x1 + 4 x2 + 9 x3 + 16 x4 + 25 x5 + 36 x6 + 49 x7 = 1;
4 x1 + 9 x2 + 16 x3 + 25 x4 + 36 x5 + 49 x6 + 64 x7= 12;
9 x1 + 16 x2 + 25 x3 + 36 x4 + 49 x5 + 64 x6 + 81 x7= 123.
Find 16 x1 + 25 x2 + 36 x3 + 49 x4 + 64 x5 + 81 x6 + 100 x7.
| 1. Find sqrt(1 + 28·29·30·31). |
| 2. 10 points lie on a circle. How many distinct convex polygons can be formed by connected some or all of the points? |
| 3. For some digit d we have 0.d25d25d25 ... = n/810, where n is a positive integer. Find n. |
| 4. Given five consecutive positive integers whose sum is a cube and such that the sum of the middle three is a square, find the smallest possible middle integer. |
| 5. A coin has probability p of coming up heads. If it is tossed five times, the probability of just two heads is the same as the probability of just one head. Find the probability of just three heads in five tosses. [The original question asked for m+n, where the probability is m/n in lowest terms.] |
| 6. C and D are 100m apart. C runs in a straight line at 8m/s at an angle of 60o to the ray towards D. D runs in a straight line at 7m/s at an angle which gives the earliest possible meeting with C. How far has C run when he meets D? |
| 7. k is a positive integer such that 36 + k, 300 + k, 596 + k are the squares of three consecutive terms of an arithmetic progression. Find k |
|
8. Given that:
x1 + 4 x2 + 9 x3 + 16 x4 + 25 x5 + 36 x6 + 49 x7 = 1; 4 x1 + 9 x2 + 16 x3 + 25 x4 + 36 x5 + 49 x6 + 64 x7= 12; 9 x1 + 16 x2 + 25 x3 + 36 x4 + 49 x5 + 64 x6 + 81 x7= 123. Find 16 x1 + 25 x2 + 36 x3 + 49 x4 + 64 x5 + 81 x6 + 100 x7. |
| 9. Given that 1335 + 1105 + 845 + 275 = k5, with k an integer, find k. |
| 10. The triangle ABC has AB = c, BC = a, CA = b as usual. Find cot C/(cot A + cot B) if a2 + b2 = 1989 c2. |
| 11. a1, a2, ... , a121 is a sequence of positive integers not exceeding 1000. The value n occurs more frequently than any other, and m is the arithmetic mean of the terms of the sequence. What is the largest possible value of [m - n]? |
|
12. A tetrahedron has the edge lengths shown. Find the square of the distance between the midpoints of the sides length 41 and 13.
|
| 13. Find the largest possible number of elements of a subset of {1, 2, 3, ... , 1989} with the property that no two elements of the subset have difference 4 or 7. |
| 14. Any number of the form M + Ni with M and N integers may be written in the complex base (i - n) as am(i - n)m + am-1(i - n)m-1 + ... + a1(i - n) + a0 for some m >= 0, where the digits ak lie in the range 0, 1, 2, ... , n2. Find the sum of all ordinary integers which can be written to base i - 3 as 4-digit numbers. |
|
15. In the triangle ABC, the segments have the lengths shown and x + y = 20. Find its area.
|
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
AIME home
© John Scholes
jscholes@kalva.demon.co.uk
14 July 2003
Last updated/corrected 14 Jul 03