A two-digit number is 6 more than 7 times the sum of its digits. Its tens digit is 2 more than twice its ones digit. How many such two-digit numbers are there?
A steel cuboid is reshaped into a cube. Initially, its length, breadth, and depth were 270 cm, 100 cm, and 64 cm respectively. Find the sum of digits of the surface area of the cube.
Assume no steel is lost or added and neither shape contains voids. Use the numerical surface area in square centimeters.
The product of two or more consecutive positive integers is a six-digit palindrome whose first two digits are 47. What is the largest integer in the product? A palindrome reads the same backwards and forwards. Use positive integers, as intended by the source.
40 people are in a movie theatre which offers two snack options: popcorn and chips. 26 people like popcorn but do not like chips. 32 people like popcorn. Everyone in the theatre likes at least one of the two snack options. Find the number of people who like both.
Several people were surveyed about their preference among four drinks A,B,C,D. Each person chose one drink, and everyone surveyed is represented in the graph. If the fraction preferring A is $m/n$ in lowest terms, find $m+n$.
The cardboard pattern consists of six numbered squares and is folded to form a cube. What is the sum of the digits in the greatest product of the numbers on three faces that meet at a common vertex?
Andy and Brett begin at opposite points on a circular track of circumference 300 feet and jog in the directions shown. Andy jogs at 4.3 feet per second and Brett at 5.5 feet per second. How many seconds will it take for them to meet for the first time?
The spinner is divided into five equal parts. Spin once. If it lands on a perfect square number, flip a fair coin once. If it lands on a prime number, do not flip the coin. What is the probability that you flipped a coin and it landed heads up? If that probability is $p/q$ in simplest form, choose the value of $p+q$.
A cricket climbs a staircase by jumping either 1 step or 2 steps each time. In how many different ways can it climb a 7-step staircase? For example, 3 steps can be climbed in exactly three ways: 1+1+1, 1+2, or 2+1. The order of jumps matters.
Find all ordered pairs of integers $(x,y)$ satisfying $x^y=16$. Sum the x-coordinates of all those pairs and separately sum their y-coordinates, counting one coordinate per pair. Find the positive difference between these two sums.
In triangle ABC, point D lies on side AC with $AD=4$ and $CD=7$. Triangle ABD has area 20 square units. What is the area of triangle ABC, in square units?
Thirty-six points are arranged in a unit-square array as shown. Figure $ABCDEFG$ is composed entirely of straight-line segments, with vertices $A,B,C,D,E,F,G$. Find its area in square units.
