Olympiad problems

2006 MOP Homework, 1

Determine all positive real numbers $a$ such that there exists a positive integer $n$ and partition $A_1$, $A_2$, ..., $A_n$ of infinity sets of the set of the integers satisfying the following condition: for every set $A_i$, the positive difference of any pair of elements in $A_i$ is at least $a^i$.

2012 Moldova Team Selection Test, 1

Tags: polynomial , algebra

Prove that polynomial $x^8+98x^4+1$ can be factorized in $Z[X]$.

2007 Thailand Mathematical Olympiad, 18

Tags: prime , remainder , number theory , prime number

Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.

2001 All-Russian Olympiad Regional Round, 9.4

Tags: equilateral , combinatorics , combinatorial geometry

The target is a triangle divided by three families of parallel lines into $100$ equal regular triangles with single sides. A sniper shoots at a target. He aims at triangle and hits either it or one of the sides adjacent to it. He sees the results of his shooting and can choose when stop shooting. What is the greatest number of triangles he can with a guarantee of hitting five times?

2010 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , sum , algebra , absolute value

Let $a_1, a_2, ..., a_n$ real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $| a_1 | + | a_2 | + ... + | a_n | = 1$. Show that: $| a _ 1 + 2 a _ 2 + ... + n a _ n | \le \frac {n-1} {2}$.

LMT Team Rounds 2010-20, B25

Tags: number theory , algebra

Emmy goes to buy radishes at the market. Radishes are sold in bundles of $3$ for $\$5$and bundles of $5$ for $\$7$. What is the least number of dollars Emmy needs to buy exactly $100$ radishes?

2018 Vietnam National Olympiad, 4

Tags: analytic geometry , inequalities

On the Cartesian plane the curve $(C)$ has equation $x^2=y^3$. A line $d$ varies on the plane such that $d$ always cut $(C)$ at three distinct points with $x$-coordinates $x_1,\, x_2,\, x_3$. a. Prove that the following quantity is a constant: $$\sqrt[3]{\frac{x_1x_2}{x_3^2}}+\sqrt[3]{\frac{x_2x_3}{x_1^2}}+\sqrt[3]{\frac{x_3x_1}{x_2^2}}.$$ b. Prove the following inequality: $$\sqrt[3]{\frac{x_1^2}{x_2x_3}}+\sqrt[3]{\frac{x_2^2}{x_3x_1}}+\sqrt[3]{\frac{x_3^2}{x_3x_1}}<-\frac{15}{4}.$$

2006 Turkey Team Selection Test, 3

Tags: combinatorics

Each one of 2006 students makes a list with 12 schools among 2006. If we take any 6 students, there are two schools which at least one of them is included in each of 6 lists. A list which includes at least one school from all lists is a good list. a) Prove that we can always find a good list with 12 elements, whatever the lists are; b) Prove that students can make lists such that no shorter list is good.

2000 Austrian-Polish Competition, 6

Tags: 3d geometry , cube , geometry , combinatorics

Consider the solid $Q$ obtained by attaching unit cubes $Q_1...Q_6$ at the six faces of a unit cube $Q$. Prove or disprove that the space can be filled up with such solids so that no two of them have a common interior point.

1991 Romania Team Selection Test, 1

Tags: geometry

Let $M=\{A_{1},A_{2},\ldots,A_{5}\}$ be a set of five points in the plane such that the area of each triangle $A_{i}A_{j}A_{k}$, is greater than 3. Prove that there exists a triangle with vertices in $M$ and having the area greater than 4. [i]Laurentiu Panaitopol[/i]

2011 Canadian Mathematical Olympiad Qualification Repechage, 3

Tags: system of equations , algebra

Determine all solutions to the system of equations: \[x^2 + y^2 + x + y = 12\]\[xy + x + y = 3\] [This is the exact form of problem that appeared on the paper, but I think it means to solve in $\mathbb R.$]

2011 ELMO Shortlist, 5

Tags: geometry , inequalities

Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that \[\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,\]where $k=2+\sqrt{3}$. [i]Victor Wang.[/i]

2023 Philippine MO, 5

Tags: string , operation , combinatorics

Silverio is very happy for the 25th year of the PMO. In his jubilation, he ends up writing a finite sequence of As and Gs on a nearby blackboard. He then performs the following operation: if he finds at least one occurrence of the string "AG", he chooses one at random and replaces it with "GAAA". He performs this operation repeatedly until there is no more "AG" string on the blackboard. Show that for any initial sequence of As and Gs, Silverio will eventually be unable to continue doing the operation.

Ukraine Correspondence MO - geometry, 2007.9

Tags: geometry , perimeter , min , geometric inequality , equal angles

In triangle $ABC$, the lengths of all sides are integers, $\angle B=2 \angle A$ and $\angle C> 90^o$. Find the smallest possible perimeter of this triangle.

2005 China Team Selection Test, 1

Tags: geometry , projective geometry , concurrency

Point $P$ lies inside triangle $ABC$. Let the projections of $P$ onto sides $BC$,$CA$,$AB$ be $D$, $E$, $F$ respectively. Let the projections from $A$ to the lines $BP$ and $CP$ be $M$ and $N$ respectively. Prove that $ME$, $NF$ and $BC$ are concurrent.

2023 Iran Team Selection Test, 6

Tags: subset , combinatorics

Suppose that we have $2n$ non-empty subset of $ \big\{0,1,2,...,2n-1\big\} $ that sum of the elements of these subsets is $ \binom{2n+1}{2}$ . Prove that we can choose one element from every subset that some of them is $ \binom{2n}{2}$ [i]Proposed by Morteza Saghafian and Afrouz Jabalameli [/i]

2001 Regional Competition For Advanced Students, 4

Tags: sequence , set , algebra

Let $A_o =\{1, 2\}$ and for $n> 0, A_n$ results from $A_{n-1}$ by adding the natural numbers to $A_{n-1}$ which can be represented as the sum of two different numbers from $A_{n-1}$. Let $a_n = |A_n |$ be the number of numbers in $A_n$. Determine $a_n$ as a function of $n$.

2016 Hong Kong TST, 1

Tags: number theory

Find all natural numbers $n$ such that $n$, $n^2+10$, $n^2-2$, $n^3+6$, and $n^5+36$ are all prime numbers.

2012 Tournament of Towns, 5

Tags: weighing , combinatorics

Among $239$ coins identical in appearance there are two counterfeit coins. Both counterfeit coins have the same weight different from the weight of a genuine coin. Using a simple balance, determine in three weighings whether the counterfeit coin is heavier or lighter than the genuine coin. A simple balance shows if both sides are in equilibrium or left side is heavier or lighter. It is not required to find the counterfeit coins.

1973 IMO Shortlist, 16

Tags: algebra , polynomial , trigonometry , geometry , factorization

Given $a, \theta \in \mathbb R, m \in \mathbb N$, and $P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials.

1972 AMC 12/AHSME, 17

Tags: probability

A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least $x$ times as large as the shorter piece is $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{2}{x}\qquad\textbf{(C) }\frac{1}{x+1}\qquad\textbf{(D) }\frac{1}{x}\qquad \textbf{(E) }\frac{2}{x+1}$

2023 Czech-Polish-Slovak Junior Match, 6

Tags: geometry , rectangle , area

Given a rectangle $ABCD$. Points $E$ and $F$ lie on sides $BC$ and $CD$ respectively so that the area of triangles $ABE$, $ECF$, $FDA$ is equal to $1$. Determine the area of triangle $AEF$.

2012 France Team Selection Test, 3

Tags: geometry , parallelogram , circumcircle , perpendicular bisector , power of a point

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

2019 Hong Kong TST, 2

Tags: number theory

Let $p$ be a prime number greater than 10. Prove that there exist positive integers $m$ and $n$ such that $m+n < p$ and $5^m 7^n-1$ is divisible by $p$.

IMSC 2024, 2

Tags: geometry , imsc

Let $ABC$ be an acute angled triangle and let $P, Q$ be points on $AB, AC$ respectively, such that $PQ$ is parallel to $BC$. Points $X, Y$ are given on line segments $BQ, CP$ respectively, such that $\angle AXP = \angle XCB$ and $\angle AYQ = \angle YBC$. Prove that $AX = AY$. [i]Proposed by Ervin Maci$\acute{c},$ Bosnia and Herzegovina[/i]