Olympiad problems

2020 Dutch IMO TST, 2

Tags: algebra , polynomial

Determine all polynomials $P (x)$ with real coefficients that apply $P (x^2) + 2P (x) = P (x)^2 + 2$.

2025 India National Olympiad, P3

Tags: construction geo , incenter , excenter

Euclid has a tool called splitter which can only do the following two types of operations : • Given three non-collinear marked points $X,Y,Z$ it can draw the line which forms the interior angle bisector of $\angle{XYZ}$. • It can mark the intersection point of two previously drawn non-parallel lines . Suppose Euclid is only given three non-collinear marked points $A,B,C$ in the plane . Prove that Euclid can use the splitter several times to draw the centre of circle passing through $A,B$ and $C$. [i]Proposed by Shankhadeep Ghosh[/i]

1979 IMO Shortlist, 6

Tags: parametric equation

Find the real values of $p$ for which the equation \[\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}\] in $x$ has exactly two real distinct roots.($\sqrt t $ means the positive square root of $t$).

2022 Balkan MO Shortlist, C2

Tags: combinatorics , geometry

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

2010 Romanian Masters In Mathematics, 2

Tags: function , inequalities , cauchy inequality , absolute value , algebra

For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

2004 South East Mathematical Olympiad, 6

Tags: circumcircle , trigonometry , cyclic quadrilateral , geometry

ABC is an isosceles triangle with AB=AC. Point D lies on side BC. Point F is inside $\triangle$ABC and lies on the circumcircle of triangle ADC. The circumcircle of triangle BDF intersects side AB at point E. Prove that $CD\cdot EF+DF\cdot AE=BD\cdot AF$.

2008 Hong kong National Olympiad, 1

Tags: function , polynomial , induction , algebra

Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$. (a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$. (b) Show that $ a_{2008} \neq 0$

2021 CCA Math Bonanza, L1.3

Tags:

A coin is flipped $20$ times. Let $p$ be the probability that each of the following sequences of flips occur exactly twice: [list] [*] one head, two tails, one head [*] one head, one tails, two heads. [/list] Given that $p$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $\gcd (m,n)$. [i]2021 CCA Math Bonanza Lightning Round #1.3[/i]

2014 Argentina Cono Sur TST, 4

Tags: number theory

Find all pairs of positive prime numbers $(p,q)$ such that $p^5+p^3+2=q^2-q$

2017 239 Open Mathematical Olympiad, 6

Tags: combinatorics , blackboard

The natural numbers $y>x$ are written on the board. Vassya decides to write the reminder of one number on the board to some other non-zero number in each step. Prove that Vassya can find a natural number $k$ such that if $y>k$ then the number distinct numbers on the board after arbitrary number of steps does not exceed $\frac{y}{1000000}.$

PEN O Problems, 51

Tags: number theory , relatively prime

Prove the among $16$ consecutive integers it is always possible to find one which is relatively prime to all the rest.

1965 Kurschak Competition, 2

Tags: combinatorics , combinatorial geometry

$D$ is a closed disk radius $R$. Show that among any $8$ points of $D$ one can always find two whose distance apart is less than $R$.

2002 Hungary-Israel Binational, 3

Tags: polynomial , algebra

Let $p(x)$ be a polynomial with rational coefficients, of degree at least $2$. Suppose that a sequence $(r_{n})$ of rational numbers satisﬁes $r_{n}= p(r_{n+1})$ for every $n\geq 1$. Prove that the sequence $(r_{n})$ is periodic.

1991 Tournament Of Towns, (289) 5

Tags: combinatorics

There are $8$ cities in a certain kingdom. The king wants to have a system of roads constructed so that one can go along those roads from any city to any other one without going through more than one intermediate city and so that no more than $k$ roads go out of any city. For what values of $k$ is this possible? (D. Fomin, Leningrad)

2000 Putnam, 3

Tags: geometry , rectangle , analytic geometry , quadratic , geometric transformation , reflection

The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle with the vertices around the circumference in the given order. Given that the polygon $P_1P_3P_5P_7$ is a square of area $5$, and the polygon $P_2P_4P_6P_8$ is a rectangle of area $4$, find the maximum possible area of the octagon.

2018-IMOC, G1

Tags: point , geometry , combinatorial geometry , polygon

Given an integer $n \ge 3$. Find the largest positive integer $k $ with the following property: For $n$ points in general position, there exists $k$ ways to draw a non-intersecting polygon with those $n$ points as it’s vertices. [hide=Different wording]Given $n$, find the maximum $k$ so that for every general position of $n$ points , there are at least $k$ ways of connecting the points to form a polygon.[/hide]

2010 CHMMC Fall, 1

Tags: number theory , digit

The numbers $25$ and $76$ have the property that when squared in base $10$, their squares also end in the same two digits. A positive integer is called [i]amazing [/i] if it has at most $3$ digits when expressed in base $21$ and also has the property that its square expressed in base $21$ ends in the same $3$ digits. (For this problem, the last three digits of a one-digit number b are 00b, and the last three digits of a two-digit number $\underline{ab}$ are $0\underline{ab}$.) Compute the sum of all amazing numbers. Express your answer in base $21$.

2016 IFYM, Sozopol, 1

Tags: combinatorics , graph

There are $2^{2n+1}$ towns with $2n+1$ companies and each two towns are connected with airlines from one of the companies. What’s the greatest number $k$ with the following property: We can close $k$ of the companies and their airlines in such way that we can still reach each town from any other (connected graph).

2011 Puerto Rico Team Selection Test, 1

Tags:

A set of ten two-digit numbers is given. Prove that one can always choose two disjoint subsets of this set such that the sum of their elements is the same. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

1962 Poland - Second Round, 1

Tags: algebra , system of equations

Prove that if the numbers $ x $, $ y $, $ z $ satisfy the equationw $$x + y + z = a,$$ $$ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{a},$$ then at least one of them is equal to $ a $.

2018 PUMaC Combinatorics A, 5

Tags: combinatorics , geometry , geometric transformation , rotation

How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.

2011 Morocco National Olympiad, 4

Tags: perpendicular bisector , geometry

$ (C)$ and $(C')$ are two circles which intersect in $A$ and $B$. $(D)$ is a line that moves and passes through $A$, intersecting $(C)$ in P and $(C')$ in P'. Prove that the bisector of $[PP']$ passes through a non-moving point.

2006 Romania National Olympiad, 1

Tags: linear algebra , matrix , vector

Let $A$ be a $n\times n$ matrix with complex elements and let $A^\star$ be the classical adjoint of $A$. Prove that if there exists a positive integer $m$ such that $(A^\star)^m = 0_n$ then $(A^\star)^2 = 0_n$. [i]Marian Ionescu, Pitesti[/i]

1992 IberoAmerican, 2

Tags: trapezoid , trigonometry , rhombus , circumcircle , perpendicular bisector , geometry

Given a circle $\Gamma$ and the positive numbers $h$ and $m$, construct with straight edge and compass a trapezoid inscribed in $\Gamma$, such that it has altitude $h$ and the sum of its parallel sides is $m$.

2012 Grigore Moisil Intercounty, 4

Tags: vector , geometry

Let $ \Delta ABC$ be a triangle with $M$ the middle of the side $[BC]$. On the line $BC$, to the left and to the right of the point $M,$ at the same distance from $M,$ let us consider $d_1$ and $d_2,$ which are perpendicular to the line BC. The perpendicular line from $M$ to $AB$ intersects $d_1$ in $P,$ and the perpendicular line from $M$ to $AC$ intersects $d_2$ in $Q.$ Prove that \[AM\perp PQ.\] [b]Author: Marin Bancoș Regional Mathematical Contest GRIGORE MOISIL, Romania, Baia Mare, 2012, 9th grade[/b]