Challenge Definition and 911 Threads

Here is a thread of challenges made especially for high school students and first year university students. All the following problems can be solved with algebra, trigonometry, analytic geometry, precalculus and single-variable calculus. That does not mean that the question are all easy. For...

Summer, July, hot weather: every reason is good enough for some new challenge questions. NEW: ranking can be found here: https://www.physicsforums.com/threads/micromass-big-challenge-ranking.879070/ For high school and first year university students, there is a special challenge thread for you...

List of challenges: Integral Challenge: https://www.physicsforums.com/threads/micromass-big-integral-challenge.867904/ Counterexample Challenge: https://www.physicsforums.com/threads/micromass-big-counterexample-challenge.869194/ Counterexample Challenge 2...

Given that $a,\,b$ and $c$ are positive real numbers. Prove that $$\frac{a^3+b^3+c^3}{3abc}+\frac{8abc}{(a+b)(b+c)(c+a)}\ge 2$$.

Given that $x,\,y$ and $z$ are positive real numbers such that $xy + yz + zx = 3xyz.$ Prove that $x^2y + y^2z + z^2x\ge 2(x + y + z) − 3$.

In this thread, I present a few challenging problems from all kinds of mathematical disciplines. RULES: In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. It is fine to use nontrivial results without proof as long as you cite...

Prove that $m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$ is never equal to 33.

I recently decided to take a whack at this problem. Came up with an interesting approach, thought it would make a good conversation topic. Anyone else tried to do this? What were your results?

Given that $$\frac{\sin 3x}{\sin x}=\frac{6}{5}$$, what is the ratio of $$\frac{\sin 5x}{\sin x}$$?

Find $$\left\lfloor{S}\right\rfloor$$ if $$S=1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{80}}$$.

Let $a,\,b$ and $c$ be positive real numbers, prove that $$\frac{a}{2a+b+c}+\frac{b}{a+2b+c}+\frac{c}{a+b+2c}\le \frac{3}{4}$$.

We had integrals, so we have to have series as well. Here are 10 easy to difficult series and infinite products. Up to you to find out the exact sum. Rules: The answer must be a finite expression. The only expressions allowed are integers written in base 10, the elementary arithmetic...

Factorize $x^4+y^4+z^4-xyz(x+y+z)$.

For positive reals $a,\,b,\,c$, prove that $$\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}\gt 2$$.

Let's do some simulations. Below are 5 simulation problems that require a computer to solve. Use any language you want. Post the answers (including graphics) and code here! Any use of outside sources is allowed, but do not look up the question directly. For example, it is ok to go check...

Let the real $x\in \left(0,\,\dfrac{\pi}{2}\right)$, prove that $\dfrac{\sin^3 x}{5}+\dfrac{\cos^3 x}{12}≥ \dfrac{1}{13}$.

Prove that $x^2 + y^2+ z^2\le xyz + 2$ where the reals $x,\,y,\, z\in [0,1]$.

Let $f(x) = Ax^2 + Bx +C$ where A,B,C are real numbers. prove that if $f(x)$ is integer for all integers x then $2A, A + B, C$ are integers. prove the converse as well.

If we're having a thread about probability theory, then we must have one on statistics too! The following questions are all very open-ended and thus multiple answers may seem possible. Your goal is to find a strategy to find the answer to the questions. Furthermore, you must provide some kind of...

Probability theory is very nice. It contains many questions which are very easy to state, but not so easily solved. Let's see if you can solve these questions. For an answer to count, not only the answer must be given but also a detailed explanation. Any use of outside sources is allowed, but...

Homework Statement One side of the roof of a house slopes up at 37.0°. A roofer kicks a round, flat rock that has been thrown onto the roof by a neighborhood child. The rock slides straight up the incline with an initial speed of 15.0 m/s. The coefficient of kinetic friction between the rock...

Original source: https://twitter.com/DrReneHeller/status/724935476327624704 Instructions (copied and pasted from original source): This is a call for a fun scientific challenge. Suppose a telescope on Earth receives a series of pulses from a fixed, unresolved source beyond the solar system...

Homework Statement Homework Equations 1/do+1/di=1/f The Attempt at a Solution I tried finding the distance of image at 27m and then at 30.5m and taking the difference but that didn't work.

And we continue our parade of counterexamples! Most of them are again in the field of real analysis, but I put some other stuff in there as well. This time the format is a bit different. We present 10 statements that are all of the nature ##P## if and only if ##Q##. As it turns out, only one of...

Well, the last thread of counterexamples was pretty fun. So why not do it again! Again, I present you a list with 10 mathematical statements. The only rub now is that only ##9## are false, thus one of the statements is true. Provide a counterexample to the false statements and a proof for the...

I adore counterexamples. They're one of the most beautiful things about math: a clevery found ugly counterexample to a plausible claim. Below I have listed 10 statements about basic analysis which are all false. Your job is to find the correct counterexample. Some are easy, some are not so easy...

Let $\alpha,\,\beta$ and $\gamma$ be the interior angles of an acute triangle. Prove that if $\alpha \lt \beta \lt \gamma$, then $\sin 2\alpha \gt \sin 2\beta \gt \sin 2\gamma$.

There are many apparent paradoxes in quantum mechanics. Luckily, a careful application of math reveals that all is well. But can you figure out why the following ##7## challenges are not paradoxes? Rules: Do not look at paper [1] before answering. It contains all the answers in detail. Any...

I'm up against this Laplace transform integral: $$F(s) ~:=~ \int^\infty_0 \exp\left( -sx + \frac{i\omega}{1+\lambda x} \right) \, dx $$where ##s## is complex, ##\omega## is a real constant, and ##\lambda## is a positive real constant. By inspection, I think it should converge, at least for some...

Great thread!: https://www.physicsforums.com/threads/micromass-big-integral-challenge.867904/#post-5450007 I have a 16GB i7 windows 7 box with Firefox v45.0.2. I have NO add-ons to Firefox. Every time I open the one thread, it takes longer and longer - literally a minute the last time. No...

Integrals are pretty interesting, and there are a lot of different methods to solve them. In this thread, I will give as a challenge 10 integrals. Here are the rules: For a solution to count, the answer must not only be correct, but a detailed solution must also be given. A correct answer...

Let $ABCD$ be a convex quadrilateral such that $AB=BC,\,AC=BD,\,\angle CBD=20^{\circ},\,\angle ABD=80^{\circ}$. Find $\angle BCD.$

Prove that $\arg[(a+bi)(c+di)]=\arg(a+bi)+\arg(c+di)$.

In a triangle $ABC$ with side lengths $a,\,b$ and $c$, it's given that $17a^2+b^2+9c^2=2ab+24ac$. Evaluate $\cos \angle B$.

Prove, with no knowledge of the decimal value of $\pi$ should be assumed or used that $$1\lt \int_{3}^{5} \frac{1}{\sqrt{-x^2+8x-12}}\,dx \lt \frac{2\sqrt{3}}{3}$$.

Homework Statement Find i and v_x Given answer: 2. Homework Equations 3. The Attempt at a Solution I have attempted the problem in two different manners, and I consistently reach a different answer. I would appreciate if someone would be willing to take the time to attempt the problem and...

Explain the physical process of the tongue action of a dog drinking water. State yr answer then watch the clip to see if you got it correct. Secret Life of Dogs: Alsatian dog drinking water …:

Maximize $\sin x \cos y+\sin y \cos z+\sin z \cos x$ for all real $x,\,y$ and $z$.

Prove $$\frac{a-\sqrt{bc}}{a+2b+2c}+\frac{b-\sqrt{ca}}{b+2c+2a}+\frac{c-\sqrt{ab}}{c+2a+2b}\ge 0$$ holds for all positive real $a,\,b$ and $c$.

The CMS or cosmological background permates the whole universe. It's mentioned so often in study halls and on documentaries that NOTHING can peak further back into the universe evolution beyond the CMS. Just want to hear all of your ideas on how YOU would solve this problem if you were on this...

$n\in N,n\geq 2$ prove: $ \sum_{1}^{n}(\dfrac{1}{2n-1}-\dfrac{1}{2n})>\dfrac {2n}{3n+1}$

Factorize $\cos^2 x+\cos^2 2x+\cos^2 3x+\cos 2x+\cos 4x+\cos 6x$.

Find the number of distinct real solutions of the equation $(x − 1)(x − 3)(x − 5) · · · (x − 2017) = (x − 2)(x − 4)(x − 6) · · · (x − 2016)$.

$x$ and $y$ are two real numbers such that $x^3-y^3=2$ and $x^5-y^5\ge 4$. Prove that $x^2+y^2\gt 2$.

Let $a\,b$ and $c$ be the sides of a triangle. Prove that $$\frac{a}{b+c-a}+\frac{b}{a+c-b}+\frac{c}{a+b-c}\ge 3$$.

Prove $$\frac{ab}{a+b+ab}+\frac{bc}{b+c+bc}+\frac{ca}{c+a+ca}\le \frac{a^2+b^2+c^2+6}{9}$$ for all positive real $a,\,b$ and $c$ and $a,\,b,\,c\ne 0$.

Let $1\lt a \lt 2$, $a$ is a root of the equation $x^5-x-2=0$. Prove that $\large a>\sqrt[9]{8}$.

Let $a,\,b,\,c$ and $d$ be positive real numbers such that $(a^2+b^2)^3=c^2+d^2$, prove that $\dfrac{a^3}{c}+\dfrac{b^3}{d}\ge 1$.

Let $a,\,b,\,c$ be real numbers such that $a\ge b\ge c>0$. Prove that $$\frac{a^2-b^2}{c}+\frac{c^2-b^2}{a}+\frac{a^2-c^2}{b}\ge 3a-4b+c$$.