Use all nine digits 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once across the numerator and denominator to create a fraction equal to $\frac13$. What is the numerator? The conditions do not determine a unique numerator; enter any one valid numerator.
**Input:** Enter one valid numerator as an integer, without the denominator.
Look at the distance–time graph. Which listed interval has the highest average rate of change of distance? The vertical axis is distance (km); the horizontal axis is time (minutes).
A tessellation covers a plane without gaps or overlaps. The triangular example below can continue infinitely in horizontal and vertical directions. Consider three identical regular pentagons. In how many ways can three such pentagons be arranged so that they form a tessellation?
Interpret this as asking whether an arrangement can extend to a tiling of the plane using only identical regular pentagons.
A mother makes $x$ cakes in 2 hours and $y$ pizzas in 3 hours. How many cakes and pizzas altogether can she make in 6 hours? Assume she makes both together, each at the same constant rate as when making it separately.
The absolute value of $N$, symbolized by $|N|$, represents the distance from the signed number $N$ to the origin, without regard to sign. For example, $|+5|=5$ and $|-3|=3$. Find the sum of the integer values of $N$ that satisfy $2<|N-3|<5$.
Using the digits 3, 4, 5, 6, 7 exactly once each, form the largest and the smallest numbers. What is the difference between the place values of the digit 7 in these two numbers?
A palindrome reads the same forwards and backwards. The number 2017102 is a 7-digit palindrome. Let $A$ be the least palindrome greater than 2017102, and $B$ the greatest palindrome less than 2017102. Find $(A-B)/10$.
Suppose $\frac2N$, $\frac3N$ and $\frac5N$ are all fractions in lowest terms. How many composite whole-number values of $N$ are there from 20 to 150 inclusive?
A car travels on a road beside a train track at a constant 22 km/h. A train is 250 metres long and travels at a constant 17 km/h. From the moment the car passes the rear of the train, it takes M minutes to reach the front. Find M. Assume the car and train move in the same direction; track the same point on the car from the rear to the front of the train, so car length is not included.
The counting numbers are arranged five at a time in every row, as shown. What is the least counting number in the row in which 214 will eventually appear?
For example, $2^5=2\times2\times2\times2\times2$. If $4096=a^b$, find the sum of all possible values of $a$, where $a$ is a positive integer and $b$ is a whole number greater than 1.
A lattice point in a Cartesian coordinate system has integer $x$- and $y$-coordinates. How many lattice points are on the line through $(2,4)$ and $(3,6)$? The line extends infinitely in both directions.
