# New exercises and problems in Mathematics

April
2002

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

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

**C.** **670.** We have arranged the first nine positive
integers in a 3x3 table, using each of them only once. Suppose that
the seven three-digit numbers represented by the rows left to right,
by the columns top to bottom, and by the diagonal from the upper left
corner are all divisible by 11. What may be the value of the
three-digit number represented by the diagonal starting at the upper
right corner? (Suggested by *S.* *Kiss,* Nyíregyháza)

**C.** **671.** We have placed two jam jars into a
cooking pot of diameter 36 cm. The radii of the jars are
6 cm and 12 cm. What is the maximum possible radius of a
third jam jar to be placed next to them in the pot?

**C.** **672.** The lengths of the edges from vertex
*A* of a cuboid are 1, 2, and 3 units. Consider the triangle
formed by the other endpoints of these edges. Determine the distance
of point *A* from the plane of the triangle.

**C.** **673.** A number is selected out of 0, 1, 2, 3,
4, 5, 6, 7, 8, 9 at random, and then a number is selected again. (It
is allowed to select the same number twice.) Which is greater, the
probability that the sum of the two numbers is divisible by 3, or
the probability that their difference is divisible by 3?

**C.** **674.** Solve the following equation:

\(\displaystyle {x\over20}=\left({5\over2}\right)^{\log_x50}.\)

## New problems in April 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.** **3542.** Prove that if a number of the form
111...1 is divisible by 7 then it is also divisible
by 37. (*3* *points*)

**B.** **3543.** There is a square hole in a square
sheet of paper. Each side of the hole is extended in the
counterclockwise direction. Suppose that the rays obtained in this way
cut the sheet into four quadrangles, as shown in the figure. Prove
that it is possible to assemble the four pieces in a different way to
form a square with a square hole in it.

(*4* *points*)

(Based on the idea of *P.* *Gál*)

**B.** **3544.** Prove that every triangle has an
escribed circle whose radius is at least three times the radius of the
incircle. (*3* *points*)

**B.** **3545.** Prove that

\(\displaystyle {n^2-n\over2}\leq\big\{\sqrt1\big\}+\big\{\sqrt2\big\}+\ldots+\big\{\sqrt{n^2}\big\}\leq{n^2-1\over2},\)

where {*x*} denotes the fractional part of *x*
(i.e. the difference of *x* and the greatest integer not
greater than *x*). (*4* *points*)

**B.** **3546.** A plane intersects a cube in a hexagon
*ABCDEF*. Given that the diagonals *AD*, *BE*,
*CF* of the hexagon are concurrent, prove that the plane passes
through the centre of the cube. (*5* *points*)

**B.** **3547.** Prove that if

\(\displaystyle f(x+1)+f(x-1)=\sqrt2f(x)\)

is true for every real *x* then the function *f* is
periodic. (*4* *points*)

**B.** **3548.** Each term of an infinite arithmetic
sequence of different positive integers is divided by its greatest
prime factor. Is it possible to obtain a bounded sequence in this way?
(*4* *points*)

**B.** **3549.** Prove that for every real number
*x*,

cos cos *x\(\displaystyle ge\)*|sin *x*|.

(*4* *points*) (Suggested by:
*L.* *Szobonya,* Budapest)

**B.** **3550.** The altitudes from the vertices
*A* and *B* of triangle *ABC* intersect each other at
*M*, and they intersect the opposite sides at
*A*_{1} and *B*_{1}, respectively. The line
*A*_{1}*B*_{1} intersects the side *AB*
at *D*. Prove that the line *DM* is perpendicular to the
median from vertex *C*. (*5* *points*)

**B.** **3551.** Let *a*, *b*, *c*,
*d* be positive integers, such that

*a*^{2}+*b*^{2}+*ab*=*c*^{2}+*d*^{2}+*cd*.

Prove that *a*+*b*+*c*+*d* is a composite
number. (*5* *points*)

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

**A.** **290.** Given a finite number of square sheets
of paper with a total area of 4 units, prove that it is possible to
cover a unit square with them. (*Allan Wilson,* England)

**A.** **291.** Solve the equation

\(\displaystyle x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}.\)

**A.** **292.** In a metropolis, there are *n*
underground lines (*n*>4). At most three lines meet at any
station, and for any two lines there is a third line such that one can
make a transfer to it from each of the two lines. Prove that there are
at least \(\displaystyle {5\over6}(n-5)\) underground stations.

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary