Found problems: 85335
2000 National Olympiad First Round, 35
If every $k-$element subset of $S=\{1,2,\dots , 32\}$ contains three different elements $a,b,c$ such that $a$ divides $b$, and $b$ divides $c$, $k$ must be at least ?
$ \textbf{(A)}\ 17
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 25
\qquad\textbf{(D)}\ 29
\qquad\textbf{(E)}\ \text{None}
$
2003 Junior Balkan MO, 2
Suppose there are $n$ points in a plane no three of which are collinear with the property that if we label these points as $A_1,A_2,\ldots,A_n$ in any way whatsoever, the broken line $A_1A_2\ldots A_n$ does not intersect itself. Find the maximum value of $n$.
[i]Dinu Serbanescu, Romania[/i]
2016 India Regional Mathematical Olympiad, 5
Let $x,y,z$ be non-negative real numbers such that $xyz=1$. Prove that $$(x^3+2y)(y^3+2z)(z^3+2x) \ge 27.$$
Kvant 2020, M2598
Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$?
Mikhail Evdokimov
2011 Sharygin Geometry Olympiad, 6
Let $BB_1$ and $CC_1$ be the altitudes of acute-angled triangle $ABC$, and $A_0$ is the midpoint of $BC$. Lines $A_0B_1$ and $A_0C_1$ meet the line passing through $A$ and parallel to $BC$ in points $P$ and $Q$. Prove that the incenter of triangle $PA_0Q$ lies on the altitude of triangle $ABC$.
1970 IMO Longlists, 56
A square hole of depth $h$ whose base is of length $a$ is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length $L >\sqrt{2a^2+h^2}$, and walks on the ground around the hole. The edges of the hole are smooth, so that the rope can freely slide along it. Find the shape and area of the territory accessible to the dog (whose size is neglected).
2022 AIME Problems, 12
For any finite set $X$, let $|X|$ denote the number of elements in $X.$ Define $$S_n = \sum |A \cap B|,$$ where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\{1, 2, 3, …, n\}$ with $|A| = |B|.$ For example, $S_2 = 4$ because the sum is taken over the pairs of subsets $$(A, B) \in \{ (\emptyset, \emptyset), (\{1\}, \{1\}), (\{1\}, \{2\}), (\{2\}, \{1\}), (\{2\}, \{2\}), (\{1, 2\}, \{1, 2\})\},$$ giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4.$ Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by $1000.$
2019 Tournament Of Towns, 7
Peter has a wooden square stamp divided into a grid. He coated some $102$ cells of this grid with black ink. After that, he pressed this stamp $100$ times on a list of paper so that each time just those $102$ cells left a black imprint on the paper. Is it possible that after his actions the imprint on the list is a square $101 \times 101$ such that all the cells except one corner cell are black?
(Alexsandr Gribalko)
2004 Spain Mathematical Olympiad, Problem 3
Represent for $\mathbb {Z}$ the set of all integers. Find all of the functions ${f:}$ $ \mathbb{Z} \rightarrow \mathbb{Z}$ such that for any ${x,y}$ integers, they satisfy:
${f(x + f(y)) = f(x) - y.}$
2022 MMATHS, 8
In the number puzzle below, each cell contains a digit, each cell in the same bolded region has the same digit, and cells in different bolded regions have different digits. The answers to the clues are to be read as three-, four-, or five-digit numbers. Find the unique solution to the puzzle, given that no answer to any clue has a leading $0$.
[img]https://cdn.artofproblemsolving.com/attachments/b/a/23514673819aea46c30fd2947f8c82710a1fb3.png[/img]
2021 Miklós Schweitzer, 5
Let $f(x)=\frac{1+\cos(2 \pi x)}{2}$, for $x \in \mathbb{R}$, and $f^n=\underbrace{ f \circ \cdots \circ f}_{n}$. Is it true that for Lebesgue almost every $x$, $\lim_{n \to \infty} f^n(x)=1$?
2024 Pan-African, 6
Find all integers $n$ for which $n^7-41$ is the square of an integer
1974 Poland - Second Round, 3
Prove that the orthogonal projections of the vertex $ D $ of the tetrahedron $ ABCD $ onto the bisectors of the internal and external dihedral angles at the edges $ \overline{AB} $, $ \overline{BC} $ and $ \overline{CA} $ belong to one plane .
2014 District Olympiad, 3
The medians $AD, BE$ and $CF$ of triangle $ABC$ intersect at $G$. Let $P$ be a point lying in the interior of the triangle, not belonging to any of its medians. The line through $P$ parallel to $AD$ intersects the side $BC$ at $A_{1}$. Similarly one defines the points $B_{1}$ and $C_{1}$. Prove that
\[ \overline{A_{1}D}+\overline{B_{1}E}+\overline{C_{1}F}=\frac{3}{2}\overline{PG} \]
2014 Spain Mathematical Olympiad, 3
$60$ points are on the interior of a unit circle (a circle with radius $1$). Show that there exists a point $V$ on the circumference of the circle such that the sum of the distances from $V$ to the $60$ points is less than or equal to $80$.
2023 South East Mathematical Olympiad, 5
Let $AB$ be a chord of the semicircle $O$ (not the diameter). $M$ is the midpoint of $AB$, and $D$ is a point lies on line $OM$ ($D$ is outside semicircle $O$). Line $l$ passes through $D$ and is parallel to $AB$. $P, Q$ are two points lie on $l$ and $PO$ meets semicircle $O$ at $C$.
If $\angle PCD=\angle DMC$, and $M$ is the orthocentre of $\triangle OPQ$. Prove that the intersection of $AQ$ and $PB$ lies on semicircle $O$.
1992 All Soviet Union Mathematical Olympiad, 565
An $m \times n$ rectangle is divided into mn unit squares by lines parallel to its sides. A gnomon is the figure of three unit squares formed by deleting one unit square from a $2 \times 2$ square. For what $m, n$ can we divide the rectangle into gnomons so that no two gnomons form a rectangle and no vertex is in four gnomons?
2019 Greece Team Selection Test, 3
Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied:
[list=1]
[*] Each number in the table is congruent to $1$ modulo $n$.
[*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$.
[/list]
Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.
2022 Brazil EGMO TST, 8
Find all pairs $(a,b)$ of positive integers, such that for [b]every[/b] $n$ positive integer, the equality $a^n+b^n=c_n^{n+1}$ is true, for some $c_n$ positive integer.
2019 Online Math Open Problems, 7
At a concert $10$ singers will perform. For each singer $x$, either there is a singer $y$ such that $x$ wishes to perform right after $y$, or $x$ has no preferences at all. Suppose that there are $n$ ways to order the singers such that no singer has an unsatisfied preference, and let $p$ be the product of all possible nonzero values of $n$. Compute the largest nonnegative integer $k$ such that $2^k$ divides $p$.
[i]Proposed by Gopal Goel[/i]
2019 Abels Math Contest (Norwegian MO) Final, 1
You have an $n \times n$ grid of empty squares. You place a cross in all the squares, one at a time. When you place a cross in an empty square, you receive $i+j$ points if there were $i$ crosses in the same row and $j$ crosses in the same column before you placed the new cross. Which are the possible total scores you can get?
2024 MMATHS, 1
Let $f$ be a function over the domain of all positive real numbers such that $$f(x)=\frac{1-\sqrt{x}}{1+\sqrt{x}}$$ Now, let $g(x)$ be a function given by $$g(x)=f(x)^{\tfrac{2f\left(\tfrac{1}{x}\right)}{f(x)}}$$ $g(100)$ can be expressed as a fraction $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime integers. What is the sum of $a$ and $b$?
1987 Tournament Of Towns, (154) 5
We are given three non-negative numbers $A , B$ and $C$ about which it is known that $$A^4 + B^4 + C^4 \le 2(A^2B^2 + B^2C^2 + C^2A^2)$$
(a) Prove that each of $A, B$ and $C$ is not greater than the sum of the others.
(b) Prove that $A^2 + B^2 + C^2 \le 2(AB + BC + CA)$ .
(c) Does the original inequality follow from the one in (b)?
(V.A. Senderov , Moscow)
2014 Middle European Mathematical Olympiad, 1
Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\]
holds for all $x,y \in \mathbb{R}$.
2011 QEDMO 9th, 7
Find all functions $f: R\to R$, such that $f(xy + x + y) + f(xy-x-y)=2f (x) + 2f (y)$ for all $x, y \in R$.