Combinatorics | Number Theory Answers

Given non-negative integers such that (108^a) ⋅ (288^b) ⋅ (36^c) divided by 6^220. Find the smallest possible value of a + b + c.

Ten non-negative numbers are such that their sum is 3, and the sum of their squares is 5.8. What is the greatest value of the largest of these numbers?

Vasya has cubes of three colors. He builds a tower from them, placing each next cube on the previous one. It is forbidden to use more than 6 cubes of each color. Vasya finishes building the tower as soon as it contains 6 cubes of some two colors. How many different towers can Vasya build?

Let λ be a real number. Suppose that if ABCD is any convex cyclic quadrilateral such that AC = 4, BD = 5, and AB ⊥ CD, then the area of ABCD is at least λ. Then the greatest possible value of λ is m/n, where m and n are positive integers with gcd(m,n) = 1. Compute 100m+n.

On a 9×9 square lake composed of unit squares, there is a 2×4 rectangular iceberg also composed of unit squares (it could be in either orientation; that is, it could be 4×2 as well). The sides of the iceberg are parallel to the sides of the lake. Also, the iceberg is invisible. Lily is trying to sink the iceberg by firing missiles through the lake. Each missile fires through a row or column, destroying anything that lies in its row or column. In particular, if Lily hits the iceberg with any missile, she succeeds. Lily has bought missiles and will fire all of them at once. Let N be the smallest possible value of n such that Lily can guarantee that she hits the iceberg. Let M be the number of ways for Lily to fire N missiles and guarantee that she hits the iceberg. Compute 100M+N

Compute the number of ordered triples of integers (a,b,c) between 1 and 12, inclusive, such that, if q =) a + ( 1 / b ) − ( 1 / (b + 1 / c )), then q is a positive rational number and, when q is written in lowest terms, the numerator is divisible by 13.

An alien from the planet OMO Centauri writes the first ten prime numbers in arbitrary order as U,W, XW, ZZ, V, Y, ZV, ZW, ZY, and X. Each letter represents a nonzero digit. Each letter represents the same digit everywhere it appears, and different letters represent different digits. Also, the alien is using a base other than base ten. The alien writes another number as UZWX. Compute this number (expressed in base ten, with the usual, human digits).

Compute the number of ways to write the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the cells of a 3 by 3 grid such that
•each cell has exactly one number
,•each number goes in exactly one cell,
•the numbers in each row are increasing from left to right
,•the numbers in each column are increasing from top to bottom, and
•the numbers in the diagonal from the upper-right corner cell to the lower-left corner cell are increasing from upper-right to lower-left.

In a certain country, telephone numbers have 9 digits. The first two digits are the area code and are the same within a given area. The last seven digits are the local number and can’t begin with 0. How many different numbers are possible within a given area code in this country?

Let us consider two irreducible fractions. The denominator of the first one is equal to 8200,and the denominator of the second to 4300. What is the smallest possible denominator of a fraction equal to the sum of these fractions, after the fraction is reduced? (For example, (2/3) + (8/15) = (18/15) = (6/5), and the denominator after the reduction is equal to 5.)