Olympiad problems

1988 Irish Math Olympiad, 12

Tags: trigonometry

Prove that if $n$ is a positive integer ,then \[cos^4\frac{\pi}{2n+1}+cos^4\frac{2\pi}{2n+1}+\cdots+cos^4\frac{n\pi}{2n+1}=\frac{6n-5}{16}.\]

2006 Romania Team Selection Test, 3

Tags: modular arithmetic , induction , number theory

For which pairs of positive integers $(m,n)$ there exists a set $A$ such that for all positive integers $x,y$, if $|x-y|=m$, then at least one of the numbers $x,y$ belongs to the set $A$, and if $|x-y|=n$, then at least one of the numbers $x,y$ does not belong to the set $A$? [i]Adapted by Dan Schwarz from A.M.M.[/i]

1984 IMO Longlists, 27

Tags: function , algebra

The function $f(n)$ is defined on the nonnegative integers $n$ by: $f(0) = 0, f(1) = 1$, and \[f(n) = f\left(n -\frac{1}{2}m(m - 1)\right)-f\left(\frac{1}{2}m(m+ 1)-n\right)\] for $\frac{1}{2}m(m - 1) < n \le \frac{1}{2}m(m+ 1), m \ge 2$. Find the smallest integer $n$ for which $f(n) = 5$.

1950 Miklós Schweitzer, 8

Tags: probability , probability and stats

A coastal battery sights an enemy cruiser lying one kilometer off the coast and opens fire on it at the rate of one round per minute. After the first shot, the cruiser begins to move away at a speed of $ 60$ kilometers an hour. Let the probability of a hit be $ 0.75x^{ \minus{} 2}$, where $ x$ denotes the distance (in kilometers) between the cruiser and the coast ($ x\geq 1$), and suppose that the battery goes on firing till the cruiser either sinks or disappears. Further, let the probability of the cruiser sinking after $ n$ hits be $ 1 \minus{} \frac {1}{4^n}$ ($ n \equal{} 0,1,...$). Show that the probability of the cruiser escaping is $ \frac {2\sqrt {2}}{3\pi}$

2010 Today's Calculation Of Integral, 608

Tags: calculus , integration , calculus computations

For $a>0$, find the minimum value of $\int_0^1 \frac{ax^2+(a^2+2a)x+2a^2-2a+4}{(x+a)(x+2)}dx.$ 2010 Gakusyuin University entrance exam/Science

2014 PUMaC Algebra A, 7

Tags: inequalities

$x$, $y$, and $z$ are positive real numbers that satisfy $x^3+2y^3+6z^3=1$. Let $k$ be the maximum possible value of $2x+y+3z$. Let $n$ be the smallest positive integer such that $k^n$ is an integer. Find the value of $k^n+n$.

2009 Puerto Rico Team Selection Test, 5

Tags: algebra , number theory

The [i]weird [/i] mean of two numbers $ a$ and $ b$ is defined as $ \sqrt {\frac {2a^2 + 3b^2}{5}}$. $ 2009$ positive integers are placed around a circle such that each number is equal to the the weird mean of the two numbers beside it. Show that these $ 2009$ numbers must be equal.

2007 Harvard-MIT Mathematics Tournament, 23

Tags: trigonometry , trig identities , law of sines , geometry

In triangle $ABC$, $\angle ABC$ is obtuse. Point $D$ lies on side $AC$ such that $\angle ABD$ is right, and point $E$ lies on side $AC$ between $A$ and $D$ such that $BD$ bisects $\angle EBC$. Find $CE$ given that $AC=35$, $BC=7$, and $BE=5$.

1967 IMO Shortlist, 6

Tags: geometry , 3d geometry , sphere , angle

Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).

2007 Today's Calculation Of Integral, 168

Tags: calculus , integration , trigonometry , calculus computations

Prove that $\sum_{n=1}^{\infty}\int_{\frac{1}{n+1}}^{\frac{1}{n}}{\left|\frac{1}{x}\sin \frac{\pi}{x}\right| dx}$ diverge for $x>0.$