MathDB

1

Part of 2006 Germany Team Selection Test

Problems(5)

Confusing but simple cyclic hexagon (ae || bd, bc || df)

Source: 1st German pre-TST 2006, problem 1

12/5/2005
Let AA, BB, CC, DD, EE, FF be six points on a circle such that AE∥BDAE\parallel BD and BC∥DFBC\parallel DF. Let XX be the reflection of the point DD in the line CECE. Prove that the distance from the point XX to the line EFEF equals to the distance from the point BB to the line ACAC.
geometrygeometric transformationreflectiongeometry proposed
Same set of prime divisors of n

Source: 3rd German TST 2006, 12 march 2006, problem 1

12/29/2006
Does there exist a natural number nn in whose decimal representation each digit occurs at least 20062006 times and which has the property that you can find two different digits in its decimal representation such that the number obtained from nn by interchanging these two digits is different from nn and has the same set of prime divisors as nn ?
number theoryprime numbersnumber theory proposed
W(n) < w(n+1) < w(n+2) for infinitely many n

Source: 4th German TST 2006, written on 1 April 2006, problem 1

4/3/2006
For any positive integer nn, let w(n)w\left(n\right) denote the number of different prime divisors of the number nn. (For instance, w(12)=2w\left(12\right)=2.) Show that there exist infinitely many positive integers nn such that w(n)<w(n+1)<w(n+2)w\left(n\right)<w\left(n+1\right)<w\left(n+2\right).
Eulernumber theory proposednumber theory
On hans-dietrich's calculator, the sin and cos buttons clamp

Source:

6/15/2006
Find all real solutions xx of the equation cos⁡cos⁡cos⁡cos⁡x=sin⁡sin⁡sin⁡sin⁡x\cos\cos\cos\cos x=\sin\sin\sin\sin x. (Angles are measured in radians.)
trigonometrypercentfunctionalgebra proposedalgebra
Angles of 15°, 20° and 25° in equilateral triangle

Source: 6th German TST 2006, 21 may 2006, problem 1

12/29/2006
Let ABC ABC be an equilateral triangle, and P,Q,R P,Q,R three points in its interior satisfying \measuredangle PCA \equal{} \measuredangle CAR \equal{} 15^{\circ},\ \measuredangle RBC \equal{} \measuredangle BCQ \equal{} 20^{\circ},\ \measuredangle QAB \equal{} \measuredangle ABP \equal{} 25^{\circ}. Compute the angles of triangle PQR PQR.
geometry proposedgeometry