14th ASU 1980

14th ASU 1980 problems


1. All two digit numbers from 19 to 80 inclusive are written down one after the other as a single number N = 192021...7980. Is N divisible by 1980?
2. A square is divided into n parallel strips (parallel to the bottom side of the square). The width of each strip is integral. The total width of the strips with odd width equals the total width of the strips with even width. A diagonal of the square is drawn which divides each strip into a left part and a right part. Show that the sum of the areas of the left parts of the odd strips equals the sum of the areas of the right parts of the even strips.
3. 35 containers of total weight 18 must be taken to a space station. One flight can take any collection of containers weighing 3 or less. It is possible to take any subset of 34 containers in 7 flights. Show that it must be possible to take all 35 containers in 7 flights.
4. ABCD is a convex quadrilateral. M is the midpoint of BC and N is the midpoint of CD. If k = AM + AN show that the area of ABCD is less than k²/2.
5. Are there any solutions in positive integers to a⁴ = b³ + c²?
6. Given a point P on the diameter AC of the circle K, find the chord BD through P which maximises the area of ABCD.
7. There are several settlements around Big Lake. Some pairs of settlements are directly connected by a regular shipping service. For all A ≠ B, settlement A is directly connected to X iff B is not directly connected to Y, where B is the next settlement to A counterclockwise and Y is the next settlement to X counterclockwise. Show that you can move between any two settlements with at most 3 trips.
8. A six digit (decimal) number has six different digits, none of them 0, and is divisible by 37. Show that you can obtain at least 23 other numbers which are divisible by 37 by permuting the digits.
9. Find all real solutions to: sin x + 2 sin(x+y+z) = 0 sin y + 3 sin(x+y+z) = 0 sin z + 4 sin(x+y+z) = 0
10. Given 1980 vectors in the plane. The sum of every 1979 vectors is a multiple of the other vector. Not all the vectors are multiples of each other. Show that the sum of all the vectors is zero.
11. Let f(n) be the sum of n and its digits. For example, f(34) = 41. Is there an integer such that f(n) = 1980? Show that given any positive integer m we can find n such that f(n) = m or m+1.
12. Some unit squares in an infinite sheet of squared paper are colored red so that every 2 x 3 and 3 x 2 rectangle contains exactly two red squares. How many red squares are there in a 9 x 11 rectangle?
13. There is a flu epidemic in elf city. The course of the disease is always the same. An elf is infected one day, he is sick the next, recovered and immune the third, recovered but not immune thereafter. Every day every elf who is not sick visits all his sick friends. If he is not immune he is sure to catch flu if he visits a sick elf. On day 1 no one is immune and one or more elves are infected from some external source. Thereafter there is no further external infection and the epidemic spreads as described above. Show that it is sure to die out (irrespective of the number of elves, the number of friends each has, and the number infected on day 1). Show that if one or more elves is immune on day 1, then it is possible for the epidemic to continue indefinitely.
14. Define the sequence a_n of positive integers as follows. a₁ = m. a_n+1 = a_n plus the product of the digits of a_n. For example, if m = 5, we have 5, 10, 10, ... . Is there an m for which the sequence is unbounded?
15. ABC is equilateral. A line parallel to AC meets AB at M and BC at P. D is the center of the equilateral triangle BMP. E is the midpoint of AP. Find the angles of DEC.
16. A rectangular box has sides x < y < z. Its perimeter is p = 4(x + y + z), its surface area is s = 2(xy + yz + zx) and its main diagonal has length d = √(x² + y² + z²). Show that 3x < (p/4 - √(d² - s/2) and 3z > (p/4 + √(d² - s/2).
17. S is a set of integers. Its smallest element is 1 and its largest element is 100. Every element of S except 1 is the sum of two distinct members of the set or double a member of the set. What is the smallest possible number of integers in S?
18. Show that there are infinitely many positive integers n such that [a^3/2] + [b^3/2] = n has at least 1980 integer solutions.
19. ABCD is a tetrahedron. Angles ACB and ADB are 90 deg. Let k be the angle between the lines AC and BD. Show that cos k < CD/AB.
20. x₀ is a real number in the interval (0, 1) with decimal representation 0.d₁d₂d₃... . We obtain the sequence x_n as follows. x_n+1 is obtained from x_n by rearranging the 5 digits d_n+1, d_n+2, d_n+3, d_n+4, d_n+5. Show that the sequence x_n converges. Can the limit be irrational if x₀ is rational? Find a number x₀ so that every member of the sequence is irrational, no matter how the rearrangements are carried out.