Olympiad problems

1975 AMC 12/AHSME, 11

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Let $ P$ be an interior point of circle $ K$ other than the center of $ K$. Form all chords of $ K$ which pass through $ P$, and determine their midpoints. The locus of these midpoints is $ \textbf{(A)}\ \text{a circle with one point deleted} \qquad$ $ \textbf{(B)}\ \text{a circle if the distance from } P \text{ to the center of } K \text{ is less than}$ $ \text{one half the radius of } K \text{; otherwise a circular arc of less than}$ $ 360^{\circ}\qquad$ $ \textbf{(C)}\ \text{a semicircle with one point deleted} \qquad$ $ \textbf{(D)}\ \text{a semicircle} \qquad$ $ \textbf{(E)}\ \text{a circle}$

2000 Moldova National Olympiad, Problem 8

Tags: geometry , 3d geometry

A rectangular parallelepiped has dimensions $a,b,c$ that satisfy the relation $3a+4b+10c=500$, and the length of the main diagonal $20\sqrt5$. Find the volume and the total area of the surface of the parallelepiped.

2018 Harvard-MIT Mathematics Tournament, 9

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Evan has a simple graph with $v$ vertices and $e$ edges. Show that he can delete at least $\frac{e-v+1}{2}$ edges so that each vertex still has at least half of its original degree.

2002 IMO Shortlist, 2

Tags: combinatorics , tiling , dissection

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

2010 Germany Team Selection Test, 3

Tags: algebra , polynomial , number theory , permutation

Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

2015 ASDAN Math Tournament, 3

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For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tournament can generate.

2014 Brazil Team Selection Test, 2

Tags: number theory , prime divisor , polynomial

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

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

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

1988 Putnam, A5

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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]