For which integer value of a does the inequality \( \frac{1}{6} < \frac{a}{12} < \frac{1}{3} \) hold true?
The sum of three consecutive integers is 99. What is the smallest of these three numbers?
A motorcyclist traveled 30 km in 30 minutes and returned in 1 hour. What was the average speed for the entire trip?
A book has 300 pages. How many digits are used to number all pages starting from 1?
A large \( 3 \times 3 \times 3 \) cube is made of 27 smaller cubes and painted on the outside. How many small cubes have no paint on any side?
Find the number of three-digit numbers where the sum of the digits is a perfect cube.
An isosceles triangle has a perimeter of 23. How many such triangles exist with integer side lengths?
How many two-digit numbers have a tens digit that is greater than the units digit?
A regular 12-sided polygon (dodecagon) is drawn. What is the maximum number of diagonals that can be drawn such that they do not intersect inside the polygon?
In a triangle with integer side lengths and a perimeter of 49, what is the smallest possible length for the longest side?
Find the area of the shape drawn on the squared notebook paper. Consider each small grid square of the notebook paper as 1 unit of area (\( 1 \times 1 \)).
Find the sum of all integer values of x that satisfy the inequality: