Found problems: 85335
2025 Romania National Olympiad, 1
Let $N \geq 1$ be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer $k$ (not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are $x$ and $y$ respectively, we replace them with $x+k+1$ and $y+k^2+2$, which we color blue and red (in this order). We keep doing this procedure until the blue number is at least $N$.
Determine the minimum value of the red number at the end of this procedure.
2005 Turkey Team Selection Test, 1
Show that for any integer $n\geq2$ and all integers $a_{1},a_{2},...,a_{n}$ the product $\prod_{i<j}{(a_{j}-a_{i})}$ is divisible by $\prod_{i<j}{(j-i)}$ .
2022 Brazil Team Selection Test, 2
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
2006 China Team Selection Test, 2
Given positive integers $m$, $a$, $b$, $(a,b)=1$. $A$ is a non-empty subset of the set of all positive integers, so that for every positive integer $n$ there is $an \in A$ and $bn \in A$. For all $A$ that satisfy the above condition, find the minimum of the value of $\left| A \cap \{ 1,2, \cdots,m \} \right|$
2005 Kazakhstan National Olympiad, 4
Find all functions $f :\mathbb{R}\to\mathbb{R}$, satisfying the condition
$f(f(x)+x+y)=2x+f(y)$
for any real $x$ and $y$.
1987 IMO Longlists, 71
To every natural number $k, k \geq 2$, there corresponds a sequence $a_n(k)$ according to the following rule:
\[a_0 = k, \qquad a_n = \tau(a_{n-1}) \quad \forall n \geq 1,\]
in which $\tau(a)$ is the number of different divisors of $a$. Find all $k$ for which the sequence $a_n(k)$ does not contain the square of an integer.
2010 SEEMOUS, Problem 4
Suppose that $A$ and $B$ are $n\times n$ matrices with integer entries, and $\det B\ne0$. Prove that there exists $m\in\mathbb N$ such that the product $AB^{-1}$ can be represented as
$$AB^{-1}=\sum_{k=1}^mN_k^{-1},$$where $N_k$ are $n\times n$ matrices with integer entries for all $k=1,\ldots,m$, and $N_i\ne N_j$ for $i\ne j$.
2012 India IMO Training Camp, 3
Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$
[i]Proposed by Igor Voronovich, Belarus[/i]
2009 Turkey MO (2nd round), 3
If $1<k_1<k_2<...<k_n$ and $a_1,a_2,...,a_n$ are integers such that for every integer $N,$ $k_i \mid N-a_i$ for some $1 \leq i \leq n,$ find the smallest possible value of $n.$
2007 Oral Moscow Geometry Olympiad, 5
At the base of the quadrangular pyramid $SABCD$ lies the quadrangle $ABCD$. whose diagonals are perpendicular and intersect at point $P$, and $SP$ is the altitude of the pyramid. Prove that the projections of the point $P$ onto the lateral faces of the pyramid lie on the same circle.
(A. Zaslavsky)
1993 Baltic Way, 18
In the triangle $ABC$, $|AB|=15,|BC|=12,|AC|=13$. Let the median $AM$ and bisector $BK$ intersect at point $O$, where $M\in BC,K\in AC$. Let $OL\perp AB,L\in AB$. Prove that $\angle OLK=\angle OLM$.
2020 Caucasus Mathematical Olympiad, 8
Let real $a$, $b$, and $c$ satisfy $$abc+a+b+c=ab+bc+ca+5.$$ Find the least possible value of $a^2+b^2+c^2$.
2023 CIIM, 6
Let $n$ be a positive integer. We define $f(n)$ as the number of finite sequences $(a_1, a_2, \ldots , a_k)$ of positive integers such that $a_1 < a_2 < a_3 < \cdots < a_k$ and $$a_1+a_2^2+a_3^3+\cdots + a_k^k \leq n.$$ Determine the positive constants $\alpha$ and $C$ such that $$\lim\limits_{n\rightarrow \infty} \frac{f(n)}{n^\alpha}=C.$$
2009 Rioplatense Mathematical Olympiad, Level 3, 3
Call a permutation of the integers $(1,2,\ldots,n)$ [i]$d$-ordered[/i] if it does not contains a decreasing subsequence of length $d$. Prove that for every $d=2,3,\ldots,n$, the number of $d$-ordered permutations of $(1,2,\ldots,n)$ is at most $(d-1)^{2n}$.
1963 Putnam, B2
Let $S$ be the set of all numbers of the form $2^m 3^n$, where $m$ and $n$ are integers. Is $S$ dense in the set of positive real numbers?
2007 IMO Shortlist, 5
Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$.
[i]Author: Christopher Bradley, United Kingdom [/i]
IMSC 2024, 6
Let $a\equiv 1\pmod{4}$ be a positive integer. Show that any polynomial $Q\in\mathbb{Z}[X]$ with all positive coefficients such that
$$Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})$$
is a perfect square for any $n\in\mathbb{N}^{\ast}$ must be a constant polynomial.
[i]Proposed by Vlad Matei, Romania[/i]
1954 Czech and Slovak Olympiad III A, 3
Show that $$\log_2\pi+\log_4\pi<\frac52.$$
2007 Nicolae Păun, 2
The bisector of $ \angle BAC $ of a triangle $ ABC $ meet the segment $ BC $ at $ D. $ Through the midpoint of $ AD $ passes aline that intersects $ AB,AC $ at $ M,N, $ respectively. Show that:
$$ \frac{1}{MA}+\frac{1}{NA} =2\left( \frac{1}{AB} +\frac{1}{AC} \right) $$
[i]Toni Mihalcea[/i]
2024 Taiwan TST Round 2, C
Find all functions $f:\mathbb{N}\to\mathbb{N}$ s.t. for all $A\subset \mathbb{N}$ with 2024 elements, the set $$S_A:=\{f^{(k)}(x)\mid k=1,...,2024,x\in A\}$$ also has 2024 elements. ($f^{(k)}=f\circ f\circ...\circ f$ is the $k$-th iteration of $f$.)
1966 IMO Shortlist, 16
We are given a circle $K$ with center $S$ and radius $1$ and a square $Q$ with center $M$ and side $2$. Let $XY$ be the hypotenuse of an isosceles right triangle $XY Z$. Describe the locus of points $Z$ as $X$ varies along $K$ and $Y$ varies along the boundary of $Q.$
1984 All Soviet Union Mathematical Olympiad, 379
Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.
2021 Israel National Olympiad, P1
Sophie wrote on a piece of paper every integer number from 1 to 1000 in decimal notation (including both endpoints).
[b]a)[/b] Which digit did Sophie write the most?
[b]b)[/b] Which digit did Sophie write the least?
2018 MOAA, 6
Consider an $m \times n$ grid of unit squares. Let $R$ be the total number of rectangles of any size, and let $S$ be the total number of squares of any size. Assume that the sides of the rectangles and squares are parallel to the sides of the $m \times n$ grid. If $\frac{R}{S} =\frac{759}{50}$ , then determine $mn$.
2024 China National Olympiad, 5
In acute $\triangle {ABC}$, ${K}$ is on the extention of segment $BC$. $P, Q$ are two points such that $KP \parallel AB, BK=BP$ and $KQ\parallel AC, CK=CQ$. The circumcircle of $\triangle KPQ$ intersects $AK$ again at ${T}$. Prove that:
(1) $\angle BTC+\angle APB=\angle CQA$.
(2) $AP \cdot BT \cdot CQ=AQ \cdot CT \cdot BP$.
Proposed by [i]Yijie He[/i] and [i]Yijuan Yao[/i]