# 2010 PUMaC Problems

## Contents

## Division A

### Algebra

Find the sum of the coefficients of the polynomial .

Calculate where is the largest integer less than or equal to .

Let be the sum of all real such that . Find the nearest integer to .

Define . Find the smallest integer such that .

Let . If the roots of are given by , , and , find

Assume that , and that . Find .

The expression is equal to , where is an integer. Find .

Let be a polynomial with integer coefficients such that , , and . Find an integer such that .

### Combinatorics

PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, the probability that the 4th customer orders a previously unordered item is , where and are relatively prime positive integers. Find .

Let represent the three-digit number with hundreds digit , tens digit , and units digit , and similarly let represent the two-digit number with tens digit and units digit . How many three-digit numbers , none of whose digits are 0, are there such that ?

Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region). What is the maximum number of resulting regions?

Erick stands in the square in the 2nd row and 2nd column of a 5 by 5 chessboard. There are $1 bills in the top left and bottom right squares, and there are $5 bills in the top right and bottom left squares, as shown below. Every second, Erick randomly chooses a square adjacent to the one he currently stands in (that is, a square sharing an edge with the one he currently stands in) and moves to that square. When Erick reaches a square with money on it, he takes it and quits. The expected value of Erick's winnings in dollars is , where and are relatively prime positive integers. Find .

We say that a rook is ``attacking" another rook on a chessboard if the two rooks are in the same row or column of the chessboard and there is no piece directly between them. Let be the maximum number of rooks that can be placed on a chessboard such that each rook is attacking at most one other. How many ways can rooks be placed on a chessboard such that each rook is attacking at most one other?

All the diagonals of a regular decagon are drawn. A regular decagon satisfies the property that if three diagonals concur, then one of the three diagonals is a diameter of the circumcircle of the decagon. How many distinct intersection points of diagonals are in the interior of the decagon?

Matt is asked to write the numbers from 1 to 10 in order, but he forgets how to count. He writes a permutation of the numbers across his paper such that:

- The leftmost number is 1.
- The rightmost number is 10.
- Exactly one number (not including 1 or 10) is less than both the number to its immediate left and the number to its immediate right.[/list]

How many such permutations are there?

Let be the sum of all binomial coefficients such that and are nonnegative integers and is an even integer less than 100. Find the remainder when is divided by 144. (Note: if , and .)

### Geometry

As in the following diagram, square and square are placed side by side (i.e. is between and and is between and ). If , , compute the minimal area of .

In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners , he starts at and walks to . Then, he walks to the midpoint of side , say . Then, he walks to the midpoint of side say , and then the midpoint of which is . He continues in this fashion, indefinitely. The total length of his path if and is of the form . Find .

Triangle has , , and . An angle bisector is drawn from angle , and meets at . What is the nearest integer to ?

In regular hexagon , , are two diagonals. Points , are on , respectively and satisfy . Suppose are collinear, find . (diagram goes here) Solution

A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle (see picture). Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is . Find . (diagram goes here)

In the following diagram, a semicircle is folded along a chord and intersects its diameter at . Given that and . If , find . Solution

Square is divided into four rectangles by and . is parallel to and parallel to . . and meet at point . The area of rectangle is twice that of rectangle . Given that the value of in degrees is , find the nearest integer to . Solution

There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?

### Number Theory

Find the smallest positive integer such that is composite.

Find the largest positive integer such that , where is the sum of the divisors of , including .

Find the sum of the first 5 positive integers such that is the product of 3 distinct primes.

Find the largest positive integer such that is a perfect square. ( is the number of integers , that are relatively prime to )

Given that , are positive integers with , but neither nor divides either of or , and as small as possible, find .

Find the numerator of when reduced.

Let be the number of polynomial functions from the integers modulo to the integers modulo . can be written as , where the s are (not necessarily distinct) primes. Find .

A consecutive pythagorean triple is a pythagorean triple of the form , and positive integers. Given that , , and form the third consecutive pythagorean triple, find .

## Division B

### Algebra

Let the operation be defined by . Calculate .

Let . Find the fourth smallest prime such that is divisble by for some integer .

Write , with , , , , and integers. Find .

Let be the sum of all real such that . Find the nearest integer to .

Let be a real root of the polynomial . Find .

Define . Find the smallest integer such that .

Let be a function such that and . Find the last two digits of .

The expression is equal to , where is an integer. Find .

### Combinatorics

The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs?

PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, the probability that the 4th customer orders a previously unordered item is , where and are relatively prime positive integers. Find . % Yeah, I stole the PUMaCDonalds joke from another round (the team round?). We can change the problem statement to "A newly-opened fast food restaurant has 5 menu items..." if that's better. Let represent the three-digit number with hundreds digit , tens digit , and units digit , and similarly let represent the two-digit number with tens digit and units digit . How many three-digit numbers , none of whose digits are 0, are there such that ?

Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region). What is the maximum number of resulting regions?

people take part in a chess tournament: girls and boys. Each participant plays with each of the others exactly once. There were no ties and the number of games won by the girls is the number of games won by the boys. How many people took part in the tournament?

A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of the region it's in, with an equal probability of choosing each edge, and crosses that edge into another region. If the ant enters an unbounded region, it explodes. After first leaving the central region of the pentagon, let be the expected number of times the ant re-enters the central region before it explodes. Find the closest integer to .

We say that a rook is ``attacking" another rook on a chessboard if the two rooks are in the same row or column of the chessboard and there is no piece directly between them. Let be the maximum number of rooks that can be placed on a chessboard such that each rook is attacking at most one other. How many ways can rooks be placed on a chessboard such that each rook is attacking at most one other?

Matt is asked to write the numbers from 1 to 10 in order, but he forgets how to count. He writes a permutation of the numbers across his paper such that:

- The leftmost number is 1.
- The rightmost number is 10.
- Exactly one number (not including 1 or 10) is less than both the number to its immediate left and the number to its immediate right.

How many such permutations are there?

### Geometry

In a polygon, every external angle is one sixth of its corresponding internal angle. How many sides does the polygon have?

On rectangular coordinates, point , . is on -axis. Given that is chosen such that is minimized, compute .

As in the following diagram, square and square are placed side by side (i.e. is between and and is between and ). Now , , compute the minimal area of . (diagram goes here)

Unit square is divided into four rectangles by and , with . is parallel to and parallel to . and meet at point . Suppose , calculate the nearest integer to the degree of . (diagram goes here)

In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners , he starts at and walks to . Then, he walks to the midpoint of side , say . Then, he walks to the midpoint of side say , and then the midpoint of which is . He continues in this fashion, indefinitely. The total length of his path if and is of the form . Find .

In regular hexagon , , are two diagonals. Points , are on , respectively and satisfy . Suppose are collinear, find . (diagram goes here)

A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle (see picture). Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is . Find . (diagram goes here)

Point is in the interior of . The side lengths of are , , . The three foots of perpendicular lines from to sides , , are , , respectively. Suppose the minimal value of can be written as , where and is square free, calculate . (diagram goes here)

### Number Theory

Find the positive integer less than 18 with the most positive divisors.

Let be the sum of the digits of . Find .

Find the smallest positive integer such that is composite.

Find the sum of the first 5 positive integers such that is the product of 3 distinct primes.

Given that , , and are positive integers such that . Find the sum of all possible values.

Given that , are positive integers with , but neither nor divides either of or , and as small as possible, find .

Find the numerator of when reduced.

Let be the number of (positive) divisors of ending in the digit . What is the remainder when is divided by 2010?