# New exercises and problems in Mathematics

October
2003

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

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

**C. 730.** How many solutions does the equation \(\displaystyle \left[\frac{x}{10}\right]=
\left[\frac{x}{11}\right]+1\) have in the set of integers?

**C. 731.** The circle whose diameter is the base *AB* of the trapezium *ABCD* touches the base *CD* and bisects the sides *AD* and *BC*. Find the angles of the trapezium. (12th International Hungarian Mathematics Competition)

**C. 732.** Prove that the following inequality is true for all non-negative real numbers *a* and *b*: \(\displaystyle a+b+\frac{1}{2}\ge\sqrt{a}+
\sqrt{b}\).

**C. 733.** Going around the triangle in the same direction, we have divided each side of an equilateral triangle in a *p*:*q* ratio to obtain another triangle formed by the division points. The area of the new triangle is \(\displaystyle \frac{19}{64}\) of the area of the original triangle. What is the ratio *p*:*q*. (Suggested by *L. Koncz,* Budapest)

**C. 734.** In the coordinate plane, plot the set of points *P*(*x*,*y*) whose coordinates satisfy the inequality \(\displaystyle \frac{2+y}{x}<
\frac{4-x}{y}\).

## 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. 3662.** A pocket radio is operated by two AA batteries. In the drawer, we have 8 batteries, four of which are flat. Unfortunately, we do not know which four. The only way we can test the batteries is by inserting two of them in the radio set. If the radio is operating, both batteries are good, otherwise at least one of them is flat. What is the minimum number of trials needed in order to be certain that the radio will operate? (*5 points*)

**B. 3663.** Are there any odd numbers *a*, *b*, *c*, such that (*a*+*b*)^{2}+ (*a*+*c*)^{2}= (*b*+*c*)^{2}? (*3 points*)

**B. 3664.** Find the polynomial *p*(*x*) of the lowest possible degree with integer coefficients and with leading coefficient 1, such that *p*(0)=0, *p*(1)=1 and *p*(-1)=3. (*3 points*)

**B. 3665.** Given thirteen points in the plane, every five of which include four that are concyclic, prove that at least six of the given points lie on a circle. (*4 points*)

**B. 3666.** Find a square in the interior of each face of a cube, such that every face of the convex polyhedron determined by the vertices of the squares is a regular polygon. (*4 points*)

**B. 3667.** The points *A*, *B*, *C* are the vertices of an equilateral triangle. In the plane of the triangle, find the locus of the points *P*, such that *a*) *PA*^{2}+*PB*^{2}=*PC*^{2}, *b*) *PA*^{2}+*PB*^{2}=2*PC*^{2}. (*3 points*)

**B. 3668.** Find the real numbers *a* and *b* such that

\(\displaystyle |x^2-ax-b|\le\frac{1}{8} \)

holds for every *x*\(\displaystyle \in\)[0;1]? (*4 points*)

**B. 3669.** Find the right circular cone of unit volume and the minimum surface area. (*4 points*)

**B. 3670.** The radii of the escribed circles of a triangle are *r*_{a}, *r*_{b} and *r*_{c}, and the radius of its circumscribed circle is *R*. Given that *r*_{a}+*r*_{b}=3*R* and *r*_{b}+*r*_{c}=2*R*, find the angles of the triangle. (Suggested by *S. Kiss,* (Satu Mare, Romania)) (*5 points*)

**B. 3671.** Solve the equation (*x*^{2}+*y*)(*x*+*y*^{2})= (*x*-*y*)^{3} on the set of integers. (*5 points*)

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

**A. 326.** Assume that *x*_{1},*x*_{2},...,*x*_{n} are integers with no common divisor greater than 1, and let *s*_{k}= *x*_{1}^{k}+ ...+*x*_{n}^{k} for all positive integer *k*. Prove that \(\displaystyle \mathop{\rm gcd}\,(s_1,s_2,\dots,s_n)\) divides \(\displaystyle \mathop{\rm lcm}\,(1,2,\dots,n)\).

**A. 327.** Let *f*(*z*)=*a*_{n}*z*^{n}+*a*_{n-1}*z*^{n-1}+...+*a*_{1}*z*+*a*_{0} be a polynomial of degree *n* (*n*\(\displaystyle \ge\)3) with real coefficients. Prove that if all (real and complex) roots of *f* lie in the left half-plane \(\displaystyle \{z\in\mathbb{C}\colon\mathop{\rm Re}z<0\}\) then *a*_{k}*a*_{k+3}<*a*_{k+1}*a*_{k+2} holds for every *k*=0,1,...,*n*-3. (IMC 10, Cluj-Napoca, 2003)

**A. 328.** Find all functions \(\displaystyle f\colon(0,\infty)\to(0,\infty)\) such that *f*(*f*(*x*)+*y*)=*xf*(1+*xy*) for all positive real numbers *x*, *y*.

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary