POTD_2008

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?

2008-9-22 by kmh
Probability snack:
John’s father has 2 children. What’s the probablilty for John to have a brother?

2008-9-18 by kmh
Determine the number of subspaces of $\mathbb{Z}_p^n$

2008-8-27 by kmh
The number 916238457 is a nine-digit number which contains each of the digits 1 to 9 exactly. It also has the property that the digits 1 to 5 occur in their correct order, while the digits 1 to 6 do not. How many numbers like this exist?

2008-8-11 by lhrrwcc
Given a circle C of center O and another circle C’ that pass trought O and cuts C at points A,B and let X != O be a point of C’ interior to C, the line AX cuts C at D, show that XD = XB

2008-8-1 by karlo
A calculus professor wants to assign his students a problem of the form, “A pipe of diameter ___ meets another pipe of diameter ___ with a right-angle bend. What’s the longest pin that can make it around that bend?” He decides to make each diameter a positive integer, chosen so that the correct answer will also be a positive integer. What values should he choose?

2008-7-27 by kmh
Consider a line segment $\overline{AB}$ and pick an arbiraty point $C$ on it, that is different from the midpoint. Now consider some circle $k$ with radius $r$ which touches $\overline{AB}$ in $C$ . The tangents from $A$ and $B$ on $k$ intersect in a point $P$ . If you modify the radius $r$ , you’ll get a new position for $P$ or more precisely if you let $r$ run from 0 to $\infty$ then $P$ moves a long a certain curve. Determine that curve.
(original source: de.rec.denksport)

2008-7-22 by dsster
Compute $\displaystyle{ \int \arcsin(\sqrt{x})dx }$ .

2008-7-17 by kmh
Show that for all power of 3 the 2nd last digit is even. For instance we have $3^5=729$ and 2 is even.
(original source: de.rec.denksport)
Solution

2008-6-29 by kmh
Draw the function $f(x)=x^2$ in a coordinate system and erase its axis’. Now reconstruct the coordinate axis’ by using ruler and compass only.

2008-6-20 by kmh
Compute $\int_0^1 \ln(x)\cdot\ln(1-x) dx$
(Original source: The Red Book of Mathematical Problems)

2008-6-10 by ChaosTheory
Compute $\displaystyle{ \int_0^\infty \frac{\sin(x)^2}{x^2} dx}$ .
Solution

2008-6-5 June Special: The Octahedron of Horrors by kmh
This problem is taken from the central Abitur finals of North Rhine-Westphalia of 2008. Though not necessarily that difficult it nevertheless caused such havoc among highschool students for some reason, that the media subsequently dubbed it the “octahedron of horrors“.

An Octahedron is inscribed into a cube as shown in the the figure. The following coordinates are given $A=(13,-5,3)$ , $B=(11,3,1)$ , $C=(5,3,7)$ , $S_1=(13,1,9)$

a) The thickness of an octahedron is defined as the distance between 2 parallel faces. Compute the thickness of the given octahedron.

b) Compute the coordinates of $P_6$ and $P_8$

c) Let $M_{\overline{AB}}$ be the midpoint of $\overline{AB}$ and $M_{\overline{CD}}$ the midpoint of $\overline{CD}$ . The line g connects $M_{\overline{AB}}$ and $M_{\overline{CD}}$ . The octahedron is now rotated around g such that $A$ gets mapped to $A^\prime=(12+2\sqrt{2}, -1+\sqrt{2}, 2+2\sqrt{2})$ . Show that the rotation angle is $\alpha=90^\circ$ and compute the coordinates of $B^\prime$ being the image of B under the rotation.

d) Let $E_a: 2x_1+x_2+2x_3+9\cdot(2a-5)=0$ a set of planes and $h=S_1S_2$ a line connecting $S_1$ and $S_2$ ( $S_2=(5,-3,1)$ ). Show that $E_a$ is orthogonal to $h$ and compute their intersection $P_a$ (control value $P_a=(13-4a,1-2a,9-4a)$ ). For $0\leq a\leq 1$ $E_a$ cuts off a pyramid with $S_1$ as top from the octahedron. Compute the volume $V_a$ of that pyramid.

e) Now cut off the pyramid of d) with volume $V_a$ from every corner of the octahedron. The resulting figure is called $R_a$ ( $0\leq a\leq\frac{1}{2}$ ). Describe $R_a$ for $a=\frac{1}{3}$ and $a =\frac{1}{2}$ regarding the shape and number of its faces.

2008-5-30 by kmh
Determine the size of the dark area.

Solution

2008-5-26 by slikrs
How can 2n+1 items be arranged in a circle n different ways, such that no item has the same neighbor twice?

2008-5-22 by kmh
Find all values of $\sin(x^8-x^6-x^4+x^2)$ for $x\in\mathbb{N}$ and $x^8-x^6-x^4+x^2$ denoting an angle in degrees.
Solution

2008-5-19 by kmh
Show that $\frac{\binom{2n}{n}}{n+1} \in \mathbb{N}$ for all $n\in\mathbb{N}$ .
(Original source: The Red Book of Mathematical Problems)
Solution

2008-5-15 by karlo
The game of Set has 81 cards, each with 4 attributes (number, symbol, shading, color) that can have any of three values. (All 3^4 possible combinations are represented, once each.) Three cards form a valid triple if, for each attribute, the three cards have all different values or all identical values for that attribute. For example, (2-oval-solid-red, 1-oval-empty-red, 3-oval-hashed-red) is a valid triple. What is the maximum number of cards one can have with no valid triples among them?

2008-5-11 by kmh
Consider a convex quadrangle with sides a, b, c, d. Show that if the quadrangle has an incircle and a circumcircle then its area is $\sqrt{abcd}$ .

2008-5-7 by peaceofmind
How many 10 digits numbers can be formed from the digits 1,2,3,such that the digit 3 occurs exactly 3 times in the 10 digits number? How many of those numbers are divisible by 6?

2008-5-4 by karlo
A polygon is “treble” if the number of edges bounding it is a multiple of three (triangle, hexagon, etc.) A polyhedron is treble if all of its faces are treble. Prove or disprove: Any convex polyhedron can be made treble by trimming it a finite number of times, where the trim operation cuts off one corner. (If that corner had n edges, then the trimmed polyhedron has one more (n-sided) face, n more edges, and n more vertices. Additional trimming may be done at either a new vertex or an old one.)

2008-4-29 by kmh
Compute $\displaystyle \lim_{n\rightarrow\infty} \sum_{k=n}^{2n} \frac{1}{k}$
( original source: Internet Math Olympiad Israel 2008 )
Solution

2008-4-14 by Karlo
Given the graph of an arbitrary ellipse E and some point P outside of E. Construct the 2 tangents from P on E by using ruler and compass only.

2008-4-11 by kmh
Prove that the equation $y^2=x^3+23$ has no solutions in integers x and y.
(Source: The Red Book of Mathmatical Problems (Dover))

2008-4-7 by phi
A 2008×2008 matrix has only elements from the set {0, 1}. Given that every two lines differ from each other in a half of the positions, prove that every two columns also differ in a half of the positions.
( original source: Internet Math Olympiad Israel 2008 )
2008-3-31 by Karlo Construct the foci of arbitrary ellipse given as a graph using ruler and compass only.

Solution

2008-3-29 by kmh

$P$ is a point on a circle and $A$ is point outside the circle. The tangents from $A$ onto the circle meet it in $B$ and $C$ . Now construct the perpendiculars from P on the 2 tangents and the line $BC$ , they intersect those lines in $A_1, B_1, C_1$ . Show that $|PA_1|^2=|PB_1|\cdot |PC_1|$ .

(Original source: Coxeter/Greitzer: Geometry Revisited)

2008-3-23 by phi

Let A, B, C, D be 4 distinct spheres in space. Suppose the spheres A and B intersect along a circle which belongs to plane P. The spheres B and C intersect along a circle which belongs to plane Q, the spheres C and D intersect along a circle which belongs to plane S and spheres D and A intersect along a circle which belongs to plane T. Prove that the planes P, Q, S, T are either parallel to the same line or have a common point.

(original source: Internet Math Olympiad Israel 2008)

2008-3-20 The Cossack and the Goat by kmh

Two roads are intersecting in a $30^\circ$ angle. On the first road there is cossack riding his horse with a speed 15 km/h towards the crossing and on the second road there is a goat trotting towards the crossing with a speed of 2 km/h.The cossack is still 500 m away from the crossing and the goat only 40 m. The cossack realizes that he might not get close enough to grab the goat and therefore plans to catch her with a lasso. Determine the minimal distance the lasso has to cover, so that the cossack can figure out whether his rope is long enough to catch the goat without leaving his own road.

2008-3-12 by lhrrwcc

Find all rings between $\mathbb{Z}$ (integers) and $\mathbb{Q}$ (rationals).

2008-3-4 by rogue

Let $P$ be an interior point of an acute triangle $\triangle ABC$ . The line $BP$ meets the line $AC$ at $E$ , and the line $CP$ meets the line $AB$ at $F$ . The lines $AP$ and $EF$ intersect each other at $D$ . Let $K$ be the foot of the perpendicular from the point $D$ to the line $BC$ . Show that $KD$ bisects angle $\angle EKF$

2008-2-28 by yourshadow

Determine the maximum of $f(x,y,z)=xy^2z^3$ given the following additional conditions: $x+2y+3z=a$ , $x> 0$ , $y>0$ , $z>0$ .

2008-2-25 by rogue
Let $a, b, c, d, e$ be 5 strictly positive real numbers with the following propreties:
$a^2 + b^2 + c^2 = d^2 + e^2$ , $a^4 + b^4 + c^4 = d^4 + e^4$ .
Compare: $a^3 + b^3 + c^3$ and $d^3 + e^3$ .

2008-2-21 by efne1
Determine whether $\displaystyle \sum_{n=0}^\infty \frac{n^n}{(n+1)^{n+1}}$ converges or diverges.

2008-2-19 by icosagon

2008-2-16 by karlo
A rod is placed in a hemispherical bowl whose diameter is the length of the rod. How much of its length is inside the bowl when it’s at rest?
The rest position will be when the center of the rod is as low as possible.

2008-2-13 by black_hole
In how many ways can you fill a (n+1)x(n+1) matrix with 0s and 1s so that the sum on each column and each line is even?

2008-2-11 by kmh
What is the probability that the equation $x^2+bx+c=0$ has a (real) solution?

2008-2-5 by Karlo
Three friends are taking me out for my birthday. The product of their ages is 2450. The sum of their ages is my cousin’s age. I could tell you my cousin’s age, but to find the ages of my friends, you’d also need to know that each of the three is younger than I am.

How old am I?
Solution

2008-2-1 by kmh
An arbitrary circle contains the origin of the coordinate system in its interior and therefore it is partitioned into 8 different areas by the coordinate axis´ and the bisectors of the quadrants. Now color the areas with 2 colors white and green, such that 2 neighbouring areas are of a different color (see graphic). Prove that green area equals the white area.

circlerjpg

2008-1-29 Truth or Dare by Karlo
You’re given three constants T, L, and N, and you enter a room containing T truth tellers, L liars, and N normals (who have no constraints on their answers to questions), all of whom know each other’s truth class and the information in this paragraph. They will answer any yes-or-no questions. (But if you ask anyone a question for which neither answer is logically permitted, he will kill you.) For which (T, L, N) triples is it always possible to identify everyone’s truth class, even if they’re trying to prevent you from doing so?

2008-1-17 by yfk
Let $x_1, x_2,\ldots, x_n \in \{0,1\}$ and $\overline{x}= \frac{1}{n} \displaystyle \sum_{i=1}^n x_i$ . Show that $\frac{1}{n} \displaystyle \sum_{i=1}^n (x_i-\overline{x})^2 \leq\frac{1}{4}$ .
Solution

2008-1-13 by sniffle
Let $x_1, x_2,\ldots, x_n$ be strictly positive real numbers and $S = x_1 + x_2 +\ldots + x_n$ . Prove that $\frac{S}{S - x_1} +\ldots+ \frac{S}{S - x_n} \geq \frac{ n^2}{n-1}$ .

2008-1-10 by kmh
Consider an equilateral triangle $\triangle ABC$ and its circumcircle c. Let Q be an arbitrary point on $\overline{AB}$ and the line QC intersects the circumcircle c in P. Show that $\frac{1}{|\overline{AP}|}+\frac{1}{|\overline{BP}|}=\frac{1}{|\overline{PQ}|}$

2008-1-8 by lhrrwcc
Let $A$ be a noetherian ring. Prove that if $f:A\rightarrow A$ is a surjective ring homomorphism then $f$ is bijective.
Solution

2008-1-6 by kmh
Show that: $a_n\rightarrow a \quad \Rightarrow \quad \displaystyle \sum_{k=0}^n \frac{1}{2^n} \binom{n}{k} a_k \rightarrow a$