A council of 2,000 deputies decided to approve a state budget containing 200 items of expenditure. Each deputy prepared his draft budget, which indicated for each item the maximum allowable, in his opinion, amount of expenditure, ensuring that the total amount of expenditure did not exceed the set value of \(S\). For each item, the board approves the largest amount of expenditure that is agreed to be allocated by no fewer than \(k\) deputies. What is the smallest value of \(k\) for which we can ensure that the total amount of approved expenditures does not exceed \(S\)?
Aladdin visited all of the points on the equator, moving to the east, then to the west, and sometimes instantly moving to the diametrically opposite point on Earth. Prove that there was a period of time during which the difference in distances traversed by Aladdin to the east and to the west was not less than half the length of the equator.
Izzy wrote a correct equality on the board: \(35 + 10 - 41 = 42 + 12 - 50\), and then subtracted 4 from both parts: \(35 + 10 - 45 = 42 + 12 - 54\). She noticed that on the left hand side of the equation all of the numbers are divisible by 5, and on the right hand side by 6. Then she took 5 outside of the brackets on the left hand side and 6 on the right hand side and got \(5(7 + 2 - 9)4 = 6(7 + 2 - 9)\). Having simplified both sides by a common multiplier, Izzy found that \(5 = 6\). Where did she go wrong?
A carpet of size 4 m by 4 m has had 15 holes made in it by a moth. Is it always possible to cut out a 1 m \(\times\) 1 m area of carpet that doesn’t contain any holes? The holes are considered to be points.
Prove that in any group of 2001 whole numbers there will be two whose difference is divisible by 2000.
Prove that there is a number of the form
a) \(1989 \dots 19890 \dots 0\) (the number 1989 is repeated several times, and then there are a few zeros), which is divisible by 1988;
b) \(1988 \dots 1988\), which is divisible by 1989.
The natural number \(a\) was increased by 1, and its square increased by 1001. What is \(a\)?
In a basket, there are 30 red and green apples. Among any 12 apples there is at least one red one, and among any 20 apples there is at least one green one. How many red apples and how many green apples are there in the basket?
In the numbers of MEXAILO and LOMONOSOV, each letter denotes a number (different letters correspond to different numbers). It is known that the products of the numbers of these two words are equal. Can both numbers be odd?
The numbers \(a\) and \(b\) are such that the first equation of the system \[\begin{aligned} \sin x + a & = bx \\ \cos x &= b \end{aligned}\] has exactly two solutions. Prove that the system has at least one solution.