# 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

2017-8-19 Yet Another Geometry Snack by kmh
Determine the length of the curve enclosing two tangent circles of radius 3 and 9 (see drawing).

2017-7-19 Geometry Snack by kmh
Determine the radius of the inscribed circle in terms of $r$ (see drawing).

2017-6-2 Number Theory Snack by kmh
Show that the number $ababab$ with digits $a, b \in \{1,\ldots , 9\}$ is always divisible by 7.

2016-12-28 The Limit that might answer it all by kmh
$\displaystyle \lim_{x\rightarrow\infty}7\cdot \left( \left( \frac{1}{x} \right)^{\frac{1}{x}-1}\cdot \sin\left(\frac{1}{x}\right)+\frac{\tan(\frac{\pi}{4}+\frac{1}{x})-1}{\frac{1}{x}} +\ln\left(\left(1+\frac{3}{x}\right)^x \right)\right)$

2016-9-23 Diagonals and Polygons by karlo
In a regular 18-gon, find four diagonals, none passing through the center, that meet at an interior point.

2016-9-6 Fair Triangle Partition by kmh
Consider a triangle ABC with a point X on AB. Construct a point Y in AC, such that the line segment XY divide the trainle ABC into equal parts. Y should be constructed with ruler and compass only without any supporting calculations.

2016-8-9 The First One of 2016 at last – Symmetry by karlo
If $x + y + z = 0$, and $x^2 + y^2 + z^2 = 14$, what is $x^4 + y^4 + z^4$?

2015-8-1 Canadian IMO problem by kmh
$DEB$ is a chord of a circle such that $DE=3$ and $EB=5$. Let $O$ be the center of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$. Given $EC=1$, find the radius of the circle.

2015-7-28 by kmh
The three altitudes of a triangle have the followings lengths: 3cm, 4cm and 5cm.
a) Construct the triangle with ruler and compass.
b) Determine the length for all three sides.

2015-7-20 by hershol
Compute $\displaystyle \sum_{k=1}^{2015} \lceil \sqrt{k} \rceil$ without the use of calculators or software.

2015-7-8 by kmh
Determine $\displaystyle \lim_{x\rightarrow 0} \left( \frac{(1+x)^\frac{1}{x}}{e} \right)^\frac{1}{x}$

2015-6-19 by karlo, kmh & musk
Musk’s favoured question: “You’re standing on the surface of the Earth. You walk one mile south, one mile west, and one mile north. You end up exactly where you started. Where are you?”
Hint: Nobody is asking about bears and the color of their furs and there is more than one location.

2015-6-5 by kmh
Let $f(x)=arctan(x)+arccot(x)$ and determine $f(0)+f(1)+f(\sqrt{2})+f(\sqrt{3})$

2015-4-20 by karlo
What is the longest string of nonzero digits that can appear at the end of (a) a square, (b) a power of two?

2015-4-15 Singaporean Logic by kmh
Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.

May 15 May 16 May 19
June 17 June 18
July 14 July 16
August 14 August 15 August 17

Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively.

Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.
Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.
Albert: Then I also know when Cheryl’s birthday is.

So when is Cheryl’s birthday?

2015-4-10 by kmh
Consider an equilateral triangle ABC and a point P in its interior. Show that d(P,AB)+d(P,BC)+d(P,AC)=3r, where r is the radius of the incircle and d(P,x) denotes the distance between point P and side x.

2015-3-20 by kmh
Construct a parallelogram ABCD with diagonals AC=6.8 cm, BD=4.8 cm and an angle in A of 65 degrees using only compass, ruler and protractor (“Geodreieck”).

2015-2-22 Some Basic Geometry by kmh
a) Consider a line $g$ and a point $P$ with a distance of 4 cm to the line $g$. Construct with ruler and compass only all circles with a radius of 6 cm that go through $P$ and have $g$ as a tangent.
b) Construct a $30^\circ$ angle with ruler and compass utilizing the theorem of Thales.

2014-11-30 Spiegel Online Geometry Snack by kmh
Determine the radius of the half circle in the graphic on the right.

2014-8-12 Back with Power by karlo
Determine the last 2 digits of $\displaystyle 2^{3^{4^{5^{6^{7^{8^9}}}}}}$.

2014-4-20 Integer Easter Egg Snack by kmh
Show that $\tfrac{1}{n+1} \binom{2n}{n}$ is an integer for all $n \in \mathbb{N}$

2014-3-27 Of Angles and Triangles inspired by klong
Consider a right angle triangle ABC with a right able in A. Furthermore for its sides the following identity holds: a+b=3c. Determine all angles of the triangle ABC.

2014-3-12 Of Angles and Quadrangles by kmh
Determine all angles of the quadrilateral ABCD (see graphic on the right), given that the angle in E is $98.87^o$ and angle in M $141.91^o$.

2014-2-21 Of Circles and Quadrangles by kmh
Construct with ruler and compass a quadrangle, which is possesses an incircle and a circumcircle but that is not a square. In other words its vertices are placed on one circle and its edges are tangents to another.

2014-2-17 A Touch of 3n+1 by karlo (taken from NCS Team Contest, Nov. 2013)
A sequence is defined by $a_{n+1} = a_n + 2013$ if $a_n$ is odd, or $a_{n+1} = a_n / 2$ if $a_n$ is even. Prove that for any starting value, the sequence is eventually cyclic.

2014-2-3 Stolen from freenode by kmh
Determine $\displaystyle{\lim_{p \rightarrow \infty} \int_{p-7}^{p+9} \left( \frac{2x+1}{2x}\right)^x \, dx}$

2014-1-27 New Year’s First by kmh
Consider a parallelogram ABCD with $|AC|=6.8cm$, $|BD|=4.8cm$ and $\alpha=65^o$.
a) Compute all angles and sides of the parallelogram.
b) Construct the parallelogram with ruler and compass only and consider the angle $\alpha$ seeded. Or alternatively use a protractor in addition to ruler and compass, but do not compute any parts or use the results from a).

Determine $\displaystyle \int_0^2 \frac{\ln(x+1)}{x^2-x+1} dx$

2013-12-17 You got that right, it’s still a limit by Y0uRShAD0W
Determine the limit:
$\displaystyle \lim_{n \rightarrow \infty} \prod_{k=1}^n \left(1+\ln\left(\frac{k+\sqrt{k^2+n^2}}{n}\right)^{\frac{1}{n}}\right)$

2013-12-8 It’s a Limit again by kmh
Determine the limit:
$\displaystyle \lim_{x \rightarrow \infty} \sqrt[3]{x^3+x^2} - \sqrt[3]{x^3-x^2}$

2013-11-30 Yet another Limit by kmh
Determine the limit:
$\displaystyle \lim_{x \rightarrow \infty} x^{\frac{3}{2}}(\sqrt{x+1}+\sqrt{x-1}-2\sqrt{x})$

2013-11-24 Series Snack by kmh
Show that $\displaystyle \sum_{n=1}^\infty \frac{1}{n^3}\leq \frac{5}{4}$.

2013-11-17 Angles in Triangles by kmh
In the drawing below, you have $|CD|=|BD|$. Determine the angles $\alpha$, $\beta$ and $\delta$.

2013-8-27 Weather: ’tis no blur in the mind by Karlo
Two innumerate weathermen are arguing about the weather report. The one who uses Fahrenheit says that it’s twice as warm as yesterday. The one who uses Celsius says that it’s half as cold as yesterday. What is the temperature?

Compute the product of two complex numbers $(a+b \cdot i) \cdot (c+d \cdot i)$ with only three real multiplications and five real additions.

2013-7-30 Vexing Convexities by karlo
a) Prove that for any triangle, the largest rectangle contained in it has $\tfrac{1}{2}$ the area of the triangle.
b) Find a positive constant c such that every convex region of area 1 contains a rectangle of area c.

2013-5-22 Stolen from a T-shirt by kmh
What’s going on here?
$\sqrt{-1}\,2^3\,\sum\, \pi$ and it was delicious.

Prove that the last two digits the power tower $\displaystyle 7^{7^{7^{...}}}$ are 43, that is any element of sequence $\displaystyle 7^7, 7^{7^7}, \ldots$ has 43 as its last two digits.

2013-5-16 Improper Integrals by kmh
Determine the following improper integrals:
a)$\displaystyle \int_0^\infty \frac{x}{e^x-1} dx$
b)$\displaystyle \int_0^\infty \frac{x^3}{e^x-1} dx$

2013-4-11 Inscribed Trapezoid of Maximal Area by Moonies

In the diagram to the left the areas of triangle ABC is 1. Trapezoid DEFG is constructed so that G is to the left of F, DE is parallel to BC, EF is parallel to AB and DG is parallel to AC. Determine the maximal possible area of the trapezoid DEFG.

2013-3-27 Treasure Map by kmh

An old treasure map contains the following description:

From the gallows walk towards the old oak west of it. At the oak turn to the left in a right angle and walk the same distance into the new direction to arrive at a point g’. Now move back  to gallows and walk from it to the pine tree east of it. At the pine tree turn to the right in a right angle and walk the same distance into the new direction to arrive at a point g”. The treasure is buried at the midpoint of g’and g”.

When Jack Sparrow arrives at scene however the gallows has vanished and only the oak the pine tree still remain. Can he recover the treasure nevertheless?

Compute $\displaystyle \lim_{n \to \infty} \sin(2\pi\sqrt[3]{n^3-n^2+1})$.

2013-2-11 High school math contest question by Karlo & HisShadow
Suppose $f: R \to R^+$ satisfies $f(x) \ln(f(x)) = e^x$.
Compute $\displaystyle \lim_{x \to \infty} \left(1 + \frac{\ln(x)}{f(x)}\right)^{\frac{f(x)}{x}}$.

2013-2-3 by kmh
Consider a triangle ABC with vertex A at (0,0), vertex B on the x-axis and vertex C on the line y=3x. In addition the side BC passes through the point (1,1) and all vertices are located in the first quadrant.

a) Determine the triangle with the minimal area.
b) Determine the triangle with the minimal circumference.
c) Determine the triangle with the minimal side length for BC.
d) Find a “geometrical solution” for a), b) and c).

2013-1-5 by kmh
Show $(\sin \theta)^p\leq\sin(p\theta)$ for $0 < \theta \leq \pi /2,\,0< p< 1$

2012-12-16 Calculus snack by kmh
Determine a power series for $\displaystyle f(x)=\frac{x^2+x}{(1-x)^3}$.

2012-12-8 Maximally Scalene Triangles by Karlo
In any triangle, there are two sides whose ratio is at most s. What is the smallest value of s for which this is true?

2012-12-6 by boro
Prove that (3, 5, 7) are the only 3 consecutive odd numbers that are prime.

Determine $\displaystyle \int_0^\infty \frac{1}{(1+x^2)(1+x^e)} dx$

2012-11-5 by karlo & kmh
A duck is swimming in the center of a circular lake and there is a fox at the shore of the lake. The duck wants to leave the lake, but it cannot take off into the air from the water but only from land, so to leave the lake it needs to reach the shore without being caught by the fox.

a) Can the duck leave the lake, when the fox runs 4 times as fast as the duck swims?
b) Determine the lowest upper bound for the fox’s speed, that still allows the duck to escape.

2012-11-5 by kmh
Consider three identical cones with a height of 12 cm and a radius of 5 cm. They are places in such a way that their base circles are touching. Now a sphere is placed between the cones, such that it touches the three cones and its highest point has the same altitude as the vertices of the cones. Determine the radius of that sphere.

2012-9-30 by kmh
Let ABCD be a parallogram and AC its diagonal. Let E be a point on AC and the line through BE intersecting AD in F and the extension of CD in G. Determine the length of FG given that |BE|=24 and |EF|=18.

2012-5-26 by karlo
Suppose $x$ satisfies the cubic $x^3 + ax^2 + bx + c = 0$, and $y = x^2$ satisfies $y^3 + Ay^2 + By + C = 0$. What are $(A,B,C)$ in terms of $(a,b,c)$?

2012-5-3 by wutawutawhat
For the quadrangle ABCD (see drawing) you have BE=3, AB=BC and right angles ABC, BEA, CDA. Determine its area.

Show that:
$\displaystyle{\frac{\pi}{5}\leq\int_0^1 x^x dx\leq \frac{\pi}{4}}$

2012-4-27 Interview question from Hell? by Karlo
You have a source of independent random numbers distributed uniformly on (0,1]. You continue drawing numbers from this source until their sum exceeds 1. What’s the expected number of draws required?

2012-4-23 Tricky Series by Y0UrShAD0W & Karlo
Determine:
a) $\displaystyle \sum_{k=1}^\infty \frac{\sin(k)}{k}$
b) $\displaystyle \sum_{k=1}^\infty (-1)^{k+1} \frac{\sin(k)}{k}$

2012-4-5 Constant in Circles  by Y0UrShAD0W & kmh
Consider a square $\square ABCD$ with an inscribed circle of radius 1. Let $P$ be a point on the circle.

a) Prove $|PA|^2\cdot|PC|^2+|PB|^2\cdot|PD|^2=10$
b) Prove $|PA|^2+|PC|^2+|PB|^2+|PD|^2=12$

2012-4-2 Ruler Geometry by kmh
Consider a circle k with center C and straight line g through C. Now let P be an arbitrary point outside the circle k. Construct the perpendicular line from P on g using only a ruler. The ruler has no tick marks, i. e. it cannot measure distances but only draw straight lines.

2012-3-31 A Funny Inequality by Y0UrShAD0W
Prove the following inequality:
$\log_{15}(25)\le 15^{\tfrac{1}{5}}$

2012-2-29 Leap of Logic by Karlo
“My wife and I were married on February 29th, and we’ve had three children since then”, said my host. “We only celebrate our anniversary when the date actually occurs, so this is only our fifth party. Usually, I ask visiting mathematicians to figure the ages of our children given the sum and product — but since Professor Miller failed earlier tonight, and Professor Klein failed at our last party, I suppose I’ll let you off the hook.”

“Oh, there’s no need for that,” I replied. “You’ve already given me all the information I need.”

How old are the children?

2012-1-22 Touching Triangles by kmh
Consider an arbitrary triangle $\triangle ABC$ with its circumcircle $k$. Let $P$ be an arbitrary point on $k$ and $t$ be the tangent through $P$ on $k$. Reflect $t$ on each of the triangle sides to get three new lines $t_a$, $t_b$ and $t_c$. Show that the circumcircle of the triangle formed by $t_a$, $t_b$ and $t_c$ is tangent to $k$.

2012-1-3 May your 2012 be the greatest! by Karlo
Choose a positive integer $n$, and positive integers $x_1, x_2, ..., x_n$ satisfying $x_1 + x_2 + ... + x_n = 2012$, so as to maximize the product $x_1 x_2 ... x_n$.

2011-12-8 Putnam Problem by Karlo
Define a growing spiral in the plane to be a set of points with integer coordinates $P_0=(0,0), P_1, ..., P_n$ such that $n \ge 2$, the directed line segments $P_{0}P_{1}, P_{1}P_{2}$, … are in the successive coordinate directions east, north, west, south, east, etc., and the lengths of the line segments are positive and strictly increasing.

How many of the points $(x,y)$ with integer coordinates $0 \le x \le 2011, 0 \le y \le 2011$ cannot be the last point $P_n$ of any growing spiral?

Consider a triangle $\triangle ABC$ and let $M$ be the intersection of its medians (centroid). Now consider an arbitrary line through $M$ that intersects the sides $AC$ in $P$ and $BC$ in $Q$.
Show that $A(\triangle PQC)=A(\triangle AQP)+A(\triangle BQP)$ where $A()$ denotes the area of a triangle.

2011-11-14 Sine after Sine by Y0UrShAD0W
Let $0 < u_0 < \pi$, and $u_{n+1} = \sin(u_n)$ for $n \ge 0$.
Calculate $\displaystyle \lim_{n \to \infty} u_n \sqrt{n}$.

2011-10-16 Yet Another Limit by kmh
Determine $\displaystyle \lim_{n\to\infty} \frac{n}{2^n} \sum_{k=1}^n \frac{2^k}{k}$
Solution

2011-10-12 Random Drunks by Y0UrShAD0W & Korolla
At midnight the pub closes and kicks out five drunks. The drunks stumble onto the street and start walking off. Now lets assume the drunks stay on that street and make a step to left or right with a probability $p=\tfrac{1}{2}$.

a) What’s the probability that all five men meet again after five steps?
b) Now you have n drunks, find the probability that they meet again after r steps.

Let h, k be positive integers with k < h.
Determine $\displaystyle \lim_{n\to \infty} \prod_{r=kn+1}^{hn} 1-\frac{r}{n^2}$

Determine $\displaystyle \lim_{n\to \infty} \sum_{k=1}^n \frac{k^2}{(2k)^3+n^3}$
Solution

2011-08-17 Fermat’s First Theorem? by Karlo
Prove that n^x + n^y = n^z has no solutions in positive integers for n > 2.

2011-06-23 Pythagorean Triangles by Karlo
Prove that in any Pythagorean triangle, one of the sides is divisible by 3, one is divisible by 4, and one is divisible by 5. (They need not be different sides.)

2011-06-03 Honoring National Donut Day by Karlo
Cut a donut with one stroke such that the intersection with the surface is

(a) Two disjoint circles.
(b) Two concentric circles.
(c) Two intersecting circles.

[A donut is a torus such as (sqrt(x^2+y^2)-5)^2 + z^2 = 9; the cut is a plane]

2011-5-25 Golden Snack by kmh
Let $a=\tfrac{1+\sqrt{5}}{2}$. Now show that $\displaystyle a = \sum_{k=1}^\infty\frac{1}{a^k}$

2011-4-15 Take Ten by Karlo
Prove that in any set of 10 consecutive integers, there is one that is co-prime to all of the others.

2011-3-11 Triangle Altitudes by kmh
The three altitudes of a triangle are $h_a=5cm$, $h_b=6cm$, $h_a=7cm$.

a) Determine all three side lengths.
b) Give a ruler & compass construction based on the the altitudes.

2011-3-11 Boxing Curve by Karlo
Show that any curve of length 1 can be contained in a rectangle of area 1/4.

2011-2-25 by Karlo
Two integers, x and y, satisfy 1 < x <= y < 100; S knows the sum x+y, P knows the product xy, and each knows the information contained in this sentence.

S says "I know that you don't know the numbers."

P says "I now know the numbers."

S says "I now know the numbers."

What are the numbers?

2011-2-21 by kmh
Determine the line, that divides both rectangles into equal parts.

2011-2-16 A Little Mooning by kmh
The crescent is formed by a white disc overlapping a black disc, such that the circle arc of the white disc passes through the end points of the diameter of the black disc. Determine how much of the black disc is covered by the white disc, if the center of the white disc (C)  lies on the edge of the black disc.

2010-12-31 Last Limits of 2010 by Y0uRShAD0W
Compute the following limits:

a) $\displaystyle \lim_{x\to 0} \frac{\sin(x^n)-\sin(x)^n}{x^{n+2}}$
b) $\displaystyle \sum_{n=0}^\infty \frac{1}{(2n+1)\cdot q^{2n+1}} \qquad$ with $q > 1$
c) $\displaystyle \sum_{n=1}^\infty \frac{1}{n\cdot q^{n}} \qquad$ with $q > 1$

2010-12-31 Series Special by kmh
Compute the following limits:

a) $\displaystyle \sum_{n=1}^\infty \frac{n^4}{2^n}$
b) $\displaystyle \sum_{n=0}^\infty \frac{cos(n)}{3^n}$
c) $\displaystyle \sum_{n=1}^\infty \frac{n^2+7n+11}{n!}$

2010-12-6 Number Theory Quickie by Stumbo
Let $n, k$ be 2 positive integers with $n\geq 2k$.
Show that either $\binom{n}{k}$ or $\binom{n-k}{k}$ is divisble by 2.
Solution

2010-11-20 Math around the Clock by Karlo
On an idealized standard clock, at what time(s) of the day are the hour, minute, and second hand closest to partitioning the clock face into three equal sectors?

(The three angles will be 120°, (120+epsilon)°, and (120-epsilon)°, with epsilon made as small as possible.)

2010-11-10 by Stumbo
A co-ed chess tournament is held, round-robin style (i.e., each player plays each other player exactly once), with the usual scoring system (1 point for a win, 0.5 for a draw, 0 for a loss). Afterwards, it is noticed that each player earned as many total points against male opponents as against female opponents.

Prove that the total number of players is a perfect square.

2010-10-20 by foocraft
Prove or disprove:

Every natural number $n\in \mathbb{N}$ divides some Fibonacci number $F_k, \, k\in \mathbb{N}$

Consider the Fibonacci number $F_n$ with $F_{n+2}=F_{n+1}+F_n$ and $F_0=0$, $F_1=1$.

Now determine $\displaystyle \frac{1}{2}+ \sum_{i=1}^{\infty} \frac{(-1)^{i+1}}{F_{i}\cdot F_{i+1}}$.

2010-8-3 Robber Logistics by karlo
Each of three robbers has a bag of loot, with values $30K,$50K, and \$80K. They’ve come to a river which has a small boat on the bank. The boat can carry two robbers, or one robber and one bag. Although they all want to reach the other side, and would prefer to stay together, none of them can be trusted with excess loot; if any robber or robbers are left alone with more than their original share, they will abandon the original plan and run off with what they have, no matter which side of the river they’re on. How can they cross the river?

2010-7-15 by MLF
The warden has agreed to let his 100 prisoners go free, if each can successfully find his own name from among the 100 boxes (one name in each) that he’s lined up, with each prisoner getting 50 attempts; if any of them fail, then none will go free. The prisoners will not be allowed to communicate with each other after they’ve begun. The warden thinks that their probability of success will be a negligible 2-100, but the prisoners know a strategy that will free them all with probability at least 30%. How do they do it?

2010-6-29 by kmh
A candy company is packaging its candies with pictures of the players of the national soccer team. How many candies do you have to buy in average to get the complete national team, if it consists of 25 members?

2010-6-22 Number Theory Quickie by kmh
Determine the smallest odd positive integer n which is a multiple of 7 and for which the division by 6, 5, 4 and 3 yields a remainder of 1.

2010-6-19 Number Theory Quickie by lovetruth

Show that any (integer) power of 10 is the sum of two squares. In other words for any $\displaystyle{10^n,\, n \in \mathbb{N}}$ there exists $\displaystyle{ a, b \in \mathbb{N}}$ such that $\displaystyle{10^n=a^2+b^2}$.
Solution (pdf)

Consider  triangles ABC, where the side AB is fixed and C is a variable point on a line parallel to AB. Let H be the intersection of the altitudes (orthocenter) of any such triangle. Describe the geometrical locus of all such points H.

Determine $\displaystyle{\lim_{x\rightarrow 0} \frac{\int_{x}^{sin(x)} \exp(t^2) \, dt}{x\cdot \ln(x+1)}}$

2010-6-4 Geometry Quickie by kmh
Let c be a circle and P be some point outside of it. The two tangents from P on c, touch c in A and B and |PA|=|PB|=10 cm. Now let C be an arbitrary point on the arc AB (the smaller arc close to P). The tangent on c at C intersects the tangent segments PA and PB in Q and R. Determine the perimeter of the triangle  PQR.

2010-5-19 by stefys
Consider a sequence $(x_n)$ with

$x_n:=\begin{cases} \sqrt{n^2+x_{n-1}} & \text{if}\; n>0 \\ 0 & \text{if}\; n=0 \end{cases}$

Determine:

a)$\displaystyle{\lim_{n\rightarrow \infty} x_n-n}$
b)$\displaystyle{\lim_{n\rightarrow \infty} \frac{x_n}{n}}$

2010-4-26 by kmh
Determine the smallest M such that
$ab(a^2-b^2)+bc(b^2-c^2)+ca(c^2-a^2)\leq M (a^2+b^2+c^2)^2$
holds for all $a, b, c \in \mathbb{R}$

2010-4-10 by phalmy
Explain the errors in this proof. How is the supposed contradiction achieved?

(1) $a=1+a^2,\,a\neq 0,\,a\in\mathbb{R}$
(2) $1=\frac{1}{a}+a$
(3) $a=\frac{1}{a}+a+a^2$ | substitute (2) in (1)
(4) $0=\frac{1}{a}+a^2$
(5) $-\frac{1}{a}=a^2$
(6) $-1=a^3$
(7) $-1=a$
(8) $-1=2$ | substitute (7) in (1)

What would change if you assume $a \in \mathbb{C}$ instead of $a \in \mathbb{R}$ in (1)

2010-3-31 by zoma
Let $a, b, c$ be positive integers with $abc+ab+2ac+6bc+2a+6b+12c=158$. Determine $a, b, c$.

2010-3-23 by kmh
Does the improper integral $\displaystyle{ \int_0^\infty\frac{|\sin(x)|}{x^{\frac{3}{2}}}dx}$ exist?

2010-3-10 by kmh
Let A, B be 2 arbitrary distinct points in the plane. Describe the location of all points P for which the ratio of the distances to A and B is constant. In other words describe the set $M:=\{P |\frac{|PA|}{|PB|}=c\}$ with being a positive constant.

2010-2-22 by Kmh
Let g be the graph of $f(x)=\frac{1}{x}\,(x>0)$ and P a point with coordinates (4,-1).

a) Determine the number of tangents on g through P.
b) Compute those tangents analytically (line as equation or 2D curve).
c) Construct those tangents geometrically using only compass and ruler.

2010-2-16 by seth1
Show that if $x,y \in [0,1]$ and $x^3+y^3=x-y$ then $x^2+4\cdot y^2\leq1$

2010-2-12 by b0ro

2010-2-5 Whacky Dice by karlo
Find all possible ways to label the 12 faces of two dice with positive integers such that the probability of rolling a sum of 2, 3, …, 12 is the same as it would be for standard dice.

2010-1-22 Combinatorial Quickie by rio
Show that $\displaystyle{\sum_{k=0}^{min(i,j)} \binom{i}{k}\cdot\binom{j}{k}=\binom{i+j}{i}}$.

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}.

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 $p,q\in\mathbb{R}$

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?

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 nampes 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.

Former Problems

POTD 2008

POTD 2007

/em