Olympiad problems

2024 Thailand October Camp, 5

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

2022 Oral Moscow Geometry Olympiad, 1

Tags: geometry , trapezoid , circumcircle , Concyclic

Given an isosceles trapezoid $ABCD$. The bisector of angle $B$ intersects the base $AD$ at point $L$. Prove that the center of the circle circumscribed around triangle $BLD$ lies on the circle circumscribed around the trapezoid. (Yu. Blinkov)

2005 South africa National Olympiad, 5

Tags: inequalities , induction , inequalities unsolved , n-variable inequality

Let $x_1,x_2,\dots,x_n$ be positive numbers with product equal to 1. Prove that there exists a $k\in\{1,2,\dots,n\}$ such that \[\frac{x_k}{k+x_1+x_2+\cdots+x_k}\ge 1-\frac{1}{\sqrt[n]{2}}.\]

2005 May Olympiad, 3

Tags: combinatorics

A segment $AB$ of length $100$ is divided into $100$ little segments of length $1$ by $99$ intermediate points. Endpoint $A$ is assigned $0$ and endpoint $B$ is assigned $1$. Gustavo assigns each of the $99$ intermediate points a $0$ or a $1$, at his choice, and then color each segment of length $1$ blue or red, respecting the following rule: [i]The segments that have the same number at their ends are red, and the segments that have different numbers at their ends are blue. [/i] Determine if Gustavo can assign the $0$'s and $1$'s so as to get exactly $30$ blue segments. And $35$ blue segments? (In each case, if the answer is yes, show a distribution of $0$'s and $1$'s, and if the answer is no, explain why).

2011 Bosnia Herzegovina Team Selection Test, 1

Tags: combinatorics proposed , combinatorics

Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$.

1990 Austrian-Polish Competition, 3

Tags: algebra , system of equations

Show that there are two real solutions to: $$\begin{cases} x + y^2 + z^4 = 0 \\ y + z^2 + x^4 = 0 \\ z + x^2 + y^5 = 0\end {cases}$$

2014 JBMO Shortlist, 1

Tags: combinatorics

There are some real numbers on the board (at least two). In every step we choose two of them, for example $a$ and $b$, and then we replace them with $\frac{ab}{a+b}$. We continue until there is one number. Prove that the last number does not depend on which order we choose the numbers to erase.

2013 Sharygin Geometry Olympiad, 14

Tags: geometry , trapezoid , circumcircle , analytic geometry , quadratics , Asymptote , trigonometry

Let $M$, $N$ be the midpoints of diagonals $AC$, $BD$ of a right-angled trapezoid $ABCD$ ($\measuredangle A=\measuredangle D = 90^\circ$). The circumcircles of triangles $ABN$, $CDM$ meet the line $BC$ in the points $Q$, $R$. Prove that the distances from $Q$, $R$ to the midpoint of $MN$ are equal.

2014 Saudi Arabia BMO TST, 5

Tags: geometry unsolved , geometry

Let $ABC$ be a triangle. Circle $\Omega$ passes through points $B$ and $C$. Circle $\omega$ is tangent internally to $\Omega$ and also to sides $AB$ and $AC$ at $T,~ P,$ and $Q$, respectively. Let $M$ be midpoint of arc $\widehat{BC}$ (containing T) of $\Omega$. Prove that lines $P Q,~ BC,$ and $MT$ are concurrent.

2004 China Team Selection Test, 1

Tags: function , quadratics , algebra unsolved , algebra

Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.

1998 AMC 12/AHSME, 7

Tags: LaTeX

If $ N > 1$, then ${ \sqrt [3] {N \sqrt [3] {N \sqrt [3] {N}}}} =$ $ \textbf{(A)}\ N^{\frac {1}{27}}\qquad \textbf{(B)}\ N^{\frac {1}{9}}\qquad \textbf{(C)}\ N^{\frac {1}{3}}\qquad \textbf{(D)}\ N^{\frac {13}{27}}\qquad \textbf{(E)}\ N$

1987 Traian Lălescu, 2.2

Tags: function , calculus , integration , inequalities

Let $ f:[0,1]\longrightarrow\mathbb{R} $ a continuous function. Prove that $$ \int_0^1 f^2\left( x^2 \right) dx\ge \frac{3}{4}\left( \int_0^1 f(x)dx \right)^2 , $$ and find the circumstances under which equality happens.

2013 Iran MO (2nd Round), 3

Tags: geometry , circumcircle , trapezoid , parallelogram , trigonometry , geometric transformation , reflection

Let $M$ be the midpoint of (the smaller) arc $BC$ in circumcircle of triangle $ABC$. Suppose that the altitude drawn from $A$ intersects the circle at $N$. Draw two lines through circumcenter $O$ of $ABC$ paralell to $MB$ and $MC$, which intersect $AB$ and $AC$ at $K$ and $L$, respectively. Prove that $NK=NL$.

1973 IMO, 3

Tags: algebra , polynomial , roots , minimum value , IMO , IMO 1973 , coefficients

Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.

2015 Dutch BxMO/EGMO TST, 2

Tags: Integer sequence , Sequence , algebra , number theory

Given are positive integers $r$ and $k$ and an infinite sequence of positive integers $a_1 \le a_2 \le ...$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a $t$ satisfying $\frac{t}{a_t}=k$.

1967 Dutch Mathematical Olympiad, 2

Tags: perfect cube , number theory , arithmetic sequence

Consider arithmetic sequences where all terms are natural numbers. If the first term of such a sequence is $1$, prove that that sequence contains infinitely many terms that are the cube of a natural number. Give an example of such a sequence in which no term is the cube of a natural number and show the correctness of this example.

PEN I Problems, 13

Tags: floor function , function

Suppose that $n \ge 2$. Prove that \[\sum_{k=2}^{n}\left\lfloor \frac{n^{2}}{k}\right\rfloor = \sum_{k=n+1}^{n^{2}}\left\lfloor \frac{n^{2}}{k}\right\rfloor.\]

1993 AMC 8, 16

Tags: AMC

$\dfrac{1}{1+\dfrac{1}{2+\dfrac{1}{3}}} =$ $\text{(A)}\ \dfrac{1}{6} \qquad \text{(B)}\ \dfrac{3}{10} \qquad \text{(C)}\ \dfrac{7}{10} \qquad \text{(D)}\ \dfrac{5}{6} \qquad \text{(E)}\ \dfrac{10}{3}$

2021 Serbia Team Selection Test, P4

Tags: combinatorics

Given that $a_1, a_2, \ldots,a_{2020}$ are integers, find the maximal number of subsequences $a_i,a_{i+1}, ..., a_j$ ($0<i\leq j<2021$) with with sum $2021$

2002 VJIMC, Problem 2

Tags: Ring Theory , abstract algebra

A ring $R$ (not necessarily commutative) contains at least one non-zero zero divisor and the number of zero divisors is finite. Prove that $R$ is finite.

2006 ISI B.Stat Entrance Exam, 10

Tags: function , ratio , probability , induction , geometric sequence , algebra proposed , algebra

Consider a function $f$ on nonnegative integers such that $f(0)=1, f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \ge 2$. Show that \[\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}\]

2017 Sharygin Geometry Olympiad, 5

Tags: geometry , Tangents , midpoints , Squares

A square $ABCD$ is given. Two circles are inscribed into angles $A$ and $B$, and the sum of their diameters is equal to the sidelength of the square. Prove that one of their common tangents passes through the midpoint of $AB$.

2012 Stars of Mathematics, 4

Tags: combinatorics , boolean lattice method

Consider a set $X$ with $|X| = n\geq 1$ elements. A family $\mathcal{F}$ of distinct subsets of $X$ is said to have property $\mathcal{P}$ if there exist $A,B \in \mathcal{F}$ so that $A\subset B$ and $|B\setminus A| = 1$. i) Determine the least value $m$, so that any family $\mathcal{F}$ with $|\mathcal{F}| > m$ has property $\mathcal{P}$. ii) Describe all families $\mathcal{F}$ with $|\mathcal{F}| = m$, and not having property $\mathcal{P}$. ([i]Dan Schwarz[/i])

1991 Spain Mathematical Olympiad, 3

Tags: algebra , polynomial , triangle inequality

What condition must be satisfied by the coefficients $u,v,w$ if the roots of the polynomial $x^3 -ux^2+vx-w$ are the sides of a triangle

2012 India Regional Mathematical Olympiad, 6

Tags: polynomial , polynomial equation , Real Roots , algebra

Let $a$ and $b$ be real numbers such that $a \ne 0$. Prove that not all the roots of $ax^4 + bx^3 + x^2 + x + 1 = 0$ can be real.