Olympiad problems

2021 Yasinsky Geometry Olympiad, 5

Construct an equilateral trapezoid given the height and the midline, if it is known that the midline is divided by diagonals into three equal parts. (Grigory Filippovsky)

1980 IMO, 2

Tags: algebra , polynomial , analytic geometry , geometry

In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

2006 Sharygin Geometry Olympiad, 8.3

Tags: parallelogram , equal segments , circles , geometry

A parallelogram $ABCD$ is given. Two circles with centers at the vertices $A$ and $C$ pass through $B$. The straight line $\ell$ that passes through $B$ and crosses the circles at second time at points $X, Y$ respectively. Prove that $DX = DY$.

2024 Harvard-MIT Mathematics Tournament, 1

Tags: guts

Compute the sum of all integers $n$ such that $n^2-3000$ is a perfect square.

2010 Belarus Team Selection Test, 3.3

Tags: polynomial , size , number theory

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

1991 Arnold's Trivium, 99

Tags: probability

One player conceals a $10$ or $20$ copeck coin, and the other guesses its value. If he is right he gets the coin, if wrong he pays $15$ copecks. Is this a fair game? What are the optimal mixed strategies for both players?

2014 ASDAN Math Tournament, 1

Tags: algebra test

A college math class has $N$ teaching assistants. It takes the teaching assistants $5$ hours to grade homework assignments. One day, another teaching assistant joins them in grading and all homework assignments take only $4$ hours to grade. Assuming everyone did the same amount of work, compute the number of hours it would take for $1$ teaching assistant to grade all the homework assignments.

2010 Tournament Of Towns, 1

Tags: geometry

There are $100$ points on the plane. All $4950$ pairwise distances between two points have been recorded. $(a)$ A single record has been erased. Is it always possible to restore it using the remaining records? $(b)$ Suppose no three points are on a line, and $k$ records were erased. What is the maximum value of $k$ such that restoration of all the erased records is always possible?

2024 Francophone Mathematical Olympiad, 1

Tags: game , polynomial , algebra

Let $d$ and $m$ be two fixed positive integers. Pinocchio and Geppetto know the values of $d$ and $m$ and play the following game: In the beginning, Pinocchio chooses a polynomial $P$ of degree at most $d$ with integer coefficients. Then Geppetto asks him questions of the following form "What is the value of $P(n)$?'' for $n \in \mathbb{Z}$. Pinocchio usually says the truth, but he can lie up to $m$ times. What is, as a function of $d$ and $m$, the minimal number of questions that Geppetto needs to ask to be sure to determine $P$, no matter how Pinocchio chooses to reply?

2017 USAMO, 5

Tags:

Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $ ( x, y ) \in \mathbf{Z}^2$ with positive integers for which: [list] [*] only finitely many distinct labels occur, and [*] for each label $i$, the distance between any two points labeled $i$ is at least $c^i$. [/list] [i]Proposed by Ricky Liu[/i]

1961 Putnam, A3

Tags: limit , series

Evaluate $$\lim_{n\to \infty} \sum_{j=1}^{n^{2}} \frac{n}{n^2 +j^2 }.$$

1987 Tournament Of Towns, (152) 3

Tags: game strategy , game , combinatorics

In a game two players alternately choose larger natural numbers. At each turn the difference between the new and the old number must be greater than zero but smaller than the old number. The original number is 2. The winner is considered to be the player who chooses the number $1987$. In a perfect game, which player wins?

2018-2019 SDML (High School), 4

Tags:

How many $3$-element subsets of $\left\{1, 2, 3, \dots, 11\right\}$ are there, such that the sum of the three elements is a multiple of $3$?

2020 Belarusian National Olympiad, 11.5

Tags: number theory

All divisors of a positive integer $n$ are listed in the ascending order: $1=d_1<d_2< \ldots < d_k=n$. It turned out that the amount of pairs $(d_i,d_{i+1})$ of adjacent divisors such that $d_{i+1}$ is a multiple of $d_i$, is odd. Prove that $n=pm^2$, where $p$ is the smallest prime divisor of $n$, and $m$ is a positive integer.

1962 IMO Shortlist, 4

Tags: equation , trigonometry , algebra , trigonometric equation

Solve the equation $\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1$

2018 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: prime number , perfect square , number theory

Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime

2007 IMO Shortlist, 3

Tags: trapezoid , geometry , homothety , ddit

The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

1967 IMO Shortlist, 2

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

Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.

1988 IMO Shortlist, 14

Tags: matrix , combinatorics , extremal combinatorics , linear algebra

For what values of $ n$ does there exist an $ n \times n$ array of entries -1, 0 or 1 such that the $ 2 \cdot n$ sums obtained by summing the elements of the rows and the columns are all different?

1996 Turkey Team Selection Test, 3

Tags: limit , algebra

Determine all ordered pairs of positive real numbers $(a, b)$ such that every sequence $(x_{n})$ satisfying $\lim_{n \rightarrow \infty}{(ax_{n+1} - bx_{n})} = 0$ must have $\lim_{n \rightarrow \infty} x_n = 0$.

1985 All Soviet Union Mathematical Olympiad, 405

Tags: periodical , sequence

The sequence $a_1, a_2, ... , a_k, ...$ is constructed according to the rules: $$a_{2n} = a_n,a_{4n+1} = 1,a_{4n+3} = 0$$Prove that it is non-periodical sequence.

1992 IMTS, 1

Tags: trapezoid , geometry

In trapezoid $ABCD$, the diagonals intersect at $E$, the area of $\triangle ABE$ is 72 and the area of $\triangle CDE$ is 50. What is the area of trapezoid $ABCD$?

LMT Speed Rounds, 2010.9

Tags:

Let $ABC$ and $BCD$ be equilateral triangles, such that $AB=1,$ and $A \neq D.$ Find the area of triangle $ABD.$

2009 All-Russian Olympiad Regional Round, 10.1

Tags: polynomial , algebra

Square trinomial $f(x)$ is such that the polynomial $(f(x)) ^3- f(x)$ has exactly three real roots. Find the ordinate of the vertex of the graph of this trinomial.

2024 All-Russian Olympiad Regional Round, 10.10

Tags: combinatorics

There are $2024$ people, which are knights and liars and some of them are friends. Every person is asked for the number of their friends and the answers were $0,1, \ldots, 2023$. Every knight answered truthfully, while every liar changed the real answer by exactly $1$. What is the minimal number of liars?