It is given that the 4-digit number $\overline{abab}$ has 6 factors. What is the smallest possible value of $\overline{abab}$?
**Editorial note for this self-study exercise:** The overline denotes the four decimal digits a, b, a, b; a is nonzero. Count positive integer divisors.
The points A, B, C, D and E are arranged as shown so that no three lie on a straight line. How many different triangles can be formed by connecting three of these points as vertices?
The same 20 contestants, on each of 3 days, answered 5 questions in order to win a prize. Each contestant received 1 point for each correct answer. The number of contestants receiving a given score on each day was shown in the table below.
**Number of Contestants by Score and Day**
| Day | 5 out 5 | 4 out 5 | 3 out 5 | 2 out 5 | 1 out 5 | 0 out 5 | Total |
|---|---|---|---|---|---|---|---|
| Day 1 | 2 | 3 | 4 | 6 | 2 | 3 | 20 |
| Day 2 | 2 | 3 | 5 | 5 | 4 | 1 | 20 |
| Day 3 | 3 | 3 | 4 | 5 | 3 | 2 | 20 |
| Total | 7 | 9 | 13 | 6 | 9 | 6 | 60 |
What was the mean score of the contestants on Day 1?
**Editorial note for this self-study exercise:** Use the Day 1 row for the requested mean. The source table is preserved, including an erroneous bottom-row total in the two-point column; that total is not needed for this question. Use a decimal point (.) if you enter a decimal.
Define $n!=n\times(n-1)\times\cdots\times2\times1$. For example, $4!=4\times3\times2\times1=24$ and $3!=3\times2\times1=6$. Simplify
$$\frac{29!\times29!}{28!\times28!}.$$
Mel sells jumbo cheesy pretzel nuggets in a 4-nugget “diet” size, a 7-nugget “share” size and a 17-nugget “party” size. What is the greatest total number of nuggets that cannot be purchased? For example, 8 nuggets can be purchased, but 9 cannot. Assume any nonnegative whole number of each pack may be bought, and packs cannot be split.
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?
Find the value of A. (Round off to the nearest whole number)
**Editorial note for this self-study exercise:** There are ten products: the first factor runs from 1.01 to 1.10 in steps of 0.01, while the second runs from 1.19 down to 1.10. Enter the rounded whole number.
A square is inscribed in a circle with diameter 2. Four smaller circles are then constructed with their diameters on each of the sides of the square. Determine the area of the shaded region.
**Editorial note for this self-study exercise:** Only the four exterior crescent-shaped regions shown are shaded. Enter their total area in square units as a number; the diameter has unspecified length units.
A ladder has 7 rungs. An ant climbs from A, the middle of the bottom rung, to B, the middle of the top rung, without retracing any part of its path. How many different paths are possible? Count routes along the rungs and side rails only, ignoring pauses; no segment may be traversed twice.
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.
Mrs. Rama bought a box of paper clips. She shared the paper clips among her 4 children. Alice received 190 paper clips. Rita received 56 more paper clips than Jim. Jim received 25% of the paper clips while Tom received 20% of the paper clips. How many paper clips did Mrs. Rama buy altogether?
**Editorial note for this self-study exercise:** Assume all the paper clips are distributed among exactly these four children, with none left over.
William puts some red, blue and white marbles into a box. The number of red marbles is 1/3 of the number of blue marbles. The number of white marbles is 1/4 of the number of red marbles. If there are 144 more blue marbles than red marbles, how many marbles are there in the box?
Town A and Town B are 435km apart. At 9 : 15 a.m, a car left Town A and traveled toward Town B at a constant speed of 80km/h. At the same time, a lorry left B and traveled toward Town A at a constant speed. When they met, the car traveled a distance of 240km. Find the speed of the lorry.
**Editorial note for this self-study exercise:** Both vehicles use the same route, travel without stopping, and meet between A and B. Enter the lorry speed in km/h, without units.
