Number Theory
Find gcd(42, 70), the greatest common divisor of 42 and 70.
What is 14?
Solution:
42 = 2 * 3 * 7
70 = 2 * 5 * 7
Common prime factors: 2 and 7
gcd(42, 70) = 2 * 7 = 14
In a right triangle, angle A is 30 degrees and the hypotenuse is 12.
Find the length of the side opposite angle A.
What is 6?
In a right triangle, sin(angle) = opposite / hypotenuse.
So sin(30 degrees) = opposite / 12.
We know sin(30 degrees) = 1/2.
So 1/2 = opposite / 12.
Multiply both sides by 12: opposite = 12 * (1/2) = 6.
Solve for x:
3(x - 2) = 2x + 7
What is 13?
How many different orders can the letters A, B, and C be arranged in?
What is 6?
A student’s first three test scores are 80, 90, and 95.
What score must the student get on the fourth test to have an average of 90 over all four tests?
What is 95?
Find the smallest positive integer N that leaves a remainder of 1 when divided by 2, 3, and 4.
What is 13?
N leaves remainder 1 when divided by 2, 3, and 4, so N - 1 is divisible by 2, 3, and 4.
So N - 1 is a multiple of lcm(2, 3, 4).
lcm(2, 3, 4) = 12.
Smallest positive multiple of 12 is 12, so N - 1 = 12, and therefore N = 13.
An isosceles right triangle has hypotenuse 10. The two legs are equal.
Find the length of each leg and the area of the triangle.
What is 25?
Let each leg have length x.
By the Pythagorean theorem:
x^2 + x^2 = 10^2
2x^2 = 100
x^2 = 50
x = sqrt(50) = 5 * sqrt(2).
Area of a right triangle = (1/2) * leg1 * leg2
= (1/2) * x * x
= (1/2) * x^2
= (1/2) * 50
= 25.
Solve the system of equations:
2x - y = 5
x + 2y = 11
What is x = 21/5, y = 17/5 ?
From a group of 5 students, how many different 3-student committees can be formed?
What is 10?
We are choosing 3 students out of 5, order does not matter.
Number of combinations = C(5, 3) = 5! / (3! * 2!)
= (5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (2 * 1))
= 120 / (6 * 2)
= 120 / 12
= 10
A car travels from town A to town B at 60 miles per hour and returns from B to A along the same route at 40 miles per hour.
What is the average speed for the entire round trip?
What is 48?
Time going from A to B: D / 60 hours.
Time coming back from B to A: D / 40 hours.
Total distance = 2D.
Total time = D/60 + D/40.
Compute the total time:
D/60 + D/40 = (2D/120) + (3D/120) = 5D/120 = D/24.
Average speed = (total distance) / (total time)
= (2D) / (D/24)
= 2D * (24 / D)
= 48.
Find the last digit of 7^2025.
What is 7?
7^1 = 7 -> last digit 7
7^2 = 49 -> last digit 9
7^3 = 343 -> last digit 3
7^4 = 2401 -> last digit 1
The pattern of last digits repeats every 4 powers: 7, 9, 3, 1.
Now compute 2025 mod 4.
2024 is divisible by 4, so 2025 mod 4 = 1.
So 7^2025 has the same last digit as 7^1.
A circle has radius 10. Find the area of a sector with central angle 72 degrees.
Give your answer in terms of pi.
What is 20pi?
Full circle area = pi * r^2 = pi * 10^2 = 100 * pi.
The sector is 72/360 of the circle (since 72 degrees of 360).
Fraction of circle = 72/360 = 1/5.
Sector area = (1/5) * 100 * pi = 20 * pi
Answer: 20 * pi
Solve for x:
(x - 1)(x - 3) = 10
What is x = 2 + sqrt(11) or x = 2 - sqrt(11) ?
How many distinct 3-digit numbers can be formed using the digits 1, 2, 3, 4, and 5 if no digit is repeated?
What is 60?
For a 3-digit number (no repetition):
First digit: 5 choices (1, 2, 3, 4, 5)
Second digit: 4 choices (remaining digits)
Third digit: 3 choices
Total = 5 * 4 * 3 = 60
A two-digit number has a digit sum of 11. The number is 45 more than the number formed by reversing its digits.
What is the original two-digit number?
What is 83?
The original number is 10a + b.
The reversed number is 10b + a.
We are told:
a + b = 11
10a + b = (10b + a) + 45
Simplify equation 2:
10a + b = 10b + a + 45
10a - a + b - 10b = 45
9a - 9b = 45
Divide both sides by 9:
a - b = 5.
Now we have the system:
a + b = 11
a - b = 5
Add the equations:
( a + b ) + ( a - b ) = 11 + 5
2a = 16
a = 8.
Then b = 11 - a = 11 - 8 = 3.
So the original number is 10a + b = 10 * 8 + 3 = 83.
How many positive integers n with 1 ≤ n ≤ 100 are divisible by 3 or 5 but NOT by 7?
What is 41?
Step 1: Count numbers from 1 to 100 that are divisible by 3 or 5.
Multiples of 3 up to 100: floor(100 / 3) = 33.
Multiples of 5 up to 100: floor(100 / 5) = 20.
Multiples of both 3 and 5 (i.e., 15) up to 100: floor(100 / 15) = 6.
By inclusion-exclusion,
count(divisible by 3 or 5) = 33 + 20 - 6 = 47.
Step 2: Subtract those that are also divisible by 7.
These are numbers divisible by 7 and (3 or 5).
So they are multiples of 21 (37) or 35 (57).
Multiples of 21 up to 100: 21, 42, 63, 84 → 4 numbers.
Multiples of 35 up to 100: 35, 70 → 2 numbers.
LCM(21, 35) = 105, which is greater than 100, so there is no overlap.
Total to subtract = 4 + 2 = 6.
So the count we want is
47 - 6 = 41.
A triangle has side lengths 13, 14, and 15.
Find the area of the triangle.
What is 84?
Use Heron’s formula.
First compute the semiperimeter s:
s = (13 + 14 + 15) / 2 = 42 / 2 = 21.
Area = sqrt[ s(s - a)(s - b)(s - c) ]
= sqrt[ 21 * (21 - 13) * (21 - 14) * (21 - 15) ]
= sqrt[ 21 * 8 * 7 * 6 ].
Compute step by step:
21 * 8 = 168
7 * 6 = 42
168 * 42 = 7056
So area = sqrt(7056).
84^2 = 7056, so sqrt(7056) = 84.
Real numbers x and y satisfy
x + y = 5
and
xy = 6.
Find the value of x^3 + y^3.
What is 35?
Use the identity:
x^3 + y^3 = (x + y)^3 - 3xy(x + y).
We know x + y = 5 and xy = 6.
Compute (x + y)^3:
5^3 = 125.
Compute 3xy(x + y):
3 * 6 * 5 = 90.
So
x^3 + y^3 = 125 - 90 = 35.
A class has 6 boys and 4 girls.
How many different 3-student teams can be formed that include at least one girl?
What is 100?
Total number of 3-student teams from 10 students:
C(10, 3) = 10! / (3! * 7!) = 120
Number of teams with no girls = all boys: choose 3 from 6 boys:
C(6, 3) = 6! / (3! * 3!) = 20
Teams with at least one girl = total teams - all-boy teams
= 120 - 20
= 100
there exist four circles that are tangent to each other. the sum of their reciprocals is 18. Find the sum of the squares of their reciprocals.
After 3 minutes, the answer will be shown, and no team will get the point.
What is 162
This is stated by the Descartes Circle Theorem. (a+b+c+d)^2 = 2(a^2+b^2+c^2+d^2), where a,b,c, and d are the reciprocals of the radius of the circles.
However, as the question is too hard, the answer "David" is also acceptable.
Find the smallest positive integer n such that:
n ≡ 2 (mod 5)
n ≡ 3 (mod 7)
n ≡ 4 (mod 9)
What is 157?
First combine the congruences modulo 5 and 7.
Solve
n ≡ 2 (mod 5)
n ≡ 3 (mod 7)
Write n = 5k + 2.
Substitute into the second congruence:
5k + 2 ≡ 3 (mod 7)
5k ≡ 1 (mod 7)
We need the inverse of 5 modulo 7.
5 * 3 = 15 ≡ 1 (mod 7), so 3 is the inverse.
Thus k ≡ 3 * 1 = 3 (mod 7).
So k = 7t + 3 for some integer t.
Then
n = 5k + 2 = 5(7t + 3) + 2 = 35t + 15 + 2 = 35t + 17.
So n ≡ 17 (mod 35).
Now use n ≡ 17 (mod 35) and n ≡ 4 (mod 9).
Write n = 35m + 17.
Substitute into n ≡ 4 (mod 9):
35m + 17 ≡ 4 (mod 9)
Note 35 ≡ 35 - 27 = 8 (mod 9), so:
8m + 17 ≡ 4 (mod 9)
8m ≡ 4 - 17 ≡ -13 (mod 9)
Since -13 ≡ -13 + 18 = 5 (mod 9), we get
8m ≡ 5 (mod 9).
Find inverse of 8 modulo 9.
8 * 8 = 64 ≡ 1 (mod 9), so inverse of 8 is 8.
Thus
m ≡ 8 * 5 ≡ 40 (mod 9).
40 ≡ 40 - 36 = 4 (mod 9).
So m = 9k + 4.
Then
n = 35m + 17 = 35(9k + 4) + 17 = 315k + 140 + 17 = 315k + 157.
A triangle has side lengths 5, 5, and 6.
Find the radius R of the circumscribed circle (the circumradius).
What is 25/8?
Solution:
Step 1: Find the area using Heron’s formula.
Side lengths: a = 5, b = 5, c = 6.
Semiperimeter:
s = (5 + 5 + 6) / 2 = 16 / 2 = 8.
Area = sqrt[ s(s - a)(s - b)(s - c) ]
= sqrt[ 8 * (8 - 5) * (8 - 5) * (8 - 6) ]
= sqrt[ 8 * 3 * 3 * 2 ]
= sqrt[ 8 * 18 ]
= sqrt[ 144 ]
= 12.
Step 2: Use formula for circumradius:
R = (a * b * c) / (4 * area).
So
R = (5 * 5 * 6) / (4 * 12)
= 150 / 48
Simplify: divide numerator and denominator by 6:
150 / 48 = 25 / 8.
Suppose x + 1/x = 3.
Find the value of x^2 + 1/x^2.
What is 7?
Find x^2 + 1/x^2.
We have
(x + 1/x)^2 = x^2 + 2 + 1/x^2.
So
x^2 + 1/x^2 = (x + 1/x)^2 - 2
= S1^2 - 2
= 3^2 - 2
= 9 - 2
= 7.
A 5-character password is formed using the digits 0 through 9. Digits may repeat.
How many such passwords contain at least one digit 7?
What is 40951?
Total passwords with 5 positions and 10 choices each:
10^5 = 100000
Passwords with no 7: for each position, 9 choices (any digit except 7).
So number with no 7 is 9^5 = 59049
Passwords with at least one 7 = total - no 7
= 100000 - 59049
= 40951
How many 4-digit positive integers have a digit sum equal to 7?
(For example, 1006 has digit sum 1 + 0 + 0 + 6 = 7.)
What is 84?
Let the 4-digit number have digits d1, d2, d3, d4, where d1 is the thousands digit.
So d1 is from 1 to 9 and d2, d3, d4 are from 0 to 9.
We want
d1 + d2 + d3 + d4 = 7.
Since d1 ≥ 1, set d1 = y1 + 1, where y1 ≥ 0.
Then:
(y1 + 1) + d2 + d3 + d4 = 7
y1 + d2 + d3 + d4 = 6.
Now y1, d2, d3, d4 are all nonnegative integers.
Also, because their sum is 6, none of them can exceed 6, so the digit upper bounds (9) do not cause any extra restrictions.
We are counting the number of nonnegative integer solutions to:
y1 + d2 + d3 + d4 = 6.
This is a standard “stars and bars” problem.
The number of solutions in nonnegative integers to
x1 + x2 + x3 + x4 = 6
is C(6 + 4 - 1, 4 - 1) = C(9, 3).
Compute C(9, 3):
C(9, 3) = 9 * 8 * 7 / (3 * 2 * 1) = 84.