Olympiad problems

2011 Turkey Junior National Olympiad, 1

Tags: algebra , inequalities

Show that \[1 \leq \frac{(x+y)(x^3+y^3)}{(x^2+y^2)^2} \leq \frac98\] holds for all positive real numbers $x,y$.

Kvant 2024, M2813

Tags: geometry

The quadrilateral $ABCD$ is described around a circle centered on $I$. Let the diagonals $AC$ and $BD$ intersect at point $E$. The perpendicular bisectors to the segments $AC$ and $BD$ intersect at the point $P$ lying inside the triangle $BEC$. The circumscribed circles of the triangles $APC$ and $BPD$ intersect at points $P$ and $Q$. Prove that $I$ lies on the line $PQ$. [i] Proposed by Tran Quang Hung [/i]

2014 Federal Competition For Advanced Students, P2, 1

Tags: number of divisors , divisor , number theory

For each positive natural number $n$ let $d (n)$ be the number of its divisors including $1$ and $n$. For which positive natural numbers $n$, for every divisor $t$ of $n$, that $d (t)$ is a divisor of $d (n)$?

2023 Yasinsky Geometry Olympiad, 3

Tags: geometry , incircle

Points $K$ and $N$ are the midpoints of sides $AC$ and $AB$ of triangle $ABC$. The inscribed circle $\omega$ of the triangle $AKN$ is tangent to $BC$. Find $BC$ if $AC + AB = n$. (Oleksii Karliuchenko)

2007 Singapore Junior Math Olympiad, 5

Tags: algebra , function

For any positive integer $n$, let $f(n)$ denote the $n$- th positive nonsquare integer, i.e., $f(1) = 2, f(2) = 3, f(3) = 5, f(4) = 6$, etc. Prove that $f(n)=n +\{\sqrt{n}\}$ where $\{x\}$ denotes the integer closest to $x$. (For example, $\{\sqrt{1}\} = 1, \{\sqrt{2}\} = 1, \{\sqrt{3}\} = 2, \{\sqrt{4}\} = 2$.)

2008 Postal Coaching, 1

Tags: incenter , circumcircle , right angle , arc midpoint , angle chasing , angle , geometry

In triangle $ABC,\angle B > \angle C, T$ is the midpoint of arc $BAC$ of the circumcicle of $ABC$, and $I$ is the incentre of $ABC$. Let $E$ be point such that $\angle AEI = 90^0$ and $AE$ is parallel to $BC$. If $TE$ intersects the circumcircle of $ABC$ at $P(\ne T)$ and $\angle B = \angle IPB$, determine $\angle A$.

2018 Costa Rica - Final Round, 4

Tags: functional , functional inequality , algebra

Determine if there exists a function f: $N^*\to N^*$ that satisfies that for all $n \in N^*$, $$10^{f (n)} <10n + 1 <10^{f (n) +1}.$$ Justify your answer. Note: $N^*$ denotes the set of positive integers.

1980 IMO Shortlist, 11

Tags: combinatorics , system of equations , counting

Ten gamblers started playing with the same amount of money. Each turn they cast (threw) five dice. At each stage the gambler who had thrown paid to each of his 9 opponents $\frac{1}{n}$ times the amount which that opponent owned at that moment. They threw and paid one after the other. At the 10th round (i.e. when each gambler has cast the five dice once), the dice showed a total of 12, and after payment it turned out that every player had exactly the same sum as he had at the beginning. Is it possible to determine the total shown by the dice at the nine former rounds ?

2015 Kazakhstan National Olympiad, 6

Tags: geometry

The quadrilateral $ABCD$ has an incircle of diameter $d$ which touches $BC$ at $K$ and touches $DA$ at $L$. Is it always true that the harmonic mean of $AB$ and $CD$ is equal to $KL$ if and only if the geometric mean of $AB$ and $CD$ is equal to $d$?

1965 All Russian Mathematical Olympiad, 058

Tags: geometry , incenter , circumcenter

A circle is circumscribed around the triangle $ABC$. Chords, from the midpoint of the arc $AC$ to the midpoints of the arcs $AB$ and $BC$, intersect sides $[AB]$ and $[BC]$ in the points $D$ and $E$. Prove that $(DE)$ is parallel to $(AC)$ and passes through the centre of the inscribed circle.

V Soros Olympiad 1998 - 99 (Russia), 10.2

Tags: algebra , trigonometry

Solve the equation $$ |\cos 3x - tgt| + |\cos 3x + tgt| = |tg^2t -3|.$$

2017 Switzerland - Final Round, 8

Tags: midpoint , isosceles , equal segments , geometry

Let $ABC$ be an isosceles triangle with vertex $A$ and $AB> BC$. Let $k$ be the circle with center $A$ passsing through $B$ and $C$. Let $H$ be the second intersection of $k$ with the altitude of the triangle $ABC$ through $B$. Further let $G$ be the second intersection of $k$ with the median through $B$ in triangle $ABC$. Let $X$ be the intersection of the lines $AC$ and $GH$. Show that $C$ is the midpoint of $AX$.

2022 Bulgarian Spring Math Competition, Problem 8.3

Tags: algebra , inequality , inequalities

Given the inequalities: $a)$ $\left(\frac{2a}{b+c}\right)^2+\left(\frac{2b}{a+c}\right)^2+\left(\frac{2c}{a+b}\right)^2\geq \frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ $b)$ $\left(\frac{a+b}{c}\right)^2+\left(\frac{b+c}{a}\right)^2+\left(\frac{c+a}{b}\right)^2\geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+9$ For each of them either prove that it holds for all positive real numbers $a$, $b$, $c$ or present a counterexample $(a,b,c)$ which doesn't satisfy the inequality.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.7

Tags: inequalities , algebra

Without using a calculator, prove that $$2^{1995} >5^{854},$$

2001 Singapore MO Open, 3

Tags: combinatorics , difference

Suppose that there are $2001$ golf balls which are numbered from $1$ to $2001$ respectively, and some of these golf balls are placed inside a box. It is known that the difference between the two numbers of any two golf balls inside the box is neither $5$ nor $8$. How many such golf balls the box can contain at most? Justify your answer.

1961 Czech and Slovak Olympiad III A, 1

Tags: sequence , consecutive , positive integer

Consider an infinite sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, \ldots, \underbrace{n,\ldots,n}_{n\text{ times}},\ldots.$$ Find the 1000th term of the sequence.

2008 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory , divisibility , extremal combinatorics , additive combinatorics

Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]

1951 AMC 12/AHSME, 8

Tags:

The price of an article is cut $ 10\%$. To restore it to its former value, the new price must be increased by: $ \textbf{(A)}\ 10\% \qquad\textbf{(B)}\ 9\% \qquad\textbf{(C)}\ 11\frac {1}{9}\% \qquad\textbf{(D)}\ 11\% \qquad\textbf{(E)}\ \text{none of these answers}$

1999 Miklós Schweitzer, 7

Tags: real analysis , function

let $f:R\to R$ be a continuous function tf(t)>0 for $t\neq 0$. Prove that there exists a non-zero differentiable function $y:[0,\infty)\to R$ such that $y'(t)=f(y(t-1))\,\forall t>1$ and the roots of y are bounded.

2021 Kosovo National Mathematical Olympiad, 3

Tags: algebra

Find all real numbers $a,b,c$ and $d$ such that: $a^2+b^2+c^2+d^2=a+b+c+d-ab=3.$

2014 Kurschak Competition, 3

Tags: geometry , geometric transformation , reflection , perimeter

Let $K$ be a closed convex polygonal region, and let $X$ be a point in the plane of $K$. Show that there exists a finite sequence of reflections in the sides of $K$, such that $K$ contains the image of $X$ after these reflections.

2022 Harvard-MIT Mathematics Tournament, 7

Tags: geometry

Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $O_2O_3$, $O_3O_4$, $O_4O_1$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

2008 Stars Of Mathematics, 3

Tags: geometry , incenter , geometric transformation , homothety , parallelogram , ellipse , conic

Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals. [i]Dan Schwarz[/i]

2014 AIME Problems, 8

Tags: trigonometry , asymptote , geometry , geometric transformation , reflection , pythagorean theorem

Circle $C$ with radius $2$ has diameter $\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C,$ externally tangent to circle $D,$ and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$ and can be written in the form $\sqrt{m} - n,$ where $m$ and $n$ are positive integers. Find $m+n$.

2012 AIME Problems, 2

Tags: ratio

Two geometric sequences $ a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3\ldots $have the same common ratio, with $a_1=27$,$b_1=99$, and $a_{15}=b_{11}$. Find $a_9.$