instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find the smallest positive integer $n$ such that $n^2$ is a palindrome and contains exactly three digits. | nvidia |
Find the sum of all real solutions to the equation \[\frac{x-2}{x^2+4x+1} = \frac{x-5}{x^2-10x}.\] | nvidia |
Let $a$ be a real number such that $a > 1$ and
\[\sum_{n=1}^{\infty} \frac{n^2}{a^n} = \frac{18}{5}.\]Find $a$. | nvidia |
Evaluate the product \[ (a-10) \cdot (a-9) \cdot \dotsm \cdot (a-1) \cdot a, \] where $a=2$. | nvidia |
A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Determine the possible values for the other two zeros.
Answer: The other two zeros must be complex conjugates, and their product must be a rational number. This is because the polynomial has ... | nvidia |
Consider the function
\[h(x)=f(\ln x)\]
where $f(x)$ is defined by the graph below.
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=5.25,ymin=-3.25,ymax=4.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+li... | nvidia |
Let $n$ be the least common multiple of the positive integers from 1 to 15. Find the largest number $m$ such that $m^2$ divides $n$. | nvidia |
Let $T$ be the set of all positive real numbers. Let $g : T \to T$ be a function such that
\[g(x) + g(y) = g(x^y + y^x)\]for all $x,$ $y \in T$. Find the largest possible value of $g(3)$. | nvidia |
Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \[\begin{aligned} xy+x+y&=71 \\ x^2y+xy^2 &= 880.\end{aligned}\] | nvidia |
The function $g(n)$ is defined on the positive integers such that $g(g(n)) = 2n + 1$ and $g(2n + 2) = 4n + 1$ for all positive integers $n.$ Find $g(250).$ | nvidia |
Let $\mathbf{P}$ be a $2 \times 2$ matrix such that $\det \mathbf{P} = -1.$ Given that $\det (\mathbf{P}^n) = 256$ for some positive integer $n,$ find $n.$ | nvidia |
Find the sum of all numbers $x$ satisfying $x+25/x = 10.$ | nvidia |
Find the smallest possible value of $n$ for which the sum of $n$ consecutive positive integers is a perfect square, and the largest term in the sum is less than $2n$. | nvidia |
For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 50$.
(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? | nvidia |
Find the number of integer solutions to the inequality $|2x-5| \geq |x+1|$ that satisfy the condition $-10 \leq x \leq 10$. | nvidia |
Find the number of ordered pairs of non-negative integers $(a,b)$ such that $a^3 + b^2 < 100$ and $a < b$. | nvidia |
Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\sin x + \cos x = \sqrt{2}.$ Enter all the solutions, separated by commas. | nvidia |
Determine the value of $k$ such that the polynomial $x^3-kx^2-3x+8$ is divisible by $x+2$. | nvidia |
A sequence is defined by $b_n = 2^n + 3^n$. Find the smallest positive integer $n$ such that $b_n$ is divisible by $13$. | nvidia |
Find the maximum value of
\[\frac{x^2 + 11}{\sqrt{x^2 + 5}}\]over all real numbers $x.$ | nvidia |
Find the maximum value of $|(z^2-4)^2(z^2+4)|$, given that $z$ is a complex number with $|z|=\sqrt{3}$. | nvidia |
Let $P(x)$ be a polynomial of degree 6 such that
\[P(x) = P(2016-x).\]
Find the maximum number of real roots that $P(x) - 1$ can have. | nvidia |
How many different $4\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?
Problem:
How many different $4\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each... | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that
\[a^2b^2 + b^2c^2 + c^2a^2 = a^2b^2c^2.\]
Find the maximum value of $a + b + c.$ | nvidia |
Find the number of ordered pairs of positive integers $(a, b)$ such that the equation $ax^3 + bx^2 + 1 = 0$ has at least one rational root. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a + b = 2.$ Find set of all possible values of $\frac{1}{a} + \frac{1}{b}.$ | nvidia |
Let $a$ be a real number such that
\[a^3 + |a|^3 = 28.\]Find $a$. | nvidia |
Find the number of positive integers $n \le 500$ such that $24n$ is a perfect cube. | nvidia |
Find the largest positive integer $n$ such that $n^2$ is divisible by 72, $n^3$ is divisible by 168, and $n^4$ is divisible by 400. | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2+2x+a^2=0$$
are such that
$x_1^2+x_2^2=12$.
Determine $a$. | nvidia |
Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2=1$. What is the minimum value of the expression \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+a+b+c?\] | nvidia |
A quadratic equation in the form $x^2+bx+c=0$ has one root that is twice the other root. Find the sum of all possible values of $b$. | nvidia |
In the equation $x^3 + ax^2 + bx + c = 0$, the sum of the roots is $-2$, the sum of the pairwise products of the roots is $5$, and the product of the roots is $-6$. Find the sum of the cubes of the roots. | nvidia |
Find the largest positive real number $c$ such that for all real numbers $a$ and $b$,
$$|a^3 + b^3| + |a^2b + ab^2| + |ab(a + b)| \ge c|a + b|^3.$$ | nvidia |
For how many ordered pairs of positive integers $(x,y),$ with $y<x\le 100,$ are both $\frac xy$ and $\frac{x+1}{y+1}$ integers?
| nvidia |
A right triangle with a perimeter of 60 cm has integer side lengths. If the area of the triangle is 150 cm^2, find the length of the hypotenuse. | nvidia |
Determine the number of integers between 10 and 100 that have at least one digit equal to 5. | nvidia |
In the expansion of $(a+b)^n$, what is the coefficient of the term that contains the factor $a^3b^2$, given that $n$ is a positive integer greater than or equal to 5? | nvidia |
Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}.$ Find the number of possible values of $S.$
| nvidia |
Find the volume of the tetrahedron whose vertices are $(0,0,0)$, $(1,0,0)$, $(0,2,0)$ and $(x,y,z)$, where $x,y,z$ are positive real numbers and the point $(x,y,z)$ lies on the surface defined by the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. The numbers $a,b,c$ are positive and are given by $a=1, b=... | nvidia |
Determine the real values of $m$ for which the cubic equation $$x^3-mx^2-mx+m^2=0$$ has only one positive root. | nvidia |
Let $x$ be a positive real number such that $\lfloor x\rfloor+\lceil 1/x\rceil=11$. How many possible values of $x$ are there? | nvidia |
Let $z$ be a complex number such that $z^2 + z + 1$ is a multiple of $5$. Determine the smallest positive integer $m$ such that $z^3 + m$ is also a multiple of $5$. | nvidia |
The volume of a cube with edge length $x$ inches is twice the volume of a cube with edge length $y$ inches. Find the edge length of the larger cube in the form $c\sqrt[3]{d}$, where $c$ and $d$ are positive integers and $d$ has no perfect cube factors. Compute $c+d$. | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xy) + f(xz) - f(x) f(yz) \ge 1\]for all real numbers $x,$ $y,$ and $z.$ | nvidia |
Let $f(x)$ be a function defined on the set of positive integers, such that $f(x)$ is the sum of the digits of $x$ when $x$ is a perfect square, and $f(x) = f(x - 1) + 2$ otherwise. Find the maximum value of $f(x)$ for $1 \le x \le 100$. | nvidia |
A hexagon $ABCDEF$ is inscribed in a circle such that $\angle ADB = \angle BEC = \angle CFD = 120^\circ$. The sum of the areas of triangles $ABD$, $BCE$, and $CFD$ is equal to $\frac{m}{n}\pi$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | nvidia |
Let $k$ be the product of all positive integers less than $6!$ that are coprime to $6!$. Determine the remainder when $k$ is divided by $6!$. | nvidia |
Find all values of x such that x^2 + 3x + 2 = 0 and the sum of the roots is equal to -3. | nvidia |
Three players roll a standard dice in succession. Find the probability that at least one of the players will roll a number that is greater than the number obtained by the previous player. | nvidia |
In the five-digit numeral $ABCBA$, $A$ is a prime number and the sum of all digits is a multiple of 4. Determine the largest possible value of $A$ that makes the number $ABCBA$ a multiple of 5. | nvidia |
What is the smallest positive integer n such that the sum of the elements in the nth row of Pascal's triangle is a perfect square? | nvidia |
Let $a_n$ be the smallest positive integer such that $a_n!$ is divisible by $2^n$.
Find the largest value of $n$ for which $a_n<1992$. | nvidia |
Which of the following quantities is the largest? (Write $A$, $B$, or $C$.) \[ A.\ \ \frac{2006}{2005}+\frac{2006}{2007} \qquad B.\ \ \frac{2006}{2007}+\frac{2008}{2007} \qquad C.\ \ \frac{2007}{2006}+\frac{2007}{2008} \] | nvidia |
Find $x$, such that $4^{\log_7x}=16$. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that
\[x^2+y^2+z^2=3\sqrt[3]{xyz}.\]
If $T=x+y+z$, determine the maximum value of $T^3$.
[Your response should end here] | nvidia |
Find the smallest positive integer that is a perfect cube and has exactly 4 distinct prime factors. | nvidia |
Determine the smallest positive integer $m$ such that
\[\tan^m x + \cot^m x \le 2\]
for all $x$ in the interval $(\pi/4, \pi/2)$. | nvidia |
A circular pizza is cut into two pieces along a chord, and the ratio of the areas of the two pieces is $3:5$. The pizza is then cut along a second chord that is parallel to the first, and the ratio of the areas of the two pieces on one side of the second chord is $2:3$. If the diameter of the pizza is $30$ inches, find... | nvidia |
Find the minimum value of the expression $x^2+y^2-8x-6y+22$ for real $x$ and $y$. | nvidia |
Let $a$, $b$, and $c$ be real numbers such that $\sqrt{a}+\sqrt{b}+\sqrt{c} = 5$. Determine the minimum value of $a+b+c$ such that all square roots involved are defined. | nvidia |
Find the sum of all integral values of $d$ with $d\ge -20$ for which the equation $z=x^3-5x-d$ has exactly one real root. | nvidia |
Let $P(x) = ax^4 + bx^3 + cx^2 + dx + e$ be a quartic polynomial such that the sum of its coefficients is equal to zero. Find the sum of squares of its coefficients if it's known that
$P(-1)=-7$,
$P(1)=5$. | 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 |
Find the length of the shortest path that connects the point $(-2,3)$ to the line $y=2x-1$ and then to the point $(4,1)$, given that the path must form a right angle where it meets the line. | nvidia |
A natural number $a$ is such that $\sqrt{a^2+16}$ is an integer,
and $a$ is not divisible by 4.
Find $a$. | nvidia |
Find the difference between the sum of the squares of the first 10 prime numbers and the sum of the cubes of the first 7 prime numbers. | nvidia |
Find the positive real number x such that [ x ] ⋅ x = 30 . Express x as a decimal. (Here [ x ] denotes the greatest integer less than or equal to x .) | nvidia |
How many of the 200 smallest positive integers are such that the sum of their digits is a multiple of 3? | nvidia |
Quadrilateral $ABCD$ is a parallelogram with center $O$. Let the feet of the perpendiculars from $O$ to $\overline{AB}$ and $\overline{BC}$ be $E$ and $F$, respectively. Given that $AB = 121$ and $AD = 40$, and that the length of one of the diagonals of the parallelogram is $71$, find the area of the parallelogram. | nvidia |
Determine the number of positive four-digit integers of the form $\_\_99$ that have a remainder of 3 when divided by 7 and have a sum of digits that is divisible by 9. | nvidia |
There are 2011 non-negative integers whose sum is equal to 2012 and the sum of their squares is equal to 2013. Let $x$ be one of these integers. Find the maximum value of $x + \sqrt{x}$ | nvidia |
The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the the smallest possible value of $m+n.$ | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$$\frac{1}{x_1}+\frac{1}{x_2}=\frac{1}{2}.$$
Determine $a$. | nvidia |
Find the sum of all positive integers $k$ that have a unique expansion of the form
\[k = (a_3a_2a_1a_0)_i\]
where $a_3 \neq 0$ and the base $i$ satisfies $i^2 = -1$. | nvidia |
The vertices of a convex pentagon lie on a circle. What is the largest possible value of the sum of the interior angles of the pentagon that are formed by three consecutive vertices? | nvidia |
The real function $g$ has the property that, whenever $a,$ $b,$ $n$ are positive integers such that $a \times b = 2^n,$ the equation
\[g(a) \times g(b) = n^3\]holds. What is $g(8)$? | nvidia |
Find the sum of the solutions of the equation: $\left| x-3 \right|+\left| x-5 \right|=4$ | nvidia |
Find the smallest positive real number $y$ such that
\[\lceil y^3 \rceil - y^2 \lceil y \rceil = 12.\] | nvidia |
Compute the sum of the infinite series $1 + 2x + 3x^2 + 4x^3 +...$, where $x$ is a real number such that $|x| < 1$. | nvidia |
Find the maximum number of consecutive integers whose sum is equal to 2022 and for which the sum of their squares is less than or equal to 2022 times 999. | nvidia |
Compute the number of distinct subsets of size $3$ that can be formed from a set containing $10$ red balls and $15$ blue balls. | nvidia |
Find the equation of the circle that passes through the points $(0,0)$, $(3,4)$, and $(6,8)$. | nvidia |
What is the smallest positive integer $n$ such that $\frac{n}{n+101}$ is equal to a terminating decimal? | nvidia |
Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$. | nvidia |
Find all values of $x$ so that $\arccos x > \arcsin x.$ | nvidia |
A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$? | nvidia |
The sum of the squares of two consecutive integers is 17 times the square of the larger integer. Determine the value of the smaller integer. | nvidia |
A certain sequence of integers begins with 0 and ends with 1000, and contains exactly one instance of every positive integer with exactly three digits. The sequence is formed by writing the integers in a specific order, such that the last two digits of each integer are the first two digits of the next integer in the se... | nvidia |
If $x$ and $y$ are real numbers such that $x^2 + y^2 = 25$ and $xy = 12$, what is the sum of all possible values of $x$? | nvidia |
Find the smallest positive $a$ such that $a$ is a multiple of $4$ and $a$ is a multiple of $14.$
Another problem inspired by this one:
Find the smallest positive integer $n$ such that $n$ is a multiple of $6$ and $n+3$ is a multiple of $9$. | nvidia |
Find the real roots of the equation: $$\sqrt[3]{x+\frac{1}{x}}=\sqrt{x}-1$$ | nvidia |
Determine the greatest number of distinct non-zero digits that can occur in the decimal representation of the product of two 3-digit numbers. | nvidia |
Milton spilled some ink on his homework paper. He can't read the coefficient of $x$, but he knows that the equation has two distinct negative, integer solutions. What is the sum of all of the distinct possible integers that could be under the ink stain?
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)-... | nvidia |
Find the product of the possible values of $y$ that satisfy the equation $\frac{y-1}{y+1} \cdot \frac{y+2}{y-2} = 3$. | nvidia |
Find the area of the triangle with vertices $(-1,4),$ $(7,0),$ and $(11,5).$ | nvidia |
Let $x$ be a positive integer such that its cube $x^3$ ends with four zeros.
What is the least possible number of zeros that the integer $x$ can have? | nvidia |
The sum of the squares of two positive integers is 117. The positive difference of their squares is 45. What is the product of the two integers? | nvidia |
Let $A$ be the set of all positive integers less than $20$ that are either a multiple of $4$ or a multiple of $5$. Let $B$ be the set of all positive integers less than $20$ that are a multiple of $4$ and a multiple of $5$. Compute $(|A| - |B|)^2$. | nvidia |
Let $Q$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $Q$ and the point $\left(\frac{3}{4}, \frac{1}{4} \right)$ is less than or equal to $-\frac{1}{3}$ can be written as $\frac{p}{q}... | nvidia |
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