COUNTING
In a class of 18 students, a teacher always selects two students to clean up the room after class.
If each student can be selected at most twice, for how many weeks can the teacher select students for cleanup without repetitions?
18.
A graph with 18 vertices, each vertex has two edges. The number of edges = number of vertices = 18.
How many perfect cube divisors does 1,000,000,000 have?
16
109 = 29*59. For perfect cubes, we need combinations of (20, 23, 26, 29) with (50, 53, 56, 59).
What is the remainder when 456,564,465,645 is divided by 6?
3
It is divisible by 3 (from the sum of digits) but not even, so its remainder (mod 6) has to be 3.
What is the measure of the interior angle in a regular pentagram?
36o
Either by the Inscribed Angle Theorem (half of the central angle 360o/5), or by isosceles + interior angle of a regular pentagon.
The sequence 1, 1, 2, 3, 5, 8, . . . is called the Fibonacci sequence. Each successive term is the sum of the previous two terms.
How many odd numbers are there among the first 1,000 terms of the sequence?
667
The sequence goes O, O, E, O, O, E, so there are 666 odd numbers among the first 999 and the 1000th is also odd.
In how many ways can a chess knight move from the lower left corner of a standard chess board to the upper right corner, if it can only move up and to the right?
0.
Each knight move adds either 2+1 or 1+2 to the sum of horizontal and vertical positions. We start at (1,1) and end at (8,8). We cannot move from sum=2 to sum=16 by steps of 3.
What is the greatest divisor of 1800 that is not a multiple of 30?
225
1800 = 23*32*52. We must exclude divisors that have a 2, a 3, and a 5 all present, so the choice is between “biggest without 2”, “biggest without 3” and “biggest without 5”.
A679B is a five-digit number divisible by 72.
What is the value of A+B?
5
We need divisibility by 8 and 9, so the last 3 digits must be divisible by 8, which gives B=2. The sum of digits must be divisible by 9, which gives us the value of A = 3.
What is the largest number of interior right angles that 7-gon can have?
5
If it has 6, the sum of angles would be less than 6*90o+360o=900o but it must be 5*180o=900o
What is the sum of all positive divisors of 80?
961
400=24*52, so the divisors are 1, 2, 4, 8, 16 and 5*those, and 25*those.
Hence the sum is (1+2+16)*(1+5+25)=312=961.
In Harry Potter’s world, wizarding money comes in three coins: bronze Knuts, silver Sickles, and golden Galleons.
29 Knuts make up one Sickle, and there are 17 Sickles in a Galleon.
In how many ways can 2 Galleons be made as a combination of Galleons, Sickles and Knuts?
Split into cases, by the number of Galleons used.
1 + 18 + 35 = 54
What is the sum of the three numbers less than 1000 that have exactly five positive integer divisors?
722
If the number has 5 divisors, it must have only one prime factor with power 4. 16 + 81 + 625 = 722.
How many ordered triples of three prime numbers exist for which the sum of the members of the triple is 24?
15
One of the primes has to be 2 (otherwise the sum would be odd).
(2,3,19) x 6; (2,5,17) x 6; (2,11,11) x3
What is the area of a regular 12-gon inscribed in a circle with radius 10?
300
Split into 6 sectors. For each sector, the area of the quadrilateral is half the product of its diagonals (as they are orthogonal). Both diagonals have length 10 (regular hexagon).
In how many ways can you cover a 2*5 shape with dominoes?
(You are not allowed to rotate, so ==| and |== are considered distinct.)
8
Consider smaller cases:
2*1 – one way (|)
2*2 – two ways (|| or =)
2*3 – start with | → number of ways to do 2*2; start with = → number of ways to do 2*1; so 2+1=3
2*4 = ways for 2*3 + ways for 2*2 ; so 3+2=5
2*5 ; same idea: 5+3=8