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Determine all prime numbers \( p \) such that \( p^2 + 2 \) is also a prime number. | automath |
Determine all real values of \( c \) for which the equation \( x^5 - 5x = c \) has exactly three real roots. | automath |
Describe all numbers \( x \) that are at a distance of 4 from the number 8. | automath |
Prove that for all integers \( n \geq 4 \), the inequality \( 2^n < n! \) holds true. | automath |
Prove the inequality \(2^n < n!\) for all integers \(n \geq 4\) using mathematical induction. | automath |
What are the two numbers whose sum is 53, and for which three times the smaller number equals 19 more than the larger number? | automath |
What is the smallest positive integer \( n \) such that the sum of its digits in base 10 is divisible by 9, but \( n \) itself is not divisible by 9? | automath |
How many distinct poker hands can be formed that consist of two pairs and one card of a different rank? | automath |
Determine all positive integers \( D \) and \( N \) such that \( 2^D = N^2 + 1 \) or \( 2^D = N^2 - 1 \). | automath |
What is the maximum number of $k$-element subsets that can be formed from a set of $n$ points such that any two subsets have a pairwise intersection of size at most one? Specifically, for $k=3$, what is the maximum out of $k$-element sets for various values of $n$ for $n = 3, 4, 5, \ldots, 9$? | automath |
Prove that for all integers \( n \geq 1 \), the expression \( 6^n - 1 \) is divisible by \( 5 \). | automath |
What are the two different forms of the antiderivative of the function \(\sec^2 x \tan x\) that lead to the discrepancy mentioned, and how can you verify that these forms are equivalent by examining their constant differences? | automath |
Prove that for all integers \( n \geq 4 \), the inequality \( 3^n > n^3 \) holds true. What is the smallest integer \( n \) for which the inequality \( 3^n > n^3 \) is valid? | automath |
Given a square matrix \( A \in \mathbb{R}^{n \times n} \), what is the total number of unique factorizations \( C \) such that \( CB = A \) can be constructed, where \( B \) is any invertible matrix? Consider the condition that \( n \) is a positive integer. | automath |
In any arrangement of the integers from 1 to 10 placed in a circle, is it true that there must exist at least one set of three adjacent numbers that sum to 18 or more? If so, can you provide a mathematical proof of this statement? | automath |
What are the convex polyhedra that can be formed with exactly 6 faces, and can you provide a verification of whether all possible configurations have been accounted for, including naming any that might be missing? | automath |
What is the largest integer \( n \) such that there exists a partition of \( n \) into parts that can generate all integers from \( 1 \) to \( n-1 \) through sums of those parts? | automath |
Prove that if a quadrilateral has two pairs of opposite sides that are both parallel and congruent, then it is a parallelogram. | automath |
In a cone-shaped container with a semi-vertical angle of \(30^\circ\), liquid is flowing in at a rate of \(\frac{\sqrt{3}\pi}{4} \text{ cm}^3/\text{s}\). When the radius of the circular surface of the liquid is \(5 \text{ cm}\), what are the rates of increase of (a) the radius of the circular surface and (b) the area o... | automath |
How many distinct arrangements of the letters in "WONDERING" can be made such that there are exactly two consecutive vowels? | automath |
What is the total number of distinct triplets \((a, b, c)\) such that \(a + b + c\) is divisible by \(a\), \(b\), and \(c\), where each of \(a\), \(b\), and \(c\) is a positive integer less than or equal to 50? | automath |
Prove that any second-order contravariant tensor \( T^{ij} \) can be uniquely expressed as the sum of a symmetric tensor \( S^{ij} \) and an antisymmetric tensor \( A^{ij} \), where \( S^{ij} = S^{ji} \) and \( A^{ij} = -A^{ji} \). What are the dimensions of the subspaces spanned by the symmetric and antisymmetric part... | automath |
Given positive integers \(a = 30\), \(b = 72\), and \(N\) such that the product of any two of them is divisible by the third, what is the smallest possible value of \(N\)? | automath |
What is the maximum number of unit squares (2x2) that can fit inside a right isosceles triangle with legs of length \(L\)? | automath |
Consider a chessboard where each of the 64 squares is filled with a positive integer such that each integer is the average of the integers in its neighboring squares (sharing a common edge or vertex). Can you demonstrate that all 64 integers on the chessboard are equal? | automath |
Given that the probability that student $A$ will fail a certain statistics examination is $0.5$, the probability that student $B$ will fail is $0.2$, and the probability that both student $A$ and student $B$ will fail is $0.1$, what is the probability that exactly one of the two students will fail the examination? | automath |
Find all complex numbers \( z \) such that the equation \( 3z + i\bar{z} = 4 + i \) holds true. What is the value of \( z \) in the form \( a + bi \), where \( a \) and \( b \) are real numbers? | automath |
Prove that for any odd integer \( m \) and any even integer \( n \), the product \( m \cdot n \) is even. What is the structure of your proof, and can you summarize it clearly? | automath |
Prove that any second-order tensor can be uniquely decomposed into a symmetric tensor and an antisymmetric tensor, and provide the expression for this decomposition. | automath |
Prove that there exist infinitely many positive integers \( n \) such that \( 2^n - 1 \) is not a prime number. | automath |
A circle is centered at the point (200, 200) with a radius of 150. If the angle is measured clockwise from the positive y-axis, what are the coordinates of the point on the circumference at an angle of 128 degrees? | automath |
Let \( x \) and \( y \) be positive real numbers that satisfy the following simultaneous equations:
\[
2 \log y = \log 2 + \log x
\]
\[
2^y = 4^x
\]
What are the values of \( x \) and \( y \)? | automath |
Prove that the function \( f(x) = x^2 \) is continuous at every point \( x = a \) in its domain using the definition of continuity. | automath |
What is the probability that the sum \( w + x + y + z \) is odd, given that \( w, x, y, z \) are independent random variables taking on the values \( 0 \) and \( 1 \) with equal probability? If you can generalize, what would be the probability that the sum is odd for \( n \) such independent random variables? | automath |
Prove that an integer \( n \) can be expressed as the squared norm \( \| A \|^2 \) of a \( 2 \times 2 \) integer matrix \( A \) if and only if \( n \) can be expressed as the sum of two squares. Additionally, determine the structure of integers that can be written in this form. | automath |
In a kite, the measures of the angles are given as \(3x\), \(75^\circ\), \(90^\circ\), and \(120^\circ\). Find the value of \(x\) and determine which angles are congruent. | automath |
Given that the interior angles of an irregular hexagon are 80°, 130°, 102°, 36°, x°, and 146°, calculate the value of x and determine the sum of the exterior angles for this hexagon. | automath |
How many positive integers less than or equal to 2345 have all unique digits? | automath |
Evaluate the sum of the series
\[
\sum_{n=1}^{39} \frac{1}{n(n+1)}.
\]
Select the correct answer from the following options:
a) \(\frac{31}{40}\)
b) \(\frac{33}{40}\)
c) \(\frac{37}{40}\)
d) \(\frac{39}{40}\) | automath |
If \( x \) is a positive integer, what are all the possible values of \( x-1 \) such that \( x-1 \) is a factor of 104? | automath |
In a 6-by-5 array containing 30 distinct items, how many unique ways can you choose a set of three items such that no two items share the same row or the same column? | automath |
Prove that for every integer \( a \), the expression \( a^3 - a \) is divisible by 3, using mathematical induction. | automath |
Prove that for a fixed positive integer \( k \), there are only finitely many trees with \( k \) leaves and no vertices of degree 2. | automath |
What is the number of partitions of the integer \( n \) such that no part is a perfect square? | automath |
Prove that any contravariant tensor of second rank can be uniquely expressed as the sum of a symmetric tensor and an antisymmetric tensor. Additionally, in a 3-dimensional space, if \(T_{ij}\) is a given tensor, what are the explicit forms of its symmetric \(S_{ij}\) and antisymmetric \(A_{ij}\) parts? | automath |
What is the value of \( n \) for which the sum of the cubes of the three consecutive natural numbers \( n, n+1, n+2 \) is equal to 27? | automath |
What is the set of all real numbers \( x \) such that the inequality \( |x - 1| + |x - 6| \leq 11 \) holds true? Please specify the interval or intervals that contain all such \( x \). | automath |
What is the smallest positive odd integer that can be the least of four consecutive odd integers whose sum is a perfect cube? | automath |
Given that \( x \) is a positive integer, determine the number of distinct positive integer values of \( x \) such that \( x - 1 \) is a factor of 104. | automath |
What is the total number of multisets of size 4 that can be constructed from \( n \) distinct elements, ensuring that at least one element occurs exactly twice? | automath |
In triangle \( ABC \), points \( D \) and \( E \) are located on sides \( AB \) and \( AC \) respectively, such that \( DE \) is parallel to \( BC \). Given that the area of triangle \( PDE \) is \( \frac{3}{7} \) of the area of triangle \( ADE \), what is the ratio of the lengths of \( BC \) to \( DE \)? | automath |
Is it possible to find a set of 5 binary sequences of length 8 such that the Hamming distance between any two sequences in the set is at least 5? If so, provide an example of such a set; if not, prove that no such set exists. | automath |
What is the probability that four rooks, placed randomly on a $4 \times 4$ chessboard with the condition that no two rooks can attack each other, will be arranged such that none of the rooks are on the main diagonal of the chessboard? | automath |
Find all integer solutions \( x \) such that \( x^2 \equiv 1 \pmod{91} \). What are these solutions, and how many distinct solutions are there? | automath |
What is the probability that the sum of the four largest numbers among 15 randomly selected numbers from a uniform distribution on the interval \( (0, 1) \) exceeds \( 3.5 \)? | automath |
For which positive integers \( n \) is it possible to fill an \( n \times n \) table with the numbers \( 1, 2, \ldots, n \) such that every number appears exactly once in each row, each column, and each diagonal (including both main diagonals and minor diagonals)? Additionally, can you prove whether \( n=4 \) and \( n=... | automath |
Prove that any two empty sets are equal, and thus have the same cardinality. | automath |
What are all the possible real number pairs \((p, q)\) such that
\[
\frac{3 + 5i}{1 + pi} = q + 4i?
\]
Provide the pairs in the format \((p, q)\). | automath |
What are all the values of \( x \) that satisfy the equation
\[
\sin^{8}{x} + \cos^{8}{x} = \frac{17}{32}
\]
in the form \( x = \frac{n\pi}{4} \pm \frac{\pi}{8} \) for integers \( n \)? Include all possible values of \( n \). | automath |
How many distinct ways can a student complete a 12-question multiple choice test, where each question has 5 choices, such that at least one pair of consecutive questions has the same answer? | automath |
Prove that for any positive integer \( n \), the equation
\[
\sum_{i=1}^n i + \sum_{i=1}^{n-1} i = n^2
\]
holds true, and provide a complete proof for it. | automath |
How many distinct sums can be formed using at least 1 and up to all of the 15 cards, where there are 8 cards with the number 10, 5 cards with the number 100, and 2 cards with the number 500? | automath |
Given a rhombus with a perimeter of 80 cm, calculate the length of each side and then determine the area of the rhombus if the height from one base to the opposite side is 10 cm. What is the area of the rhombus? | automath |
What are the two numbers whose sum is 53 and for which three times the smaller number is equal to 19 more than the larger number? | automath |
Prove that if a quadrilateral has one pair of opposite sides that are equal in length and parallel, then the quadrilateral is a parallelogram. How would you demonstrate this mathematically? | automath |
What is the value of the sum \(\sum_{i=0}^{n} \binom{n+i}{i}\) in terms of \(n\), and can you prove that it equals \(\binom{2n+1}{n+1}\)? | automath |
Given a set of ordered integers from \(1\) to \(n\), describe the algorithm that removes the lowest number in the set at each step. If the number is odd, it is simply removed; if it is even, half of the number is added back to the set. How many steps does it take for the algorithm to completely empty the set? Provide a... | automath |
Let \( x, y, z \) be real numbers such that \( x+y+z = 1 \).
1. What is the maximum possible value of \( xy + yz + zx \)?
2. Is there a specific set of real numbers \( x, y, z \) that achieves this maximum? If so, provide these values.
3. Prove that \( xy + yz + zx < \frac{1}{2} \) for all real numbers \( x, y, z... | automath |
How many isosceles right triangles with vertices at the corners of a square of side length $n$ can be formed, considering both types described: those with hypotenuses along the sides of the square and those with vertices at the corners? Please provide a formula for the total number of such triangles in terms of $n$. | automath |
Prove that \( n^3 + 5n \) is always divisible by 6 for all positive integers \( n \) using the principle of mathematical induction. What specific steps must be demonstrated to complete the proof? | automath |
Given a sequence \(\{x_n\}\) such that \(x_{2k} \rightarrow x\) and \(x_{2k-1} \rightarrow x\) as \(k \rightarrow \infty\), prove that \(x_k \rightarrow x\). Specifically, provide a rigorous proof that demonstrates the convergence of the sequence \(\{x_n\}\) to \(x\) by leveraging the properties of subsequences. | automath |
Given the definitions of the functions \( f(x) = \max\{x, -x\} \) and \( g(x) = \max\{x, 1/x\} \), show that the function \( g(x) \) satisfies the following property for all positive real numbers \( x \) and \( y \):
\[
g(xy) \leq g(x) g(y).
\]
Additionally, determine under what conditions \( g(x) = 1 \). | automath |
Prove that for all integers \( n > 0 \), the following equation holds:
\[
\sum_{k=1}^{n} \frac{k}{2^k} = 2 - \frac{n+2}{2^n}.
\]
What is the value of the left-hand side when \( n = 5 \)? | automath |
What is the least number \( k \) such that every natural number can be expressed as a sum of at most \( k \) square numbers? | automath |
What are the coordinates of the point that divides the line segment connecting points A(2, 4) and B(-3, 9) in the ratio 1:4 internally? | automath |
Starting with the number \(1\) and using a calculator that has only two operations, \(+1\) and \(\times 2\), what is the minimum number of steps required to exactly reach the number \(200\)? Provide a detailed explanation or proof of your reasoning for why this number of steps is optimal. | automath |
For all positive integers \( n \), show that the expression \( 7^n + 2^{2n+1} \) is divisible by 3. Can you provide a proof or reasoning to demonstrate this? | automath |
Prove the formula relating Euler's totient function:
$$\varphi(mn) = \varphi(m)\varphi(n) \frac{d}{\varphi(d)}$$
where \( m \) and \( n \) are positive integers, \( d = \gcd(m, n) \), and \( \varphi \) is Euler's totient function. Specifically, demonstrate that the index of the image of the natural map \( U_{mn} \to... | automath |
In a rectangle, if the lengths of the two diagonals are denoted as \(d_1\) and \(d_2\) and the area of the rectangle can be expressed as \(A = \frac{1}{2} d_1 d_2\), what is the relationship between the lengths of the diagonals and the area of the rectangle if one diagonal is known to be 10 units long and the rectangle... | automath |
Show that there exists a bijection between the interval \((0, 1)\) and the closed interval \([0, 1]\). Provide a specific function that establishes this bijection and verify that it is both injective and surjective. | automath |
What is the sum of the number of digits of the numbers \(2^N\) and \(5^N\) for a positive integer \(N\)? | automath |
Four integers are randomly selected from the set \(\{-1, 0, 1\}\) with repetitions allowed. What is the probability that the product of the four integers selected is equal to \(-1\)?
A. \(\frac{1}{81}\)
B. \(\frac{4}{81}\)
C. \(\frac{8}{81}\)
D. \(\frac{10}{81}\)
E. \(\frac{16}{81}\) | automath |
Given a right triangle with legs of lengths \( a \) and \( b \), and the hypotenuse of length \( c \), calculate the total area of the three triangles formed outside the triangle within a configuration of two squares constructed on each leg of the triangle and the hypotenuse, assuming all triangles formed have the same... | automath |
Given a set \( S = \{1, 2, \ldots, k\} \), two persons independently choose subsets from this set such that neither chooses the empty set. What is the probability that the cardinality of the intersection of the subsets chosen by both persons is exactly one? | automath |
Let \( (G, +) \) be an abelian group, and consider the elements \( a_1, a_2, \ldots, a_n \in G \). Prove that the sum \( a_1 + a_2 + \cdots + a_n \) is independent of the order in which the elements are added, where the sum is defined through iterated applications of the group operation. Additionally, show that the pla... | automath |
Calculate the number of digits in the number \( 2^{2^{2^{2^{2}}} } \). | automath |
Show that for a positive integer \( n \), the following equation holds:
\[
2 + \sum_{k=1}^{n-1} \binom{n}{k} = (1+1)^n
\]
Provide a detailed explanation or proof for your answer, including any combinatorial interpretations or algebraic manipulations that justify the equality. | automath |
Let \( x \) and \( y \) be real numbers greater than 1. Prove that the following inequality holds:
\[
\log_2(x) + \log_x(y) + \log_y(8) \geq 3
\]
Determine under what conditions on \( x \) and \( y \) this inequality is equality. | automath |
Given two positive integers \( L \) and \( R \), determine the number of positive integers \( N \) in the range \([L, R]\) such that \( N \) can be expressed in the form \( N = 2^a \cdot 3^b \) where \( a \geq 1 \) and \( b \geq 0 \). Additionally, include \( N = 1 \) in your count if it lies within the range. Calculat... | automath |
A and B play a game where they alternately roll a fair six-sided die, with A starting first. A player wins if, at the end of their turn, the sum of scores from both players equals 7, and that player rolled last. What is the probability that A wins the game? | automath |
Prove that the equality \( 0.999\dots = 1 \) holds by showing that the limit of the series defined by the decimal expansion \( 0.999\dots \) converges to 1. Specifically, use the definition of the limit of a sequence and demonstrate that for any \(\varepsilon > 0\), there exists an \(N \in \mathbb{N}\) such that for al... | automath |
In the game TRIPOLLO, marbles numbered from 1 to 33 are placed in an 11-column board with 3 rows, where the player wins if the sum of the numbers in the first two rows of each column equals three times the number in the third row. Prove that it is impossible to achieve a winning placement with these conditions on the g... | automath |
How many four-digit passwords can be formed using a pool of 95 ASCII characters, ensuring that each password contains at least one uppercase letter and at least one digit? | automath |
Prove or disprove the following statements concerning the behavior of sequences:
1. There exist sequences \((x_n)\) and \((y_n)\) such that both diverge, yet their sum \((x_n + y_n)\) converges. If so, provide an example.
2. There exist sequences \((x_n)\) and \((y_n)\) such that \((x_n)\) converges, \((y_n)\) diverg... | automath |
Is it true that for any given positive real number \( \varepsilon > 0 \) and any integer \( n \geq 2 \), there exists a polynomial \( P(z) \) of degree \( n \) with \( n \) distinct roots such that there exists a root \( z_a \) of its derivative \( P'(z) \) and a root \( z_b \) of \( P(z) \) satisfying the condition \(... | automath |
Given the differential equation \( y' = 3y^{2/3} \), determine the number of distinct solutions that satisfy the initial condition \( y(0) = 0 \). How can the solutions be characterized, and what implications does this have for the uniqueness of solutions in the context of the Lipschitz condition? | automath |
Evaluate the integral
\[
\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 1} \, dx.
\]
Provide the exact numerical value of the integral. | automath |
Prove that the two integrals
\[
\int \sin^3 x \, \mathrm{d}x = \frac{1}{3} \cos^3 x - \cos x + C
\]
and
\[
\int \sin^3 x \, \mathrm{d}x = \frac{1}{12} \cos(3x) - \frac{3}{4} \cos x + C
\]
are indeed equal for all \(x \in \mathbb{R}\) by showing that the difference between the two expressions is a constant. | automath |
What are the dimensions of the rectangle with the greatest possible area that can be inscribed in a right triangle with sides of lengths 3 cm, 4 cm, and 5 cm? | automath |
Given a number \( N \), what is the maximum possible value of \( N \) such that \( N \) can be represented as the sum of two distinct semi-primes, assuming the largest prime number used in the semi-prime calculations is 100? | automath |
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