# 2015 AMC 10A Problems

## Problem 1

What is the value of $(2^0-1+5^2-0)^{-1}\times5?$

## Problem 3

Ann made a 3-step staircase using 18 toothpicks. How many toothpicks does she need to add to complete a 5-step staircase?

(A) 9 (B) 18 (C) 20 (D) 22 (E) 24

## Problem 4

Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?

## Problem 6

The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?

## Problem 8

Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$?

## Problem 20

A rectangle has area $A$ $\text{cm}^2$ and perimeter $P$ $\text{cm}$, where $A$ and $P$ are positive integers. Which of the following numbers cannot equal $A+P$?

$\textbf{(A) }100\qquad\textbf{(B) }102\qquad\textbf{(C) }104\qquad\textbf{(D) }106\qquad\textbf{(E) }108$

## Problem 23

The zeros of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of the possible values of $a$?

$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$