Found problems: 85335
1990 Greece National Olympiad, 4
Froa nay real $x$, we denote $[x]$, the integer part of $x$ and with $\{x\}$ the fractional part of $x$, such that $x=[x]+\{x\}$.
a) Find at least one real $x$ such that$\{x\}+\left\{\frac{1}{x}\right\}=1$
b) Find all rationals $x$ such that $\{x\}+\left\{\frac{1}{x}\right\}=1$
2006 Cezar Ivănescu, 3
[b]a)[/b] Prove that the function $ f:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} , $ given as $ f(n)=n+(-1)^n $ is bijective.
[b]b)[/b] Find all surjective functions $ g:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} $ that have the property that $ g(n)\ge n+(-1)^n , $ for any nonnegative integer.
2019 HMNT, 7
Consider sequences $a$ of the form $a = (a_1, a_2, ... , a_{20})$ such that each term $a_i$ is either $0$ or $1$. For each such sequence $a$, we can produce a sequence $b = (b_1, b_2, ..., b_{20})$, where $$b_i\begin{cases} a_i + a_{i+1} & i = 1 \\ a_{i-1} + a_i + a_{i+1} & 1 < i < 20\\ a_{i-1} + a_i &i = 20 \end{cases}$$
2015 District Olympiad, 1
Let $ f:[0,1]\longrightarrow [0,1] $ a function with the property that, for all $ y\in [0,1] $ and $ \varepsilon >0, $ there exists a $ x\in [0,1] $ such that $ |f(x)-y|<\varepsilon . $
[b]a)[/b] Prove that if $ \left. f\right|_{[0,1]} $ is continuos, then $ f $ is surjective.
[b]b)[/b] Give an example of a function with the given property, but which isn´t surjective.
1964 AMC 12/AHSME, 26
In a ten-mile race First beats Second by $2$ miles and First beats Third by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 2\frac{1}{4}\qquad\textbf{(C)}\ 2\frac{1}{2}\qquad\textbf{(D)}\ 2\frac{3}{4}\qquad\textbf{(E)}\ 3 $
1995 Miklós Schweitzer, 1
Prove that a harmonic function that is not identically zero in the plane cannot vanish on a two-dimensional positive-measure set.
2018 Yasinsky Geometry Olympiad, 6
Let $O$ and $I$ be the centers of the circumscribed and inscribed circle the acute-angled triangle $ABC$, respectively. It is known that line $OI$ is parallel to the side $BC$ of this triangle. Line $MI$, where $M$ is the midpoint of $BC$, intersects the altitude $AH$ at the point $T$. Find the length of the segment $IT$, if the radius of the circle inscribed in the triangle $ABC$ is equal to $r$.
(Grigory Filippovsky)
2004 Estonia National Olympiad, 2
Albert and Brita play a game with a bar of $19$ adjacent squares. Initially, there is a button on the middle square of the bar. At every turn Albert mentions one positive integer less than $5$, and Brita moves button a number of squares in the direction of her choice - while doing so however, Brita must not move the button more than twice in one direction order. Prove that Albert can choose the numbers so that by the $19$th turn, Brita to be forced to move the button out of the bar.
2005 South East Mathematical Olympiad, 6
Let $P(A)$ be the arithmetic-means of all elements of set $A = \{ a_1, a_2, \ldots, a_n \}$, namely $P(A) = \frac{1}{n} \sum^{n}_{i=1}a_i$. We denote $B$ "balanced subset" of $A$, if $B$ is a non-empty subset of $A$ and $P(B) = P(A)$. Let set $M = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$.
Find the number of all "balanced subset" of $M$.
1995 USAMO, 1
Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is defined as follows: $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$ is the least positive integer that does not form an arithmetic sequence of length $\, p \,$ with any of the preceding terms. Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained by writing $\, n \,$ in base $\, p-1 \,$ and reading the result in base $\, p$.
2023 Mongolian Mathematical Olympiad, 2
There are $n$ students in a class, and some pairs of these students are friends. Among any six students, there are two of them that are not friends, and for any pair of students that are not friends there is a student among the remaining four that is friends with both of them. Find the maximum value of $n$.
2013 QEDMO 13th or 12th, 7
Let $X_1, X_2,...,X_n$ be points in the plane. For every $i$, let $A_i$ be the list of $n-1$ distances from $X_i$ to the remaining points. Find all arrangements of the $n$ points such all of these lists are the same, except for the order.
2013 Thailand Mathematical Olympiad, 10
Find all pairs of positive integers $(x, y)$ such that $\frac{xy^3}{x+y}$ is the cube of a prime.
2012 Bundeswettbewerb Mathematik, 4
A rectangle with the side lengths $a$ and $b$ with $a <b$ should be placed in a right-angled coordinate system so that there is no point with integer coordinates in its interior or on its edge.
Under what necessary and at the same time sufficient conditions for $a$ and $b$ is this possible?
2018 Harvard-MIT Mathematics Tournament, 10
Lily has a $300\times 300$ grid of squares. She now removes $100 \times 100$ squares from each of the four corners and colors each of the remaining $50000$ squares black and white. Given that no $2\times 2$ square is colored in a checkerboard pattern, find the maximum possible number of (unordered) pairs of squares such that one is black, one is white and the squares share an edge.
2017 Harvard-MIT Mathematics Tournament, 17
Sean is a biologist, and is looking at a strng of length $66$ composed of the letters $A$, $T$, $C$, $G$. A [i]substring[/i] of a string is a contiguous sequence of letters in the string. For example, the string $AGTC$ has $10$ substrings: $A$, $G$, $T$, $C$, $AG$, $GT$, $TC$, $AGT$, $GTC$, $AGTC$. What is the maximum number of distinct substrings of the string Sean is looking at?
2012 Today's Calculation Of Integral, 826
Let $G$ be a hyper elementary abelian $p-$group and let $f : G \rightarrow G$ be a homomorphism. Then prove that $\ker f$ is isomorphic to $\mathrm{coker} f$.
1999 Harvard-MIT Mathematics Tournament, 3
In a cube with side length $6$, what is the volume of the tetrahedron formed by any vertex and the three vertices connected to that vertex by edges of the cube?
2023 MMATHS, 1
Lucy has $8$ children, each of whom has a distinct favorite integer from $1$ to $10,$ inclusive. The smallest number that is a perfect multiple of all of these favorite numbers is $1260,$ and the average of these favorite numbers is at most $5.$ Find the sum of the four largest numbers.
1990 Putnam, A2
Is $ \sqrt{2} $ the limit of a sequence of numbers of the form $ \sqrt[3]{n} - \sqrt[3]{m} $, where $ n, m = 0, 1, 2, \cdots $.
2013 VJIMC, Problem 2
An $n$-dimensional cube is given. Consider all the segments connecting any two different vertices of the cube. How many distinct intersection points do these segments have (excluding the vertices)?
2018 ASDAN Math Tournament, 1
Each vertex on a cube is colored black or white independently at random with equal probability. What is the expected number of edges on the cube that connect vertices of different colors?
2010 Contests, 1
$a)$ Let $p$ and $q$ be distinct prime numbers such that $p+q^2$ divides $p^2+q$. Prove that $p+q^2$ divides $pq-1$.
$b)$ Find all prime numbers $p$ such that $p+121$ divides $p^2+11$.
2022 Durer Math Competition Finals, 5
Benedek draws circles with the same center in the following way. The first circle he draws has radius $1$. Next, he draws a second circle such that the ring between the first and second circles has twice the area of the first circle. Next, he draws a third circle such that the ring between the second and third circles is three times the area of the first circle, and so on (see the diagram).
What is the smallest $n$ fow which the radius of the $n$-th circle is an integer greater than $1$?
[img]https://cdn.artofproblemsolving.com/attachments/e/2/afa6d5ead6f2252aa821028370a3768912e674.png[/img]
PEN H Problems, 82
Find all triples $(a, b, c)$ of positive integers to the equation \[a! b! = a!+b!+c!.\]