# New exercises and problems in Mathematics

September
2001

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

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

**C. 635. **In an arithmetic class, pupils are asked
how many legs a hen, six dogs and seven palpigradis have in
total. (``Palpigradi'' is the Latin name of a certain animal). Alex
says 46, Ben says 52, Cecilia says 66, Dora says 78, and Edith says
82. Who is right?

*Based on a problem of a Zrínyi Competition*

**C. 636. **In the Cartesian coordinate system,
represent the set of points whose coordinates *x* and *y*
satisfy -2\(\displaystyle le\)*x*2, -3\(\displaystyle le\)*y*\(\displaystyle le\)3, {*x*}\(\displaystyle le\){*y*} ({*z*} denotes the difference between the
number *z* and the greatest integer less than or equal to
*z*.)

**C. 637. **Find the positive integer that has two
digits both in base 10 and in base 8, and for which the sum of the
digits is fourteen in both number systems.

**C. 638. **How many different square prisms are
possible in which the lengths of the edges in cm are integers and the
measure of the surface area in cm^{2} equals the measure of
the volume in cm^{3}?

**C. 639. **How many pairs (*a*;*b*) of
positive integers are there with 1\(\displaystyle le\)*a*\(\displaystyle le\)2001,
1\(\displaystyle le\)*b*2001, where the lowest common
multiple of *a* and *b* is 2001?

## New problems in September 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. 3472. **How many time instants are there between
6 a.m. and 6 p.m. when it is not possible to tell the time
if we cannot distinguish the long and short hands of the clock?
(3 points)

*Kvant*

**B. 3473. **The area of parallelogram *ABCD* is
2 units. The line parallel to *AD* intersects the boundary
of the parallelogram at *P* and *R*, and the line parallel
to *AB* intersects it at *S* and *Q*, as shown in the
figure. What is the total area of the triangles *AQR*,
*BSR*, *DPQ* and *CSP*? (3 points)

**B. 3474. **What is the seventy-third digit from the
back in the number \(\displaystyle (\underbrace{111\dots1}_{112})^2\)? (4 points)

**B. 3475. **The angles of a triangle with sides
*a*, *b*, *c* are \(\displaystyle alpha\), \(\displaystyle beta\), \(\displaystyle gamma\). Prove that if 3\(\displaystyle alpha\)+2\(\displaystyle beta\)=180^{o}, then
*a*^{2}+*bc*-*c*^{2}=0.
(4 points)

*International Hungarian Mathematics Competition,*
2001

**B. 3476. **An interior point of a regular decagon is
connected to each vertex to form 10 triangles. The triangles are
coloured red and blue alternately. Prove that the total blue area
equals the total red area. (4 points)

**B. 3477. **What is the maximum possible value of the
sine of 2^{n} degrees if *n* is a positive
integer? (4 points)

**B. 3478. **The centres of the exscribed circles of
triangle *ABC* are *O*_{1}, *O*_{2},
*O*_{3}. Prove that the area of triangle
*O*_{1}*O*_{2}*O*_{3} is at
least four times the area of triangle *ABC*. (4 points)

Proposed by: *Á. Besenyei,* Budapest

**B. 3479. **Find the minimum value of the expression
6*t*^{2}+3*s*^{2}-4*st*-8*t*+6*s*+5.
(4 points)

**B. 3480. **Given are the points *A*, *B*,
*C* and *D* in this order on the line *e*. What is the
locus of those points *P* in the plane for which the angles
*APB* and *CPD* are equal? (5 points)

*Gillis--Turán Mathematics Competition*

**B. 3481. **Let \(\displaystyle f_1(x)=-{{2x+7}\over{x+3}}\),
*f*_{n+1}(*x*)=*f*_{1}(*f*_{n}(*x*)). Evaluate
*f*_{2001}(2002). (4 points)

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

**A. 269. **A round hole is to be completely covered
with two square boards. The sides of the squares are 1 metre. In what
interval may the diameter of the hole vary?

**A. 270. **Prove that if *a*, *b*,
*c*, *d* are positive numbers then .

**A. 271. **Prove that for any prime, *p*5, the number is divisible by
*p*^{2}.

*Vojtech Jarnik Mathematics Competition,* Ostrava,
2001

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary