Find the smallest positive integer \(x\) that satisfies the following system of congruences:\(x \equiv 3 \pmod 5\)\(x \equiv 5 \pmod 7\)\(x \equiv 7 \pmod{11}\)
A school club has 15 members, consisting of 8 boys and 7 girls. They need to form a committee of 4 members.a) How many different committees can be formed?b) How many different committees can be formed if it must consist of 2 boys and 2 girls?
Compute the sum of the first 16 terms of the geometric sequence \(2, 6, 18, \dots\).
Let \( a, b, c \) be integers. Prove that if \( a \) divides \( b \) and \( b \) divides \( c \), then \( a \) divides \( c \).
Find all polynomials \( P(x) \) such that \( P(x^2) = (P(x))^2 \) for all real \( x \).
6 friends are lining up side-by-side for a group photo.In how many different orders can they arrange themselves?
Jessica has 7 distinct books she wants to arrange on a bookshelf.a) In how many different ways can she arrange all 7 books?b) If 3 of the books are history books and must be kept together, in how many ways can she arrange the books?
In a circle, an inscribed angle subtending an arc is equal to half the measure of the corresponding central angle.
Find all real solutions to the equation \(x \ln x = 1\). Express your answer using the Lambert W function.
Prove that for points lying on the sides of a triangle and collinear, the product of the ratios of the segments equals one.
Let \(a, b, c\) be positive real numbers. Prove that:$$\frac{\sqrt{2a+b} + \sqrt{2b+c} + \sqrt{2c+a}}{\sqrt{a+b+c}} \le 3$$
Find all functions \(f: \mathbb{R} \to \mathbb{R}\) such that \(f(x+y) = f(x) + f(y)\) for all \(x,y \in \mathbb{R}\), and \(f(xy) = f(x)f(y)\) for all \(x,y \in \mathbb{R}\).
A triangle has side lengths 5, 12, and 13. Compute its area.
Let a sequence be defined by \( a_1 = 1 \) and \( a_{n+1} = \sqrt{2 + a_n} \). Prove the sequence converges and find \( \lim_{n \to \infty} a_n \).
Use mathematical induction to prove the formula for the sum of the first \( n \) terms of an arithmetic series, which is given by $$ S_n = \sum_{k=1}^n (a + (k-1)d) = \frac{n}{2}(2a + (n-1)d) $$ for all positive integers \( n \ge 1 \), where \( a \) is the first term and \( d \) is the common difference.
Evaluate the indefinite integral \( \int \tan(2x) \, dx \).
A quiz consists of 5 multiple-choice questions. Each question has 4 possible answers (A, B, C, D), and only one is correct.How many different ways can a student answer all 5 questions, regardless of correctness?
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles.If you pick 3 marbles at random, how many ways can you pick exactly 1 red, 1 blue, and 1 green marble?
Prove that the area of a triangle with sides a, b, c can be computed as the square root of s(s-a)(s-b)(s-c), where s is the semiperimeter.
Prove that the sum of the measures of the interior angles of any convex quadrilateral is \( 360^\circ \).
Factor the polynomial \(x^3 - 3x^2 + 3x - 1\).
A certain state's license plates consist of 3 letters followed by 3 digits.If repetition of letters and digits is allowed, how many different license plates are possible? Assume there are 26 possible letters and 10 possible digits (0-9).
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Find the indefinite integral:$$ \int \sec^3 x \, dx $$
Solve the equation: \n $$ (x-1)^4 + (x-5)^4 = 82 $$
For any two sets \( A \) and \( B \), prove that \( (A \setminus B) \cup B = A \cup B \).
Prove that the sum of two odd integers is always an even integer.
Solve for \( x \) in the infinite nested radical: \n $$ \sqrt{x + \sqrt{x + \sqrt{x + \dots}}} = 5 $$
Find the remainder when \\( 2^{100} \\) is divided by 13.
Find all real solutions to the system: \n $$ x_1^2 = x_2 + 2, \quad x_2^2 = x_3 + 2, \quad x_3^2 = x_1 + 2 $$
Evaluate the definite integral:$$ \int_{0}^{\pi/2} \cos^2 x \, dx $$
Given a triangle \( \triangle ABC \), with \( D \) a point on the line containing \( BC \) such that \( C \) is between \( B \) and \( D \). Prove that the measure of the exterior angle \( \angle ACD \) is equal to the sum of the measures of the two opposite interior angles, \( \angle BAC + \angle ABC \).
There are 4 different meetings that need to be scheduled for Monday afternoon, one after another.In how many different orders can these meetings be scheduled?
Evaluate the definite integral \( \int_0^{\pi/4} \sec^2 x \, dx \).
Solve the following system of equations for real numbers \(x, y, z\):$$ x+y+z = 6 $$$$ x^2+y^2+z^2 = 14 $$$$ x^3+y^3+z^3 = 36 $$
A flag has 3 horizontal stripes. There are 5 different colors available to choose from.How many different flags can be designed if adjacent stripes must have different colors?
In a circle, prove that the angle subtended by a diameter is always a right angle.
Calculate the limit \( \lim_{x \to \pi/2} \frac{\cos x}{\pi/2 - x} \).
A tourist wants to visit 3 specific cities out of a list of 7 possible cities. The order in which the tourist visits these cities matters.How many different travel itineraries are possible?
Let \(a, b, c\) be positive real numbers. Prove that: i) \(a^2 + b^2 + c^2 \ge ab + bc + ca\) ii) \(a^3 + b^3 + c^3 + ab^2 + bc^2 + ca^2 \ge 2(a^2b + b^2c + c^2a)\)
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Let \( a_n \) be a sequence defined by \( a_1=1, a_2=2 \), and \( a_{n+2} = 2a_{n+1} + a_n \) for \( n \ge 1 \).Prove by induction that \( a_n \) is even if and only if \( n \) is a multiple of 3.
A computer manufacturer allows customers to customize their orders. There are 3 CPU options, 2 RAM options, and 4 storage options.How many different custom computer configurations are possible?
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Let \(a, b, c\) be positive real numbers. Prove that:$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} \ge \frac{3}{2}$$
A bag contains 5 red marbles and 3 blue marbles. If two marbles are drawn at random without replacement, what is the probability that both are red?
Find the real solution for the equation $$ xe^x = 3 $$ Express your answer using the Lambert W function.
Calculate the limit \( \lim_{x \to 0} \frac{\sin(3x)}{\tan(5x)} \).
Prove that the difference between the squares of any two consecutive integers is always an odd integer.
Prove that the sum of a rational number and an irrational number is always an irrational number.
7 friends are going to watch a movie and sit in a single row of 7 seats.In how many different orders can they sit?
How many distinct 5-letter words can be formed using the letters A, B, C, D, E, F, G, H if each letter can be used at most once and the word must contain the letter A?
Prove that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side.
Prove that the sum of the first n squares is a perfect square only for n = 1 and n = 24. That is, show that $$1^2 + 2^2 + \dots + n^2 = m^2$$ has solutions only for these values of n.
Find the set of all real numbers \(x\) such that the following inequalities hold simultaneously:$$ x^2 - 4x + 3 < 0 $$$$ \frac{x-1}{x+2} \geq 0 $$
Use the principle of mathematical induction to prove that for every positive integer \(n \geq 1\), the expression$$3^{2n+2} - 8n - 9$$is divisible by 64.
Prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
Prove that the centroid, circumcenter, and orthocenter of a non-equilateral triangle lie on a single line called the Euler line.
Let \( F_n = 2^{2^n} + 1 \) be the \( n \)-th Fermat number for \( n \ge 0 \).a) Prove by induction that for any \( n \ge 1 \), \( F_n = F_0 \cdot F_1 \cdot \dots \cdot F_{n-1} + 2 \).b) Use the result from part (a) to show that any two distinct Fermat numbers \( F_m \) and \( F_n \) (where \( m \ne n \)) are relatively prime, i.e., \( \gcd(F_m, F_n) = 1 \).
Find a closed-form expression for the sum \( \sum_{k=1}^{N} \sin(kx) \).
Compute the sum of squares of the first 1000 positive integers: $$1^2 + 2^2 + \dots + 1000^2.$$
Let \( a, b, c \) be non-negative real numbers such that \( a+b+c=1 \). Find the maximum value of: \n $$ ab + bc + ca - 2abc $$
Given triangle ABC, the perpendicular bisectors of its sides intersect at point O. Prove that O is equidistant from A, B, and C.
A pizza restaurant offers 8 different toppings. A customer wants to choose exactly 3 different toppings for their pizza.How many different combinations of toppings are possible?
Find the smallest positive integer \(x\) such that \(17x \equiv 1 \pmod{101}\).
Find all positive integer solutions \((x,y)\) to the equation $$ \frac{1}{x} + \frac{1}{y} = \frac{1}{3} $$
Let \( F_k \) be the \( k \)-th Fibonacci number defined by \( F_1 = 1 \), \( F_2 = 1 \), and \( F_k = F_{k-1} + F_{k-2} \) for \( k \geq 3 \).Prove by mathematical induction that for all positive integers \( n \):$$ \sum_{i=1}^n F_i = F_{n+2} - 1 $$
Find the derivative of the function \(f(x) = \frac{\sin x}{x}\) with respect to \(x\).
Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Find the smallest positive integer \(n\) that satisfies the following congruences:\(n \equiv 2 \pmod{5}\)\(n \equiv 3 \pmod{7}\)\(n \equiv 4 \pmod{11}\).
Find the sum of the series:$$ \sum_{n=1}^{N} \arctan\left(\frac{1}{n^2+n+1}\right) $$
Let \(p\) be a prime number. If \(p\) divides \(n^2 + 1\) for some integer \(n\), prove that \(p \equiv 1 \pmod{4}\) or \(p=2\).
Find the area of the triangle with vertices \((1,2)\), \((4,5)\), and \((6,1)\)
Two fair six-sided dice are rolled. What is the probability that the sum of the numbers rolled is 7, given that at least one die shows a 3?
An ice cream shop has 12 different flavors available.How many ways can a customer choose 2 different scoops for their cone, if the order of the scoops does not matter?
Find the derivative of the function \(f(x) = \ln(\tan x)\) with respect to \(x\).
In triangle ABC, the internal angle bisector of angle A meets side BC at point D. Prove that $$\frac{BD}{DC} = \frac{AB}{AC}$$.
In a right triangle with legs \(a\) and \(b\), compute the length of the median from the right angle.
Evaluate the indefinite integral \( \int \cos^3 x \, dx \).
A bank requires its customers to create a 4-digit Personal Identification Number (PIN) using digits from 0 to 9.a) How many different 4-digit PINs are possible if digits can be repeated?b) How many different 4-digit PINs are possible if digits cannot be repeated?c) How many different 4-digit PINs are possible if the first digit cannot be 0 and digits cannot be repeated?