# New exercises and problems in Mathematics

September
2002

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

## New exercisesMaximum score for each exercise (sign "C") is 5 points. |

**C.** **680.** A committee of nine members was set up
to choose one out of three candidates. They used the following
procedure: every member of the committee ranked the candidates, giving
3 points to the first one, 2 to the second one, and 1 to the
third one. When the points were added, the sums were all different,
which clearly established the order of the candidates. However, one
member of the committee noticed that if each of them had selected only
one candidate and given 1 point to that candidate, the final
order of the candidates would have been reversed. How many points did
each candidate (originally) get?

**C.** **681.** The ``pyramid'' in the *figure* is
built out of three layers of 1 cm^{3} cubes, its surface
area is 42 cm^{2}. Using the same technique, we also
built a larger ``pyramid'' with total surface area of
2352 cm^{2}. How many layers does it consist of?

**C.** **682.** In 2002, the tax paid by those with an
annual gross income of more than 1,050,000 forints (HUF) was 40% of
the amount over 1,050,000 forints plus 267,000 forints. For what gross
annual income was the tax equal to 30% of the income?
(*E.* *Fried,* Budapest)

**C.** **683.** In the isosceles right triangle
*ABC*, *AC*=*BC*. the angle bisector drawn from vertex
*A* intersects the leg *BC* at the point *P*. Prove
that the length of the line segment *PB* is equal to the diameter
of the inscribed circle of *ABC*.

**C.** **684.** Solve the simultaneous equations

*x*-*xy*+*y*=1, *x*^{2}+*y*^{2}=17.

## New problemsThe 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.** **3562.** For her birthday, an eight-year-old
little girl was given a cake shaped like a right prism of a convex
heptagonal base. The cake is cut into pieces with vertical cuts along
some of its diagonals. The little girl wants an octagonal piece. Is
that possible? (*3* *points*)

**B.** **3563.** For what minimum value of *t* is
the inequality \(\displaystyle \sqrt{xy}\leq t(2x+3y)\) true for all non-negative real *x*,
*y*? (*3* *points*)

**B.** **3564.** Move the point *A*(0;1) of the
(*x*,*y*) coordinate plane onto the point *B*(3;2) by
an appropriate composition of reflections about the *x*-axis and
rotations about the point *A*. (*4* *points*)

**B.** **3565.** We are given 2002 boxes with a few
pebbles in each, and we are also given a large (inexhaustible) heap of
pebbles. In each step, we are allowed to put one pebble into each of
any set of *k* boxes. Is it possible to achieve that there are
the same number of pebbles in every box if

*a*) *k*=8;

*b*) *k*=9? (*4* *points*)

**B.** **3566.** For every interior point *P* in a
given triangle *ABC*, a triangle can be constructed out of the
line segments *PA*, *PB* and *PC*. Show that the
triangle *ABC* is equilateral. (*4* *points*)

**B.** **3567.** Given that each side of a convex
pentagon is parallel to one of its diagonals, what may be the ratio of
the length of a side to that of the parallel diagonal?
(*5* *points*)

**B.** **3568.** Find the angle , given that
for every non-negative integer *k*, cos2^{k}0. (*5* *points*)

**B.** **3569.** Given that any two out of 100 given
cubes have a common interior point, and each of the edges is parallel
to one of three given lines, show that the cubes have a common
interior point. Give an example of 100 cubes, such that any three of
them have a common interior point but there is no point contained in
every cube. (*5* *points*)

**B.** **3570.** *f* is a continuous function,
*f*(1000)=999 and *f*(*x*)^{.}*f*(*f*(*x*))=1 for
every real number *x*. Find the value of
*f*(500). (*4* *points*)

**B.** **3571.** A cup shaped like a right truncated
cone is rolled around on a round table, so that the cup never touches
the edge of the table. The diameter of the table is
1.6 metres. The diameter of the base of the cup is 5 cm,
that of the top is 6.5 cm. What is the height of the cup?
(*4* *points*)

## New advanced problemsMaximum score for each advanced problem (sign "A") is 5 points. |

**A.** **296.** Alice has chosen one of the numbers
1,2,...,16. Bob can ask seven yes-no questions to find it out. Alice
is allowed to give at most one wrong answer. Help Bob to find out the
number in all cases.

**A.** **297.** Let
*a*_{0},*a*_{1},... be positive integers
such that *a*_{0}=1, *a*_{1}>1 and

\(\displaystyle a_{n+1}=\frac{a_1\cdot\ldots\cdot a_n}{a_{[n/2]}}+1\)

for all *n*=1,2,.... ([*x*] denotes the integer part of
*x*.) Show that \(\displaystyle \sum_{n=1}^\infty\frac{1}{a_{n+1}a_{[n/2]}}\)
is a rational number.

**A.** **298.** The angles of a spherical triangle are
\(\displaystyle alpha\), and , the lengths
of the opposite side arcs are *a*, *b* and *c*,
respectively. Prove that *a*^{.}cos+*b*^{.}cos\(\displaystyle alpha\)<*c*.

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary