Olympiad problems

1968 Polish MO Finals, 5

Given $n \ge 4$ points in the plane such that any four of them are the vertices of a convex quadrilateral, prove that these points are the vertices of a convex polygon.

2015 Romania Team Selection Tests, 1

Tags: sum , divisibility , number theory , gcd

Let $a$ be an integer and $n$ a positive integer . Show that the sum : $$\sum_{k=1}^{n} a^{(k,n)}$$ is divisible by $n$ , where $(x,y)$ is the greatest common divisor of the numbers $x$ and $y$ .

2022 Israel Olympic Revenge, 3

Tags: algebra , inequalities

Determine if there exist positive real numbers $x, \alpha$, so that for any non-empty finite set of positive integers $S$, the inequality \[\left|x-\sum_{s\in S}\frac{1}{s}\right|>\frac{1}{\max(S)^\alpha}\] holds, where $\max(S)$ is defined as the maximum element of the finite set $S$.

2010 AIME Problems, 11

Tags: linear algebra , matrix , rotation

Define a [i]T-grid[/i] to be a $ 3\times3$ matrix which satisfies the following two properties: (1) Exactly five of the entries are $ 1$'s, and the remaining four entries are $ 0$'s. (2) Among the eight rows, columns, and long diagonals (the long diagonals are $ \{a_{13},a_{22},a_{31}\}$ and $ \{a_{11},a_{22},a_{33}\}$, no more than one of the eight has all three entries equal. Find the number of distinct T-grids.

1992 Taiwan National Olympiad, 1

Tags: trigonometry , quadratic , geometry

Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$. The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\hat{BAC}<\frac{\pi}{8}$ then $S_{ABC}<2S_{A'B'C'}$.

2022 Ecuador NMO (OMEC), 6

Tags: number theory , prime

Prove that for all prime $p \ge 5$, there exist an odd prime $q \not= p$ such that $q$ divides $(p-1)^p + 1$

2016 Canadian Mathematical Olympiad Qualification, 2

Tags: geometry , algebra , perimeter , area

Let $P = (7, 1)$ and let $O = (0, 0)$. (a) If $S$ is a point on the line $y = x$ and $T$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $ST$, determine the minimum possible area of triangle $OST$. (b) If $U$ is a point on the line $y = x$ and $V$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $UV$, determine the minimum possible perimeter of triangle $OUV$.

2021 MIG, 15

Tags:

Which of the following answer choices is the closest approximation to \[\dfrac34+\dfrac78+\dfrac{15}{16}+\cdots+\dfrac{1023}{1024} = \dfrac{2^2-1}{2^2}+\dfrac{2^3-1}{2^3}+\cdots+\dfrac{2^{10}-1}{2^{10}}?\] $\textbf{(A) }\dfrac{15}{2}\qquad\textbf{(B) }8\qquad\textbf{(C) }\dfrac{17}{2}\qquad\textbf{(D) }9\qquad\textbf{(E) }\dfrac{19}{2}$

2007 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory

We call a real number $x$ with $0 < x < 1$ [i]interesting[/i] if $x$ is irrational and if in its decimal writing the first four decimals are equal. Determine the least positive integer $n$ with the property that every real number $t$ with $0 < t < 1$ can be written as the sum of $n$ pairwise distinct interesting numbers.

1995 Singapore MO Open, 3

Tags: geometric inequalities , geometry , perpendicular

Let $P$ be a point inside $\vartriangle ABC$. Let $D, E, F$ be the feet of the perpendiculars from $P$ to the lines $BC, CA$ and $AB$, respectively (see Fig. ). Show that (i) $EF = AP \sin A$, (ii) $PA+ PB + PC \ge 2(PE+ PD+ PF)$ [img]https://cdn.artofproblemsolving.com/attachments/d/f/f37d8764fc7d99c2c3f4d16f66223ef39dfd09.png[/img]

2020 Final Mathematical Cup, 3

Tags: combinatorics , invariant

Given a paper on which the numbers $1,2,3\dots ,14,15$ are written. Andy and Bobby are bored and perform the following operations, Andy chooses any two numbers (say $x$ and $y$) on the paper, erases them, and writes the sum of the numbers on the initial paper. Meanwhile, Bobby writes the value of $xy(x+y)$ in his book. They were so bored that they both performed the operation until only $1$ number remained. Then Bobby adds up all the numbers he wrote in his book, let’s call $k$ as the sum. $a$. Prove that $k$ is constant which means it does not matter how they perform the operation, $b$. Find the value of $k$.

2007 Moldova National Olympiad, 12.1

Tags: polynomial , algebra

For $a\in C^{*}$ find all $n\in N$ such that $X^{2}(X^{2}-aX+a^{2})^{2}$ divides $(X^{2}+a^{2})^{n}-X^{2n}-a^{2n}$

1993 Czech And Slovak Olympiad IIIA, 6

Tags: combinatorial geometry , 3d geometry , tetrahedron , similar , congruent

Show that there exists a tetrahedron which can be partitioned into eight congruent tetrahedra, each of which is similar to the original one.

1997 AMC 8, 5

Tags:

There are many two-digit multiples of 7, but only two of the multiples have a digit sum of 10. The sum of these two multiples of 7 is $\textbf{(A)}\ 119 \qquad \textbf{(B)}\ 126 \qquad \textbf{(C)}\ 140 \qquad \textbf{(D)}\ 175 \qquad \textbf{(E)}\ 189$

2016 PUMaC Team, 2

Tags: geometry

Temerant is a spherical planet with radius $1000$ kilometers. The government wants to build twelve towers of the same height on the equator of Temerant, so that every point on the equator can be seen from at least one tower. The minimum possible height of the towers can be written, in kilometers, as a $\sqrt{b} - c\sqrt{d} - e$ for positive integers $a, b, c, d$, and $e$ (with $b$ and $d$ not divisible by the square of any prime). Compute $a + b + c + d + e$.

2004 AMC 10, 25

Tags: geometry , 3d geometry , sphere , tetrahedron , ratio

Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere? $ \textbf{(A)}\; 3+\frac{\sqrt{30}}2\qquad \textbf{(B)}\; 3+\frac{\sqrt{69}}3\qquad \textbf{(C)}\; 3+\frac{\sqrt{123}}4\qquad \textbf{(D)}\; \frac{52}9\qquad \textbf{(E)}\; 3+2\sqrt{2} $

Indonesia Regional MO OSP SMA - geometry, 2007.1

Tags: geometry , parallel , equal segments

Let $ABCD$ be a quadrilateral with $AB = BC = CD = DA$. (a) Prove that point A must be outside of triangle $BCD$. (b) Prove that each pair of opposite sides on $ABCD$ is always parallel.

1998 All-Russian Olympiad, 2

Tags: circumcircle , projective geometry , geometry

Let $ABC$ be a triangle with circumcircle $w$. Let $D$ be the midpoint of arc $BC$ that contains $A$. Define $E$ and $F$ similarly. Let the incircle of $ABC$ touches $BC,CA,AB$ at $K,L,M$ respectively. Prove that $DK,EL,FM$ are concurrent.

2010 Iran MO (3rd Round), 5

Tags: combinatorics

suppose that $\mathcal F\subseteq p(X)$ and $|X|=n$. prove that if $|\mathcal F|>\sum_{i=0}^{k-1}\dbinom{n}{i}$ then there exist $Y\subseteq X$ with $|Y|=k$ such that $p(Y)=\mathcal F\cap Y$ that $\mathcal F\cap Y=\{F\cap Y:F\in \mathcal F\}$(20 points) you can see this problem also here: COMBINATORIAL PROBLEMS AND EXERCISES-SECOND EDITION-by LASZLO LOVASZ-AMS CHELSEA PUBLISHING- chapter 13- problem 10(c)!!!

2014 Balkan MO, 1

Tags: algebra , inequality

Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]

2002 China Team Selection Test, 1

Tags: inequalities , trigonometry , geometry

In acute triangle $ ABC$, show that: $ \sin^3{A}\cos^2{(B \minus{} C)} \plus{} \sin^3{B}\cos^2{(C \minus{} A)} \plus{} \sin^3{C}\cos^2{(A \minus{} B)} \leq 3\sin{A} \sin{B} \sin{C}$ and find out when the equality holds.

1998 Romania National Olympiad, 3

Tags: abstract algebra , rings

A ring $A$ is called Boolean if $x^2 = x$ for every $x \in A$. Prove that: a) A finite set $A$ with $n \geq 2$ elements can be equipped with the structure of a Boolean ring if and only if $n = 2^k$ for some positive integer $k$. b) The set of nonnegative integers can be equipped with the structure of a Boolean ring.

2014 Peru IMO TST, 5

Tags: geometry , combinatorics

$n$ vertices from a regular polygon with $2n$ sides are chosen and coloured red. The other $n$ vertices are coloured blue. Afterwards, the $\binom{n}{2}$ lengths of the segments formed with all pairs of red vertices are ordered in a non-decreasing sequence, and the same procedure is done with the $\binom{n}{2}$ lengths of the segments formed with all pairs of blue vertices. Prove that both sequences are identical.

2022 Saudi Arabia IMO TST, 1

Tags: number theory , divide , number theory with sequences , recurrence relation

Let $(a_n)$ be the integer sequence which is defined by $a_1= 1$ and $$ a_{n+1}=a_n^2 + n \cdot a_n \,\, , \,\, \forall n \ge 1.$$ Let $S$ be the set of all primes $p$ such that there exists an index $i$ such that $p|a_i$. Prove that the set $S$ is an infinite set and it is not equal to the set of all primes.

2015 IFYM, Sozopol, 6

Tags: algebra , functional equation

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that for $\forall$ $x,y\in \mathbb{R}$ : $f(x+f(x+y))+xy=yf(x)+f(x)+f(y)+x$.