Tagged Questions
1
vote
1answer
42 views
Trigonometric Sums - URSS
Calculate the value of the sums:
(a) $\cos x+\binom{n}{1}\cos 2x +\cdots+\binom{n}{n} \cos (n+1)x $;
(b) $\sin x+\binom{n}{1}\sin 2x +\cdots+\binom{n}{n} \sin (n+1)x $.
2
votes
1answer
138 views
Trigonometry / Geometry Puzzle with a Circle Inscribed within a Square
Point P is any point on the inscribed circle. You must prove that
(tan(a))^2 + (tan(B))^2 = 8
I first moved point P down to the point where the square would be tangent to the curve to make the ...
13
votes
4answers
438 views
Showing that $ |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$
For every nonnegative integer $n$ and every real number $ x$ prove the inequality:
$$\sum_{k=0}^n|\cos(2^kx)|= |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$$
4
votes
1answer
81 views
Can the distance from the vertices of a square of integer width to an inscribed circle all be integer?
I'm looking for solutions to the following British Mathematical Olympiad question:
Suppose that $ABCD$ is a square and that $P$ is a point which is on the circle inscribed in the square. Determine ...
6
votes
2answers
258 views
Help me solve this olympiad challenge?
Given:
$$p(x) = x^4 - 5773x^3 - 46464x^2 - 5773x + 46$$
What is the sum of all arctan of all the roots of $p(x)$?
1
vote
4answers
255 views
Geometry Prove - two perpendicular lines in a circle
In a circle of radius r, two lines (AB and CD) are perpendicular to each other and meet at X.
Show that:
0
votes
1answer
75 views
Finding a diagonal of a trapezoid that touches 3 points on a circle
In the image below:
- AB and AD are tangent to the circle
- BC and AD are parallel
What is the length of AC?
Thank you very much in advance!
2
votes
3answers
418 views
Finding a diagonal in a trapezoid given the other diagonal and three sides
The figure below is a trapezoid, what is the length of the red line?
Thank you very much in advance!
8
votes
1answer
397 views
Hard math contest trigonometry type problem
How to solve this problem:
Also, most people would use trigonometry, but is there a way to use derivative to solve this too?
1
vote
2answers
85 views
When does $f(x;\alpha)=\cos(\alpha x)-\sin^2x-1$ has unique zero?
This is a contest math question that I don't remember the reference.
When does $f(x;\alpha)=\cos(\alpha x)-\sin^2x-1$ has unique zero?
Obviously, $f(0;\alpha)=0$ for all $\alpha\in{\mathbb R}$. ...
4
votes
1answer
260 views
How to find all rational numbers satisfy this equation?
Find all rational number $a,b,c$ satisfy:
$$a+b+c=abc$$
I try to change this in different forms like $(ab-1)c = a+b$, $(ac-1)b = a+c$, $(cb-1)a = b+c$ etc but it won't help...
0
votes
2answers
320 views
Solving $\arctan(a) + \arctan(b) + \arctan(c) = \pi$ for $0 < a < b < c < 10$
This is a trigonometry math contest problem.
Which ordered triple of numbers $(a,b,c)$ with $0 < a < b < c < 10$ satisfies the equation
$$\arctan(a) + \arctan(b) + \arctan(c) = ...
4
votes
2answers
220 views
Prove this trigonometric identity in quadrilateral
If $\alpha,\beta,\gamma,\delta$ are angles in quadrilateral different from $90^\circ$, prove the following:
$$ ...
1
vote
1answer
244 views
Another Trigonometry problem, sum of products of sine function over partitions of N
I don't know how to write the summation symbol so I'm providing you the original link to problem http://www.codechef.com/OCT11/problems/PARSIN .My approach to solve this problem is first reduce the ...
2
votes
1answer
403 views
Average and minimum Values of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$, $\forall x \in \mathbb{R}$
A problem was asked at Putnam Competition in 2003 (Problem 3), about finding the minimum Value of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$ where $x$ is Real.
the question paper and ...
4
votes
1answer
660 views
Finding equal sums of constrained trig functions
Problem 63 of the 2001 St. Petersburg Mathematical Olympiad, Second Round, 11th grade:
Are there three different numbers $x, y, z$ in $[0,\pi/2]$ such that the numbers $\sin x$, $\sin y$, $\sin z$, ...
