AMC 10 specimen paper

One round of a process adds 9 to a two-digit positive integer and then reverses the resulting digits. The process is allowed only if both the sum and its reversal have exactly two digits. How many unordered pairs of distinct two-digit integers form a two-round cycle, meaning each maps to the other in one round?

A school has 360 pupils. Exactly half the seniors and three-quarters of the non-seniors play an instrument; in all, two-thirds of the pupils play. Then 24 instrument-playing non-seniors become seniors, retaining their instrument status. What fraction of the remaining non-seniors play an instrument?

A bag has 9 red, 7 blue, 5 green and 3 yellow counters. Without replacement, what is the least number of draws guaranteeing that two distinct colours have each appeared at least 4 times?

A 24-hour clock displays four digits HHMM, including leading zeroes, from 00:00 through 23:59. How many displays have four distinct digits and a digit sum divisible by 7?

Let S be the sum of 7 + 5j for integer j from 0 through 48, and let T be the sum of 7 + 5j for integer j from 2 through 50. Find T − S.

A box has side lengths 2 t, 3 t and 5 t centimetres, where t is a positive integer. Its volume is at most 2000 cubic centimetres. Find the greatest possible surface area, in square centimetres.

Find all positive integers n for which 3/4 + 1/n + 1/(2n) is an integer.

Two affine functions satisfy f(-5) = 8, f(7) = 38, g(0) = -3 and g(20) = 7. How many integers n from -100 through 100 satisfy both f(g(n)) >= n and g(f(n)) < n?

List all distinct arrangements of the multiset A, A, B, B, C in alphabetical order, starting with position 1. What is the position of BACAB?

Positive integers a < b are the first and last terms of both a four-term arithmetic progression of integers and a five-term geometric progression of integers with positive common ratio. Suppose gcd(a, b) = 1 and b <= 625. What is the sum of the common differences of all possible arithmetic progressions?

A sequence satisfies v(n + 2) = 3v(n + 1) − v(n). Given v(2) = 4 and v(5) = 76, find v(4).

Two readers independently choose uniformly random four-book subsets from books labelled 1 through 9. Given that both include book 1 and that their overlap contains at least two books, what is the probability that their union contains exactly six books?

How many primes p with 30 <= p <= 200 have p + 6 prime and ((p + 6)^3 - p^3)/6 divisible by 7?

From an allowance A, food costs one quarter of what remains after travel. Travel costs one fifth of what remains after food, plus a fixed $2fee.Thetotalfoodandtravelcostis$12. Find A in dollars.

Let A = (0, 0), B = (12, 8), C = (0, 7), D = (10, 0), and let E be the intersection of A B and C D. How many integer lattice points lie strictly between A and E?

Tokens X, Y and Z have values 1, 4 and 16. An exchange replaces one X by four Y tokens, one Y by four Z tokens, or one Z by four X tokens. Starting with one X, which of the total values 100, 102 and 104 can be reached? Any finite number of exchanges is allowed.

Label the eight vertices of a cube with 1 through 8, using each label once, so that opposite vertices have labels summing to 9. Labelings related by a rotation of the cube are equivalent; reflections are distinct. How many equivalence classes are there?

A cube is centred at the origin with faces at x, y and z equal to ±3 centimetres. A square bore of side 2 runs along the entire x-axis, a square bore of side 4 along the entire y-axis, and a square bore of side 2 along the entire z-axis. Every bore is centred on its axis and has sides parallel to the cube edges. Find the volume left.

How many distinct positive integers divisible by 3 can be written as a + 5 b + 25 c + 125 d, with each coefficient in {-2, -1, 0, 1, 2} and exactly two coefficients nonzero?

Let a_1 = 2, a_2 = 3, a_3 = 5, and for n at least 4 define a_n = a_(n-3) + a_(n-2) - a_(n-1) + 1. Find the least index n for which a_n is at least 300.

In triangle OAB, P lies on AB with AP: PB = 2: 3. The line through P parallel to OA meets OB at Q. The line through Q parallel to AB meets OA at R. Find area(OQR)/area(OAB).

For n from 2 through 180, let S(n) be the sum of (-1)^i * i * j over all integers 1 <= i < j <= n with j - i >= 2. How many n make S(n) divisible by 21?

For each integer t with 3 ≤ t ≤ 30 choose a positive real a satisfying a + 1/a = t. For how many t is a⁶ + a^( − 6) an integer divisible by 7? The count must not depend on which positive root a is chosen.

Each face of a cube has a track joining the midpoints of one pair of opposite edges. On the top and bottom faces, the orientation parallel to the x-axis is selected independently with probability 2/3, and the y-axis orientation otherwise. On each of the other four faces its two orientations are selected independently with probability 1/2. What is the probability at least one continuous track loops around the cube?

Use the Gregorian calendar, in which century years are leap years only when divisible by 400. Given that 1 January 1900 was Monday, how many years from 1900 through 2100 inclusive have 29 February on a Sunday?