Found problems: 85335
2007 German National Olympiad, 5
Determine all finite sets $M$ of real numbers such that $M$ contains at least $2$ numbers and any two elements of $M$ belong to an arithmetic progression of elements of $M$ with three terms.
2021 Miklós Schweitzer, 8
Prove that for a $2$-dimensional Riemannian manifold there is a metric linear connection with zero curvature if and only if the Gaussian curvature of the Riemannian manifold can be written as the divergence of a vector field.
JBMO Geometry Collection, 2013
Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.
2008 Romania Team Selection Test, 3
Show that each convex pentagon has a vertex from which the distance to the opposite side of the pentagon is strictly less than the sum of the distances from the two adjacent vertices to the same side.
[i]Note[/i]. If the pentagon is labeled $ ABCDE$, the adjacent vertices of $ A$ are $ B$ and $ E$, the ones of $ B$ are $ A$ and $ C$ etc.
1946 Moscow Mathematical Olympiad, 117
Prove that for any integers $x$ and $y$ we have $x^5 + 3x^4y - 5x^3y^2 - 15x^2y^3 + 4xy^4 + 12y^5 \ne 33$.
2010 Sharygin Geometry Olympiad, 9
A point inside a triangle is called "[i]good[/i]" if three cevians passing through it are equal. Assume for an isosceles triangle $ABC \ (AB=BC)$ the total number of "[i]good[/i]" points is odd. Find all possible values of this number.
2012 Indonesia TST, 2
The positive integers are colored with black and white such that:
- There exists a bijection from the black numbers to the white numbers,
- The sum of three black numbers is a black number, and
- The sum of three white numbers is a white number.
Find the number of possible colorings that satisfies the above conditions.
2014 HMNT, 1
What is the smallest positive integer $n$ which cannot be written in any of the following forms?
$\bullet$ $n = 1 + 2 +... + k$ for a positive integer $k$.
$\bullet$ $n = p^k$ for a prime number $p$ and integer $k$.
$\bullet$ $n = p + 1$ for a prime number $p$.
2017 Azerbaijan EGMO TST, 4
Find all positive integers $m$ and $n$ such that $(2^{2^{n}}+1)(2^{2^{m}}+1) $ is divisible by $m\cdot n $ .
2022 Belarusian National Olympiad, 10.1
Prove that for any positive integer one can place all it's divisor on a circle such that among any two neighbours one is a multiple of the other
1969 IMO Shortlist, 50
$(NET 5)$ The bisectors of the exterior angles of a pentagon $B_1B_2B_3B_4B_5$ form another pentagon $A_1A_2A_3A_4A_5.$ Construct $B_1B_2B_3B_4B_5$ from the given pentagon $A_1A_2A_3A_4A_5.$
2020 Estonia Team Selection Test, 2
There are 2020 inhabitants in a town. Before Christmas, they are all happy; but if an inhabitant does not receive any Christmas card from any other inhabitant, he or she will become sad. Unfortunately, there is only one post company which offers only one kind of service: before Christmas, each inhabitant may appoint two different other inhabitants, among which the company chooses one to whom to send a Christmas card on behalf of that inhabitant. It is known that the company makes the choices in such a way that as many inhabitants as possible will become sad. Find the least possible number of inhabitants who will become sad.
2013 Spain Mathematical Olympiad, 6
Let $ABCD$ a convex quadrilateral where:
$|AB|+|CD|=\sqrt{2} |AC|$ and $|BC|+|DA|=\sqrt{2} |BD|$
What form does the quadrilateral have?
2004 Purple Comet Problems, 14
Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are the centers of the circles?
2004 Tuymaada Olympiad, 1
50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine.
[i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]
2020 Kosovo National Mathematical Olympiad, 3
Find all prime numbers $p$ such that $3^p + 5^p -1$ is a prime number.
1970 Spain Mathematical Olympiad, 7
Calculate the values of the cosines of the angles $x$ that satisfy the next equation:
$$\sin^2 x - 2 \cos^2 x +\frac12 \sin 2x = 0.$$
2008 Putnam, A6
Prove that there exists a constant $ c>0$ such that in every nontrivial finite group $ G$ there exists a sequence of length at most $ c\ln |G|$ with the property that each element of $ G$ equals the product of some subsequence. (The elements of $ G$ in the sequence are not required to be distinct. A [i]subsequence[/i] of a sequence is obtained by selecting some of the terms, not necessarily consecutive, without reordering them; for example, $ 4,4,2$ is a subesequence of $ 2,4,6,4,2,$ but $ 2,2,4$ is not.)
2022 CMIMC Integration Bee, 14
\[\int_2^\infty \frac{\pi(x)}{x^3 - x}\,dx\]
[i]Proposed by Vlad Oleksenko[/i]
2006 Swedish Mathematical Competition, 5
In each square of an $m \times n$ rectangular board there is a nought or a cross. Let $f(m,n)$ be the number of such arrangements that contain a row or a column consisting of noughts only. Let $g(m,n)$ be the number of arrangements that contain a row consisting of noughts only, or a column consisting of crosses only. Which of the numbers $f(m,n)$ and $g(m,n)$ is larger?
2015 Irish Math Olympiad, 2
A regular polygon with $n \ge 3$ sides is given. Each vertex is coloured either red, green or blue, and no two adjacent vertices of the polygon are the same colour. There is at least one vertex of each colour.
Prove that it is possible to draw certain diagonals of the polygon in such a way that they intersect only at the vertices of the polygon and they divide the polygon into triangles so that each such triangle has vertices of three different colours.
2012 Junior Balkan Team Selection Tests - Romania, 4
Consider the set $A = \{1, 2, 3, ..., 2n - 1\}$, where $n \ge 2$ is a positive integer. We remove from the set $A$ at least $n - 1$ elements such that:
• if $a \in A$ has been removed, and $2a \in A$, then $2a$ has also been removed,
• if $a, b \in A (a \ne b)$ have been removed and $a + b \in A$, then $a + b$ has also been removed.
Which numbers have to be removed such that the sum of the remaining numbers is maximum?
2023 Bulgaria JBMO TST, 2
On the coast of a circular island there are eight different cities. Initially there are no routes between the cities. We have to construct five straight two-way routes, which do not intersect, so that from each city there are one or two routes. In how many ways can this happen?
2024-IMOC, N2
Find all positive integers $(m,n)$ such that
$$11^n+2^n+6=m^3$$
2003 China Team Selection Test, 2
In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.