Olympiad problems

2013 German National Olympiad, 1

Find all positive integers $n$ such that $n^{2}+2^{n}$ is square of an integer.

2008 Princeton University Math Competition, A10

Tags: trigonometry , 3d geometry , sphere , combinatorial geometry , geometry

A cuboctahedron is the convex hull of (smallest convex set containing) the $12$ points $(\pm 1, \pm 1, 0), (\pm 1, 0, \pm 1), (0, \pm 1, \pm 1)$. Find the cosine of the solid angle of one of the triangular faces, as viewed from the origin. (Take a figure and consider the set of points on the unit sphere centered on the origin such that the ray from the origin through the point intersects the figure. The area of that set is the solid angle of the figure as viewed from the origin.)

VI Soros Olympiad 1999 - 2000 (Russia), 8.6

Tags: algebra , number theory

Two players take turns writing down all proper non-decreasing fractions with denominators from $1 $ to $1999$ and at the same time writing a "$+$" sign before each fraction. After all such fractions are written out, their sum is found. If this amount is an integer number, then the one who made the entry last wins, otherwise his opponent wins. Who will be able to secure a win?

2005 China Girls Math Olympiad, 3

Tags: 3d geometry , tetrahedron , geometry

Determine if there exists a convex polyhedron such that (1) it has 12 edges, 6 faces and 8 vertices; (2) it has 4 faces with each pair of them sharing a common edge of the polyhedron.

2015 Spain Mathematical Olympiad, 2

Tags: geometry , incenter , circumcircle

In triangle $ABC$, let $A'$ is the symmetrical of $A$ with respect to the circumcenter $O$ of $ABC$. Prove that: [b]a)[/b] The sum of the squares of the tangents segments drawn from $A$ and $A'$ to the incircle of $ABC$ equals $$4R^2-4Rr-2r^2$$ where $R$ and $r$ are the radii of the circumscribed and inscribed circles of $ABC$ respectively. [b]b)[/b] The circle with center $A'$ and radius $A'I$ intersects the circumcircle of $ABC$ in a point $L$ such that $$AL=\sqrt{ AB.AC}$$ where $I$ is the centre of the inscribed circle of $ABC$.

2013 IMO Shortlist, G3

Tags: rhombus , trigonometry , geometry

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$.

1977 Putnam, B1

Tags:

Evaluate the infinite product $$\prod_{n=2}^{\infty} \frac{n^3-1}{n^3+1}.$$

2010 Contests, 2

Tags: number theory

Let $k$ be an integer $\geq 3$. Sequence $\{a_n\}$ satisfies that $a_k = 2k$ and for all $n > k$, we have \[a_n = \begin{cases} a_{n-1}+1 & \text{if } (a_{n-1},n) = 1 \\ 2n & \text{if } (a_{n-1},n) > 1 \end{cases} \] Prove that there are infinitely many primes in the sequence $\{a_n - a_{n-1}\}$.

2009 Miklós Schweitzer, 4

Tags: superior algebra , algebra , polynomial

Prove that the polynomial \[ f(x) \equal{} \frac {x^n \plus{} x^m \minus{} 2}{x^{\gcd(m,n)} \minus{} 1}\] is irreducible over $ \mathbb{Q}$ for all integers $ n > m > 0$.

2021 Sharygin Geometry Olympiad, 8.4

Tags: circumcenter , distance , geometry

Let $A_1$ and $C_1$ be the feet of altitudes $AH$ and $CH$ of an acute-angled triangle $ABC$. Points $A_2$ and $C_2$ are the reflections of $A_1$ and $C_1$ about $AC$. Prove that the distance between the circumcenters of triangles $C_2HA_1$ and $C_1HA_2$ equals $AC$.

2009 Jozsef Wildt International Math Competition, W. 27

Tags: perfect number , number theory , inequalities

Let $a$, $n$ be positive integers such that $a^n$ is a perfect number. Prove that $$a^{\frac{n}{\mu}}> \frac{\mu}{2}$$ where $\mu$ denotes the number of distinct prime divisors of $a^n$

2015 239 Open Mathematical Olympiad, 7

Tags: combinatorics

There is a closed polyline with $n$ edges on the plane. We build a new polyline which edges connect the midpoints of two adjacent edges of the previous polyline. Then we erase previous polyline and start over and over. Also we know that each polyline satisfy that all vertices are different and not all of them are collinear. For which $n$ we can get a polyline that is a сonvex polygon?

Russian TST 2019, P2

Tags: polyhedron , edge , combinatorics

Two ants are moving along the edges of a convex polyhedron. The route of every ant ends in its starting point, so that one ant does not pass through the same point twice along its way. On every face $F$ of the polyhedron are written the number of edges of $F$ belonging to the route of the first ant and the number of edges of $F$ belonging to the route of the second ant. Is there a polyhedron and a pair of routes described as above, such that only one face contains a pair of distinct numbers? [i]Proposed by Nikolai Beluhov[/i]

1999 Romania Team Selection Test, 13

Tags: combinatorics

Let $n\geq 3$ and $A_1,A_2,\ldots,A_n$ be points on a circle. Find the largest number of acute triangles that can be considered with vertices in these points. [i]G. Eckstein[/i]

2011 Princeton University Math Competition, A2 / B3

Tags: combinatorics

A set of $n$ dominoes, each colored with one white square and one black square, is used to cover a $2 \times n$ board of squares. For $n = 6$, how many different patterns of colors can the board have? (For $n = 2$, this number is $6$.)

2017 Danube Mathematical Olympiad, 2

Tags: combinatorics

Let $n\geq 3$ be a positive integer. Consider an $n\times n$ square. In each cell of the square, one of the numbers from the set $M=\{1,2,\ldots,2n-1\}$ is to be written. One such filling is called [i]good[/i] if, for every index $1\leq i\leq n,$ row no. $i$ and column no. $i,$ together, contain all the elements of $M$. [list=a] [*]Prove that there exists $n\geq 3$ for which a good filling exists. [*]Prove that for $n=2017$ there is no good filling of the $n\times n$ square. [/list]

2007 Purple Comet Problems, 5

Tags: induction

$F(0)=3$ and $F(n)=F(n-1)+4$ when $n$ is positive. Find $F(F(F(5)))$.

Kvant 2021, M2635

Tags: inequalities , inradius , geometry

In the triangle $ABC$, the lengths of the sides $BC, CA$ and $AB$ are $a,b$ and $c{}$ respectively. Several segments are drawn from the vertex $C{}$, which cut the triangle $ABC$ into several triangles. Find the smallest number $M{}$ for which, with each such cut, the sum of the radii of the circles inscribed in triangles does not exceed $M{}$. [i]Porposed by O. Titov[/i]

2006 Peru MO (ONEM), 2

Tags: equal area , combinatorial geometry , geometry

Find all values of $k$ by which it is possible to divide any triangular region in $k$ quadrilaterals of equal area.

2001 Junior Balkan Team Selection Tests - Moldova, 6

Tags: inequalities , algebra

Let the nonnegative numbers $a_1, a_2,... a_9$, where $a_1 = a_9 = 0$ and let at least one of the numbers is nonzero. Denote the sentence $(P)$: '' For $2 \le i \le 8$ there is a number $a_i$, such that $a_{i - 1} + a_{i + 1} <ka_i $”. a) Show that the sentence $(P)$ is true for $k = 2$. b) Determine whether is the sentence $(P)$ true for $k = \frac{19}{10}$

MathLinks Contest 2nd, 1.1

Tags: inequalities

Let $x, y, z$ be positive numbers such that $xyz \le 2$ and $\frac{1}{x^2}+ \frac{1}{y^2}+ \frac{1}{z^2}< k$, for some real $k \ge 2$. Find all values of $k$ such that the conditions above imply that there exist a triangle having the side-lengths $x, y, z$.

1987 Federal Competition For Advanced Students, P2, 4

Tags: number theory

Find all triples $ (x,y,z)$ of natural numbers satisfying $ 2xz\equal{}y^2$ and $ x\plus{}z\equal{}1987$.

1972 AMC 12/AHSME, 4

Tags:

The number of solutions to $\{1,~2\}\subseteq~X~\subseteq~\{1,~2,~3,~4,~5\}$, where $X$ is a subset of $\{1,~2,~3,~4,~5\}$ is $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }\text{None of these}$

1951 Putnam, B4

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Investigate, in any way which yields significant results, the existence, in the plane, of the configuration consisting of an ellipse simultaneously tangent to four distinct concentric circles.

2018 Dutch IMO TST, 1

Tags: combinatorics , grid

Suppose a grid with $2m$ rows and $2n$ columns is given, where $m$ and $n$ are positive integers. You may place one pawn on any square of this grid, except the bottom left one or the top right one. After placing the pawn, a snail wants to undertake a journey on the grid. Starting from the bottom left square, it wants to visit every square exactly once, except the one with the pawn on it, which the snail wants to avoid. Moreover, it wants to finish in the top right square. It can only move horizontally or vertically on the grid. On which squares can you put the pawn for the snail to be able to finish its journey?