Found problems: 85335
2022 Romania National Olympiad, P3
Let $f,g:\mathbb{R}\to\mathbb{R}$ be two nondecreasing functions.
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[*]Show that for any $a\in\mathbb{R},$ $b\in[f(a-0),f(a+0)]$ and $x\in\mathbb{R},$ the following inequality holds \[\int_a^xf(t) \ dt\geq b(x-a).\]
[*]Given that $[f(a-0),f(a+0)]\cap[g(a-0),g(a+0)]\neq\emptyset$ for any $a\in\mathbb{R},$ prove that for any real numbers $a<b$\[\int_a^b f(t) \ dt=\int_a^b g(t) \ dt.\]
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[i]Note: $h(a-0)$ and $h(a+0)$ denote the limits to the left and to the right respectively of a function $h$ at point $a\in\mathbb{R}.$[/i]
2007 Estonia Team Selection Test, 4
In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$
2016 Purple Comet Problems, 1
Two integers have a sum of 2016 and a difference of 500. Find the larger of the two integers.
2005 MOP Homework, 5
A group consists of $n$ tourists. Among every three of them there are at least two that are not familiar. For any partition of the group into two busses, there are at least two familiar tourists in one bus. Prove that there is a tourist who is familiar with at most two fifth of all the tourists.
2019 European Mathematical Cup, 4
Let $u$ be a positive rational number and $m$ be a positive integer. Define a sequence $q_1,q_2,q_3,\dotsc$ such that $q_1=u$ and for $n\geqslant 2$:
$$\text{if }q_{n-1}=\frac{a}{b}\text{ for some relatively prime positive integers }a\text{ and }b, \text{ then }q_n=\frac{a+mb}{b+1}.$$
Determine all positive integers $m$ such that the sequence $q_1,q_2,q_3,\dotsc$ is eventually periodic for any positive rational number $u$.
[i]Remark:[/i] A sequence $x_1,x_2,x_3,\dotsc $ is [i]eventually periodic[/i] if there are positive integers $c$ and $t$ such that $x_n=x_{n+t}$ for all $n\geqslant c$.
[i]Proposed by Petar Nizié-Nikolac[/i]
2016 Croatia Team Selection Test, Problem 2
Let $N$ be a positive integer. Consider a $N \times N$ array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A [i]move[/i] consists of choosing a row or a column and changing the color of every cell in the chosen row or column.
What is the minimal number of additional cells that one has to color black such that, after a finite number of moves, a completely black board can be reached?
MathLinks Contest 6th, 6.2
A $n \times n$ matrix is filled with non-negative real numbers such that on each line and column the sum of the elements is $1$. Prove that one can choose n positive entries from the matrix, such that each of them lies on a different line and different column.
1984 IMO Shortlist, 1
Find all solutions of the following system of $n$ equations in $n$ variables:
\[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\]
where $a$ is a given number.
1995 Flanders Math Olympiad, 4
Given a regular $n$-gon inscribed in a circle of radius 1, where $n > 3$.
Define $G(n)$ as the average length of the diagonals of this $n$-gon.
Prove that if $n \rightarrow \infty, G(n) \rightarrow \frac{4}{\pi}$.
2018 VJIMC, 3
Let $n$ be a positive integer and let $x_1,\dotsc,x_n$ be positive real numbers satisfying $\vert x_i-x_j\vert \le 1$ for all pairs $(i,j)$ with $1 \le i<j \le n$. Prove that
\[\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1} \ge \frac{x_2+1}{x_1+1}+\frac{x_3+1}{x_2+1}+\dots+\frac{x_n+1}{x_{n-1}+1}+\frac{x_1+1}{x_n+1}.\]
2022-2023 OMMC, 11
Positive real numbers $x,y$ satisfy $$\left \lfloor xy \right \rfloor - \lfloor x \rfloor \lfloor y \rfloor = 8.$$ Find the sum of all possible values of the quantity $\left \lfloor 2xy \right \rfloor - \lfloor 2x \rfloor \lfloor y \rfloor.$
2010 Finnish National High School Mathematics Competition, 4
In a football season, even number $n$ of teams plays a simple series, i.e. each team plays once against each other team. Show that ona can group the series into $n-1$ rounds such that in every round every team plays exactly one match.
2021 Miklós Schweitzer, 9
For a given natural number $n$, two players randomly (uniformly distributed) select a common number $0 \le j \le n$, and then each of them independently randomly selects a subset of $\{1,2, \cdots, n \}$ with $j$ elements. Let $p_n$ be the probability that the same set was chosen. Prove that
\[ \sum_{k=1}^{n} p_k = 2 \log{n} + 2 \gamma - 1 + o(1), \quad (n \to \infty),\]
where $\gamma$ is the Euler constant.
2000 Manhattan Mathematical Olympiad, 3
Suppose one has an unlimited supply of identical tiles in the shape of a right triangle
[asy]
draw((0,0)--(3,0)--(3,2)--(0,0));
label("$A$",(0,0),SW);
label("$B$",(3,0),SE);
label("$C$",(3,2),NE);
size(100);
[/asy]
such that, if we measure the sides $AB$ and $AC$ (in inches) their lengths are integers. Prove that one can pave a square completely (without overlaps) with a number of these tiles, exactly when $BC$ has integer length.
2021 Canadian Mathematical Olympiad Qualification, 1
Determine all real polynomials $p$ such that $p(x+p(x))=x^2p(x)$ for all $x$.
1971 Canada National Olympiad, 5
Let \[ p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0, \] where the coefficients $a_i$ are integers. If $p(0)$ and $p(1)$ are both odd, show that $p(x)$ has no integral roots.
2014 Saint Petersburg Mathematical Olympiad, 5
$M$ is infinite set of natural numbers. If $a,b, a\neq b$ are in $M$, then $a^b+2$ or $a^b-2$ ( or both) are in $M$. Prove that there is composite number in $M$
2020 AMC 12/AHSME, 16
A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0),$ $(2020, 0),$ $(2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth$?$
$\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$
2003 Costa Rica - Final Round, 6
Say a number is [i]tico[/i] if the sum of it's digits is a multiple of $2003$.
$\text{(i)}$ Show that there exists a positive integer $N$ such that the first $2003$ multiples, $N,2N,3N,\ldots 2003N$ are all tico.
$\text{(ii)}$ Does there exist a positive integer $N$ such that all it's multiples are tico?
2014 Online Math Open Problems, 14
What is the greatest common factor of $12345678987654321$ and $12345654321$?
[i]Proposed by Evan Chen[/i]
2003 Pan African, 2
Find all positive integers $n$ such that $21$ divides $2^{2^n}+2^n+1$.
1994 Abels Math Contest (Norwegian MO), 1b
Let $C$ be a point on the extension of the diameter $AB$ of a circle. A line through $C$ is tangent to the circle at point $N$. The bisector of $\angle ACN$ meets the lines $AN$ and $BN$ at $P$ and $Q$ respectively. Prove that $PN = QN$.
2007 Grigore Moisil Intercounty, 1
For a point $ P $ situated in the plane determined by a triangle $ ABC, $ prove the following inequality:
$$ BC\cdot PB\cdot PC+AC\cdot PC\cdot PA +AB\cdot PA\cdot PB\ge AB\cdot BC\cdot CA $$
1997 China Team Selection Test, 1
Given a real number $\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\bigtriangleup ABC$. Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \lambda BA$, $CV = \lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \lambda UV$. Find the locus of point $Q$.
2008 Turkey MO (2nd round), 3
Let a.b.c be positive reals such that their sum is 1. Prove that
$ \frac{a^{2}b^{2}}{c^{3}(a^{2}\minus{}ab\plus{}b^{2})}\plus{}\frac{b^{2}c^{2}}{a^{3}(b^{2}\minus{}bc\plus{}c^{2})}\plus{}\frac{a^{2}c^{2}}{b^{3}(a^{2}\minus{}ac\plus{}c^{2})}\geq \frac{3}{ab\plus{}bc\plus{}ac}$