POTD « Undernetmath’s Weblog

POTD

This page contains collection of problems that have been suggested, solved, discussed in the channel by various chatters. While in rare cases it is possible, that a problem was stated for the first time in this channel, it is usually safe to assume, that it might have been published on the internet or in literature before. If you have a problem you consider a good problem of the day (potd) please feel free to suggest it to a channel operator or post directly to this blog (see the About page).

Current Problem of the Day

2009-12-3 by Indig0696
Long John has captured a treasure map from Adam Mcbones. Adam has buried the treasure at the point (x,y) with integer co-ordinates(not necessarily positive). He has indicated on the map the values of x^2+y and x+y^2, and these numbers are distinct. Prove that Long John has to dig only in one place to find the treasure.

2009-11-19 by kmh
In a bag there are 6 red and 8 green marbles. 5 marbles are drawn randonlmy and are put in a red box. the remaining marbles are put in a green box. Determine the probability that the sum of the green marbles in the red box and the red marbles in the green box is not a prime number.

2009-10-31 Halloween Calculus Candy by mimis
Show that for all $x>0$ you have $\frac{1}{x+1}<ln(\frac{x+1}{x})<\frac{1}{x}$ .

2009-10-29 Of Lotteries and Fairies by kmh
Peter is playing a 6 from 49 numbers lottery every week, but he has never won anything so far. This time around however while he is musing which numbers to pick a good fairy appears and offers him one wish. Peter seizes the opportunity and asks for winning lottery numbers of the next week. The fairy remarks, that telling the lottery numbers would mean to grant six wishes rather than one and declines the request. She offers to tell Peter the sum of the winning numbers instead. Peter thinks for a moment and replies, that this would still allow thousands of possibilities. The Fairy agrees and offers him an additional hint: “If you multiply the sum with the number of lotto combinations yielding this sum, then you get a number which is roughly 1 million. Furthermore this number is the exact result of the product of all winning numbers”. After the fairy has vanished, Peter sits down and thinks for a very very very long time. Finally he writess six numbers on his lottery ticket and wins the following week. Which numbers did Peter pick?

2009-10-24 Probability Snack by kmh
Two archers A and B are playing the following game. They each take turns in shooting at a target, the first archer who hits the bull’s eye wins.
a) Let’s assume each archer hits the bull’s eye with a probability $p=0.5$ and archer A gets the first turn. Determine the probability that A will win the game.
b) Let’s assume archer A hits the bull’s eye with probability $p_1=0.4$ and still has the first turn as before. What probability $p_2$ for his aim does archer B need to make sure that the game is fair?

2009-10-11 Right All Around by Karlo
Is there a (non-degenerate) tetrahedron whose faces are all right triangles?

2009-10-5 by kmh
Compute $\displaystyle{\lim_{x\to 0} \frac{\sqrt{x^2+p^2}-p}{\sqrt{x^2+q^2}-q}}$ .
a) with $p,q\in\mathbb{R}^{+}$
b) with $p,q\in\mathbb{R}$

2009-7-26 by yourshadow
Prove that
$\frac{1}{1+x_1} + \frac{1}{1+x_1+x_2} + ... + \frac{1}{1 + x_1 + x_2 + ... + x_n} < \sqrt{\frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n}}$
for all positive reals $x_1, x_2, ..., x_n$

2009-7-21 by mocker3
Consider a triangle $\triangle ABC$ with its circumcircle $c$ . Let $P$ be an arbitrary point on $\overline{AB}$ , the line $CP$ intersects $c$ in $Q$ . Show that the follwoing inequality holds:
$\frac{|\overline{PQ}|}{|\overline{CQ}|} \leq \left(\frac{|\overline{AB}|}{|\overline{AC}|+|\overline{BC}|} \right)^2$

2009-7-14 by kmh
Consider a pentagon with an integer number placed at each vertex and the requirement that the sum of those 5 integers is positive. Perform the following procedure: If there are negative integers, then pick an arbitrary one of them and add its value to the 2 adjacent numbers and change its sign afterwards. For example you have the numbers a, b, c, d, e (in sequence) and b is negative then you get the new numbers a+b, -b, c+b, d, e. Now you repeat this step for as long as you have negative integers. Prove that this procedure will end, i.e. after a finite amount of steps all numbers will be nonnegative.

2009-7-7 by kmh
Lines $a$ and $b$ intersect in $S$ . Let $P$ be an arbitrary point, which is not on $a$ or $b$ . Let $A$ be the foot of the perpendicular from $P$ on $a$ and $B$ the foot of the perpendicular from $P$ on $b$ . Let $h$ be the perpendicular from $S$ on $\overline{AB}$ with $C$ as the foot point $\overline{AB}$ . Show that $\angle PSB = \angle ASC$ .

2009-6-30 by kmh
Let $c$ be circle with center $M$ and radius $r$ . Let $A$ be a point within the interior of $c$ with the distance $d$ from $M$ . Let $P$ be point on $c$ such that the angle $\angle MPA$ is maximal.
a) Devise a method to construct the point(s) $P$ with ruler and compass only, when $c$ , $M$ and $A$ are given.
b) Express the maximal angle $\angle MPA$ as a function of $r$ and $d$ .

2009-6-24 by kmh
The triangle with sides of 3, 4 and 5 length units has an area, that is an integer value as well. Are there any other triangles that property, i.e. that have consecutive integers as side lengths and an integer area? If so how many?
(original source: de.rec.denksport)

2009-6-11 by karlo
Consider the sequence of equations in 26 variables:
0 = Z + E + R + O
1 = O + N + E
2 = T + W + O
3 = T + H + R + E + E
…

What is the largest initial sequence that can be simultaneously solved :
(a) in the reals?
(b) In the integers?

2009-5-19 by karlo
ABC is a triangle; M is the midpoint of BC. Points X and Y are on side AC such that the lines BX and BY divide the median AM into three equal parts. XY has length 3. What’s the length of AC?

2009-5-13 by kmh
The power series $\sum_0^\infty a_nx^n$ has a radius of convergence R. Now consider power series $\sum_0^\infty a_{3n}x^n$ . What can you say about its convergence?

2009-5-1 Shortest Roadwork by Karlo
Four towns are located at the corners of a square, 4 km to a side. They wish to build some roads such that it’s possible to reach any town from any other town. They only have enough funding to build a total of 11 km of road. Can they succeed?

2009-4-24 Circle Wanted by kmh
Consider 3 arbitrary points in the plane. Determine the smallest the disc, that contains all 3 of them.

2009-4-8 A QUEER COINCIDENCE by Pisagor
Seven men, whose names were Adams, Baker, Carter, Dobson, Edwards,
Francis, and Gudgeon, were recently engaged in play. The name of the
particular game is of no consequence. They had agreed that whenever a
player won a game he should double the money of each of the other
players–that is, he was to give the players just as much money as they
had already in their pockets. They played seven games, and, strange to
say, each won a game in turn, in the order in which their names are
given. But a more curious coincidence is this–that when they had
finished play each of the seven men had exactly the same amount–two
shillings and eightpence–in his pocket. The puzzle is to find out how
much money each man had with him before he sat down to play.
(original source: Amusements in Mathematics by H.E. Dudeney (1917))

2009-3-31 Something Japanese by kmh
Consider a concyclic quadrangle $\square ABCD$ with its diagonals $\overline{AC}$ and $\overline{BD}$ . Show that the 4 centers of the incircles of the triangles $\triangle ABD$ , $\triangle ABC$ , $\triangle CDA$ and $\triangle CDB$ form a rectangle.

2009-3-26 Thermal Vortex by Karlo
Fahrenheit and Celsius temperature scales are related by 5(F-32)=9C. A temperature of 527 °F corresponds to 275 °C, in which the first digit has been rotated to the end. What is the next larger integer for which this happens?

2009-3-17 Ides of March by Karlo
Assume that Caesar’s last gasp contained one billion air molecules, which have since been randomly distributed around the world. Assume that there’s enough air in the world for a billion gasps. What’s the probability that my next gasp shares at least one molecule with Caesar’s last gasp? (a) 0-1%, (b) 1-49% , (c) 49-51% , (d) 51-99%, (e) 99-100%

2009-2-26 by Heinrich
Determine the antiderivative of $f(x)=\frac{(1+x)e^{-x}}{x^2}$

2009-2-7 power tower by karlo
For what values of x does the infinite exponentiation tower $x^{x^{x^{\ldots}}}$ converge?

2009-2-3 by wakaka
Let p be an odd prime number and $A:=\{1,2,3,\ldots,2p\}$ How many p-element subsets of A are there, such that the sum of their elements is multiple of p?

2009-1-28 by kmh
Let $\triangle ABC$ be a triangle with $|AB|=7cm$ , $|AC|=6cm$ , $|BC|=5cm$ . Determine a triangle $\triangle UVW$ , such that $U$ lies on $\overline{AB}$ , $V$ lies on $\overline{AC}$ and $W$ lies on $\overline{BC}$ ( $U\in \overline{AB},\, V\in \overline{AC},\, W\in \overline{BC}$ ) with the perimeter of $\triangle UVW$ being minimal.

2009-1-25 weather wonders by karlo
Prove that there exists a pair of antipodal points on the Earth’s surface that have exactly the same temperature and exactly the same humidity. (You may assume that both temperature and humidity are continuous functions.)

2009-1-22 by pisagor
A fair coin is tossed until 2 heads in sequence appear. How many tosses are needed in average?

2009-1-20 some ode by UniStudent
Determine $y(x)$ such that $x\cdot y(x)^2\cdot y'(x)=x^3+y(x)^3$ and $y(1)=1$ .

2009-1-16 Redistribution of Wealth by Karlo
Andy and Betty are playing a game. They have A and B money, respectively. Each round, the loser of that round must pay min(A,B) to the winner — that is, the player with less money either loses it all or doubles up. They continue playing more rounds until one player is bankrupt. If the two players are equally likely to win each individual round, what is the probability that Andy eventually wins the game?

2009-1-14 by kmh
Consider an arbitrary integer x and an integer y being a arbitrary permutation of the digits of x. Show that their difference x-y can be divided by 9.

2009-1-4 by kmh
Let $c$ be a circle with center $O$ and $P$ be an arbitrary point in the interior of $c$ . Let $\overrightarrow{OP}$ be the ray, that originates in $O$ and passes through P. The perpendicular to $\overrightarrow{OP}$ in P intersects the circle $c$ in $M$ and the tangent on $c$ though $M$ intersects $\overrightarrow{OP}$ in $Q$ . Now let $X$ be an arbitrary point on the circle $c$ . Show that the ratio $\frac{|XP|}{|XQ|}$ is constant.

2008-12-23 by kmh
Consider $n$ points being placed on a circle with any any 2 points being connected by a line. These lines partition the disk into faces/regions. Let $f(n)$ be the maximal numbers of faces that n points can create. Conjecture & proof a formula for $f(n)$ .

2008-12-13 by kmh
Consider some arbitrary parallelogram $ABCE$ and construct square over its sides. Show the the centers of thos 4 squares from a square themselves.
(original source: Coxeter/Greitzer: Geometry Revisited, p 84)

2008-12-11 by kmh
$A,B$ are 2 points with a distance of $7cm$ , $c$ is a circle with $A$ as its center and a radius of $2cm$ and finally $C, D$ are 2 points on the circle $c$ . Determine the maximal area that the triangle $\triangle BCD$ can assume.

2008-12-6 Baywatch Math by kmh
Pamela Anderson swims into a jellyfish close to the beach and faints. She has 10 more seconds left before drowning, when David Hasselhoff spots her. Hasselhoff runs 100m in 12 seconds and swims 100m in 60 seconds. The distance between Hasselhof and Anderson is 48.01m with Anderson being 8m away from the shoreline in the water and Hasselhoff being 20m away from the shoreline on the beach. Can he get to her in time before she drowns?

2008-12-3 by kmh
Consider some arbitrary ellipse $E$ and some point $P$ outside E.
a) Construct the 2 tangents from $P$ on $E$ by using ruler and compass only.
b) Compute the 2 tangents analytically (i.e. as map of the form $y=f(x)$ or $f:t\mapsto (x(t),y(t))$ .

2008-11-25 by lhrrwcc
Let A,B be sets and let U (resp V) be the subring of Real valued functions on A (resp B) st for all f in U(resp V) f>0 => Exists g in U (resp V) st f = g^2
if phi:U->V is a ring homomorphism then for all c in Reals, phi(c)=c

2008-11-19 by kmh
Compute $\displaystyle{\lim_{x\rightarrow 0} \frac{\int_0^x \ln(1+t^2)\,dt}{x^3}}$
Solution

2008-11-18 by bor0
Assume some inequality whose LHS is represented as $L(n)$ and RHS is $R(n)$ . Furthermore assume that $L(n) < R(n)$ is true and that $L(n)$ is the sum of n terms. Now prove that if $L(n+1) - L(n) < R(n+1) - R(n)$ is true, then it follows that $L(n+1) < R(n+1)$ is true as well.

2008-11-4 by lhrrwcc
Let the function $g:R\rightarrow R$ be continuous but not differentiable at $x=0$ with $g(0)=2$ . Show that $f(x)=g(x)\cdot\sin(3x)$ is differentiable at $x=0$ and compute $f^\prime(0)$ .

2008-10-30 by kmh
Show that if $100m+n$ is divisible by $7$ , then $m+4n$ is divisble by $7$ as well.
Solution

2008-10-14 by TurboBee
It takes Mr. Todd 4 hours longer to prepare an order of pies than it takes Mrs. Lovett. They bake together for 2 hours. When Mrs. Lovett leaves, Mr. Todd takes 7 additional hours to complete the work. Working alone, how long does it take Mrs. Lovett to prepare the pies?

Former Problems
POTD 2008
POTD 2007

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