Found problems: 85335
2020 Brazil National Olympiad, 3
Consider an inifinte sequence $x_1, x_2,\dots$ of positive integers such that, for every integer $n\geq 1$:
[list] [*]If $x_n$ is even, $x_{n+1}=\dfrac{x_n}{2}$;
[*]If $x_n$ is odd, $x_{n+1}=\dfrac{x_n-1}{2}+2^{k-1}$, where $2^{k-1}\leq x_n<2^k$.[/list]
Determine the smaller possible value of $x_1$ for which $2020$ is in the sequence.
2020 Princeton University Math Competition, A3
Let $n$ be a positive integer, and let $F$ be a family of subsets of $\{1, 2, ... , 2^n\}$ such that for any non-empty $ A\in F$ there exists $B \in F$ so that $|A| = |B| + 1$ and $B \subset A$. Suppose that F contains all $(2^n - 1)$-element subsets of $\{1, 2, ... , 2^n\}$ Determine the minimal possible value of $|F|$.
2024 USAMTS Problems, 5
Let $f(x) = x^2 + bx + 1$ for some real number $b$. Across all possible values of $b$, find all
possible values for the number of integers $x$ that satisfy $f(f(x) + x) < 0$.
1995 Tuymaada Olympiad, 8
Inside the triangle $ABC$ a point $M$ is given . Find the points $P,Q$ and $R$ lying on the sides $AB,BC$ and $AC$ respectively and such so that the sum $MP+PQ+QR+RM$ is the smallest.
2018 Online Math Open Problems, 7
A quadrilateral and a pentagon (both not self-intersecting) intersect each other at $N$ distinct points, where $N$ is a positive integer. What is the maximal possible value of $N$?
[i]Proposed by James Lin
2023 Iran Team Selection Test, 1
Suppose that $d(n)$ is the number of positive divisors of natural number $n$. Prove that there is a natural number $n$ such that
$$ \forall i\in \mathbb{N} , i \le 1402: \frac{d(n)}{d(n \pm i)} >1401 $$
[i]Proposed by Navid Safaei and Mohammadamin Sharifi [/i]
2018 CMIMC Individual Finals, 1
The [i]distance[/i] between two vertices in a connected graph is defined to be the length of the shortest path between them. How many graphs with the vertex set $\{0,1,2,\dots,6\}$ satisfy the following property: there are $3$ vertices of distance $1$ away from vertex $0$, $2$ of distance $2$ away, and $1$ of distance $3$ away?
2014 Olympic Revenge, 3
Let $n$ a positive integer. In a $2n\times 2n$ board, $1\times n$ and $n\times 1$ pieces are arranged without overlap.
Call an arrangement [b]maximal[/b] if it is impossible to put a new piece in the board without overlapping the previous ones.
Find the least $k$ such that there is a [b]maximal[/b] arrangement that uses $k$ pieces.
2004 Austrian-Polish Competition, 9
Given are the sequences
\[ (..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...); (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...)\]
of positive real numbers. For each integer $n$ the following inequalities hold:
\[a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1})\]
\[b_n \geq \frac{1}{2} (c_{n+1} + a_{n-1})\]
\[c_n \geq \frac{1}{2} (a_{n+1} + b_{n-1})\]
Determine $a_{2005}$, $b_{2005}$, $c_{2005}$, if $a_0 = 26, b_0 = 6, c_0 = 2004$.
2021 All-Russian Olympiad, 3
On a line $n+1$ segments are marked such that one of the points of the line is contained in all of them. Prove that one can find $2$ distinct segments $I, J$ which intersect at a segment of length at least $\frac{n-1}{n}d$, where $d$ is the length of the segment $I$.
1996 Mexico National Olympiad, 5
The numbers $1$ to $n^2$ are written in an n×n squared paper in the usual ordering. Any sequence of right and downwards steps from a square to an adjacent one (by side) starting at square $1$ and ending at square $n^2$ is called a path. Denote by $L(C)$ the sum of the numbers through which path $C$ goes.
(a) For a fixed $n$, let $M$ and $m$ be the largest and smallest $L(C)$ possible. Prove that $M-m$ is a perfect cube.
(b) Prove that for no $n$ can one find a path $C$ with $L(C ) = 1996$.
1985 Brazil National Olympiad, 2
Given $n$ points in the plane, show that we can always find three which give an angle $\le \pi / n$.
2010 Princeton University Math Competition, 1
Find the smallest positive integer $n$ such that $n^4 + (n+1)^4$ is composite.
1954 Moscow Mathematical Olympiad, 271
Do there exist points $A, B, C, D$ in space, such that $AB = CD = 8, AC = BD = 10$, and $AD = BC = 13$?
1998 Abels Math Contest (Norwegian MO), 4
Let $A,B,P$ be points on a line $\ell$, with $P$ outside the segment $AB$. Lines $a$ and $b$ pass through $A$ and $B$ and are perpendicular to $\ell$. A line $m$ through $P$, which is neither parallel nor perpendicular to $\ell$, intersects $a$ and $b$ at $Q$ and $R$, respectively. The perpendicular from $B$ to $AR$ meets $a$ and $AR$ at $S$ and $U$, and the perpendicular from $A$ to $BQ$ meets $b$ and $BQ$ at $T$ and $V$, respectively.
(a) Prove that $P,S,T$ are collinear.
(b) Prove that $P,U,V$ are collinear.
2024 IFYM, Sozopol, 7
Consider a finite undirected graph in which each edge belongs to at most three cycles. Prove that its vertices can be colored with three colors so that any two vertices connected by an edge have different colors.
[i](A cycle in a graph is a sequence of distinct vertices \( v_1, v_2, \ldots, v_k \), \( k \geq 3 \), such that \( v_i v_{i+1} \) is an edge for each \( i = 1, 2, \ldots, k \); we consider \( v_{k+1} = v_1 \). The edges \( v_i v_{i+1} \) belong to the cycle.)[/i]
1965 Miklós Schweitzer, 9
Let $ f$ be a continuous, nonconstant, real function, and assume the existence of an $ F$ such that $ f(x\plus{}y)\equal{}F[f(x),f(y)]$ for all real $ x$ and $ y$. Prove that $ f$ is strictly monotone.
2009 Princeton University Math Competition, 3
It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?
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 $10^m<B(2015^{100})<10^{m+1}$.
1960 Miklós Schweitzer, 2
[b]2.[/b] Construct a sequence $(a_n)_{n=1}^{\infty}$ of complex numbers such that, for every $l>0$, the series
$\sum_{n=1}^{\infty} \mid a_n \mid ^{l}$
be divergent, but for almost all $\theta$ in $(0,2\pi)$,
$\prod_{n=1}^{\infty} (1+a_n e^{i\theta})$
be convergent. [b](S. 11)[/b]
2006 Stanford Mathematics Tournament, 25
For positive integers $ n$ let $ D(n)$ denote the set of positive integers that divide $ n$ and let $ S(n)\equal{}\Sigma_{k \in D(n)} \frac{1}{k}$. What is $ S(2006)$? Answer with a fraction reduced to lowest terms.
2019 Saudi Arabia BMO TST, 2
Let sequences of real numbers $(x_n)$ and $(y_n)$ satisfy $x_1 = y_1 = 1$ and $x_{n+1} =\frac{x_n + 2}{x_n + 1}$ and $y_{n+1} = \frac{y_n^2 + 2}{2y_n}$ for $n = 1,2, ...$ Prove that $y_{n+1} = x_{2^n}$ holds for $n =0, 1,2, ... $
2016 Regional Olympiad of Mexico Center Zone, 5
An arithmetic sequence is a sequence of $(a_1, a_2, \dots, a_n) $ such that the difference between any two consecutive terms is the same. That is, $a_ {i + 1} -a_i = d $ for all $i \in \{1,2, \dots, n-1 \} $, where $d$ is the difference of the progression.
A sequence $(a_1, a_2, \dots, a_n) $ is [i]tlaxcalteca [/i] if for all $i \in \{1,2, \dots, n-1 \} $, there exists $m_i $ positive integer such that $a_i = \frac {1} {m_i}$. A taxcalteca arithmetic progression $(a_1, a_2, \dots, a_n )$ is said to be [i]maximal [/i] if $(a_1-d, a_1, a_2, \dots, a_n) $ and $(a_1, a_2, \dots, a_n, a_n + d) $ are not Tlaxcalan arithmetic progressions.
Is there a maximal tlaxcalteca arithmetic progression of $11$ elements?
1975 Putnam, A4
Let $m>1$ be an odd integer. Let $n=2m$ and $\theta=e^{2\pi i\slash n}$. Find integers $a_{1},\ldots,a_{k}$ such that
$\sum_{i=1}^{k}a_{i}\theta^{i}=\frac{1}{1-\theta}$.
2013 BMT Spring, P1
Prove that for all positive integers $m$ and $n$,
$$\frac1m\cdot\binom{2n}0-\frac1{m+1}\cdot\binom{2n}1+\frac1{m+2}\cdot\binom{2n}2-\ldots+\frac1{m+2n}\cdot\binom{2n}{n2}>0$$