# New exercises and problems in Mathematics

December
2003

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

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

**C. 740.** Find all positive integers which are one and a half times larger than the product of their digits.

**C. 741.** Given the sides *a* and *b* of a triangle, and given that \(\displaystyle \alpha\)=2\(\displaystyle \beta\) (in conventional notation), construct the triangle.

**C. 742.** The planks of a 2.5-metre long section of a fence are rotten. We want to replace them by new ones, 3 cm thick each. We want to set up the planks with their edges touching, the wider side facing inwards. Is it possible to do so by cutting up a cylindrical tree trunk of appropriate height and a diameter of 30 cm (assuming no loss of wood)?

**C. 743.** Solve the inequality \(\displaystyle \log_x\left(2.5-\frac{1}{x}\right\)1">.

**C. 744.** How many ways are there to place a 5 by 5 square on an 8 by 8 chessboard so that each vertex is at the centre of some field? (Solutions obtained from each other by reflection or rotation are not considered different.)

## 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. 3682.** The sum of the highest common factor and lowest common multiple of two positive integers *a* and *b* equals (*a*+*b*). Prove that one of the two numbers divides of the other one.
(*3 points*)

**B. 3683.** Find those regular polygons whose side is equal to *a* and they can be tiled with regular polugons whose sides are also equal to *a*, respectively?
(*4 points*) (*The Twelfth International Hungarian Mathematics Competition, 2003*)

**B. 3684.** Given the sides *a* and *b* of a triangle, and given that \(\displaystyle \alpha\)=3\(\displaystyle \beta\) (in conventional notation), construct the triangle.
(*3 points*)

**B. 3685.** In a video game, the players get an integer score at the end of each game. The list of the 30 best scores is displayed. Initially, the creator of the game filled up the list with dummy names with scores 30, 29, 28,..., 1, respectively. When a game is over, one's score appears on the list if it is greater than the lowest score, which in turn will be removed. In the case of equal lowest scores, the one added last will be removed. When a new score is added to the list, it will precede the smaller ones only, i.e. any previously entered score equal to the new one will precede the latter. Provided that a player qualifies himself on the list at the end of each game, at least how many games should he play in order to read his name next to each of the 30 scores?
(*4 points*)

**B. 3686.** In a tetrahedron *ABCD*, the angles *ABD* and *ACD* are obtuse. Show that *AD*>*BC*.
(*3 points*)

**B. 3687.** Solve the following equation on the set of real numbers: \(\displaystyle \root5\of{x^3+2x}=\root3\of{x^5-2x}.\)
(*5 points*)

**B. 3688.** Find an integer whose square root contains the digits 414213462 right after the decimal point.
(*4 points*)

*G. Holló,*Budapest)

**B. 3689.** Find all real numbers *x*, such that \(\displaystyle \tan\left(\frac{\pi}{12}-x\right)\), \(\displaystyle \tan\frac{\pi}{12}\) and \(\displaystyle \tan\left(\frac{\pi}{12}+x\right)\) are (in some order) three consecutive terms of a geometric progression.
(*4 points*)

**B. 3690.** Given the endpoints of the major axis of an ellipse \(\displaystyle \cal{E}\), a tangent *e* of the ellipse and a point *P* on the tangent, construct the other tangent drawn from the point *P* to the ellipse.
(*5 points*)

**B. 3691.** What is the maximum possible area of an orthogonal projection of a 3*x*4 *x*12 cuboid onto a plane?
(*5 points*)

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

**A. 332.** In a cyclic quadrilateral *ABCD*, the midpoint of diagonals *AC* and *BD* are *E* and *F*, respectively. Prove that *AEB*\(\displaystyle \angle\)= *AED*\(\displaystyle \angle\) implies *BFA*\(\displaystyle \angle\)= *BFC*\(\displaystyle \angle\).

**A. 333.** Prove that the recurrence *n*=*x*_{n}^{.}(*x*_{n-1}+*x*_{n}+*x*_{n+1}), *n*=1,2,..., *x*_{0}=0 has exactly one nonnegative solution. (*Schweitzer competition, 2003*)

**A. 334.** Let *p* and *q* be given coprime positive integers. Call a subset *S* of nonnegative integers ``ideal'' if *S* satisfies the following two conditions: *a*) 0\(\displaystyle \in\)*S*; *b*) for all *n*\(\displaystyle \in\)*S*, *n*+*p*\(\displaystyle \in\)*S* and *n*+*q*\(\displaystyle \in\)*S*. Determine the number of all ideal subsets. (Proposed by *Dobos Sándor,* Hungarian selection competition for the IMO, 2001)

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary