MathDB

p1. Two proper ordinary fractions are given. The first has a numerator that is

5

less than the denominator, and the second has a numerator that is

1998

less than the denominator. Can their sum have a numerator greater than its denominator?
p2. On New Year's Eve, geraniums, crocuses and cacti stood in a row (from left to right) on the windowsill. Every morning, Masha, wiping off the dust, swaps the places of the flower on the right and the flower in the center. During the day, Tanya, while watering flowers, swaps places between the one in the center and the one on the left. In what order will the flowers be in

365

days on the next New Year's Eve?
p3. The number

x

is such that

15\%

of it and

33\%

of it are positive integers. What is the smallest number

x

(not necessarily an integer!) with this property?
p4. In the quadrilateral

ABCD

, the extensions of opposite sides

AB

and

CD

intersect at an angle of

20^o

; the extensions of opposite sides

BC

and

AD

also intersect at an angle of

20^o

. Prove that two angles in this quadrilateral are equal and the other two differ by

40^o

.
p5. Given two positive integers

a

and

b

. Prove that

a^ab^b\ge a^ab^a.

p6. The square is divided by straight lines into

25

rectangles (fig.). The areas of some of They are indicated in the figure (not to scale). Find the area of the rectangle marked with a question mark. https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png
p7. A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly

17^o

(relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again?
p8. In expression

(a-b+c)(d+e+f)(g-h-k)(\ell +m- n)(p + q)

opened the brackets. How many members will there be? How many of them will be preceded by a minus sign?
p9. In some countries they decided to hold popular elections of the government. Two-thirds of voters in this country are urban and one-third are rural. The President must propose for approval a draft government of

100

people. It is known that the same percentage of urban (rural) residents will vote for the project as there are people from the city (rural) in the proposed project. What is the smallest number of city residents that must be included in the draft government so that more than half of the voters vote for it?
p10. Vasya and Petya play such a game on a

10 \times 10 board

. Vasya has many squares the size of one cell, Petya has many corners of three cells (fig.). They are walking one by one - first Vasya puts his square on the board, then Petya puts his corner, then Vasya puts another square, etc. (You cannot place pieces on top of others.) The one who cannot make the next move loses. Vasya claims that he can always win, no matter how hard Petya tries. Is Vasya right? https://cdn.artofproblemsolving.com/attachments/f/1/3ddec7826ff6eb92471855322e3b9f01357116.png
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.

Problem Statement