# New exercises and problems in Mathematics

September
2003

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

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

**C. 725.** A coin has been placed in each field of a 3*x*3 table, showing tails on top. At least how many coins need to be turned over, so that there are no three collinear (row, column, diagonal) heads or three collinear tails?

**C. 726.** Is there a regular polygon in which the shortest diagonal equals the radius of the circumscribed circle?

**C. 727.** Peter's telephone number (without area code) is 312837, that of Paul is 310650. If each of these numbers is divided by the same three-digit number, the remainders will be equal. That remainder is the area code of their city. What is the remainder? (Note: Area codes are two-digit numbers in Hungary.)

**C. 728.** The angles *A* and *B* of a convex quadrilateral *ABCD* are equal, and angle *C* is a right angle. The side *AD* is perpendicular to the diagonal *BD*. The lengths of sides *BC* and *CD* are equal. What is the ratio between their common length and the length of side *AD*?

**C. 729.** Solve the equation 2*x* log *x* +*x* -1 = 0 on the set of real numbers. (*Suggested by É. Gyanó,* Budapest)

## 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. 3652.** We have coloured each positive integer either red or blue. The sum of two numbers of different colours is always blue, and their product is always red. What colour is the product of two red numbers? (*3 points*)

**B. 3653.** Find the locus of those points in the plane of a given square at which the square subtends an angle of 30^{o}. (*3 points*)

**B. 3654.** Prove that if *m* and *n* are integers, *m*^{2}+*n*^{2}+*m*+*n*-1 cannot be divisible by 9. (*3 points*)

**B. 3655.** The convex hexagon *ABCDEF* is cyclic, and *AB*=*BC*=*a*, *CD*=*DE*=*b*, *EF*=*FA*=*c*. Prove that the area of the triangle *BDF* is half of the area of the hexagon. (*4 points*)

**B. 3656.** The number *F* in base-*a* notation is \(\displaystyle 0{,}3737\ldots=
0{,}\dot3\dot7\) (the dots denoting the beginning and the end of the recurring sequence of digits), and the number *G* in base-*a* notation is \(\displaystyle 0{,}7373\ldots=0{,}\dot7\dot3\). The same numbers written in base-*b* notation are \(\displaystyle F=0{,}2525\ldots=
0{,}\dot2\dot5\) and \(\displaystyle G=0{,}5252\ldots=0{,}\dot5\dot2\). Determine the numbers *a* and *b*. (*4 points*)

**B. 3657.** Is there a right-angled triangle such that the radius of the incircle and the radii of the three excircles are four consecutive terms of an arithmetic progression? (*4 points*)

**B. 3658.** The point *P* lies on the perpendicular line segment dropped from the vertex *A* of the regular tetrahedron *ABCD* onto the face *BCD*. The lines *PB*, *PC* and *PD* are pairwise perpendicular. In what ratio does *P* divide the perpendicular line segment? (*3 points*)

**B. 3659.** Given the real number *t*, write the expression *x*^{4}+*tx*^{2}+1 as a product of two quadratic factors of real coefficients. (*4 points*)

**B. 3660.** The points *X*, *Y* and *Z* divide a circle into three arcs that subtend angles of 60^{o}, 100^{o} and 200^{o} at the centre of the circle. If *A*, *B* and *C* are the vertices of a triangle, let *M*_{A} and *M*_{B} denote the intersections of the altitudes drawn from the vertices *A* and *B* with the circumscribed circle, and let *F*_{C} denote the intersection of the bisector of angle *C* with the circumscribed circle. Determine all the acute triangles *ABC* for which the points *M*_{A}, *M*_{B} and *F*_{C} coincide with the points *X*, *Y* and *Z* in some order. (*4 points*)

**B. 3661.** Let *x*_{1}=1, *y*_{1}=2, *z*_{1}=3, and let \(\displaystyle x_{n+1}=y_n+
\frac{1}{z_n}\), \(\displaystyle y_{n+1}=z_n+\frac{1}{x_n}\), \(\displaystyle z_{n+1}=x_n+
\frac{1}{y_n}\) for every positive integer *n*. Prove that at least one of the numbers *x*_{200}, *y*_{200} and *z*_{200} is greater than 20. (*5 points*)

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

**A. 323.** *I* is the isogonic point of a triangle *ABC* (the point in the interior of the triangle for which \(\displaystyle \angle\)*AIB*=\(\displaystyle \angle\)*BIC*=\(\displaystyle \angle\)*CIA*=120^{o}). Prove that the Euler lines of the triangles *ABI*, *BCI* and *CAI* are concurrent.

**A. 324.** Prove that if *a*,*b*,*c* are positive real numbers then

\(\displaystyle \frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)}\ge\frac{3}{1+abc}. \)

**A. 325.** We have selected a few 4-element subsets of an *n*-element set *A*, such that any two sets of four elements selected have at most two elements in common. Prove that there exists a subset of *A* that has at least \(\displaystyle \root3\of{6n}\) elements and does not contain any of the selected 4-tuples as a subset.

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary