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A regular tetrahedron $PQRS$ is inscribed in a sphere with center $O$ such that $OQ=7$. A point $R$ on the sphere is selected at random. Determine the probability that $PQRS$ and $OQRS$ have the same volume. | nvidia |
Find the number of positive integers that are divisors of both 10! and 15!. | nvidia |
What is the largest positive integer $n$ such that $\frac{1}{n}$ is a repeating decimal and $n$ contains the digit 7 exactly three times? | nvidia |
Determine the value of $x$ such that $$\sec \left(\frac{\pi x}{7}\right) = -2.$$ | nvidia |
Given that a, b, c, d, and e are real numbers, find all values of x that satisfy the inequality
$$\frac{\left(\sin^2x-\frac{1}{2}\right)\left(\cos^2x-\frac{1}{3}\right)}{\sin x\cos x}\le0.$$ | nvidia |
Given that $2^{30}$ is equal to $1024^y$, determine the value of $y$. | nvidia |
A sequence of five digits is called "alternating" if the sum of the first and third digits equals the sum of the second and fourth digits. How many different five-digit numbers with non-zero leading digits can be formed such that the sequence of their digits is alternating? | nvidia |
Find the number of positive integers less than 100 that can be expressed as the sum of a square and a cube. | nvidia |
For a positive integer $n$, let $f(n)$ denote the remainder when $5n!$ is divided by $7$.
If $m$ is the smallest positive integer such that $f(m)=0$, determine the value of $m$. | nvidia |
Let $g(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the largest possible integer such that $k!$ is not divisible by $n$. Given that $n$ is a power of 2, what is the smallest value of $n$ such that $g(n) < n$? | nvidia |
Given a parabola $y=ax^2+bx+c$ with the vertex at $(p,p)$ and $y$-intercept at $(0,-p)$, where $p\ne 0$, find the equation of the parabola if the tangent line to the parabola at the vertex $(p,p)$ intersects the $y$-axis at a point with coordinates $(0,2p)$. | nvidia |
Find the largest integer that must divide the sum of any 5 consecutive integers. | nvidia |
Find the smallest positive integer $z$ such that $z^2$ is a multiple of 100 and the sum of the digits of $z$ is equal to 10. | nvidia |
Given that the graphs of $y=h(x)$ and $y=j(x)$ intersect at $(2,2),$ $(4,6),$ $(6,12),$ and $(8,12),$ there is one point where the graphs of $y=h(2x)$ and $y=2j(x)$ must intersect. What is the sum of the coordinates of that point? | nvidia |
A circle is inscribed in a triangle with sides of lengths $a$, $b$, and $c$. If the radius of the circle is $r$, find the number of ways to choose three of the following six values such that their sum is equal to the semi-perimeter of the triangle: $a$, $b$, $c$, $a-r$, $b-r$, $c-r$. | nvidia |
Problem:
In the figure, $ABCD$ is a rectangle with $AB = 12$ and $BC = 8$. The points $P$ and $Q$ are on the sides $AB$ and $CD$ respectively, such that $AP = 3$. The circles with centers $P$ and $Q$ and radii $r_1$ and $r_2$ respectively are drawn such that they are tangent to each other and to the sides $BC$ and $AD$... | nvidia |
How many four-digit numbers are there such that the sum of their digits is equal to the sum of the digits of their reverse? | nvidia |
A polynomial $p(x)$ is such that $p(x)-p(x-1)=x^2$ for all real numbers $x$. Determine the value of $p(5)$, given that $p(1)=1$. | nvidia |
Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a+b+c = 64.$ Find the smallest possible value of $b.$ | nvidia |
Find the sum of the squares of the solutions of $x^2-13x+4=0$. | nvidia |
The Fibonacci Factorial Series consists of the sequence $1!+1!+2!+3!+5!+8!+13!+21!+34!+55!+89!$. Instead of finding the sum of the last two digits, consider the following problem:
Find the largest prime factor of the product $1! \cdot 1! \cdot 2! \cdot 3! \cdot 5! \cdot 8! \cdot 13! \cdot 21! \cdot 34! \cdot 55! \cdot... | nvidia |
The vertices of a triangle are located at $(x, y), (4x, y), (x, 2y)$ for positive real numbers $x$ and $y$ such that $x^2 + y^2 = 9$. Determine the largest possible area of the triangle. | nvidia |
What is the smallest positive integer $k$ such that $3k$ can be written in the form $7m+5n$, where $m$ and $n$ are positive integers? | nvidia |
Find the smallest positive integer such that when its digits are arranged in descending order, the resulting number is divisible by the sum of its digits, and when its digits are arranged in ascending order, the resulting number is divisible by the product of its digits. | nvidia |
The positive integers $x$, $y$, and $z$ are such that
$$(x^2+y^2+z^2)(xy+yz+zx)=2018.$$
Determine the maximum possible value of $x+y+z$. | nvidia |
Let $x$ be a real number such that $\lfloor x \rfloor + x = \frac{13}{3}$. Find the value of $\lceil x^2 \rceil$. | nvidia |
Let $S$ be the set of all points in the plane that are the centres of some non-degenerate rectangle whose vertices are all on the parabola $y = x^2$. What is the smallest possible area of a rectangle such that all points of $S$ are inside or on that rectangle? | nvidia |
How many different positive integers can be represented as the sum of three distinct members of the set {1, 2, 3, 4, 5, 6, 7} with the condition that the largest member is at most twice the smallest member? | nvidia |
What is the smallest positive integer $n$ such that every positive integer greater than $n$ can be expressed as a sum of distinct powers of 2 and/or 3? | nvidia |
Find the range of values for $x$ that satisfy the inequality $\sqrt{x^2+2x}+\sqrt{x^2-2x}>2$. | nvidia |
Problem:
Consider the equation \[\frac{(x+D)(Ex+F)}{(x+G)(x+H)} = 3.\]Suppose $D,$ $E,$ $F,$ $G,$ and $H$ are constants such that for any $x$, \[(x+D)(Ex+F)\equiv(x+G)(x+H)\pmod{x^2}.\] Find the minimum number of possible values of $x$ that are not solutions to this equation.
Note: The congruence notation indicates t... | nvidia |
In a circular pizza with a radius of 12 cm, four identical right-angled triangles with legs of length 8 cm are placed in the middle, leaving four circular segments unoccupied.
Each triangle is rotated by a quarter of the circle so that its right angle lies at the center of the pizza.
What is the sum of the area of th... | nvidia |
Let a be a positive real number. Consider the equation \log_a x = \frac{x}{a} that has only one real solution. Find x. | nvidia |
Find the number of integers between 1 and 1000 (inclusive) that have the property that the sum of their digits is equal to the product of their digits.
(Note: This problem requires a different approach, as it involves considering the relationships between the digits rather than their order.) | nvidia |
Find the number of positive factors of the number 24 that are also perfect squares. | nvidia |
Find all real values of $b$ such that the equation
$$x^3 - 3bx + 2b = 0$$
has at least one real solution that is also a root of the equation
$$x^2 - 2x - 1 = 0$$ | nvidia |
Find the number of ordered pairs $(a,b)$ such that $a$ is a subset of $\{1,2,3,4\}$ and $b$ is a subset of $\{2,3,4,5\}$, with the condition that $a$ and $b$ have no elements in common. | nvidia |
Let $x$ be a positive real number. Find the minimum value of $4x^5 + 5x^{-4}.$ | nvidia |
If $\det \mathbf{M} = -2,$ then find $ \det (\mathbf{M}^4)$
A related problem:
If $M$ is a square matrix such that $M^3=M$, determine all possible values of $\det M$. | nvidia |
Find the sum of the coordinates of the midpoint of the segment with endpoints $(4, 8)$ and $(-2, 3)$. | nvidia |
The sum of the squares of two consecutive odd numbers is less than 100. How many pairs of such numbers satisfy the given condition? | nvidia |
Find the number of real solutions to the equation
$$\frac{1}{x-1}+\frac{2}{x-2}+\frac{3}{x-3}+\dots+\frac{100}{x-100}=\sqrt{x}$$ | nvidia |
In the equation $$\frac{1}{a} + \frac{b}{c} = \frac{3}{4}$$, determine the largest whole number value of $b$ that makes the inequality true, given that $a=2$ and $c=5$. | nvidia |
Find the largest positive integer that is not the sum of a positive integral multiple of 28 and a positive composite integer. | nvidia |
The base of a triangle is of length $b$, and the altitude is of length $h$. A rectangle of height $x$ is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:
$\textbf{(A)}\ \frac{bx}{h}(h-x)\qquad \textbf{(B)}\ \frac{hx}{b}(b-x)\qquad \textbf{(C)}\ \frac{bx... | nvidia |
The polynomial $P(x) = 3x^3 + ax^2 + bx + c$ has the property that the sum of its zeros, the sum of the squares of its zeros, and the sum of the coefficients are in geometric progression. The $y$-intercept of the graph of $y = P(x)$ is 1. What is $a$? | nvidia |
Determine the number of positive integers $b$ less than $18$ such that the congruence $x^2 \equiv b \pmod{18}$ has a solution in $x$. | nvidia |
Let $g$ be a non-constant polynomial such that
\[g(x - 2) + g(x) + g(x + 2) = \frac{[g(x)]^2}{2015x}\]for all nonzero real numbers $x.$ Find the sum of all possible values of $g(2).$ | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x) + g(y) = g(xy) + x + y - xy - 2\]for all real numbers $x$ and $y.$
Find the number of possible values of $g(0)$ if it is known that $g(0)$ is an integer. | nvidia |
In the configuration of circles below, each circle intersects each of the others exactly twice. The tangents of the angles at the intersection points between the blue and green circles and the red and green circles are $-\frac{3}{7}$ and $\frac{4}{5}$, respectively. Find the tangent of the angle at the intersection poi... | nvidia |
The roots of the polynomial $x^4 - 12x^3 + 47x^2 - 60x + 9$ are four distinct positive real numbers. If the sum of the squares of the roots is 300, what is the sum of the reciprocals of the roots? | nvidia |
For all composite integers $n$, what is the largest integer that always divides into the difference between $n$ and the cube of $n$? | nvidia |
The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$.
| nvidia |
How many non- empty subsets $S$ of $\{1,2,3,\ldots ,15\}$ have the following two properties?
$(1)$ No two consecutive integers belong to $S$.
$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$.
$\mathrm{(A) \ } 277\qquad \mathrm{(B) \ } 311\qquad \mathrm{(C) \ } 376\qquad \mathrm{(D) \ } 377\... | nvidia |
Determine the number of integers between $1$ and $1000$ that satisfy the inequality $x^3+7x^2-8x+3 \geq 0$. | nvidia |
Find the number of ordered triples $(x,y,z)$ of real numbers such that $x + y = 2$ and $xy - z^2 = 1.$ | nvidia |
How many 5-digit numbers can be formed using the digits 0-9, with the condition that the digits in the units and tens places are even, and the digits in the hundreds and thousands places are prime numbers? | nvidia |
Let $a_1 , a_2 , \dots$ be a sequence for which $a_1=2$ , $a_2=3$, and $a_n=\frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$? | nvidia |
Let $S$ be a square and let $A$ and $B$ be two points on the sides $S$ such that the segment $AB$ has maximum length when $A$ and $B$ lie on the sides $S$.
Prove that the disk inscribed in $S$ is tangent to the segment $AB$. | nvidia |
In a set of 1000 four-digit numbers, how many numbers can be formed such that no two consecutive digits are the same, but the first and last digits are identical? | nvidia |
Let $r$ and $s$ be positive integers such that $r+s=2008$ and $12r-13s=32$.
Find the number of possible pairs $(r,s)$. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $abc=1$. Find the minimum value of $a^2+b^2+c^2+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$. | nvidia |
Find the sum of all the real numbers x such that:
\[\frac{x^3 - x}{2} = \frac{2x^2 + 1}{x + 1}.\] | nvidia |
Count the number of positive multiples of 7 that lie between \(\frac{55}{3}\) and \(\frac{88}{2}\). | nvidia |
Find the minimum value of
\[x^2 + xy + y^2\]over all real numbers $x$ and $y.$ | nvidia |
Let $\{b_k\}$ be a sequence of real numbers such that $b_1=2$ and $b_{m+n} = \frac{1}{b_m} + \frac{1}{b_n} + \frac{m}{n}b_m$, for all positive integers $m$ and $n.$ Find $b_5.$ | nvidia |
Find the number of 4-digit positive integers that are multiples of 15 but not of 35. | nvidia |
Two positive integers are in the ratio 3:4, and the sum of their squares is 2500. What are the integers? | nvidia |
In a right triangle $ABC$ with $\angle A = 90^\circ$, the tangent of angle $B$ is equal to $3/5$. If the length of side $AB$ is $15$, find the area of the triangle. | nvidia |
Find the sum of the x-intercepts of the curve
$$y=\frac{4x^3-7x^2+5x-1}{2x^2+5x-3}$$ | nvidia |
Suppose $a$ and $b$ are positive real numbers such that $a^2\cos x+b^2\sin x=\frac{1}{2}$ and $a\cos x+b\sin x=5$. Determine the value of $\sin^2 x+\cos^2 x$. | nvidia |
Find all positive values of $d$ such that the inequality $x^2-4x+d<0$ has real solutions for $x$. Express your answer in interval notation. | nvidia |
Find the sum of all integers $k$ such that $\binom{15}{6} + \binom{15}{7} = \binom{16}{k}$. | nvidia |
Find the value of x such that the fraction $\frac{x}{999}$ equals $0.\overline{162}$. | nvidia |
A fortunate number is a positive integer that is divisible by the product of its digits. What is the least positive multiple of 7 that is not a fortunate number? | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $\gcd(a,b)=1$ and $ab=c^n$ for some positive integer $n$. If $a+b$ has 14 digits, then what is the maximum number of digits that $\gcd(a,c)$ can have? | nvidia |
Let $x$, $y$, and $z$ be three real numbers such that $x+y+z=6$, $xy+xz+yz=12$ and $xyz=8$.
Find $x^2+y^2+z^2$.
Note: The new problem requires a different approach to solve, as it involves finding the sum of squares of the numbers, rather than finding the largest number. | nvidia |
Let $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are positive integers. Suppose that for every positive integer $n$, there exists a unique positive integer $m$ such that $f(n) = m^2$. Find the smallest possible value of $a$. | nvidia |
The number 0.0625 can be represented as $\frac{a}{b^c}$ where a, b and c are positive integers. Find a+b+c. | nvidia |
Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. | nvidia |
The perimeter of a rectangle is $P$ cm, where $P$ is a positive integer. If the area of the rectangle is 48 square cm, what is the sum of all possible values of $P$? | nvidia |
Find the number of four-digit numbers that are palindromes and consist only of even digits. | nvidia |
Find the number of distinct ways to arrange 12 marbles of different colors in a circle, with the restriction that no two adjacent marbles have the same color. | nvidia |
The lines with the given equations are perpendicular. Find the value of $a$ so that the lines with the given equations are perpendicular.
Find the value of $a$ so that the lines with the given equations are perpendicular.
\begin{align*}
y &= 2x+13 \\
6y+ax &= 6.
\end{align*}
The lines with the given equations are pe... | nvidia |
The product of two positive integers is equal to 1105920. The first integer can be represented as 3n^2 and the second as 2^m\*5^k. Find n+m+k, where n, m, and k are natural numbers. | nvidia |
Find $x$ such that $\log_x 4=\log_{27} 3$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2y = 1$. Find the minimum value of $x^3 + y^2 + 1/(xy)$. | nvidia |
Determine the number of solutions of the equation
$$\cos x = \frac{1}{x}$$
on the interval $(0, 2\pi)$ that are in the form of $\pi/2 + n\pi$, where $n$ is an integer. | nvidia |
Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. Express your answer in interval notation. | nvidia |
The average of two 2-digit positive integers is equal to the decimal number obtained by writing one of the two-digit integers before the decimal point and the other two-digit integer after the decimal point. What is the smaller of the two integers? | nvidia |
How many positive three-digit integers less than 500 have at least two digits that are the same? | nvidia |
Determine the smallest positive integer $n$ such that $350-n$ is divisible by $12$. | nvidia |
Let $g$ be a function defined on the positive integers, such that
\[g(x^2) = g(x) + g(x)\]for all positive integers $x.$ Given $g(4) = 6$ and $g(9) = 8,$ find $g(16).$ | nvidia |
Determine the number of positive roots of the equation
$$\sqrt[3]{x} = 1 - \sin x$$
on the interval $(0, 10 \pi)$, given that $x$ is a real number. | nvidia |
A circle is inscribed in a parabola with the equation $y=x^2$ such that the circle touches the parabola at two points and also passes through the vertex of the parabola. How many such circles exist? | nvidia |
Find the value of $a$ such that the product of the absolute values of the roots of the equation $z^2 + 2az + (a^2 + 1) = 0$ is equal to 1. | nvidia |
Find the positive real value of $a$ such that the area of the triangle formed by the points $z_1 = 3i$, $z_2 = a + 2i$, and $z_3 = a - 2i$ in the complex plane is equal to 18. | nvidia |
Let $p$ and $q$ be any two even numbers, with $q$ greater than $p$. The largest integer which divides all possible numbers of the form $q^2 - p^2$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$ | nvidia |
Given the polynomial $x^4-4x^3-4x^2+16x-8=0$, find the sum of the absolute values of the roots of the polynomial obtained by replacing $x$ with $-x$ in the original equation. | nvidia |
In a trapezoid, the lengths of the bases are 22 and 36 units, and the height is 18 units. Find the length of the segment that joins the midpoints of the legs of the trapezoid. | nvidia |
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