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What is the sum of all real numbers that are tripled when added to twice their reciprocals? | nvidia |
Find the smallest positive integer $n$ such that the decimal representation of $n^2$ is not a palindrome, but the decimal representation of $n^3$ is a palindrome. | nvidia |
Given that $$(a+b+c)(ab+ac+bc)=125$$and that $$a^2(b+c)+b^2(a+c)+c^2(a+b)=21$$for real numbers $a$, $b$, and $c$, what is the value of $abc$? | nvidia |
Find the largest positive integer n such that there are exactly 5 positive integers less than or equal to n that are both a power of 2 and a power of 3. | nvidia |
Find the number of positive real solutions of the equation $\left| x^2 - \frac{3}{2} \right| = 5$. | nvidia |
Find the polynomial $p(x),$ with real coefficients, such that
\[p(x^3) - p(x^3 - 2) = [p(x)]^2 + 12\]for all real numbers $x.$ | nvidia |
Determine the largest positive integer that must divide the sum of the squares of the first five terms of any arithmetic sequence whose terms are positive integers and have a common difference of 2. | nvidia |
Determine $k$ such that the equation
$$x^{4k}-4x^{3k}+6x^{2k}-4x^k+1=0$$
has only real roots. | nvidia |
Find the number of digits that are greater than 5 in the base-8 representation of $185_{10}$. | nvidia |
Find the smallest two-digit prime number such that reversing the digits of the number produces a composite number. | nvidia |
The graph of the function
$$f(x)=\frac{x^4-4}{x^2-1}$$
consists of two distinct parts.
Determine the integer $n$ such that
$$\left[f(x)\right]^n=16.$$
is valid for any value of $x$ except
for $-1$ and $1$. | nvidia |
Let $D$ be the domain of the function $f(x) = \cos^{-1}(\sin(\sqrt{x}))$. Find the smallest positive real number $c$ such that $D \cap [c, c+2014] \neq \emptyset$. | nvidia |
If $\sqrt{3x^2-2}=\sqrt{11}$, find the sum of all the possible values of $x$. | nvidia |
Find the greatest integer value of $y$ such that $\lvert3y-5\rvert\le 23$. | nvidia |
Find the number of positive integer solutions to the inequality
$$\frac{3}{x}+\frac{5}{y}+\frac{7}{z}>1,$$
where $x,y,z$ are pairwise distinct and relatively prime. | nvidia |
In a certain triangle, the medians intersect at a point that divides each median into two segments, one of which is twice as long as the other. The sum of the lengths of the longer segments of the medians is 120 cm. Find the perimeter of the triangle. | nvidia |
Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than $30$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$. | nvidia |
Let $P$ be a point inside a triangle $ABC$ such that the tangents from $P$ to the incircle of $\triangle ABC$ touch the circle at $D$, $E$, and $F$, where $D$, $E$, and $F$ lie on the sides $BC$, $AC$, and $AB$, respectively. Given that the sides of $\triangle ABC$ have lengths $10$, $17$, and $21$, and that the distan... | nvidia |
Compute the number of distinct subsets of size $3$ that can be formed from the elements $\left\{-\frac{1}{4}, \frac{7}{10}, \pi,\frac{5}{6},-11,49,-99,72,\sqrt{8}, 45\right\}$. | nvidia |
Find the number of distinct four-digit positive integers that have only even digits and are divisible by 4. | nvidia |
In a regular 24-gon, each vertex is labeled with a number such that the sum of the numbers on adjacent vertices is always the same. The sum of the numbers on all 24 vertices is 144. Find the sum of the numbers on the 8 vertices that form the diagonals of the 24-gon. | nvidia |
Determine the complex number $z$ satisfying the equation $2z-3\bar{z}=-2-30i$. Note that $\bar{z}$ denotes the conjugate of $z$. | nvidia |
Determine the number of ways to arrange the letters of the word LOGIC so that no two consecutive letters are vowels. | nvidia |
Let $g$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[3g(a^3 + b^3) = [g(a)]^3 + [g(b)]^3\]for all nonnegative integers $a$ and $b.$
Let $m$ be the number of possible values of $g(125),$ and let $t$ be the sum of the possible values of $g(125).$ Find $m \times t.$ | nvidia |
Find the number of four-digit numbers that are palindromes and consist only of even digits. | nvidia |
Find the number of integers that satisfy the inequality $\left|\frac{x}{2}-3\right| \leq 2.5$ and are also divisible by 3. | nvidia |
Factor $2x^2+3x-1$ into two binomials with integer coefficients. Then find the quadratic polynomial that has the same roots as $2x^2+3x-1$ but with opposite signs. | nvidia |
Find all values of $x$ such that $x^4 - 6x^2 + 8 = 0$. Enter all the solutions, separated by commas. | nvidia |
What is the maximum number of points of tangency possible for three spheres of different sizes that can be drawn so that each one is tangent to the other two and all three are tangent to the same plane? | nvidia |
Find the number of divisors of 240 that are not multiples of 3. | nvidia |
There are integers $d,e$ for which both roots of the polynomial $x^2+2x-3$ are also roots of the polynomial $x^6-dx^2-e$. Determine the sum $d+e$. | nvidia |
Problem:
In a right-angled triangle, the length of the hypotenuse is 10 cm and one of the legs is 6 cm. The ratio of the areas of the squares built on the legs is $\frac{36}{64}$. Determine the length of the other leg and calculate the ratio of the areas of the circles built on the legs.
Assume that the radius of the... | nvidia |
Find the sum of the areas of all squares with side lengths between 5 and 15, inclusive, if the side lengths are limited to positive integer values. | nvidia |
Let $\mathcal{C}$ be the set of all distinct ordered pairs $(a,b)$ of positive integers less than or equal to $10$ such that $a$ and $b$ are the geometric means of the same two positive integers $m$ and $n$. Find the largest possible value of the sum of all $ab$. | nvidia |
The function $g(x)$ satisfies
\[g(x+y) = g(x) + g(y)\]
for all real numbers $x$ and $y$, and $g(x) \neq 0$ for at least one real number $x$.
Find $g(4)$, given that $g(1) = 2$. | nvidia |
Find the value of $x$ such that the expression $\frac{x^2+5x+6}{x+2}$ is undefined. | nvidia |
In a triangle, the side lengths are given by $x-1$, $x+1$, and $\sqrt{x^2-2}$. Determine the value of $x$ such that the triangle is a right triangle. | nvidia |
Let $S$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the maximum element of $S$. | nvidia |
Determine all real numbers $c$ such that the inequality
$$\frac{(2c^2+1)x^2+(4c+1)x+c^2}{x+c}>x^2$$
holds for every $x>0$. Express your answer in interval notation. | nvidia |
Find the largest value of $x$ that satisfies the equation $|x-5|=12$. | nvidia |
A function $g$ has the property that $g(2x-3)=x^2-2x+4$ for all real numbers $x$. What is $g(1)$? | nvidia |
In the equation
$$6x^2+4x+c=0$$
determine c such that the product of the roots be equal to -2. | nvidia |
Find the number of positive divisors of 4320 that are multiples of 4. | nvidia |
Determine the sum of the squares of two distinct positive integers whose product is 12345. | nvidia |
Express the number 315 as a product of distinct prime powers, such that the sum of the exponents of these powers is minimal. What is the least possible sum of the exponents of these powers? | nvidia |
Find the sum of the first 5 terms of the geometric sequence with common ratio $r$ such that the sum of the first 3 terms is 21 and the sum of the first 6 terms is 91. | nvidia |
Find the number of positive divisors of 8!. | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$.
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
Count the number of ways to color a set of 5 beads with red, blue, and green, such that at least one bead is red or blue, and the colors are distinguishable (i.e., red-blue is different from blue-red). | nvidia |
Let $A$ be a set of positive integers such that the product of the digits of each integer in $A$ is divisible by $5$. What is the largest possible number of integers in $A$ that are divisible by neither $5$ nor $25$ and are all composed of the digits $1$, $2$, $3$, $4$, $5$ and $6$? | nvidia |
Suppose that all the positive integers are written in five columns. In each row, the numbers increase by 4 from left to right. In what column will the number 751 be written? | nvidia |
Find the number of integers between 20 and 150 that are of the form $n^2 + 1$, where $n$ is a positive integer. | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
Given the equation of a parabola in the form of $x^2+Ay^2+Bx+Cy+D=0$, determine the value of A such that the parabola has only one real focus. | nvidia |
Let $a$ and $b$ be real numbers such that
$$ab=\frac{1}{3} \mbox{ and } a^2+b^2=6.$$
Compute the value of
\[(a+b)^2\left(1+\frac{1}{(a+b)^2}\right).\] | nvidia |
Prove that the expression $10 \sin x \cos^4 x - 10 \sin^4 x \cos x$ takes values greater than 1 for some values of $x.$
(Remember that 8 from the original equation was replaced with 10.) | nvidia |
Find the smallest integer n such that the product of its positive factors is greater than 1000. | nvidia |
Let $a, b, c$ be positive real numbers such that $a + b + c = 3$. Find the minimum value of $\frac{1}{a} + \frac{4}{b} + \frac{9}{c}$. | nvidia |
Let $x,y,z$ be positive real numbers such that $x+y+z=6$ and $xy+yz+zx=8$. Let $n=\max\{xy,yz,zx\}$. Find the smallest possible value of $n$. | nvidia |
How many integers satisfy the following system of inequalities?
\[
\begin{array}{cccc}
(1) & -2x & \geq & x+5 \\
(2) & -x & \leq & 8 \\
(3) & -3x & \geq & x+11
\end{array}
\] | nvidia |
What is the sum of all real numbers that are tripled when subtracted from their reciprocals? | nvidia |
The sequences $a_n$ and $b_n$ are defined recursively for $n\geq 1$ by the equations
\[a_n= \frac{1+a_{n-1}}{1+b_{n-1}},\quad b_n= \frac{1+b_{n-1}}{1+a_{n-1}},\]
where $a_0$ and $b_0$ are both positive real numbers. Define
\[s_n=a_n+b_n,\qquad t_n=a_nb_n.\]Compute the minimum and maximum possible values of $t_{1000},... | nvidia |
Find the number of occurrences of the digit 3 in the binary representation of $5^k$, where $k$ is the largest integer such that $5^k$ can be expressed as the sum of two perfect squares. | nvidia |
Let $a$, $b$, and $c$ be three positive integers such that $a$ is the arithmetic mean of $b$ and $c$. If the sum of all possible values of $a$ is 15, and $a^2$ is equal to the sum of the squares of $b$ and $c$, find the largest possible value of $a^2$. | nvidia |
A set of real numbers $\{a,b,c,d,e,f,g,h,i,j\}$ satisfies the following equations:
\begin{align*}
a+b+c+d+e+f+g+h+i+j&=5,\\
a+b+c+d+e+f+g+h+i&=4,\\
a+b+c+d+e+f+g+h&=3,\\
a+b+c+d+e+f+g&=2.
\end{align*}Find the value of $j$. | nvidia |
Find the number of positive integers $n$ such that the base-$n$ representation of $n^2$ is a palindrome. | nvidia |
In the hyperbola $$xy = n,$$ the area of the triangle formed by the asymptotes and the line $y = 1$ is 2019 square units. Find the least positive integer value of $n.$ | nvidia |
In the equation $x^2+px+q=0$, the roots $x_1$ and $x_2$ are such that $\frac{x_1}{x_2} = \frac{1}{2}$. If $p=5$, find the value of $q$. | nvidia |
Find the sum of all positive integral values of $m$ such that $m^2 - 18m + 64 = q$ for some prime number $q$. Note that there is at least one such $m$. | nvidia |
Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fourth row is 6. The sum of the interior numbers of the fifth row is 14. What is the sum of the interior numbers of the seventh row? | nvidia |
Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability t... | nvidia |
Problem:
A point is selected at random inside the unit circle in the complex plane. Determine the probability that the distance between the selected point and the point $1+2i$ is greater than the distance between the selected point and the point $-1+i$. | nvidia |
Given that $x$ and $y$ are positive integers, find the value of $x+y$ when $x^y$ is the smallest possible value greater than 1000. | nvidia |
Find the sum of the lengths of the two longest altitudes in a triangle with sides $9$, $20$, and $22$. | nvidia |
Let $a$ and $b$ be positive real numbers such that
\[\frac{a}{a^2 + 1} + \frac{b}{b^2 + 1} = \frac{1}{2}.\]
Find the maximum value of $\frac{a}{b} + \frac{b}{a}.$ | nvidia |
Find all values of $x$ that satisfy the inequality \[\frac{x^2-2x+2}{x^2+2x-3} < 0.\]Express your answer in interval notation. | nvidia |
Find all values of $x$ that satisfy the inequality
$$\frac{x^2+1}{x-2} > \frac{x-1}{x+2}.$$
Express your answer in interval notation, simplifying any fractions that occur in your answer. | nvidia |
In a rectangle, the sum of the squares of the lengths of its two diagonals is 105, while the difference between the squares of these lengths is 21. Find the sum of the squares of the lengths of its four sides. | nvidia |
In a city, there are five different bus routes, each with a different number of stops. The number of stops on each route is such that it can be exactly divided by 2, 3, 4, 5, or 6 buses. What is the smallest number of stops that could be on all five routes? | nvidia |
Given a positive integer $n$, find the smallest positive integer $m$ such that $n \equiv m^2 \pmod{8}$. | nvidia |
Given that the polynomial $x^2+7x+m$ has only negative integer roots, find the sum of all distinct possible values of $m$. | nvidia |
Find $x$, such that $4^{\log_7x}=16$. | nvidia |
A circle is inscribed in a square so that the diameter of the circle is equal to the side length of the square. The perimeter of the square is 48 centimeters. What is the area of the region between the circle and the square? | nvidia |
The numbers $1$ through $7$ are to be arranged in a line such that all primes are consecutive and all perfect squares are consecutive. In how many ways can this be done? | nvidia |
Find the product of the roots of the equation $24=x^2-5x$. | nvidia |
Given the triangle $ABC$ with sides $a$, $b$, and $c$, find the set of all possible values of $c$ such that the perimeter of the triangle satisfies the inequality $|a+b-c| < a+b+c$. Express your answer in interval notation. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 6.$ Find the maximum value of $(x^2+1)(y^2+1)(z^2+1).$ | nvidia |
Let m be a positive integer. Determine the largest possible value of $m$ such that $m(m^2-n^2)+n$ is divisible by 8 for all positive integers n less than m. | nvidia |
Find the value of $x$ such that the terms $x^2+1, x, \frac{1}{x^2+1}$ form a geometric sequence. | nvidia |
The vertices of a regular icosahedron are labeled with integers from 1 to 12, such that the sum of the labels of any three adjacent vertices is 20. How many different labelings are possible? | nvidia |
Determine the smallest positive integer that ends in 5 and is divisible by 23. | nvidia |
Find the number of perfect cubes less than 1000 that can be expressed as the sum of three consecutive integers. | nvidia |
Find the sum of the squares of the reciprocals of the positive divisors of 28. | nvidia |
Find the sum of the absolute values of the determinants of all $3\times3$ matrices consisting of the digits $1,2,3,4,5,6,7,8,9$, each used only once, if the digits in each row form a geometric sequence. | nvidia |
Let $x$ be a positive real number such that $x + \frac{1}{x} = \sqrt{98}.$ Find $\sqrt[4]{x} + \frac{1}{\sqrt[4]{x}}.$ | 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 $f$ be a function defined on the set of positive integers that satisfies the following condition:
$f(n + m) = f(n) \times f(m) + 1$
for all positive integers $n$ and $m$. If $f(1) = 2$, find the value of $f(6)$. | nvidia |
Using only the digits 7, 8 and 9, how many positive seven-digit integers can be made that are palindromes? | nvidia |
Find the smallest positive integer congruent to 1 (mod 13) that is divisible by 7. | nvidia |
Let $a_n$ be the sum of the reciprocals of the coefficients of the polynomial $(1+x)^n$, and let $b_n$ be the sum of the products of the reciprocals of the coefficients of the polynomial $(1+x)^n$ and the corresponding powers of $x$. Find $\frac{b_n}{a_n}$. | nvidia |
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