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fresh_42

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Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other intermediate level math challenge thread!

1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month.

2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

3) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

4) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

5) Mentors, advisors and homework helpers are kindly requested not to post solutions, not even in spoiler tags, for the challenge problems, until 16th of each month. This gives the opportunity to other people including but not limited to students to feel more comfortable in dealing with / solving the challenge problems. In case of an inadvertent posting of a solution the post will be deleted by @fresh_42

**RULES:**1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month.

2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

3) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

4) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

5) Mentors, advisors and homework helpers are kindly requested not to post solutions, not even in spoiler tags, for the challenge problems, until 16th of each month. This gives the opportunity to other people including but not limited to students to feel more comfortable in dealing with / solving the challenge problems. In case of an inadvertent posting of a solution the post will be deleted by @fresh_42

**QUESTIONS:**- (solved by @Mr Davis 97 ) Show that ##\arctan \sqrt \frac{1 - x}{1 + x} + \frac{1}{2} \arcsin x = \frac{\pi}{4}##, for all ##x \in (-1, 1)##
**(by**@QuantumQuest**)**

- (solved by @mfb ) You want to drive a car around a one way circular track. The car uses a constant amount of diesel per mile driven. There are N diesel canisters mischievously placed around the track. Summing over all gas canisters there is enough fuel to make one full trip /cycle around the track. Your car initially has no gas. Does there always exist a starting position (i.e. at one of the N gas canisters) that you may choose such that you can complete one full trip around the track?
**(by**@StoneTemplePython**)**

- (solved by @ehild ) The angle bisector in a given triangle ##\bigtriangleup ABC## of the angle ##\alpha = \sphericalangle BAC## at ##A## intersects the side ##\overline{BC}## at the point ##D##. We have the information:

\begin{align*}

\overline{BD}\cdot \overline{CD}&= \overline{AD}\,^2 \\

\sphericalangle ADB &= 45°

\end{align*}

a) Determine the inner angles of ##\bigtriangleup ABC##.

b) Determine the precise ratio at which ##D## divides ##\overline{BC}##

**(by**@fresh_42**)**

- (solved by @ehild ) The beam from a lighthouse ##3## miles from a straight coastline turns at the rate of ##5## revolutions per minute. How fast is the point ##P## at which the beam hits the shore moving when that point is ##4## miles from point ##A## on the shore directly opposite the lighthouse?
**(by**@QuantumQuest**)**

- (solved by @Mr Davis 97 ) Compute the arc length ##\mathcal{L}## of the cycloid

$$\gamma\, : \,\mathbb{R} \longrightarrow \mathbb{R}^2\, , \,\gamma(t)=(t-\sin(t),1-\cos(t))$$

between two neighboring singularities.**(by**@fresh_42**)**

- (solved by @Mr Davis 97 )Let ##a## be an integer and ##b## be also an integer which is formed from ##a## by reversing the order of its digits.

Show that ##a \equiv b \mod 9\,##.**(by**@QuantumQuest**)**

- An urn contains balls, each of which has one of the following colors: red, green, blue and yellow balls. Balls are sampled randomly with replacement. Let r, g, b, and y represent the probabilities of drawing a red, green, blue or yellow.

a) (solved by @Math_QED ) what is the expected number of balls chosen before obtaining the first yellow ball?

b) (solved by @PeroK ) what is the expected number of different colors of balls obtained before getting the first yellow ball?

**(by**@StoneTemplePython**)**

- Let ##f## be a differentiable function in ##\mathbb{R}##. If ##f'## is invertible and ##(f')^{-1}## is differentiable in ##\mathbb{R}##, show that ##[I_A (f')^{-1} - f \circ [(f')^{-1} ]]' = (f')^{-1}## where ##I_A## with ##I_A(x) = x## is the identity function ##I_A : A \to A##

**(by**@QuantumQuest**)**

- (solved by @lpetrich ) At the cash desk of a shopping center five friends (Diana, Ike, Jessica, Stan, Valery) are standing in a row. They are all different in age (26, 27, 30, 33 and 35 years) and would like to buy all different tops (shirt, polo shirt, pullover, sweatshirt and T-shirt) for themselves. The tops are all different colors (blue, yellow, green, red and black) and different sizes (XS, S, M, L and XL).

Find out who is where, how old and what top to buy in which color and size. The positions in the queue can be seen from the cashier, i.e. "front" or "the first person" is right at the cash register. There are no other people in the queue and the cashier is to be ignored.

A. Diana, who wants to buy a top in size XL, stands further ahead than the person who wants to buy a black top.

B. Jessica stands in front of the person who wants to buy a polo shirt.

C. The second person in the queue wants to buy a yellow top.

D. The t-shirt is not red.

E. Stan wants to buy a sweatshirt. The person standing in front of him is older than the person standing directly behind him.

F. Ike needs a top in size L.

G. The last person in the queue is 30 years old.

H. The oldest person wants to buy the top in the smallest size.

I. The person standing directly behind Valery wants to buy a red top that is larger than size S.

J. The youngest person wants to buy a yellow top.

K. Jessica wants to buy a shirt.

L. The third person in the queue wants to buy a top in size M.

M. The polo shirt is red, yellow or green.

**(by**@fresh_42**)**

- (solved by @lpetrich ) Suppose we have two ##\text{ n x n }## commuting matrices, ##\mathbf X## and ##\mathbf Y##.

##\mathbf X## is special because there exists some positive integer ##k## where ##\mathbf X^k = \mathbf I##. On the other hand, ##\mathbf Y## is nilpotent.

a) Prove that ##\mathbf Z := \big(\mathbf X + \mathbf Y\big)## is nonsingular, and

b) compute the ##\mathbf Z^{-1}##.

**(by**@StoneTemplePython**)**

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