Olympiad problems

2023 MOAA, 14

Tags:

Let $N$ be the number of ordered triples of 3 positive integers $(a,b,c)$ such that $6a$, $10b$, and $15c$ are all perfect squares and $abc = 210^{210}$. Find the number of divisors of $N$. [i]Proposed by Andy Xu[/i]

1982 AMC 12/AHSME, 8

Tags: arithmetic sequence

By definition, $ r! \equal{} r(r \minus{} 1) \cdots 1$ and $ \binom{j}{k} \equal{} \frac {j!}{k!(j \minus{} k)!}$, where $ r,j,k$ are positive integers and $ k < j$. If $ \binom{n}{1}, \binom{n}{2}, \binom{n}{3}$ form an arithmetic progression with $ n > 3$, then $ n$ equals $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 9\qquad \textbf{(D)}\ 11\qquad \textbf{(E)}\ 12$

2006 Sharygin Geometry Olympiad, 6

Tags: geometry , construction , tangent , circles

a) Given a segment $AB$ with a point $C$ inside it, which is the chord of a circle of radius $R$. Inscribe in the formed segment a circle tangent to point $C$ and to the circle of radius $R$. b) Given a segment $AB$ with a point $C$ inside it, which is the point of tangency of a circle of radius $r$. Draw through $A$ and $B$ a circle tangent to a circle of radius $r$.

1974 Chisinau City MO, 77

Tags: algebra , arithmetic progression

Is it possible to simultaneously take away on eight three-ton vehicles $50$ stones, the weight of which is respectively equal to $416, 418, 420, .., 512, 514$ kg?

2007 Germany Team Selection Test, 3

Tags: number theory , rational , decimal representation

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

1996 Bulgaria National Olympiad, 2

Tags: geometry

Find the side length of the smallest equilateral triangle in which three discs of radii $2,3,4$ can be placed without overlap.

1988 China Team Selection Test, 3

Tags: geometry , geometric transformation , vector , analytic geometry , combinatorics , lattice

A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.

2002 Tournament Of Towns, 1

Tags: combinatorics

There are $2002$ employees in a bank. All the employees came to celebrate the bank's jubilee and were seated around one round table. It is known that the difference in salaries of any two adjacent employees is $2$ or $3$ dollars. Find the maximal difference in salaries of two employees, if it is known all salaries are different.

2003 Austrian-Polish Competition, 4

Tags: divide , divisible , product , number theory

A positive integer $m$ is alpine if $m$ divides $2^{2n+1} + 1$ for some positive integer $n$. Show that the product of two alpine numbers is alpine.

2007 Gheorghe Vranceanu, 3

Tags: limit , binom , calculus

$ \lim_{n\to\infty } \sqrt[n]{\sum_{i=0}^n\binom{n}{i}^2} $

1994 Korea National Olympiad, Problem 1

Tags: function , algebra

Let $ S$ be the set of nonnegative integers. Find all functions $ f,g,h: S\rightarrow S$ such that $ f(m\plus{}n)\equal{}g(m)\plus{}h(n),$ for all $ m,n\in S$, and $ g(1)\equal{}h(1)\equal{}1$.

1952 Moscow Mathematical Olympiad, 212

Tags: geometry , ratio , orthocenter , equilateral , altitude

Prove that if the orthocenter divides all heights of a triangle in the same proportion, the triangle is equilateral.

2007 German National Olympiad, 5

Tags: arithmetic sequence , finite , algebra , real number , set

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

Tags: manifold , differential geometry , topology

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

Tags: geometry , circumcircle , trigonometry , analytic geometry , graphing lines , slope , euler

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

Tags: inequalities , vector , trigonometry , analytic geometry , geometry

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

Tags: diophantine equation , number theory

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

Tags: geometry

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

Tags: combinatorics

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

Tags: number theory

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

Tags: number theory

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

Tags: combinatorics , number theory

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

Tags: pentagon , construction , angle bisector , geometry

$(NET 5)$ 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

Tags: combinatorics

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

Tags: geometry

Let $ABCD$ a convex quadrilateral where: $|AB|+|CD|=\sqrt{2} |AC|$ and $|BC|+|DA|=\sqrt{2} |BD|$ What form does the quadrilateral have?