Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# New exercises and problems in MathematicsSeptember 2003

## New exercises

Maximum score for each exercise (sign "C") is 5 points.

C. 725. A coin has been placed in each field of a 3x3 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 2x log x +x -1 = 0 on the set of real numbers. (Suggested by É. Gyanó, Budapest)

## New problems

The 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 30o. (3 points)

B. 3654. Prove that if m and n are integers, m2+n2+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 x4+tx2+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 60o, 100o and 200o at the centre of the circle. If A, B and C are the vertices of a triangle, let MA and MB denote the intersections of the altitudes drawn from the vertices A and B with the circumscribed circle, and let FC denote the intersection of the bisector of angle C with the circumscribed circle. Determine all the acute triangles ABC for which the points MA, MB and FC coincide with the points X, Y and Z in some order. (4 points)

B. 3661. Let x1=1, y1=2, z1=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 x200, y200 and z200 is greater than 20. (5 points)

Maximum 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=120o). 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.