f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ such that $$f(f(x)+y)-f(x)=\left( \frac{f(y)}{y}-1\right)x+f^m(y)$$
holds for all $x,y\in\mathbb{R^{+}}.$
($f^m(x) =$ $f$ applies $m$ times.)
2005 Sharygin Geometry Olympiad, 10.5
Two circles of radius $1$ intersect at points $X, Y$, the distance between which is also equal to $1$. From point $C$ of one circle, tangents $CA, CB$ are drawn to the other. Line $CB$ will cross the first circle a second time at point $A'$. Find the distance $AA'$.
2004 China Girls Math Olympiad, 7
Let $ p$ and $ q$ be two coprime positive integers, and $ n$ be a non-negative integer. Determine the number of integers that can be written in the form $ ip \plus{} jq$, where $ i$ and $ j$ are non-negative integers with $ i \plus{} j \leq n$.
1992 Czech And Slovak Olympiad IIIA, 5
The function $f : (0,1) \to R$ is defined by
$f(x) = x$ if $x$ is irrational,
$f(x) = \frac{p+1}{q}$ if $x =\frac{p}{q}$ , where $(p,q) = 1$.
Find the maximum value of $f$ on the interval $(7/8,8/9)$.
2006 ISI B.Math Entrance Exam, 4
Let $f:\mathbb{R} \to \mathbb{R}$ be a function that is a function that is differentiable $n+1$ times for some positive integer $n$ . The $i^{th}$ derivative of $f$ is denoted by $f^{(i)}$ . Suppose-
$f(1)=f(0)=f^{(1)}(0)=...=f^{(n)}(0)=0$.
Prove that $f^{(n+1)}(x)=0$ for some $x \in (0,1)$
2025 Taiwan Mathematics Olympiad, 4
Find all positive integers $n$ satisfying the following: there exists a way to fill in $1, \cdots, n^2$ into a $n \times n$ grid so that each block has exactly one number, each number appears exactly once, and:
1. For all positive integers $1 \leq i < n^2$, $i$ and $i + 1$ are neighbors (two numbers neighbor each other if and only if their blocks share a common edge.)
2. Any two numbers among $1^2, \cdots, n^2$ are not in the same row or the same column.
MathLinks Contest 1st, 3
Let $x_0 = 1$ and $x_1 = 2003$ and define the sequence $(x_n)_{n \ge 0}$ by: $x_{n+1} =\frac{x^2_n + 1}{x_{n-1}}$ , $\forall n \ge 1$
Prove that for every $n \ge 2$ the denominator of the fraction $x_n$, when $x_n$ is expressed in lowest terms is a power of $2003$.
1953 Putnam, B3
Solve the equations
$$ \frac{dy}{dx}=z(y+z)^n, \;\; \; \frac{dz}{dx} = y(y+z)^n,$$
given the initial conditions $y=1$ and $z=0$ when $x=0.$
2012 JBMO TST - Turkey, 2
Let $S=\{1,2,3,\ldots,2012\}.$ We want to partition $S$ into two disjoint sets such that both sets do not contain two different numbers whose sum is a power of $2.$ Find the number of such partitions.
2023 Romania Team Selection Test, P5
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
1962 Czech and Slovak Olympiad III A, 1
Determine all integers $x$ such that $2x^2-x-36$ is a perfect square of a prime.
2010 Austria Beginners' Competition, 3
Let $x$ and $y$ be positive real numbers with $x + y =1 $. Prove that
$$\frac{(3x-1)^2}{x}+ \frac{(3y-1)^2}{y} \ge1.$$ For which $x$ and $y$ equality holds?
(K. Czakler, GRG 21, Vienna)
2024 Middle European Mathematical Olympiad, 4
A finite sequence $x_1,\dots,x_r$ of positive integers is a [i]palindrome[/i] if $x_i=x_{r+1-i}$ for all integers
$1 \le i \le r$.
Let $a_1,a_2,\dots$ be an infinite sequence of positive integers. For a positive integer $j \ge 2$, denote by
$a[j]$ the finite subsequence $a_1,a_2,\dots,a_{j-1}$. Suppose that there exists a strictly increasing infinite
sequence $b_1,b_2,\dots$ of positive integers such that for every positive integer $n$, the subsequence
$a[b_n]$ is a palindrome and $b_{n+2} \le b_{n+1}+b_n$. Prove that there exists a positive integer $T$ such
that $a_i=a_{i+T}$ for every positive integer $i$.
2006 MOP Homework, 6
Suppose there are $18$ light houses on the Mexican gulf. Each of the lighthouses lightens an angle with size $20$ degrees. Prove that we can choose the directions of the lighthouses such that the whole gulf is lightened.
1995 Rioplatense Mathematical Olympiad, Level 3, 2
In a circle of center $O$ and radius $r$, a triangle $ABC$ of orthocenter $H$ is inscribed. It is considered a triangle $A'B'C'$ whose sides have by length the measurements of the segments $AB, CH$ and $2r$. Determine the triangle $ABC$ so that the area of the triangle $A'B'C'$ is maximum.
1993 Spain Mathematical Olympiad, 3
Prove that in every triangle the diameter of the incircle is not greater than the radius of the circumcircle.
2015 China Western Mathematical Olympiad, 8
Let $k$ be a positive integer, and $n=\left(2^k\right)!$ .Prove that $\sigma(n)$ has at least a prime divisor larger than $2^k$, where $\sigma(n)$ is the sum of all positive divisors of $n$.
2006 Thailand Mathematical Olympiad, 3
Let $P(x), Q(x)$ and $R(x)$ be polynomials satisfying the equation $2xP(x^3) + Q(-x -x^3) = (1 + x + x^2)R(x)$.
Show that $x - 1$ divides $P(x) - Q(x)$.
2007 China Team Selection Test, 3
Consider a $ 7\times 7$ numbers table $ a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.
2006 Princeton University Math Competition, 8
Evaluate the sum $$\sum_{n=0}^{\infty}\frac{5n+7}{6^n}$$
2019 Korea Junior Math Olympiad., 1
Each integer coordinates are colored with one color and at least 5 colors are used to color every integer coordinates. Two integer coordinates $(x, y)$ and $(z, w)$ are colored in the same color if $x-z$ and $y-w$ are both multiples of 3. Prove that there exists a line that passes through exactly three points when five points with different colors are chosen randomly.