Olympiad problems

1989 IMO Longlists, 78

Tags: polynomial , algebra

Let $ P(x)$ be a polynomial with integer coefficients such that \[ P(m_1) \equal{} P(m_2) \equal{} P(m_3) \equal{} P(m_4) \equal{} 7\] for given distinct integers $ m_1,m_2,m_3,$ and $ m_4.$ Show that there is no integer m such that $ P(m) \equal{} 14.$

2015 Saudi Arabia JBMO TST, 2

Tags: combinatorics

Given is a binary string $0101010101$. On a move Ali changes 0 to 1 or 1 to 0. The following conditions are fulfilled: a) All the strings obtained are different. b) All the strings obtained must have at least 5 times 1. Prove that Ali can't obtain more than 555 strings.

2021/2022 Tournament of Towns, P7

Tags: number theory , prime number , geometry

Let $p$ be a prime number and let $M$ be a convex polygon. Suppose that there are precisely $p$ ways to tile $m$ with equilateral triangles with side $1$ and squares with side $1$. Show there is some side of $M$ of length $p-1$.

1977 IMO, 3

Tags: number theory , prime number , prime factorization , dirichlet s theorem

Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

2023 JBMO TST - Turkey, 4

Tags: number theory , divisibility

For a prime number $p$. Can the number of n positive integers that make the expression \[\dfrac{n^3+np+1}{n+p+1}\] an integer be $777$?

1978 Romania Team Selection Test, 3

Tags: algebra , polynomial , analytic geometry , geometry

Let $ P[X,Y] $ be a polynomial of degree at most $ 2 .$ If $ A,B,C,A',B',C' $ are distinct roots of $ P $ such that $ A,B,C $ are not collinear and $ A',B',C' $ lie on the lines $ BC,CA, $ respectively, $ AB, $ in the planar representation of these points, show that $ P=0. $

2024 Thailand TSTST, 11

Tags: combinatorics , induction

Find the maximal number of points, such that there exist a configuration of $2023$ lines on the plane, with each lines pass at least $2$ points.

1977 Czech and Slovak Olympiad III A, 1

Tags: geometry , 3d geometry , combinatorial geometry , cube

There are given 2050 points in a unit cube. Show that there are 5 points lying in an (open) ball with the radius 1/9.

LMT Team Rounds 2021+, 6

Tags: number theory

Find the least positive integer $m$ such that $105| 9^{(p^2)} -29^p +m$ for all prime numbers $p > 3$.

1980 Austrian-Polish Competition, 6

Tags: inequalities , absolute value , sequence , algebra

Let $a_1,a_2,a_3,\dots$ be a sequence of real numbers satisfying the inequality \[ |a_{k+m}-a_k-a_m| \leq 1 \quad \text{for all} \ k,m \in \mathbb{Z}_{>0}. \] Show that the following inequality holds for all positive integers $k,m$ \[ \left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}. \]

2008 Greece Junior Math Olympiad, 4

Tags: geometry , trapezoid , isosceles , concurrency

Let $ABCD$ be a trapezoid with $AD=a, AB=2a, BC=3a$ and $\angle A=\angle B =90 ^o$. Let $E,Z$ be the midpoints of the sides $AB ,CD$ respectively and $I$ be the foot of the perpendicular from point $Z$ on $BC$. Prove that : i) triangle $BDZ$ is isosceles ii) midpoint $O$ of $EZ$ is the centroid of triangle $BDZ$ iii) lines $AZ$ and $DI$ intersect at a point lying on line $BO$

Estonia Open Senior - geometry, 2020.2.5

Tags: geometry , ratio , angle

The bisector of the interior angle at the vertex $B$ of the triangle $ABC$ and the perpendicular line on side $BC$ passing through the vertex $C$ intersects at $D$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $BD$, respectively, with $N$ on the side $AC$. Find all possibilities of the angles of the triangles $ABC$, if it is known that $\frac{| AM |}{| BC |}=\frac{|CD|}{|BD|}$. .

1992 National High School Mathematics League, 4

Tags:

In $\triangle ABC$, $b\neq1$. If $\frac{C}{A}$ and $\frac{\sin B}{\sin A}$ are solutions to equation $\log_{\sqrt{b}}x=\log_{b}(4x-4)$, then $\triangle ABC$ $\text{(A)}$is an isosceles triangle, but not right-angled triangle $\text{(B)}$is a right-angled triangle, but not isosceles triangle $\text{(C)}$is an isosceles right-angled triangle $\text{(D)}$is neither a right-angled triangle nor an isosceles triangle

2022 Saint Petersburg Mathematical Olympiad, 7

Tags: combinatorics

Given is a set of $2n$ cards numbered $1,2, \cdots, n$, each number appears twice. The cards are put on a table with the face down. A set of cards is called good if no card appears twice. Baron Munchausen claims that he can specify $80$ sets of $n$ cards, of which at least one is sure to be good. What is the maximal $n$ for which the Baron's words could be true?

2013 Pan African, 1

Tags: trapezoid , parallelogram , ratio , geometry

Let $ABCD$ be a convex quadrilateral with $AB$ parallel to $CD$. Let $P$ and $Q$ be the midpoints of $AC$ and $BD$, respectively. Prove that if $\angle ABP=\angle CBD$, then $\angle BCQ=\angle ACD$.

2009 German National Olympiad, 4

Tags: inequalities , inequality , algebra , radical

Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$, such that inequality holds $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$

2024 Tuymaada Olympiad, 2

Tags: inequalities , combinatorics

We will call a [i]hedgehog[/i] a graph in which one vertex is connected to all the others and there are no other edges; the number of vertices of this graph will be called the size of the hedgehog. A graph $G$ is given on $n$ vertices (where $n > 1$). For each edge $e$, we denote by $s(e)$ the size of the maximum hedgehog in graph $G$, which contains this edge. Prove the inequality (summation is carried out over all edges of the graph $G$): \[\sum_e \frac{1}{s(e)} \leqslant \frac{n}{2}.\] [i]Proposed by D. Malec, C. Tompkins[/i]

2010 Purple Comet Problems, 21

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Let $a$ be the sum of the numbers: $99 \times 0.9$ $999 \times 0.9$ $9999 \times 0.9$ $\vdots$ $999\cdots 9 \times 0.9$ where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$. Find the sum of the digits in the number $a$.

2016 LMT, 11

Tags:

Find all ordered triples $(a,b,c)$ of real numbers such that \[\begin{cases} a+b=c,\\ a^2+b^2=c^2-c-6,\\ a^3+b^3 = c^3-2c^2-5c. \\ \end{cases}\] [i]Proposed by Evan Fang

2022 SEEMOUS, 1

Tags: eigenvalue , linear algebra , matrices , sylvester inequality

Let $A, B \in \mathcal{M}_n(\mathbb{C})$ be such that $AB^2A = AB$. Prove that: a) $(AB)^2 = AB.$ b) $(AB - BA)^3 = O_n.$

1993 All-Russian Olympiad Regional Round, 11.6

Tags: geometry , 3d geometry , tetrahedron

Seven tetrahedra are placed on the table. For any three of them there exists a horizontal plane cutting them in triangles of equal areas. Show that there exists a plane cutting all seven tetrahedra in triangles of equal areas.

2016 ISI Entrance Examination, 7

Tags: integration , calculus , real analysis

$f$ is a differentiable function such that $f(f(x))=x$ where $x \in [0,1]$.Also $f(0)=1$.Find the value of $$\int_0^1(x-f(x))^{2016}dx$$

2019 Polish Junior MO First Round, 7

Tags: geometry , 3d geometry , pyramid

A cube $ABCDA'B'C'D'$ is given with an edge of length $2$ and vertices marked as in the figure. The point $K$ is center of the edge $AB$. The plane containing the points $B',D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$. [img]https://cdn.artofproblemsolving.com/attachments/7/9/721989193ffd830fd7ad43bdde7e177c942c76.png[/img]

1999 India Regional Mathematical Olympiad, 4

Tags:

If $p,q,r$ are the roots of the cubic equation $x^3 - 3px^2 + 3q^2 x - r^3 = 0$, then show that $p = q =r$.

1973 IMO, 1

Tags: vector , geometry , geometric inequality , inequalities

Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1.