# New exercises and problems in Mathematics

December
2001

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

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

**C. 650. ** In a picture gallery, the price of the
picture frames is proportional to the price of the paintings in them.
In order to reduce the differences between the prices of the pictures,
the manager interchanges the frames of two pairs of pictures. In one
case, one picture, that was originally five times as expensive as the
other one, is now only three times as expensive. How does the ratio
of the prices of the "Winter Landscape" and the "Naughty Boy" change
if the "Winter Landscape" originally cost nine times as much as the
"Naughty Boy"?

**C. 651. **The shaded figure in the diagram is
bounded by semicircles. The diameter *AB* has a segment of
length 1/5 units within the shaded figure. Find the perimeter and
area of the shaded figure.

**C. 652. **Let *s* be a positive integer of an
odd number of digits. Let *f* denote the number that consists of
the digits of *f*, but in opposite order. Prove that
*s*+*f* is divisible by 11 if and only if *s* is also
divisible.

**C. 653. **For how many different values of the
parameter *p* do the following simultaneous equations have
exactly one solution? *x*^{2}-*y*^{2}=0,
*xy*+*px*-*py*=*p*^{2}

**C. 654. ***f*_{a},
*f*_{b} and *f*_{c} denote the
lengths of the interior angle bisectors in a triangle of sides
*a*, *b*, *c*, and area *T* . Prove that

Proposed by:: *G. Kovács,* Budapest

## New problems in December 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. 3502. **The semicircle of diameter *AB*
intersects the altitude drawn from vertex *C* of triangle
*ABC* at point *C*_{1}, and the semicircle of
diameter *BC* intersects the altitude from *A* at point
*A*_{1}. Prove that
*BC*_{1}=*BA*_{1}. (3 points)

**B. 3503. **Given are the segments
*a*,*b*,*c*,*d* and *f*. Assuming that there
exists a quadrilateral whose sides are
*a*,*b*,*c*,*d* and the length of the segment
connecting the midpoints of two opposite sides is *f* construct
the quadrilateral. (4 points)

**B. 3504. **If *S*(*n*) denotes the sum of
the digits of the number *n* in decimal notation, and
*U*_{k}=11...1 is the number that consists of
*k* ones, find the values of *k* for which
*S*(*U*_{k}^{2})=(*S*(*U*_{k})^{2}).
(4 points)

**B. 3505. **A regular octagon is subdivided into
parallelograms. Prove that there is a rectangle among the
parallelograms. (5 points)

**B. 3506. ***f* is a polynomial for which
*f*(*x*^{2}+1)-*f*(*x*^{2}-1)=4*x*^{2}+6. Find
the polynomial
*f*(*x*^{2}+1)-*f*(*x*^{2}).
(4 points)

**B. 3507. **Let *f*(*x*) be a polynomial of
integer coefficients, *p* and *q* coprime numbers such that
*q*\(\displaystyle ne\)0.
Prove that if *p*/*q* is a root of the polynomial, then
*f*(*k*) id divisible by *p*-*kq* for every
integer *k*. Is the converse of the statement also true?
(4 points)

**B. 3508. **The triangles *ABC* and
*A*_{1}*B*_{1}*C*_{1} are
symmetric about a line. Draw a parallel through *A*_{1}
to *BC*, through *B*_{1} to *AC*, and finally,
through *C*_{1} to *AB*. Prove that the three lines
all pass through a common point. (4 points)

**B. 3509. **Prove that, if , then (3 points)

Proposed by:: *J. Balogh,* Kaposvár

**B. 3510. **Consider the planes through the vertices
of tetrahedron *ABCD* that are parallel to the opposite faces.
The four planes enclose another tetrahedron. Prove that *A*,
*B*, *C* and *D* are the centroids of the faces of the
new tetrahedron. (3 points)

**B. 3511. **For the non-negative numbers *a*,
*b*, *c* and *d*, *a\(\displaystyle le\)*1, *a*+*b*5,
*a*+*b*+*c\(\displaystyle le\)*14, *a*+*b*+*c*+*d*30. Prove that
\(\displaystyle \sqrt a+\sqrt
b+\sqrt c+\sqrt d\leq10\). (5 points)

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

**A. 278. ***P* is a point on the extension of
the diagonal *AC* of rectangle *ABCD* beyond *C*, such
that *BPD\(\displaystyle angle\)*=*CBP\(\displaystyle angle\)*. Find the ratio *PB*:*PC*.

**A. 279. **Are there such rational functions *f*
and *g* that (*f*(*x*))^{3}+(*g*(*x*))^{3}=*x*?

**A. 280. **For each positive integer *n*, let
*f*_{n}(\(\displaystyle vartheta\))=sin\(\displaystyle vartheta\)^{.}sin(2)^{.}sin(4\(\displaystyle vartheta\))^{.}...^{.}sin(2^{n}). For
all real \(\displaystyle vartheta\) and all *n*, prove that

IMC 8, Prague, 2001

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary