# New exercises and problems in Mathematics

October
2001

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

## New exercises in October 2001Maximum score for each exercise (sign "C") is 5 points. |

**C. 640. **In the Gregorian calendar, there are 97
leaping days on 29 February during the course of 400 consecutive
years. In how many years will the difference between the Gregorian
calendar and the ``true'' calendar amount to 1 day if the exact length
of the solar year is 365 days 5 hours 48 minutes and 46 seconds?

**C. 641. **A rectangular sheet of paper is folded
along its diagonal. The parts sticking out over the edge of the
overlapping area are cut off, and the sheet is unfolded. Now it has
the shape of a rhombus. The rhombus is then folded along a line
through the midpoints of two opposite sides, and the overhanging
pieces are cut off again. What should the original rectangle be like
so that the result is a regular hexagon?

**C. 642. **Find all the natural numbers with the
following property: If the last two digits of the square of the number
are interchanged, the result is the square of the next natural number.

Proposed by: *E. Fried,* Budapest

**C. 643. **The area of triangle *ABC* is
*t*, its perimeter is *k*, the radius of the circumscribed
circle is *R*. Prove that \(\displaystyle 4tR\leq\left({k\over3}\right)^3\).

**C. 644. **The angles of a triangle are , és . Prove that
\(\displaystyle {{\cos\alpha}\over{\sin\beta\sin\gamma}}+{{\cos\beta}\over{\sin\alpha\sin\gamma}}+{{\cos\gamma}\over{\sin\alpha\sin\beta}}=2\).

Proposed by: *J. L. Díaz,* Barcelona

## New problems in October 2001The 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. 3482. **Four good friends notice that if each of
them divides the number of the books he has by the sum of the digits
of that number, they all get the same integer: 13. Prove that at least
two of them have the same number of books. (4 points)

Proposed by: *L. Gerőcs,* Budapest

**B. 3483. **The midpoint of side *AB* of a
rectangle *ABCD* is *F*. *P* is a point on the angle
bisector from *C*. The orthogonal projection of *P* on the
line *BC* is *Q*. Prove that if line *PF* is
perpendicular to line *DQ* then
*AP*=*BC*. (3 points)

**B. 3484. **The angles of a triangle are , , . , \(\displaystyle \cot{{\beta}\over2}\), are consecutive integers. Find the largest
angle of the triangle. (3 points)

**B. 3485. **The elements of the set *A* are
positive integers. If *x*, *y\(\displaystyle in\)A*, *x*>*y* then . How many elements can the set *A*
have at most? (4 points)

*Vojtech Jarnik mathematics competition,* Ostrava,
2001

**B. 3486. **Given 2001 points and a circle of unit
radius in the plane, prove that there exists a point on the
circumference of the circle such that the sum of its distances from
the given points is at least 2001. (4 points)

*International Hungarian Mathematics Competition,*
2001

**B. 3487. ***f*(*x*) is a function defined
on positive real numbers, and
*f*(*x*)+2^{.}*f*(1/*x*)=3*x*+6
everywhere in the domain. Find all such functions
*f*(*x*). (3 points)

**B. 3488. **The lengths of the sides of a rectangular
billiards table *ABCD* are *AB*=150 cm and
*BC*=205 cm. There are four holes, one in each corner. At
corner *A*, a ball is struck in a direction enclosing an angle of
45^{o} with the side of the table, and it is moving away from
the hole. Whenever the ball hits the edge of the table, it rebounds
elastically. What will happen to the ball? (4 points)

**B. 3489. **Prove that . (5
points)

**B. 3490. **The benches of the Great Hall of the
Parliament of Neverenough are arranged in a rectangle of 10 rows of 10
seats each. All the 100 MPs have different salaries. Each of them asks
all his neighbours (sitting next to, in front of, or behind him, as
well as those four seated diagonally in front of or behind him, i.e. 8
members at most) how much they earn. They feel a lot of envy towards
each other: an MP is content with his salary only if he has at most
one neighbour who earns more than himself. What is the maximum
possible number of MPs who are satisfied with their salaries? (4
points)

*Kvant*

**B. 3491. **Is the following statement true or false?
``Given *n* red points in the space, it is always possible to
place 3*n* blue points in the space such that there is at least
one blue point in the interior of each tetrahedron determined by the
red points.'' (5 points)

*from the idea of Z. Füredi*

## New advanced problems in October 2001Maximum score for each advanced problem (sign "A") is 5 points. |

**A. 272. **The benches of the Great Hall of the
Parliament of Neverenough are arranged in a rectangle of 10 rows of 10
seats each. All the 100 MPs have different salaries. Each of them asks
all his neighbours (sitting next to, in front of, or behind him,
i.e. 4 members at most) how much they earn. They feel a lot of envy
towards each other: an MP is content with his salary only if he has at
most one neighbour who earns more than himself. What is the maximum
possible number of MPs who are satisfied with their salaries?

*Adapted from Kvant*

**A. 273. ** The inscribed circle of triangle
*ABC* touches the sides *AB*, *BC*, *CA* at the
points *C*_{1}, *A*_{1},
*B*_{1} respectively. Given vertex *A*, line
*A*_{1}*B*_{1}, and the line joining the
midpoint of segment *B*_{1}*C*_{1} to vertex
*B*, construct the triangle *ABC* with ruler and compasses.

Proposed by: *T. Kálmán,* Budapest

**A. 274. **Let *a*, *b*, *c* be
positive integers for which
*ac*=*b*^{2}+*b*+1. Prove that the equation
*ax*^{2}-(2*b*+1)*xy*+*cy*^{2}=1
has an integer solution.

*Polish competition problem*

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary