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What is the smallest positive integer that has exactly 10 positive divisors? | nvidia |
Find the value of $x$ such that the sum of the squares of the numbers in the set $\{6, 13, 18, 4, x\}$ is equal to 900. | nvidia |
Let $P$ be a set of numbers, where each number is a power of a prime number $p$. The set $P$ contains all powers of $p$ from $p^1$ to $p^n$, where $n$ is a large positive integer. Given that the number of digits in $p^n$ is equal to $d$ and that its first (leftmost) digit is $d_1$, find the number of elements in $P$ th... | nvidia |
Find the value of the expression
\[ (x+y+z)(x+y-z)(x-y+z)(-x+y+z), \]given that $x=2$, $y=3$, and $z=5$. | nvidia |
A rectangular garden measures $a$ meters by $b$ meters, where $a$ and $b$ are positive integers. When a path that is 1 meter wide is built around the garden, the area of the path is 15 square meters. If the garden is divided into smaller square plots of side length 1 meter, and each plot can be planted with either flow... | nvidia |
When two fair 4-sided dice are rolled, the numbers $a$ and $b$ are obtained. What is the probability that the two-digit number $ab$ (where $a$ and $b$ are digits) and $a$ and $b$ are all even numbers? | nvidia |
The function $$g(x)=\frac{1}{\lceil x^3-2x^2-5x+1\rceil}$$ has a domain that consists of several intervals. Find the sum of the lengths of these intervals. | nvidia |
Find the number of positive integers $a$ such that the roots of the cubic equation $x^3-ax^2+2a^2x-2016=0$ are three consecutive prime numbers. | nvidia |
Find the smallest positive integer $n$ such that the sum of the reciprocals of the first $n$ perfect squares is equal to the reciprocal of a positive integer $m$. | nvidia |
Find the fifth number in the row of Pascal's triangle that starts with 1 and then 15, given that the row represents the coefficients of the expansion of (1 + x)^15. | nvidia |
What is the sum of all positive integer values of $n$ such that $\frac{n+18}{n}$ is an integer? | nvidia |
Three positive integers are arranged on a circle in such a way that the difference between any two adjacent integers is always the same. If the smallest integer is 1 and the largest integer is 5, what is the value of the third integer? | nvidia |
How many positive four-digit integers less than 800 have at least two consecutive digits that are the same? | nvidia |
Let $A$ be a set of positive integers. Let $g : A \to \mathbb{Z}$ be a function such that
(i) $g(1) = 1,$
(ii) $g(x + y) = g(x)g(y)$ for all $x,$ $y \in A$ such that $x + y \in A,$ and
(iii) $g(x^2) = g(x)^2$ for all $x \in A$ such that $x^2 \in A.$
Find the number of possible functions $g(x).$ | nvidia |
Find the set of all possible values of $x$ that satisfy the inequality
$$\left(\frac{x^2+a}{x+1}\right)^n>x^2+4$$
for all positive integers $n$, where $a$ is a fixed real number. Express your answer in interval notation. | nvidia |
Find the area of the triangle formed by the three complex numbers $w$, such that
\[w^2 \overline{w}^2 + w \overline{w}^3 + \overline{w} w^3 = 400,\]
and both the real and imaginary parts of $w$ are rational numbers, when plotted in the complex plane. | nvidia |
Find two numbers such that the ratio of their sum to their product equals the ratio of the square root of the larger number to the square root of the smaller number. | nvidia |
The sum of two positive integers is 24 and their ratio is 3:5. What is the value of the larger of the two integers? | nvidia |
Find all values of $x$ that satisfy $\frac{x(x+1)}{(x-4)^2} \ge 0.$ | nvidia |
Point $P$ lies on the line $x= -3$ and is 10 units from the point $(5,2)$. Find the product of all possible $y$-coordinates that satisfy the given conditions. | nvidia |
Determine all real numbers x that satisfy the inequality
$$|x^2-2x+2|<2x.$$ | nvidia |
A bag contains ten balls, some of which are red and the rest of which are yellow. When two balls are drawn at random at the same time, the probability that both balls are red is $\frac{1}{15}$. How many balls in the bag are red? | nvidia |
Find the value of $a$ such that the inequality $\frac{x^2-4x+4}{x^2-1}>a$ holds true if and only if $x\in (0,2)$. | nvidia |
In a regular hexagon with side length 8 cm, a circle is inscribed. Find the length of the longest segment that can be drawn inside the hexagon, with one endpoint on the circle and the other endpoint on the hexagon. | nvidia |
Let $n$ be a positive integer such that its square $n^2$ has 33 factors including 1 and itself. How many whole-number factors does $n$ have? | nvidia |
Find the sum of all integers divisible by 7 between 1 and 49 (inclusive). | nvidia |
How many pairs of positive integers $(x,y)$ satisfy the equation $xy=88$ and the inequality $|x-y|<12$? | nvidia |
What is the positive integer $M$ for which $$ 33^2 \times 77^2 = 11^2 \times M^2 \? $$ | nvidia |
Let $a$ and $b$ be two positive numbers such that $\log_{a}{b}=4$ and $\log_{b}{(ab)}=7$.
Determine $a$. | nvidia |
In the equation
$$2x^2-kx+2=0$$
determine k such that the product of the roots equals the negative sum of the roots modulo 13. | nvidia |
Let $z$ be a complex number such that $\sqrt{3z}+\sqrt{2i}=5$, where $i^2=-1$. Find the real part of $z$. | nvidia |
Let $A$ and $B$ be points on the parabola $y=x^2$ such that the origin is the midpoint of $\overline{AB}$. Find the minimum value of $A$'s $y$-coordinate. | nvidia |
Let $g(x) = x^3 + ax + b$, where $a$ and $b$ are constants. Find all real numbers $a$ and $b$ such that the inverse function $g^{-1}(x)$ satisfies the condition $g^{-1}(x) = -g(x)$ for all $x$ in the domain of $g$. | nvidia |
There are four complex numbers $z$ such that
\[z \overline{z}^3 + \overline{z} z^3 = 350,\]and both the real and imaginary parts of $z$ are integers. These four complex numbers are plotted in the complex plane. Find the area of the quadrilateral formed by the four complex numbers as vertices. | nvidia |
What is the value of $a$ such that the expression $ax^2+3x-5$ can be written as the product of two binomials, one of which is $2x+1$? | nvidia |
A circle of radius $R$ is randomly chosen inside a square with side length $R$. A line is drawn through the center of the circle with a slope greater than or equal to $\frac{1}{2}$. The probability that the line intersects the circle can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are relatively prime p... | nvidia |
Given a quadratic expression $ax^2 + bx + c$ where $a$, $b$, and $c$ are integers, and $a \neq 0$, find the smallest possible value of $b^2 - 4ac$ such that the expression can be factored into the product of two quadratic factors with integer coefficients. | nvidia |
In $\triangle XYZ$, point $M$ is the midpoint of side $YZ$. Point $N$ is on $XZ$ such that $XN: NZ = 2:3$. Point $P$ is on $XM$ such that $XP: PM = 5:2$. If the area of $\triangle MNP$ is 25, determine the area of $\triangle XYZ$. | nvidia |
Find the coordinates of the point on the line segment connecting the points $(3,7)$ and $(5,1)$, such that the distance from this point to $(3,7)$ is twice the distance from this point to $(5,1)$. | nvidia |
What is the greatest integer less than or equal to\[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?\]
$\textbf{(A) }80\qquad \textbf{(B) }81 \qquad \textbf{(C) }96 \qquad \textbf{(D) }97 \qquad \textbf{(E) }625\qquad$
| nvidia |
Let $f(x)$ be a function defined on the positive reals such that for each $x,$ $f(x)$ is equal to the maximum value of the three numbers $x^2-2x+3,$ $2-x^2,$ and $x+\frac{1}{x}.$ Find the minimum value of $f(x)$. | nvidia |
What is the sum of all the positive two-digit integers divisible by both the sum and product of their digits? | nvidia |
Find the value of $b$ that satisfies the equation $542_{b}+376_{b}=90E_{b}$, where $E_{b}=14_{10}$. | nvidia |
If $x$ and $y$ are positive real numbers such that $5x^2 + 10xy = x^3 + 2x^2 y,$ what is the value of $x$? | nvidia |
Find the number of distinct positive integers that can be obtained by multiplying some or all of the prime factors of 2002, with each factor used at most once. | nvidia |
Find $y$ such that $\log_{8}2y=3$. | nvidia |
Let $a$ and $b$ be constants such that the equation
$$\frac{1}{x^2+ax+b}=\frac{x}{x^2+2ax+3b}$$holds for all real values of $x$ except the roots of the denominator. Find $b/a$. | nvidia |
Find the number of positive factors of 72 that are in the form $a^2 \cdot b$, where $a$ and $b$ are positive integers. | nvidia |
A function $g$ has the property that $g(2x+3)=x^2-2x+5$ for all real numbers $x$. What is $g(7)$? | nvidia |
Find the number of integers n that satisfy the inequality $\log_{2}(n+1) < \log_{3}(2n+1)$. | nvidia |
The sum of two positive integers is 24 and their ratio is 3:5. What is the value of the larger of the two integers? | nvidia |
Given that $\triangle ABC$ is an isosceles triangle with base $BC=50$, and point $E$ lies on line $AB$ such that $AE$ is perpendicular to $AB$. If the altitudes from $B$ and $C$ intersect at point $E$, determine the length of segment $BE$. | nvidia |
The function $f$ takes nonnegative integers to real numbers, such that $f(1) = 1,$ and
\[f(m + n) + f(m - n) = \frac{f(2m) + f(2n)}{2}\]for all nonnnegative integers $m \ge n.$ Find the sum of all possible values of $f(10).$ | nvidia |
Given the polynomial $P(x) = x^3 - 2x^2 + x + 1$, find the sum of the squares of the coefficients of the factors if $P(x)$ can be factored into the product of two polynomials with integer coefficients, one of which is linear and the other is quadratic. | nvidia |
The vertices of a triangle are given by the complex numbers $a$, $b$, and $c$. A rotation of the plane about the origin by $\frac{\pi}{2}$ takes the triangle to a new position. Find the complex number that represents the image of the midpoint of the side joining the vertices corresponding to $a$ and $b$ under this rota... | nvidia |
Let $a,$ $b,$ and $c$ be positive integers such that
$$2\sqrt{\sqrt[4]{6}-\sqrt[4]{5}}=\sqrt[4]{a}+\sqrt[4]{b}-\sqrt[4]{c}.$$Determine $a+b+c$. | nvidia |
Let $P(z)$ be a polynomial of degree $4$ with integer coefficients such that $P(3i)=P(-3i)=0$, and $P(1)=P(-1)=0$. Determine the value of $P(0)$. | nvidia |
A three-digit number $xyz$ is such that when its digits are reversed, the resulting number $zyx$ is $27$ more than the original number. If the middle digit $y$ is $2$, find the smallest possible value of the original number $xyz$. | nvidia |
The area of a square equals the square of a length of the side of the square. The perimeter of a square equals the sum of the lengths of all four sides. The sum of the areas of two squares is 65, while the difference in their areas is 33. Find the sum of their perimeters. | nvidia |
What is the sum of the seven smallest distinct positive integer multiples of 9 that are also perfect squares? | nvidia |
A function $g : \mathbb{Z}^+ \rightarrow \mathbb{R}$ has the property that for all positive integers $n$ and $m$,
\[
g(mn) + g\left(\frac{m}{n}\right) = g(m) + g(n) + 1
\]
Compute the value of $g(2007)$. | nvidia |
Let $a$ be a real number, $|a| < 1,$ and let $w$ be a complex number such that
\[w - \frac{1}{w} = a.\]Find $|w|.$ | nvidia |
Find the smallest positive integer $a$ such that $17a \equiv 23a \pmod{101}.$ | nvidia |
Find the number of times the digit 7 appears in the decimal representation of the product of all positive integers less than or equal to 10 that are not divisible by 3. | nvidia |
Let $f(x)$ be a polynomial of degree 2006 with real coefficients, and let its roots be $r_1,$ $r_2,$ $\dots,$ $r_{2006}.$ There are exactly 1006 distinct values among
\[|r_1|, |r_2|, \dots, |r_{2006}|.\]What is the minimum number of real roots that $f(x)$ can have? | nvidia |
Let $a$, $b$, $c$ be three distinct real numbers such that
$$\begin{vmatrix}
0 & 2b & 1\\
1 & b+c & c\\
1 & bc & a
\end{vmatrix}=0.$$
Find the value of $a$. | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=10$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=3$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
Determine the number of positive integers $m \le 2000$ such that $12m$ is a perfect cube. | nvidia |
Let $f$ be a continuous function defined on the entire real line, satisfying the functional equation
$$f\left(x+\frac{1}{2}\right)=f(x)+2\sin x.$$
Find the smallest positive $T$ such that $f(x+T)=f(x)$ for all $x$. | nvidia |
Let $x$ and $y$ be real numbers such that
$$x+\frac{1}{y}=2$$
and
$$y+\frac{1}{x+\frac{1}{y}}=3.$$
Compute $\displaystyle \frac{x}{y}$. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $\log_a b + \log_b c + \log_c a = 1$. Find the value of $abc$. | nvidia |
Suppose that $a$ and $b$ are positive integers such that $(a+bi)^2 = 3+4i$. What is $a+bi$? | nvidia |
A positive integer $n$ is called "prime-like" if it is a product of two distinct prime numbers. How many prime-like integers are there between $n^2$ and $n^2+n$ for any integer $n>1$? | nvidia |
Let $A$ and $B$ be two points on the $x$-axis and $y$-axis, respectively, such that $A$ is $5$ units away from the origin and $B$ is $3$ units away from the origin. Find the distance from the origin to the point on the line segment $AB$ that divides it in a $2:1$ ratio. | nvidia |
How many three-digit positive integers exist, all of whose digits are 2's and/or 5's? | nvidia |
Find $x$, such that $4^{\log_7x}=16$. | nvidia |
A set of five different numbers is chosen from the set {1, 2,..., 20}. What is the probability that there are at least two consecutive numbers among the five chosen numbers? | nvidia |
Find the greatest value of $a$ such that $a^2-10a+21 \le 0$. | nvidia |
Compute the value of $x$ that satisfies the equation $\cos 2x = \sin x$. | nvidia |
Find the minimum value of the expression
$$\frac{x^2+y^2+z^2}{xy+yz+zx}$$
over all positive real numbers $x$, $y$, and $z$ satisfying $x+y+z=6$. | nvidia |
Find the number of integers $-8 \leq x \leq 8$ that satisfy the inequality $(x-3)(x+2)(x-5) > 0$. | nvidia |
In a certain city, there are two types of streetlights: old and new. The old streetlights are less energy-efficient and have a different maintenance schedule than the new ones. The city has at most 3000 streetlights. When three streetlights are selected randomly without replacement, there is a probability of exactly 1/... | nvidia |
There are 3 complex numbers $a+bi$, $c+di$, and $e+fi$. If $b=3$, $e=-a-c$, and the sum of the numbers is $2i$, find $d+f$. | nvidia |
Consider two lines intersecting at point O. The angle between the lines varies, but they always intersect the circle centered at O with radius 4 at exactly four points A, B, C, and D.
Given that the sum of the squares of the arc lengths AB and CD is 100, find the radius of the circle tangent to lines AB and CD at thei... | nvidia |
Let $a$ and $b$ be integers such that $0<a<b$.
If $\cos(a \cdot \theta)=\cos(b \cdot \theta)$ for every angle $\theta$,
determine the minimum value of $b-a$ (expressed as a positive integer). | nvidia |
There are three real numbers $x$ that are not in the domain of $$f(x) = \frac{1}{1+\frac{1}{1+\frac 1x}}.$$ What is the sum of those three numbers? | nvidia |
Find the value of $y^4$ if the following holds true for a positive real number y: \[
|y^2-y+1|+|y^2+y+1|=4.
\] | nvidia |
Let $a$ and $b$ be positive real numbers such that
\[\frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}} = 1.\]Find the maximum value of $\frac{a}{b} + \frac{b}{a}$. | nvidia |
Find the probability that the sum of the numbers obtained from three standard dice is equal to 11. | nvidia |
Find the least positive integer n such that 6n - 1 is divisible by 11. | nvidia |
Let $z$ be a complex number such that $\frac{1+z}{1-z}=i.$ Find $\left|\frac{1+z}{1-z}\right|^4.$ | nvidia |
A circle centered at the point $(2,-7)$ with a radius of $6$ units intersects a square of side length $8$ units, which has its upper left corner at the origin $(0,0)$. The points of intersection are $A, B, C,$ and $D$, and the lengths of the segments $AB$ and $BC$ are equal. Find the area of triangle $AOC$, where $O$ i... | nvidia |
Given a 4x4 matrix where all the diagonal elements are x + a, all the elements above the diagonal are x, and all the elements below the diagonal are 0, find the value of x in terms of a, such that the determinant of this matrix is 0. | nvidia |
Let $f(x)$ be a polynomial of degree $2017$ such that the constant term is equal to $200$ and all the coefficients are integers. Prove that there exists an integer $k$ such that $f(k)$ is divisible by $1000$, and find the smallest such integer $k$. | nvidia |
Let $m$ be a positive integer, and let $a_0, a_1, \dots , a_m$ be a sequence of real numbers such that $a_0 = 37$, $a_1 = 72$, $a_m=0$, and $$ a_{k+1} = a_{k-1} - \frac{3}{a_k} $$for $k = 1,
2, \dots, m-1$. Find $m$. | nvidia |
In a right triangle with legs of length 3 and 4, a circle is inscribed such that it is tangent to all three sides of the triangle. Find the radius of the inscribed circle. | nvidia |
A square is drawn such that one of its sides coincides with the line $y = 5$, and so that the endpoints of this side lie on the parabola $y = x^2 + 3x + 2$. What is the area of the square? | nvidia |
Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 70$. Express $x$ as a decimal.
A positive real number $x$ satisfies the equation $\lfloor x \rfloor^2 + x = 100$. Determine the decimal representation of $x$. | nvidia |
Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2=1$. Find the maximum value of $xyz/(xy+yz+zx)$. | nvidia |
Find the sum of the infinite series $3+4\left(\dfrac{1}{2345}\right)+5\left(\dfrac{1}{2345}\right)^2+6\left(\dfrac{1}{2345}\right)^3+\cdots$. | nvidia |
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