Number Theory
How many positive integer divisors does the number 144 have?
First, find the prime factorization of 144, which is 2^4 × 3^2. Using the divisor formula, add 1 to each exponent and multiply: (4 + 1)(2 + 1) = 5 × 3 = 15. The answer is 15.
What is the area of a rhombus whose diagonals have lengths of 10 and 24?
The area of a rhombus is half the product of its diagonals. Area = 1/2 × 10 × 24 = 120. The answer is 120.
If 3x - y = 12 and x + 2y = 11, what is the value of x + y?
Multiply the first equation by 2 to get 6x - 2y = 24. Add this to the second equation (x + 2y = 11) to eliminate y: 7x = 35, so x = 5. Substitute x = 5 into the second equation: 5 + 2y = 11, so 2y = 6 and y = 3. Therefore, x + y = 5 + 3 = 8. The answer is 8.
In how many different ways can 5 distinct books be arranged in a single row on a shelf?
The number of ways to arrange n distinct objects is n factorial (n!). For 5 books, this is 5! = 5 × 4 × 3 × 2 × 1 = 120. The answer is 120.
Evaluate the expression: log₂(8) + log₃(81).
Find what power the base must be raised to in order to equal the argument. 2^3 = 8, so log₂(8) = 3. 3^4 = 81, so log₃(81) = 4. Add the two values together: 3 + 4 = 7. The answer is 7.
What is the units digit of 3^2026?
The units digits of powers of 3 follow a repeating 4-step cycle: 3, 9, 7, 1. Divide the exponent by 4: 2026 ÷ 4 = 506 with a remainder of 2. A remainder of 2 means the units digit is the 2nd number in the cycle. The answer is 9.
The sum of the interior angles of a convex regular polygon is 2520 degrees. How many sides does the polygon have?
The formula for the sum of interior angles is 180(n - 2), where n is the number of sides. Set up the equation: 180(n - 2) = 2520. Divide by 180 to get n - 2 = 14. Adding 2 gives n = 16. The answer is 16.
What is the sum of the squares of the roots of the quadratic equation x^2 - 5x + 3 = 0?
Let the roots be r and s. By Vieta's formulas, the sum of the roots r + s = 5, and the product rs = 3. We want to find r^2 + s^2. Using the algebraic identity r^2 + s^2 = (r + s)^2 - 2rs, substitute the values: 5^2 - 2(3) = 25 - 6 = 19. The answer is 19.
A committee of 3 students is to be chosen from a group of 8 students. How many different 3-person committees can be formed?
Since the order of selection does not matter, use combinations: "8 choose 3". The formula is 8! / (3! × 5!) = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = 56. The answer is 56.
Let i = √-1. What is the value of i^2023 + i^2024 + i^2025 + i^2026?
The powers of i cycle every 4 terms: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. The sum of any four consecutive powers of i is always i - 1 - i + 1 = 0. Since these are four consecutive powers of i, their sum is 0. The answer is 0.
How many ordered pairs of positive integers (x, y) satisfy the equation 3x + 5y = 100?
Rearrange to 5y = 100 - 3x. For y to be a positive integer, 100 - 3x must be positive and a multiple of 5. This means 3x must be a multiple of 5, so x must be a multiple of 5. Since 100 - 3x > 0, x must be less than 33.3. The valid positive integer values for x are 5, 10, 15, 20, 25, and 30. Each of these 6 values gives a valid positive integer y. The answer is 6.
An equilateral triangle is inscribed in a circle of radius 6. What is the area of the triangle?
For an equilateral triangle inscribed in a circle, the distance from the center to a vertex is the radius R, which is 2/3 of the altitude. So altitude h = 3/2 × 6 = 9. The side length s is related to the altitude by h = s√3/2, so 9 = s√3/2, meaning s = 18/√3 = 6√3. The area is (s^2)√3 / 4 = (108)√3 / 4 = 27√3. The answer is 27√3.
The first term of an infinite geometric sequence is 4 and the sum of the sequence is 12. What is the common ratio?
The sum of an infinite geometric series is S = a / (1 - r), where a is the first term and r is the common ratio. Plug in the given values: 12 = 4 / (1 - r). Divide by 4 to get 3 = 1 / (1 - r). Multiply by (1 - r) to get 3(1 - r) = 1. This simplifies to 3 - 3r = 1, so 2 = 3r. The answer is 2/3.
How many ways can 10 identical candies be distributed among 3 children so that every child receives at least 1 candy?
Use the "Stars and Bars" method for positive integers. Give 1 candy to each of the 3 children immediately to satisfy the condition, leaving 7 candies to distribute among 3 bins freely. The formula is (n - 1) choose (k - 1), where n is the original items and k is bins. For 10 items and 3 bins: (10 - 1) choose (3 - 1) = 9 choose 2. 9! / (2! × 7!) = (9 × 8) / 2 = 36. The answer is 36.
Find the value of x if x is a positive real number and
x · ⌊x⌋ = 39 (where ⌊x⌋ denotes the greatest integer less than or equal to x).
Let n = ⌊x⌋, which is an integer. Thus, n ≤ x < n + 1. Multiplying the inequality by n (since x is positive, n is positive), we get n^2 ≤ x·n < n^2 + n. This means n^2 ≤ 39 < n^2 + n. The only integer n that satisfies this is n = 6, since 6^2 = 36 and 6^2 + 6 = 42. Since we know n = 6, we can substitute it into the original equation: x · 6 = 39. Dividing by 6 gives x = 39 / 6 = 6.5. (Check: 6.5 × 6 = 39). The answer is 6.5.
What is the smallest positive integer n such that n/2 is a perfect square and n/3 is a perfect cube?
Let the prime factorization of n be 2^a × 3^b. For n/2 = 2^(a-1) × 3^b to be a square, (a-1) and b must be even. Thus, a is odd and b is even. For n/3 = 2^a × 3^(b-1) to be a cube, a and (b-1) must be multiples of 3. The smallest odd 'a' that is a multiple of 3 is 3. The smallest even 'b' such that b-1 is a multiple of 3 is 4. Thus, n = 2^3 × 3^4 = 8 × 81 = 648. The answer is 648.
A cone and a cylinder have the same base radius and the same volume. What is the ratio of the height of the cone to the height of the cylinder?
The volume of a cone is (1/3)π(r^2)(h_cone) and the volume of a cylinder is π(r^2)(h_cylinder). Set them equal: (1/3)π(r^2)(h_cone) = π(r^2)(h_cylinder). Canceling π(r^2) from both sides leaves (1/3)(h_cone) = (h_cylinder), meaning h_cone / h_cylinder = 3 / 1. The answer is 3.
If the polynomial P(x) = x^3 + ax^2 + bx + 6 is perfectly divisible by (x - 1), what is the value of a + b?
By the Factor Theorem, if a polynomial is divisible by (x - c), then P(c) = 0. Since it is divisible by (x - 1), we evaluate P(1) = 0. Plugging in x = 1 yields 1^3 + a(1^2) + b(1) + 6 = 0, which simplifies to 1 + a + b + 6 = 0, or a + b + 7 = 0. Therefore, a + b = -7. The answer is -7.
Two fair six-sided dice are rolled. What is the probability that the product of the two numbers rolled is a multiple of 4?
Total possible outcomes are 36. We count the winning outcomes by cases: 1) Both dice show even numbers: {2,4,6} × {2,4,6} = 3 × 3 = 9 ways. 2) One die shows a 4 and the other shows an odd number {1,3,5}. If the first die is 4, there are 3 ways. If the second die is 4, there are 3 ways. Total = 6 ways. Total successful outcomes = 9 + 6 = 15. The probability is 15 / 36, which simplifies to 5/12. The answer is 5/12.
If log₂(x) + log₄(x) = 12, what is the value of x?
Use the change of base property. log₄(x) can be rewritten as log₂(x) / log₂(4). Since log₂(4) = 2, this is (1/2)log₂(x). The equation becomes log₂(x) + 0.5·log₂(x) = 12, which simplifies to 1.5·log₂(x) = 12. Divide by 1.5 to get log₂(x) = 8. Rewriting in exponential form yields x = 2^8 = 256. The answer is 256.
What is the remainder when 7^100 is divided by 13?
By Fermat’s Little Theorem, since 13 is prime, 7^12 ≡ 1 (mod 13). We divide the exponent 100 by 12: 100 = 12 × 8 + 4. Therefore, 7^100 ≡ 7^4 (mod 13). Calculate 7^4 mod 13: 7^2 = 49 = 13 × 3 + 10, so 7^2 ≡ 10 ≡ -3 (mod 13). Squaring this gives 7^4 ≡ (-3)^2 = 9 (mod 13). The answer is 9.
A point P is located inside rectangle ABCD such that the distance PA = 3, PB = 4, and PC = 5. What is the distance PD?
Apply the British Flag Theorem, which states that for any point P inside a rectangle, PA^2 + PC^2 = PB^2 + PD^2. Substitute the known values: 3^2 + 5^2 = 4^2 + PD^2. This simplifies to 9 + 25 = 16 + PD^2, or 34 = 16 + PD^2. Subtracting 16 gives PD^2 = 18. Taking the square root, PD = √18 = 3√2. The answer is 3√2.
Let x and y be positive real numbers such that x + y = 10 and x^3 + y^3 = 280. What is the value of the product xy?
Use the sum of cubes factorization: x^3 + y^3 = (x + y)(x^2 - xy + y^2). This can be rewritten using the sum as (x + y)((x + y)^2 - 3xy). Substitute the known values: 280 = 10(100 - 3xy). Divide by 10 to get 28 = 100 - 3xy. Subtract 100 to get -72 = -3xy. Divide by -3 to get xy = 24. The answer is 24.
Four friends write their names on separate slips of paper and put them in a hat. They each draw one slip at random. What is the probability that exactly one person draws their own name?
First, choose the 1 person who successfully draws their own name. There are 4 ways to choose this person. The remaining 3 people must all draw a name that is not theirs. The number of such "derangements" for 3 items is 2 (specifically, if A, B, C are left, they can only draw B, C, A or C, A, B). So there are 4 × 2 = 8 successful outcomes. The total number of ways the 4 slips can be drawn is 4! = 24. The probability is 8 / 24 = 1/3. The answer is 1/3.
Let z be a complex number such that z + (1/z) = 1. What is the value of z^2024 + 1/(z^2024)?
Multiply the given equation by z to get z^2 - z + 1 = 0. Multiplying both sides by (z + 1) gives (z + 1)(z^2 - z + 1) = 0, which expands to z^3 + 1 = 0, so z^3 = -1. To find z^2024, divide 2024 by 3: 2024 = 3 × 674 + 2. Thus, z^2024 = (z^3)^674 × z^2 = (-1)^674 × z^2 = 1 × z^2 = z^2. The expression simplifies to z^2 + 1/z^2. By squaring the original equation z + (1/z) = 1, we get z^2 + 2 + 1/z^2 = 1, which means z^2 + 1/z^2 = -1. The answer is -1.