Olympiad problems

2018 Sharygin Geometry Olympiad, 3

Tags: geometry

Let $AL$ be the bisector of triangle $ABC$, $D$ be its midpoint, and $E$ be the projection of $D$ to $AB$. It is known that $AC = 3AE$. Prove that $CEL$ is an isosceles triangle.

2009 Tournament Of Towns, 1

Tags: convex polygon , convex , diagonal , combinatorial geometry , even , geometry

In a convex $2009$-gon, all diagonals are drawn. A line intersects the $2009$-gon but does not pass through any of its vertices. Prove that the line intersects an even number of diagonals.

On a circle there are $99$ natural numbers. If $a,b$ are any two neighbouring numbers on the circle, then $a-b$ is equal to $1$ or $2$ or $ \frac{a}{b}=2 $. Prove that there exists a natural number on the circle that is divisible by $3$. [i]S. Berlov[/i]

2007 Germany Team Selection Test, 1

Tags: algebra , polynomial , number theory

Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.) [b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often. [b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .

2005 Today's Calculation Of Integral, 2

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Calculate the following indefinite integrals. [1] $\int \cos \left(2x-\frac{\pi}{3}\right)dx$ [2]$\int \frac{dx}{\cos ^2 (3x+4)}$ [3]$\int (x-1)\sqrt[3]{x-2}dx$ [4]$\int x\cdot 3^{x^2+1}dx$ [5]$\int \frac{dx}{\sqrt{1-x}}dx$

2013 Tournament of Towns, 4

Tags: rook , chessboard , combinatorics

Eight rooks are placed on a $8\times 8$ chessboard, so that no two rooks attack one another. All squares of the board are divided between the rooks as follows. A square where a rook is placed belongs to it. If a square is attacked by two rooks then it belongs to the nearest rook; in case these two rooks are equidistant from this square each of them possesses a half of the square. Prove that every rook possesses the equal area.

2015 AMC 10, 16

Tags: probability

Al, Bill, and Cal will each randomly be assigned a whole number from $1$ to $10$, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's? $\textbf{(A) } \dfrac{9}{1000} \qquad\textbf{(B) } \dfrac{1}{90} \qquad\textbf{(C) } \dfrac{1}{80} \qquad\textbf{(D) } \dfrac{1}{72} \qquad\textbf{(E) } \dfrac{2}{121} $

2019 Spain Mathematical Olympiad, 1

Tags: combinatorics

An integer set [i][b]T[/b][/i] is orensan if there exist integers[b] a<b<c[/b], where [b]a [/b]and [b]c[/b] are part of [i][b]T[/b][/i], but [b]b[/b] is not part of [b][i]T[/i][/b]. Count the number of subsets [b][i]T[/i][/b] of {1,2,...,2019} which are orensan.

2019 Moldova Team Selection Test, 9

Tags: polynomial , algebra

Find all polynomials $P(X)$ with real coefficients such that if real numbers $x,y$ and $z$ satisfy $x+y+z=0,$ then the points $\left(x,P(x)\right), \left(y,P(y)\right), \left(z,P(z)\right)$ are all colinear.

1995 Tournament Of Towns, (448) 4

Tags: number theory

Can the number $a + b + c + d$ be prime if $a, b, c$ and $d$ are positive integers and $ab = cd$?

2022 AMC 12/AHSME, 5

Tags: coordinates , lattice points

Let the [i]taxicab distance[/i] between points $(x_1,y_1)$ and $(x_2,y_2)$ in the coordinate plane is given by $|x_1-x_2|+|y_1-y_2|$. For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$? $\textbf{(A) }441\qquad\textbf{(B) }761\qquad\textbf{(C) }841\qquad\textbf{(D) }921\qquad\textbf{(E) }924$

2008 ITest, 47

Tags: modular arithmetic

Find $a+b+c$, where $a,b,$ and $c$ are the hundreds, tens, and units digits of the six-digit number $123abc$, which is a multiple of $990$.

2004 Tournament Of Towns, 2

Tags: coloring , combinatorics

A box contains red, blue, and white balls, $100$ balls in total. It is known that among any $26$ of them there are always $10$ balls of the same color. Find the minimal number $N$ such that among any $N$ balls there are always $30$ balls of the same color.

2021 Bangladeshi National Mathematical Olympiad, 1

Tags: number theory , prime number , counting

How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?

2014 Kurschak Competition, 2

Tags: circumcircle , power of a point , radical axis , geometry

We are given an acute triangle $ABC$, and inside it a point $P$, which is not on any of the heights $AA_1$, $BB_1$, $CC_1$. The rays $AP$, $BP$, $CP$ intersect the circumcircle of $ABC$ at points $A_2$, $B_2$, $C_2$. Prove that the circles $AA_1A_2$, $BB_1B_2$ and $CC_1C_2$ are concurrent.

2009 Romania Team Selection Test, 2

Tags: linear algebra , matrix , combinatorics

Consider a matrix whose entries are integers. Adding a same integer to all entries on a same row, or on a same column, is called an operation. It is given that, for infinitely many positive integers $n$, one can obtain, through a finite number of operations, a matrix having all entries divisible by $n$. Prove that, through a finite number of operations, one can obtain the null matrix.

2010 Purple Comet Problems, 14

Tags: geometry , trapezoid , ratio , similar triangles

Let $ABCD$ be a trapezoid where $AB$ is parallel to $CD.$ Let $P$ be the intersection of diagonal $AC$ and diagonal $BD.$ If the area of triangle $PAB$ is $16,$ and the area of triangle $PCD$ is $25,$ find the area of the trapezoid.

1984 Miklós Schweitzer, 7

Tags: polynomial , algebra

[b]7.[/b] Let $V$ be a finite-dimensional subspace of $C[0,1]$ such that every nonzero $f\in V$ attains positive value at some point. Prove that there exists a polynomial $P$ that is strictly positive on $[0,1]$ and orthogonal to $V$, that is, for every $f \in V$, $\int_{0}^{1} f(x)P(x)dx =0$ ([b]F.39[/b]) [A. Pinkus, V. Totik]

III Soros Olympiad 1996 - 97 (Russia), 9.4

Tags: algebra , system of equations

Solve the system of equations $$\begin{cases} x^4-2x^3+x=y^2-y \\ y^4-2y^3+y=x^2-x \end{cases}$$

1991 Poland - Second Round, 2

Tags: geometry , equilateral

On the sides $ BC $, $ CA $, $ AB $ of the triangle $ ABC $, the points $ D $, $ E $, $ F $ are chosen respectively, such that $$ \frac{|DB|}{|DC|} = \frac{|EC|}{|EA|} = \frac{|FA|}{|FB|}$$ Prove that if the triangle $ DEF $ is equilateral, then the triangle $ ABC $ is also equilateral.

1993 Putnam, A2

Tags:

The sequence an of non-zero reals satisfies $a_n^2 - a_{n-1}a_{n+1} = 1$ for $n \geq 1$. Prove that there exists a real number $\alpha$ such that $a_{n+1} = \alpha a_n - a_{n-1}$ for $n \geq 1$.

2007 Irish Math Olympiad, 1

Tags: polynomial , vieta , algebra

Let $ r,s,$ and $ t$ be the roots of the cubic polynomial: $ p(x)\equal{}x^3\minus{}2007x\plus{}2002.$ Determine the value of: $ \frac{r\minus{}1}{r\plus{}1}\plus{}\frac{s\minus{}1}{s\plus{}1}\plus{}\frac{t\minus{}1}{t\plus{}1}$.

Gheorghe Țițeica 2024, P1

Tags: real analysis , convergent sequence , continuous function

Find all continuous functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any sequences $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ such that the sequence $(a_n+b_n)_{n\geq 1}$ is convergent, the sequence $(f(a_n)+g(b_n))_{n\geq 1}$ is also convergent.

2021 Latvia Baltic Way TST, P9

Tags: geometry , circumcircle

Pentagon $ABCDE$ with $CD\parallel BE$ is inscribed in circle $\omega$. Tangent to $\omega$ through $B$ intersects line $AC$ at $F$ in a way that $A$ lies between $C$ and $F$. Lines $BD$ and $AE$ intersect at $G$. Prove that $FG$ is tangent to the circumcircle of $\triangle ADG$.

2024 Simon Marais Mathematical Competition, A1

Tags: algebra

Let $a,b,c$ be real number greater than 1 satisfying $$\lfloor a\rfloor b = \lfloor b \rfloor c = \lfloor c\rfloor a.$$Prove that $a=b=c$ (Here, $\lfloor x \rfloor$ denotes the laregst integer that is less than or equal to $x$.)