Olympiad problems

1996 Iran MO (3rd Round), 2

Tags: geometry

Consider a semicircle of center $O$ and diameter $AB$. A line intersects $AB$ at $M$ and the semicircle at $C$ and $D$ s.t. $MC>MD$ and $MB<MA$. The circumcircles od the $AOC$ and $BOD$ intersect again at $K$. Prove that $MK\perp KO$.

2023 CMIMC Team, 14

Tags: team

Let $ABC$ be points such that $AB=7, BC=5, AC=10$, and $M$ be the midpoint of $AC$. Let $\omega$, $\omega_1$ be the circumcircles of $ABC$ and $BMC$. $\Omega$, $\Omega_1$ are circles through $A$ and $M$ such that $\Omega$ is tangent to $\omega_1$ and $\Omega_1$ is tangent to the line through the centers of $\omega_1$ and $\Omega$. $D, E$ be the intersection of $\Omega$ with $\omega$ and $\Omega_1$ with $\omega_1$. If $F$ is the intersection of the circumcircle of $DME$ with $BM$, find $FB$. [i]Proposed by David Tang[/i]

2006 Denmark MO - Mohr Contest, 4

Tags: combinatorics

Of the numbers $1, 2,3,..,2006$, ten different ones must be selected. Show that you can pick ten different numbers with a sum greater than $10039$ in more ways than you can select ten different numbers with a sum less than $10030$.

1998 Balkan MO, 1

Tags: floor function , number theory

Consider the finite sequence $\left\lfloor \frac{k^2}{1998} \right\rfloor$, for $k=1,2,\ldots, 1997$. How many distinct terms are there in this sequence? [i]Greece[/i]

2020 CCA Math Bonanza, I5

Tags: quadratic

Let $f(x)=x^2-kx+(k-1)^2$ for some constant $k$. What is the largest possible real value of $k$ such that $f$ has at least one real root? [i]2020 CCA Math Bonanza Individual Round #5[/i]

2017 Iran Team Selection Test, 2

Tags: geometry

Let $P$ be a point in the interior of quadrilateral $ABCD$ such that: $$\angle BPC=2\angle BAC \ \ ,\ \ \angle PCA = \angle PAD \ \ ,\ \ \angle PDA=\angle PAC$$ Prove that: $$\angle PBD= \left | \angle BCA - \angle PCA \right |$$ [i]Proposed by Ali Zamani[/i]

1991 Arnold's Trivium, 13

Tags: integration

Calculate with $5\%$ relative error \[\int_1^{10}x^xdx\]

2010 Malaysia National Olympiad, 4

Tags: geometry , angle

In the diagram, $\angle AOB = \angle BOC$ and$\angle COD = \angle DOE = \angle EOF$. Given that $\angle AOD = 82^o$ and $\angle BOE = 68^o$. Find $\angle AOF$. [img]https://cdn.artofproblemsolving.com/attachments/b/2/deba6cd740adbf033ad884fff8e13cd21d9c5a.png[/img]

1961 Poland - Second Round, 2

Tags: geometry , tetrahedron , 3d geometry , concurrency

Prove that all the heights of a tetrahedron intersect at one point if and only if the sums of the squares of the opposite edges are equal.

2005 South East Mathematical Olympiad, 8

Tags: trigonometry , calculus , derivative , function , inequalities

Let $0 < \alpha, \beta, \gamma < \frac{\pi}{2}$ and $\sin^{3} \alpha + \sin^{3} \beta + \sin^3 \gamma = 1$. Prove that \[ \tan^{2} \alpha + \tan^{2} \beta + \tan^{2} \gamma \geq \frac{3 \sqrt{3}}{2} . \]

2012 Uzbekistan National Olympiad, 1

Tags: combinatorics

Given a digits {$0,1,2,...,9$} . Find the number of numbers of 6 digits which cantain $7$ or $7$'s digit and they is permulated(For example 137456 and 314756 is one numbers).

2005 AIME Problems, 2

Tags: counting , distinguishability , probability , number theory , relatively prime

A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m+n$.

1985 Traian Lălescu, 1.3

Tags: geometry , 3d geometry

We have a parallelepiped $ ABCDA'B'C'D' $ in which the top ($ A'B'C'D' $) and the ground ($ ABCD $) are connected by four vertical edges, and $ \angle DAB=30^{\circ} . $ Through $ AB, $ a plane inersects the parallelepiped at an angle of $ 30 $ with respect to the ground, delimiting two interior sections. Find the area of these interior sections in function of the length of $ AA'. $

KoMaL A Problems 2017/2018, A. 725

Tags: function , algebra

Let $\mathbb R^+$ denote the set of positive real numbers.Find all functions $f:\mathbb R^+\rightarrow \mathbb R^+$ satisfying the following equation for all $x,y\in \mathbb R^+$: $$f(xy+f(y)^2)=f(x)f(y)+yf(y)$$

2006 Junior Balkan Team Selection Tests - Romania, 4

Tags:

Prove that the set of real numbers can be partitioned in (disjoint) sets of two elements each.

1988 Putnam, A5

Tags:

Prove that there exists a [i]unique[/i] function $f$ from the set $\mathrm{R}^+$ of positive real numbers to $\mathrm{R}^+$ such that\[ f(f(x)) = 6x-f(x) \]and\[ f(x)>0 \]for all $x>0$.

2020 BMT Fall, 11

Tags: number theory

Compute $\sum^{999}_{x=1}\gcd (x, 10x + 9)$.

2021 Saint Petersburg Mathematical Olympiad, 6

Tags: geometry , rhombus , circumcircle

A line $\ell$ passes through vertex $C$ of the rhombus $ABCD$ and meets the extensions of $AB, AD$ at points $X,Y$. Lines $DX, BY$ meet $(AXY)$ for the second time at $P,Q$. Prove that the circumcircle of $\triangle PCQ$ is tangent to $\ell$ [i]A. Kuznetsov[/i]

2004 Argentina National Olympiad, 6

Tags: arithmetic progression , algebra , arithmetic sequence

Decide if it is possible to generate an infinite sequence of positive integers $a_n$ such that in the sequence there are no three terms that are in arithmetic progression and that for all $n$ $\left |a_n-n^2\right | <\frac{n}{2}$. Clarification: Three numbers $a$, $b$, $c$ are in arithmetic progression if and only if $2b=a+c$.

2008 Sharygin Geometry Olympiad, 15

Tags: symmetry , geometry

(M.Volchkevich, 9--11) Given two circles and point $ P$ not lying on them. Draw a line through $ P$ which cuts chords of equal length from these circles.

2002 Tuymaada Olympiad, 8

Tags: tangent , geometry , area of a triangle , area

The circle with the center of $ O $ touches the sides of the angle $ A $ at the points of $ K $ and $ M $. The tangent to the circle intersects the segments $ AK $ and $ AM $ at points $ B $ and $ C $ respectively, and the line $ KM $ intersects the segments $ OB $ and $ OC $ at the points $ D $ and $ E $. Prove that the area of the triangle $ ODE $ is equal to a quarter of the area of a triangle $ BOC $ if and only if the angle $ A $ is $ 60^\circ $.

2010 Germany Team Selection Test, 2

Tags: geometry , cyclic quadrilateral , projective geometry , tangent , power of a point

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

1999 Abels Math Contest (Norwegian MO), 3

Tags: isosceles , equilateral , triangle area , area

An isosceles triangle $ABC$ with $AB = AC$ and $\angle A = 30^o$ is inscribed in a circle with center $O$. Point $D$ lies on the shorter arc $AC$ so that $\angle DOC = 30^o$, and point $G$ lies on the shorter arc $AB$ so that $DG = AC$ and $AG < BG$. The line $BG$ intersects $AC$ and $AB$ at $E$ and $F$, respectively. (a) Prove that triangle $AFG$ is equilateral. (b) Find the ratio between the areas of triangles $AFE$ and $ABC$.

2011 Iran MO (2nd Round), 1

Tags: geometry , analytic geometry , combinatorics

We have a line and $1390$ points around it such that the distance of each point to the line is less than $1$ centimeters and the distance between any two points is more than $2$ centimeters. prove that there are two points such that their distance is at least $10$ meters ($1000$ centimeters).

MOAA Team Rounds, 2021.2

Tags: team

Four students Alice, Bob, Charlie, and Diana want to arrange themselves in a line such that Alice is at either end of the line, i.e., she is not in between two students. In how many ways can the students do this? [i]Proposed by Nathan Xiong[/i]