# New exercises and problems in Mathematics

May
2002

## Please read The Conditions of the Problem Solving Competition.

## New exercises in May 2002Maximum score for each exercise (sign "C") is 5 points. |

**C. 675.** Is there a square number in
which the last two digits are both odd?

**C. 676.** The length of side *AB* of
the rectangle *ABEF* is 1 unit, and the length of side
*BE* is 3 units. The points *C* and *D* divide
side *BE* into three equal parts. Show that *BAC*+*BAD*\(\displaystyle \angle\)+*BAE*=180^{o}.

**C. 677.** Find the integers *a* and
*b*, given that
*a*^{4}+(*a*+*b*)^{4}+*b*^{4}
is a square number.

**C .678.** In a triangle *ABC*,
*AC*=1, *ABC*\(\displaystyle \angle\)=30^{o}, *BAC*=60^{o}, and *D* denotes the foot of
the altitude from vertex *C*. Find the distance between the
centres of the inscribed circles of the triangles *ACD* and
*BCD*.

**C. 679.** Given three spheres that
pairwise touch each other and also touch the plane *S*, determine
the radius of the sphere that has its centre on the plane *S*,
and that touches all the three spheres.

## New problems in May 2002The maximum scores for problems (sign "B") depend on the difficulty. It is allowed to send solutions for any number of problems, but your score will be computed from the 6 largest score in each month. |

**B. 3552.** Each member of the sequence
*a*_{1}, *a*_{2}, ...,
*a*_{2n+1} is either 2, 5 or 9. No two
consecutive members are equal, and
*a*_{1}=*a*_{2n+1}. Prove that
*a*_{1}*a*_{2}-*a*_{2}*a*_{3}+*a*_{3}*a*_{4}-*a*_{4}*a*_{5}+...-*a*_{2n}*a*_{2n+1}=0. (*3 points*)

**B. 3553.** In a triangle *ABC*, the
altitude from vertex *A*, the angle bisector from vertex
*B*, and the median from vertex *C* intersect the opposite
sides at the points *A*_{1}, *B*_{1},
*C*_{1}, respectively. Prove that if the triangle
*A*_{1}*B*_{1}*C*_{1} is
equilateral, then the triangle *ABC* is also
equilateral. (*4 points*)

**B. 3554.** The diagonals of parallelogram
*ABCD* intersect at point *M*. The circle passing through
the points *A*, *M*, *B* touches the line
*BC*. Prove that the circle through the points *B*,
*M*, *C* touches the line *CD*. (*3 points*)

**B. 3555.** A company consists of 2*n*+1
people. For every group of *n* members, there exists a member of
the company who does not belong to the group, but who knows every
member of the group. Acquaintances are considered to be mutual. Prove
that there is a member of the company who knows
everybody. (*5 points*)

**B. 3556.** Given two sides and the
bisector of the exterior angle at their common vertex, construct the
triangle. (*4 points*)

**B. 3557.** Is it possible for a perfect
cube to start with 2002 ones in decimal notation?
(*4 points*)

**B. 3558.** Is there a non-constant
polynomial of integer coefficients, such that its value is a power of
2 at every positive integer? (*4 points*)

(Inspired by a problem of the National Competition, 2002)

**B. 3559.** *e*_{1},
*e*_{2}, ..., *e*_{n} are lines in
the plane. Through an arbitrary point *P*_{1} of line
*e*_{1}, drop a perpendicular to the line
*e*_{2}, and denote the foot of the perpendicular by
*P*_{2}. Let *P*_{3} be the foot of the
perpendicular from *P*_{2} onto *e*_{3}, and
so on. Finally, let *P*_{n+1} denote the foot of
the perpendicular from *P*_{n} onto
*e*_{1}. Prove that there exists a point
*P*_{1} on the line *e*_{1}, such that the
point *P*_{n+1} obtained in this way should
coincide with *P*_{1}. (*4 points*)

**B. 3560.** Prove that no matter how we
leave 89 numbers out of the first 2002 positive integers,
the set of the remaining numbers will contain 20 elements, such
that their sum is also among the remaining
numbers. (*5 points*)

**B. 3561.** A convex polyhedron has
exactly three edges meeting at each vertex. Given that all but one of
the faces of the polyhedron are known to have circumscribed circles,
prove that all the faces have circumscribed
circles. (*5 points*)

## New advanced problems in May 2002Maximum score for each advanced problem (sign "A") is 5 points. |

**A. 293.** Prove that for every integer
*m*\(\displaystyle \ge\)2 there
exists a pair of positive integers *a* and *b*, such that in
base-m notation, *a* and *b* together contain exactly the
same number of each digit as the number
*a*^{.}*b*. (*L. Szobonya,*
Budapest)

**A. 294.** Define the sequence
*a*_{1},*a*_{2},... with the following
recursion:

\(\displaystyle a_1=1, \quad a_{n+1}={a_1a_n+a_2a_{n-1}+\dots+a_na_1\over n+1}.\)

Prove that the sequence is convergent.

**A. 295.** Positive numbers
*x*_{1},*x*_{2}...,*x*_{n}
satisfy

\(\displaystyle {1\over1+x_1}+{1\over1+x_2}+\dots+{1\over1+x_n}=1.\)

Prove that

\(\displaystyle \sqrt{x_1}+\sqrt{x_2}+\dots+\sqrt{x_n}\ge(n-1) \left({1\over\sqrt{x_1}}+{1\over\sqrt{x_2}}+\dots+ {1\over\sqrt{x_n}}\right).\)

(Vojtech Jarník Mathematical Competition, Ostrava, 2002)

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary