1
vote
2answers
44 views

$m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $

Prove that $m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $ for every $m, n, l >0$.
2
votes
2answers
38 views

An arctan problem including a diophantine equation

This is a follow-up question to An equation of the form A + B + C = ABC . I totally messed up with making the equation from the question specification . Actually the question was $$ ...
4
votes
2answers
106 views

Evaluating $\sum_{n=1}^{99}\sin(n)$ [duplicate]

I'm looking for a trick, or a quick way to evaluate the sum $\displaystyle{\sum_{n=1}^{99}\sin(n)}$. I was thinking of applying a sum to product formula, but that doesn't seem to help the situation. ...
-3
votes
1answer
61 views

Why there are two different values of θ for same quadrant?

Let Sin θ = 1/2 is function. Let us find its solution set. sine is +ve in I and II quadrant with reference angle π/6 θ = π/6 (I quadrant) Now here is my problem. We can use π-θ and (π/2)+θ to find ...
1
vote
2answers
88 views

Difficult infinite trigonometric series

Evaluate the sum of the following infinite series. $$\left(\sin{\frac{\pi}{3}}\right) + \left(\frac{1}{2}\sin{\frac{2\pi}{3}}\right) + \left(\frac{1}{3}\sin{\frac{3\pi}{3}}\right) + \ldots$$
2
votes
1answer
83 views

Putnam inspired problem

The following is a beautiful problem from Putnam 2003 minimize $|\sin x + \cos x + \tan x + \csc x + \sec x + \cot x|$ I was thinking about a small variation of the above problem minimize $|\sin ...
0
votes
1answer
60 views

Trigonometric eq.

The equation $3\sin(x)+4\cos(x)=5$ is well-known. The equation $3\sin^m(x)+4\cos^n(x)=5$ where $m$ and $n$ are non-negative integers is much more interesting.. I would like to see a nice, elementary ...
2
votes
3answers
282 views

2014 AMC 12 B problem 25

What is the sum of all positive real solutions $x$ to the following equation? $$2\cos(2x)\left( \cos(2x) - \cos{\left(\frac{2014\pi^2}{x^2}\right)} \right) = \cos(4x) - 1 $$
18
votes
2answers
875 views

Tough contest problem

I found this problem in a collection of contest problems of a Russian competition in 1995 and wasn't able to solve it. Solve for real $x$: $$ \cos (\cos (\cos (\cos(x))))=\sin (\sin (\sin (\sin ...
0
votes
0answers
30 views

Completing sets of numbers solely with trigonometric functions and an initial zero?

Last week an extra-curricular math academy I attend gave us this question as a challenge: You start with $0$, and the only functions you can do are $\sin, \cos, \tan, \sin^{-1}, \cos^{-1}, \tan^{-1}$ ...
18
votes
2answers
500 views

An upper bound on certain finite trigonometric series given a lower bound

Let $f$ be the function $f(x)=1+a\sin{x}+b\cos x+c\sin{(2x)}+d\cos{(2x)}$, where $a,b,c,d$ are arbitrary real numbers. Prove that if $f(x)>0$ for all $x\in \mathbb R$, then $f(x)<3$ for all ...
5
votes
2answers
148 views

How find this $\sum_{i=0}^{5}\frac{1}{2+\cos{\left(x+\frac{i\pi}{3}\right)}}\cdot \frac{1}{2+\cos{\left(x+\frac{(i+1)\pi}{3}\right)}}$

Find this follow function $f(x)$ range ,where $x\in R$, $$f(x)=\sum_{i=0}^{5}\dfrac{1}{2+\cos{\left(x+\dfrac{i\pi}{3}\right)}}\cdot \dfrac{1}{2+\cos{\left(x+\dfrac{(i+1)\pi}{3}\right)}}$$ or ...
0
votes
3answers
192 views

Competition Math Geometry Problem

Note: I am paraphrasing this problem Consider a quadrilateral with 3 sides of equal length, and one longer side. This quadrilateral also has equal diagonals, both of which are equal in length to the ...
2
votes
2answers
101 views

Convex Quadrilateral: $ \dfrac {\tan A + \tan B + \tan C + \tan D}{\tan A \tan B \tan C \tan D} = \cot A + \cot B + \cot C + \cot D $

Problem Let $ABCD$ be a convex quadrilateral with no right angles. Show that $$ \dfrac {\tan A + \tan B + \tan C + \tan D}{\tan A \tan B \tan C \tan D} = \cot A + \cot B + \cot C + \cot D. $$ ...
19
votes
1answer
261 views

How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?

How can I prove the following identity? $$\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$$
5
votes
1answer
94 views

Macedonia National Olympiad 2010

Problem The point O is the center of the circumscribed circle of the acute-angled triangle ABC. The line AO cuts the side BC in point N, and the line BO cuts the side AC at point M. Prove that if ...
10
votes
2answers
377 views

Another math contest problem: $\int_0^{\frac{\ln^22}4}\,\frac{\arccos\frac{\exp\sqrt x}{\sqrt2}}{1-\exp\sqrt{4\,x}}dx$

Prove: $$ {\Large\int_{0}^{\ln^{2}\left(2\right) \over4}}\, \frac{\arccos\left(\vphantom{\huge A} {\exp\left(\vphantom{\large A}\sqrt{x\,}\right) \over \sqrt{\vphantom{\large A}2\,}}\right)} ...
12
votes
1answer
319 views

Prove $\left|\sum_{k=2001}^{m}a_{k}\sin{(kx)}\right|\le 1+\pi $ ,$m\ge 2001,x\in R$

let $\{a_{n}\}$ is non-increasing postive sequence;show that if for $n\ge 2001,na_{n}\le 1$, then for any positive integer numbers $m\ge 2001,x\in R$, we have ...
1
vote
1answer
45 views

For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative?

Let $k>0$ be an integer. For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative? Since $13$ is prime, and for $\gcd(m,13)=1$, $P(2m)=P(2)=2^{-12}$ (can be shown by considering the ...
4
votes
2answers
396 views

Compute $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$

Compute the series $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$ Hint: the answer is in fact 0
3
votes
2answers
230 views

Find the value of $\sin 2013^\circ$

How do I find the value of $\sin 2013^\circ$? A precise decimal is not required, but must be expressed with $\sin 30^\circ,$ $\sin 45^\circ,$ and $\sin 60^\circ$ (cosine is also fine). Hint: Use ...
1
vote
1answer
67 views

If n=(sin^2(2x))/4cos^2(x))+1/(sec^2(x)) and x=2.01307, find 2013n^2013

If $n=\dfrac{sin^2(2x)}{4cos^2(x)+\dfrac{1}{sec^2(x)}}$ and $x=2.01307$, find 2013n^2013 Your edits are wrong! These are two separate fractions not together!anymore!
23
votes
8answers
2k views

Reasoning that $ \sin2x=2 \sin x \cos x$

In mathcounts teacher told us to use the formula $ \sin2x=2 \sin x \cos x$. What's the math behind this formula that made it true? Can someone explain?
1
vote
2answers
415 views

Sums $\sum_{k=1}^n \sin(2k-1)\theta$, $\sum_{k=1}^n \sin^2(2k-1)\theta $

To prove: $1.$ $$\sum_{k=1}^n \sin(2k-1)\theta = \frac{\sin^2 n\theta}{\sin \theta}.$$ $2.$ $$\sum_{k=1}^n \sin^2(2k-1)\theta = \frac{n}{2} - \frac{\sin 4n\theta}{4\sin 2\theta}.$$
2
votes
1answer
59 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
276 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
602 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
2answers
252 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
354 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
483 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
111 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
1k 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
701 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
153 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
406 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
421 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) = ...
5
votes
2answers
321 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
291 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
593 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
673 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 $\sin x$, $\sin y$, $\sin z$, ...