Olympiad problems

2004 Bulgaria Team Selection Test, 3

Tags: combinatorics

A table with $m$ rows and $n$ columns is given. At any move one chooses some empty cells such that any two of them lie in different rows and columns, puts a white piece in any of those cells and then puts a black piece in the cells whose rows and columns contain white pieces. The game is over if it is not possible to make a move. Find the maximum possible number of white pieces that can be put on the table.

2022 Swedish Mathematical Competition, 4

Tags: geometry , geometric inequality

Let $ABC$ be an acute triangle. Let $I$ be a point inside the triangle and let $D$ be a point on the line $AB$. The line through $D$ which is parallel to $AI$ intersects the line $AC$ at the point $E$, and the line through $D$ parallel to $BI$ intersects the line $BC$ in point $F$. prove that $$\frac{EF \cdot CI}{2} \ge area (\vartriangle ABC) $$

1999 Dutch Mathematical Olympiad, 5

Tags: algorithm , greatest common divisor , number theory , euclidean algorithm

Let $c$ be a nonnegative integer, and define $a_n = n^2 + c$ (for $n \geq 1)$. Define $d_n$ as the greatest common divisor of $a_n$ and $a_{n + 1}$. (a) Suppose that $c = 0$. Show that $d_n = 1,\ \forall n \geq 1$. (b) Suppose that $c = 1$. Show that $d_n \in \{1,5\},\ \forall n \geq 1$. (c) Show that $d_n \leq 4c + 1,\ \forall n \geq 1$.

2014 National Olympiad First Round, 27

Tags: function

Let $f$ be a function defined on positive integers such that $f(1)=4$, $f(2n)=f(n)$ and $f(2n+1)=f(n)+2$ for every positive integer $n$. For how many positive integers $k$ less than $2014$, it is $f(k)=8$? $ \textbf{(A)}\ 45 \qquad\textbf{(B)}\ 120 \qquad\textbf{(C)}\ 165 \qquad\textbf{(D)}\ 180 \qquad\textbf{(E)}\ 215 $

Denmark (Mohr) - geometry, 1999.1

Tags: parabola , tangent , conic , circles

In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.

2022 Germany Team Selection Test, 2

Tags: geometry

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

1965 Spain Mathematical Olympiad, 1

Tags: geometry

We consider an equilateral triangle with its circumscribed circle, of center $O$, and radius $4$cm. We rotate the triangle $90º$ around $O$. Compute the common area that was covered by the previous position of the triangle and is also covered by the new one.

2017 IMO Shortlist, A6

Tags: functional equation , algebra

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y$, \[ f(f(x)f(y)) + f(x+y) = f(xy). \] [i]Proposed by Dorlir Ahmeti, Albania[/i]

2017 Ecuador Juniors, 6

Tags: number theory , perfect cube

Find all primes $p$ such that $p^2- p + 1$ is a perfect cube.

2015 AMC 8, 13

Tags:

How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ so that the mean (average) of the remaining numbers is 6? $\textbf{(A)}\text{ 1}\qquad\textbf{(B)}\text{ 2}\qquad\textbf{(C)}\text{ 3}\qquad\textbf{(D)}\text{ 5}\qquad\textbf{(E)}\text{ 6}$

2002 National Olympiad First Round, 26

Tags: modular arithmetic

Which of the following is the set of all perfect squares that can be written as sum of three odd composite numbers? $\textbf{a)}\ \{(2k + 1)^2 : k \geq 0\}$ $\textbf{b)}\ \{(4k + 3)^2 : k \geq 1\}$ $\textbf{c)}\ \{(2k + 1)^2 : k \geq 3\}$ $\textbf{d)}\ \{(4k + 1)^2 : k \geq 2\}$ $\textbf{e)}\ \text{None of above}$

2009 Switzerland - Final Round, 3

Tags: algebra , inequalities

Let $a, b, c, d$ be positive real numbers. Prove the following inequality and determine all cases in which the equality holds : $$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a}+\frac{d - a}{a + b} \ge 0.$$

2009 Danube Mathematical Competition, 5

Tags: combinatorics , function , sum , inequalities , permutation

Let $\sigma, \tau$ be two permutations of the quantity $\{1, 2,. . . , n\}$. Prove that there is a function $f: \{1, 2,. . . , n\} \to \{-1, 1\}$ such that for any $1 \le i \le j \le n$, we have $\left|\sum_{k=i}^{j} f(\sigma (k)) \right| \le 2$ and $\left|\sum_{k=i}^{j} f(\tau (k))\right| \le 2$

2016 CMIMC, 1

Tags: algebra

In a race, people rode either bicycles with blue wheels or tricycles with tan wheels. Given that 15 more people rode bicycles than tricycles and there were 15 more tan wheels than blue wheels, what is the total number of people who rode in the race?

2010 Tournament Of Towns, 5

Tags: mathcamp , number theory , least common multiple , linear algebra , matrix , greatest common divisor

$33$ horsemen are riding in the same direction along a circular road. Their speeds are constant and pairwise distinct. There is a single point on the road where the horsemen can surpass one another. Can they ride in this fashion for arbitrarily long time ?

Kyiv City MO Seniors Round2 2010+ geometry, 2012.10.4

Tags: geometric inequality , angle , geometry

In the triangle $ABC$ with sides $BC> AC> AB$ the angles between altiude and median drawn from one vertex are considered. Find out at which vertex this angle is the largest of the three. (Rozhkova Maria)

2000 Balkan MO, 2

Tags: geometry

Let $ABC$ be an acute-angled triangle and $D$ the midpoint of $BC$. Let $E$ be a point on segment $AD$ and $M$ its projection on $BC$. If $N$ and $P$ are the projections of $M$ on $AB$ and $AC$ then the interior angule bisectors of $\angle NMP$ and $\angle NEP$ are parallel.

1994 Poland - First Round, 2

Tags: algebra , system of equations

Given a positive integer $n \geq 2$. Solve the following system of equations: $ \begin{cases} \ x_1|x_1| &= x_2|x_2| + (x_1-1)|x_1-1| \\ \ x_2|x_2| &= x_3|x_3| + (x_2-1)|x_2-1| \\ &\dots \\ \ x_n|x_n| &= x_1|x_1| + (x_n-1)|x_n-1|. \\ \end{cases} $

2019 Moldova Team Selection Test, 6

Tags: inequality , inequalities

Let $a,b,c \ge 0$ such that $a+b+c=1$ and $s \ge 5$. Prove that $s(a^2+b^2+c^2) \le 3(s-3)(a^3+b^3+c^3)+1$

2016 ELMO Problems, 2

Tags: geometry

Oscar is drawing diagrams with trash can lids and sticks. He draws a triangle $ABC$ and a point $D$ such that $DB$ and $DC$ are tangent to the circumcircle of $ABC$. Let $B'$ be the reflection of $B$ over $AC$ and $C'$ be the reflection of $C$ over $AB$. If $O$ is the circumcenter of $DB'C'$, help Oscar prove that $AO$ is perpendicular to $BC$. [i]James Lin[/i]

2014 Contests, 2

Tags: analytic geometry , parallelogram , circumcircle , geometry

Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.

1993 Baltic Way, 3

Tags: number theory

Let’s call a positive integer [i]interesting[/i] if it is a product of two (distinct or equal) prime numbers. What is the greatest number of consecutive positive integers all of which are interesting?

2016 Lusophon Mathematical Olympiad, 2

Tags: geometry

The circle $\omega_1$ intersects the circle $\omega_2$ in the points $A$ and $B$, a tangent line to this circles intersects $\omega_1$ and $\omega_2$ in the points $E$ and $F$ respectively. Suppose that $A$ is inside of the triangle $BEF$, let $H$ be the orthocenter of $BEF$ and $M$ is the midpoint of $BH$. Prove that the centers of the circles $\omega_1$ and $\omega_2$ and the point $M$ are collinears.

2004 AMC 12/AHSME, 1

Tags:

Alicia earns $ \$20$ per hour, of which $ 1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? $ \textbf{(A)}\ 0.0029 \qquad \textbf{(B)}\ 0.029 \qquad \textbf{(C)}\ 0.29 \qquad \textbf{(D)}\ 2.9 \qquad \textbf{(E)}\ 29$

2019 Iran Team Selection Test, 5

Tags: combinatorial geometry , combinatorics , perimeter

Let $P$ be a simple polygon completely in $C$, a circle with radius $1$, such that $P$ does not pass through the center of $C$. The perimeter of $P$ is $36$. Prove that there is a radius of $C$ that intersects $P$ at least $6$ times, or there is a circle which is concentric with $C$ and have at least $6$ common points with $P$. [i]Proposed by Seyed Reza Hosseini[/i]