Olympiad problems

2000 Irish Math Olympiad, 4

The sequence $ a_1<a_2<...<a_M$ of real numbers is called a weak arithmetic progression of length $ M$ if there exists an arithmetic progression $ x_0,x_1,...,x_M$ such that: $ x_0 \le a_1<x_1 \le a_2<x_2 \le ... \le a_M<x_M.$ $ (a)$ Prove that if $ a_1<a_2<a_3$ then $ (a_1,a_2,a_3)$ is a weak arithmetic progression. $ (b)$ Prove that any subset of $ \{ 0,1,2,...,999 \}$ with at least $ 730$ elements contains a weak arithmetic progression of length $ 10$.

1978 Germany Team Selection Test, 5

Tags: geometry , 3d geometry , pyramid , combinatorics

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

2007 Stars of Mathematics, 2

Tags: graph theory , discrete math

Let be a structure formed by $ n\ge 4 $ points in space, four by four noncoplanar, and two by two connected by a wire. If we cut the $ n-1 $ wires that connect a point to the others, the remaining point is said to be [i]isolated.[/i] The structure is said to be [i]disconnected[/i] if there are at least two points for which there isn´t a chain of wires connecting them. So, initially, it´s not disconnected. $ \text{(1)} $ Prove that, by cutting a number smaller or equal with $ n-2, $ the structure won´t become disconnected. $ \text{(2)} $ Determine the minimum number of wires that needs to be cut so that the remaining structure is disconnected, yet every point, not isloated.

2018 AMC 10, 22

Tags: number theory , greatest common divisor

Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$? $\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}$

1976 Czech and Slovak Olympiad III A, 4

Tags: algebra , system of equations , linear algebra , linear equation

Determine all solutions of the linear system of equations \begin{align*} &x_1& &-x_2& &-x_3& &-\cdots& &-x_n& &= 2a, \\ -&x_1& &+3x_2& &-x_3& &-\cdots& &-x_n& &= 4a, \\ -&x_1& &-x_2& &+7x_3& &-\cdots& &-x_n& &= 8a, \\ &&&&&&&&&&&\vdots \\ -&x_1& &-x_2& &-x_3& &-\cdots& &+\left(2^n-1\right)x_n& &= 2^na, \end{align*} with unknowns $x_1,\ldots,x_n$ and a real parameter $a.$

2013 IFYM, Sozopol, 7

Tags: set theory , greatest common divisor , set

Let $T$ be a set of natural numbers, each of which is greater than 1. A subset $S$ of $T$ is called “good”, if for each $t\in T$ there exists $s\in S$, for which $gcd(t,s)>1$. Prove that the number of "good" subsets of $T$ is odd.

2011 International Zhautykov Olympiad, 3

Tags: circumcircle , symmetry , ratio , cyclic quadrilateral , power of a point , radical axis , geometry

Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $K.$ The midpoints of diagonals $AC$ and $BD$ are $M$ and $N,$ respectively. The circumscribed circles $ADM$ and $BCM$ intersect at points $M$ and $L.$ Prove that the points $K ,L ,M,$ and $ N$ lie on a circle. (all points are supposed to be different.)

2022 Iranian Geometry Olympiad, 1

Tags: geometry , marvio

Four points $A$, $B$, $C$ and $D$ lie on a circle $\omega$ such that $AB=BC=CD$. The tangent line to $\omega$ at point $C$ intersects the tangent line to $\omega$ at $A$ and the line $AD$ at $K$ and $L$. The circle $\omega$ and the circumcircle of triangle $KLA$ intersect again at $M$. Prove that $MA=ML$. [i]Proposed by Mahdi Etesamifard[/i]

2015 JBMO Shortlist, NT4

Tags: prime number , number theory

Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\] Proposed by Moldova

CNCM Online Round 2, 2

Tags:

There is a rectangle $ABCD$ such that $AB=12$ and $BC=7$. $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$. Call $X$ the intersection of $AF$ and $DE$. What is the area of pentagon $BCFXE$? Proposed by Minseok Eli Park (wolfpack)

2004 Korea Junior Math Olympiad, 2

Tags: existence , subset , combinatorics , set

For $n\geq3$ define $S_n=\{1, 2, ..., n\}$. $A_1, A_{2}, ..., A_{n}$ are given subsets of $S_n$, each having an even number of elements. Prove that there exists a set $\{i_1, i_2, ..., i_t\}$, a nonempty subset of $S_n$ such that $$A_{i_1} \Delta A_{i_2} \Delta \ldots \Delta A_{i_t}=\emptyset$$ (For two sets $A, B$, we define $\Delta$ as $A \Delta B=(A\cup B)-(A\cap B)$)

1969 IMO Shortlist, 13

Tags: modular arithmetic , number theory , divisibility , additive number theory

$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?

1966 Miklós Schweitzer, 6

Tags: function , real analysis

A sentence of the following type if often heard in Hungarian weather reports: "Last night's minimum temperatures took all values between $ \minus{}3$ degrees and $ \plus{}5$ degrees." Show that it would suffice to say, "Both $ \minus{}3$ degrees and $ \plus{}5$ degrees occurred among last night's minimum temperatures." (Assume that temperature as a two-variable function of place and time is continuous.) [i]A.Csaszar[/i]

Today's calculation of integrals, 767

Tags: calculus , integration , trigonometry , calculus computations

For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$ Evaluate $\int_0^1 f(t)dt.$

1990 IMO Longlists, 53

Tags: function , algebra

Find the real solution(s) for the system of equations \[\begin{cases}x^3+y^3 &=1\\x^5+y^5 &=1\end{cases}\]

2012 Kazakhstan National Olympiad, 2

Tags: function , algebra

Function $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(xf(y))=yf(x)$ for any $x,y$ are real numbers. Prove that $f(-x) = -f(x)$ for all real numbers $x$.

2002 China Western Mathematical Olympiad, 1

Tags: trapezoid , circumcircle , geometry

Given a trapezoid $ ABCD$ with $ AD\parallel BC, E$ is a moving point on the side $ AB,$ let $ O_{1},O_{2}$ be the circumcenters of triangles $ AED,BEC$, respectively. Prove that the length of $ O_{1}O_{2}$ is a constant value.

2009 Cuba MO, 3

Tags: combinatorics

In each square of an $n \times n$ board with $n\ge 2$, an integer is written not null. Said board is called [i]Inca [/i] if for each square, the number written on it is equal to the difference of the numbers written on two of its neighboring squares (with a common side). For what values of $n$, can you get [i]Inca[/i] boards ?

2002 ITAMO, 2

Tags: geometry

The plan of a house has the shape of a capital $L$, obtained by suitably placing side-by-side four squares whose sides are $10$ metres long. The external walls of the house are $10$ metres high. The roof of the house has six faces, starting at the top of the six external walls, and each face forms an angle of $30^\circ$ with respect to a horizontal plane. Determine the volume of the house (that is, of the solid delimited by the six external walls, the six faces of the roof, and the base of the house).

2013 Math Prize For Girls Problems, 2

Tags:

When the binomial coefficient $\binom{125}{64}$ is written out in base 10, how many zeros are at the rightmost end?

1996 North Macedonia National Olympiad, 3

Tags: trigonometry , inequalities , geometric inequalities , angle

Prove that if $\alpha, \beta, \gamma$ are angles of a triangle, then $\frac{1}{\sin \alpha}+ \frac{1}{\sin \beta} \ge \frac{8}{ 3+2 \ cos\gamma}$ .

2003 IMO Shortlist, 2

Tags: number theory , decimal representation , algorithm , combinatorics

Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$: (i) move the last digit of $a$ to the first position to obtain the numb er $b$; (ii) square $b$ to obtain the number $c$; (iii) move the first digit of $c$ to the end to obtain the number $d$. (All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.) Find all numbers $a$ for which $d\left( a\right) =a^2$. [i]Proposed by Zoran Sunic, USA[/i]

2012 China Team Selection Test, 1

Tags: geometry , circumcircle , trigonometry

In an acute-angled $ABC$, $\angle A>60^{\circ}$, $H$ is its orthocenter. $M,N$ are two points on $AB,AC$ respectively, such that $\angle HMB=\angle HNC=60^{\circ}$. Let $O$ be the circumcenter of triangle $HMN$. $D$ is a point on the same side with $A$ of $BC$ such that $\triangle DBC$ is an equilateral triangle. Prove that $H,O,D$ are collinear.

2020 CCA Math Bonanza, L1.4

Tags: geometry

Let $ABC$ be a triangle with $AB=3$, $BC=4$, and $CA=5$. Points $A_1$, $B_1$, and $C_1$ are chosen on its incircle. Compute the maximum possible sum of the areas of triangles $A_1BC$, $AB_1C$, and $ABC_1$. [i]2020 CCA Math Bonanza Lightning Round #1.4[/i]

1980 IMO, 2

Tags: algebra , recurrence relation , sequence , inequality

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]