Olympiad problems

1969 IMO Longlists, 35

$(HUN 2)$ Prove that $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}$

2020 HK IMO Preliminary Selection Contest, 2

Tags: algebra , integer

Let $x$, $y$, $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$. Find the greatest possible value of $x+y+z$.

2018 ASDAN Math Tournament, 1

Tags: geometry

A regular hexagon $ABCDEF$ has perimeter $12$. $AB$, $CD$, and $EF$ are all extended, and the intersections of the line segments form an equilateral triangle. Compute the perimeter of the triangle.

1986 IMO Longlists, 7

Tags: geometry , point set , combinatorial geometry , straight lines

Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$ [i]Simplified version.[/i] Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$

1983 Vietnam National Olympiad, 1

Tags: modular arithmetic , greatest common divisor , number theory

Are there positive integers $a, b$ with $b \ge 2$ such that $2^a + 1$ is divisible by $2^b - 1$?

2023 MIG, 8

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Anna is buying fruits at a grocery store. If she loses a nickel, she still has enough money to buy exactly $16$ lemons. Similarly, if she loses a quarter, she has enough money to buy exactly $14$ lemons. What is the cost of each lemon? $\textbf{(A) } \$0.05\qquad\textbf{(B) } \$0.10\qquad\textbf{(C) } \$0.15\qquad\textbf{(D) } \$0.20\qquad\textbf{(E) } \$0.25$

2014 Romania Team Selection Test, 1

Tags: angle bisector , trigonometry , circumcircle , geometry , projective geometry

Let $\triangle ABC$ be an acute triangle of circumcentre $O$. Let the tangents to the circumcircle of $\triangle ABC$ in points $B$ and $C$ meet at point $P$. The circle of centre $P$ and radius $PB=PC$ meets the internal angle bisector of $\angle BAC$ inside $\triangle ABC$ at point $S$, and $OS \cap BC = D$. The projections of $S$ on $AC$ and $AB$ respectively are $E$ and $F$. Prove that $AD$, $BE$ and $CF$ are concurrent. [i]Author: Cosmin Pohoata[/i]

1988 Greece National Olympiad, 1

Tags: inequalities , algebra

Let $a>0,b>0,c>0$ and $\sqrt{1987+a}+\sqrt{1987+b}=2\sqrt{1987+c}$. Prove that $\frac{1}{2} (a+b )\ge c $.

2022 Argentina National Olympiad, 1

Tags: number theory

For every positive integer $n$, $P(n)$ is defined as follows: For each prime divisor $p$ of $n$ is considered the largest integer $k$ such that $p^k\le n$ and all the $p^k$ are added. For example, for $n=100=2^2 \cdot 5^2$, as $2^6<100<2^7$ and $5^2<100<5^3$, it turns out that $P(100)=2^6+5^2=89$ Prove that there are infinitely many positive integers $n$ such that $P(n)>n$..

2007 Korea National Olympiad, 4

Tags: induction , algebra , strong induction

Two real sequence $ \{x_{n}\}$ and $ \{y_{n}\}$ satisfies following recurrence formula; $ x_{0}\equal{} 1$, $ y_{0}\equal{} 2007$ $ x_{n\plus{}1}\equal{} x_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(y_{n}\plus{}y_{n\plus{}1})$, $ y_{n\plus{}1}\equal{} y_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(x_{n}\plus{}x_{n\plus{}1})$ Then show that for all nonnegative integer $ n$, $ {x_{n}}^{2}\leq 2007$.

2009 Junior Balkan Team Selection Test, 3

Tags: parallelogram , perpendicular bisector , geometry

Let $ ABCD$ be a convex quadrilateral, such that $ \angle CBD\equal{}2\cdot\angle ADB, \angle ABD\equal{}2\cdot\angle CDB$ and $ AB\equal{}CB$. Prove that quadrilateral $ ABCD$ is a kite.

2009 Today's Calculation Of Integral, 518

Tags: trigonometry , integration , logarithm , calculus , calculus computations

Evaluate ${ \int_0^{\frac{\pi}{8}}\frac{\cos x}{\cos (x-\frac{\pi}{8}})}\ dx$.

PEN L Problems, 11

Tags: linear recurrence

Let the sequence $\{K_{n}\}_{n \ge 1}$ be defined by \[K_{1}=2, K_{2}=8, K_{n+2}=3K_{n+1}-K_{n}+5(-1)^{n}.\] Prove that if $K_{n}$ is prime, then $n$ must be a power of $3$.

2005 IMO Shortlist, 5

Tags: inequalities , algebra

Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that \[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \] [i]Hojoo Lee, Korea[/i]

2020 Ukrainian Geometry Olympiad - December, 5

Tags: geometry , circumcircle , altitude , angle

In an acute triangle $ABC$ with an angle $\angle ACB =75^o$, altitudes $AA_3,BB_3$ intersect the circumscribed circle at points $A_1,B_1$ respectively. On the lines $BC$ and $CA$ select points $A_2$ and $B_2$ respectively suchthat the line $B_1B_2$ is parallel to the line $BC$ and the line $A_1A_2$ is parallel to the line $AC$ . Let $M$ be the midpoint of the segment $A_2B_2$. Find in degrees the measure of the angle $\angle B_3MA_3$.

2023 Peru MO (ONEM), 4

Tags: geometry , incenter

Let $ABC$ be an acute scalene triangle and $K$ be a point inside it that belongs to the bisector of the angle $\angle ABC$. Let$ P$ be the point where the line $AK$ intersects the line perpendicular to $AB$ that passes through $B$, and let $Q$ be the point where the line $CK$ intersects the line perpendicular to $CB$ that passes through $B$. Let $L$ be the foot of the perpendicular drawn from $K$ on the line $AC$. Prove that if $P Q$ is perpendicular to $BL$, then $K$ is the incenter of $ABC$.

2012 NIMO Problems, 6

Tags:

The positive numbers $a, b, c$ satisfy $4abc(a+b+c) = (a+b)^2(a+c)^2$. Prove that $a(a+b+c)=bc$. [i]Proposed by Aaron Lin[/i]

2011 Tournament of Towns, 5

Tags: number theory

Find all positive integers $a,b$ such that $b^{619}$ divides $a^{1000}+1$ and $a^{619}$ divides $b^{1000}+1$.

2002 Moldova Team Selection Test, 3

Tags: geometry , incenter , similar triangles , arc

A triangle $ABC$ is inscribed in a circle $G$. Points $M$ and $N$ are the midpoints of the arcs $BC$ and $AC$ respectively, and $D$ is an arbitrary point on the arc $AB$ (not containing $C$). Points $I_1$ and $I_2$ are the incenters of the triangles $ADC$ and $BDC$, respectively. If the circumcircle of triangle $DI_1I_2$ meets $G$ again at $P$, prove that triangles $PNI_1$ and $PMI_2$ are similar.

2012 AMC 10, 24

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Let $a,b,$ and $c$ be positive integers with $a\ge b\ge c$ such that \begin{align*} a^2-b^2-c^2+ab&=2011\text{ and}\\ a^2+3b^2+3c^2-3ab-2ac-2bc&=-1997\end{align*} What is $a$? $ \textbf{(A)}\ 249 \qquad\textbf{(B)}\ 250 \qquad\textbf{(C)}\ 251 \qquad\textbf{(D)}\ 252 \qquad\textbf{(E)}\ 253 $

2015 Romania Team Selection Tests, 2

Tags: algebra , sequence , bounded , recurrence

Let $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$ be sequences of real numbers such that $ a_0>\frac{1}{2}$ , $a_{n+1} \geq a_n$ and $b_{n+1}=a_n(b_n+b_{n+2})$ for all non-negative integers $n$ . Show that the sequence $(b_n)_{n \geq 0}$ is bounded .

2012 NIMO Problems, 6

Tags: trigonometry , algebra , polynomial

The polynomial $P(x) = x^3 + \sqrt{6} x^2 - \sqrt{2} x - \sqrt{3}$ has three distinct real roots. Compute the sum of all $0 \le \theta < 360$ such that $P(\tan \theta^\circ) = 0$. [i]Proposed by Lewis Chen[/i]

1990 Bulgaria National Olympiad, Problem 4

Tags: number theory , set

Suppose $M$ is an infinite set of natural numbers such that, whenever the sum of two natural numbers is in $M$, one of these two numbers is in $M$ as well. Prove that the elements of any finite set of natural numbers not belonging to $M$ have a common divisor greater than $1$.

2017 Purple Comet Problems, 11

Tags:

Find the greatest prime divisor of $29! + 33!$.

2005 Federal Competition For Advanced Students, Part 2, 2

Tags: inequalities

Prove that for all positive reals $a,b,c,d$, we have $\frac{a+b+c+d}{abcd}\leq \frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}+\frac{1}{d^{3}}$