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Define a positive integer $n$ to be a square root remainder if there is some positive integer $m$ such that the decimal representation of $\sqrt{m}$ has exactly $n$ digits after the decimal point. How many positive integers less than $2500$ are not square root remainders? | nvidia |
Find the value(s) of $x$ such that $8xy-12y+2x-3=0$ is true for all values of $y$. | nvidia |
In the diagram, $ACDE$ is a trapezoid such that $\overline{AC} \parallel \overline{DE}$ and $ABCD$ is a parallelogram. If the height of the trapezoid is $6$ units and the area of triangle $BCE$ is $8$ square units, then find the area of triangle $ADE$. | nvidia |
Find the area of the triangle with vertices $(3,-5),$ $(-2,0),$ and $(1,-6).$ | nvidia |
A lattice point is a point with integer coordinates. Find the number of lattice points that lie on the curve defined by the equation $x^3 + y^3 = 3^6$. | nvidia |
Determine the number of three-digit numbers that are multiples of 5 and have digits that form a geometric progression. | nvidia |
Find the smallest positive integer $a$ such that the polynomial $x^2 + (a^2 + 1)x + a^2$ has a factor of the form $x^2 + bx + c$, where $b$ and $c$ are positive integers. | nvidia |
Find the smallest positive integer that can be represented as the sum of the squares of four consecutive natural numbers. | nvidia |
Let $x$ and $y$ be positive integers such that $x^2 - y^2 = 225.$ Find the minimum value of $x + y.$ | nvidia |
Let $P(n)$ be the product of all positive integers from 1 to $n$, and let $Q(n)$ be the sum of all positive integers from 1 to $n$.
Determine the largest integer $n$ such that $P(n) + Q(n)$ has only two distinct prime factors. | nvidia |
Find the sum of all positive integer values of $k$ such that the equation $3x^3-2x^2+kx-1=0$ has at least one real root in the interval $[-1,1]$. | nvidia |
Let $f$ be a function taking the integers to the integers such that
\[f(m + n) + f(mn - 1) = f(m) f(n) + 2\]for all integers $m$ and $n.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$ | nvidia |
Find the value of $x$ such that $\lfloor x \rfloor + x^2 = 50$, where $x$ is a positive real number. | nvidia |
Determine the product of all solutions of the equation
$$\ds\sqrt {\f{x+1}{x-1}}=\f{2x+5}{x+3}$$ | nvidia |
A certain high school consists of three classes, each with 35 students. The sum of the ages of all students in the school is 2100. Given that the sum of the ages of all students in class A is equal to the sum of the ages of all students in class C, find the largest possible sum of the ages of all students in class B.
... | nvidia |
Find the value of $x$ such that
$$(x+1)(x+1)-(x-1)(x+3)$$
is equal to the square of an integer. | nvidia |
In three-dimensional space, find the number of lattice points that have a distance of 5 from the origin. | nvidia |
The circle centered at $(2,-1)$ and with radius $4$ intersects the circle centered at $(2,5)$ and with radius $\sqrt{10}$ at two points $A$ and $B$. Find $(AB)^2$. | nvidia |
Find the number of positive integers less than 1000 that have a remainder of 5 when divided by 11. | nvidia |
Determine the largest positive integer $n$ such that the difference between $n^2$ and $(n-1)^2$ is less than $100$. | nvidia |
Let $f(x)$ be a real-valued function defined for $x > 0$ such that $f(x) \geq \frac{1}{\sqrt{x}}$ for all $x$. Find the minimum value of $\frac{x^2 + 1}{f(x)}$ over all $x > 0$. | nvidia |
Find all values of $k$ such that the expression $\frac{x^2-k^2}{x^2-2kx-3k}$ is undefined for exactly two values of $x$. | nvidia |
A positive integer $x$ is defined as a "lucky number" if the sum of its digits is equal to the number of digits in $x$.
Find the smallest lucky number that is divisible by 8. | nvidia |
Let the element whose symbol begins with the letter A in the periodic table of chemical elements be denoted by a number -- the sum of the positions in the alphabet of all the letters in its Latin name. For example, carbon is denoted by the number $3+1+18+2+15+14=53$. Determine the sum of the numbers denoted to argon, s... | nvidia |
Find the largest positive integer n such that the sum of its digits is less than or equal to 10. | nvidia |
Factor the polynomial $x^4 + x^2 + 1$ into two quadratic polynomials with integer coefficients. Determine the sum of the squares of the coefficients of these polynomials. | nvidia |
Let $g(x)$ be a function defined for all real numbers satisfying the conditions $g(x) \geq 0$ for all $x$ and
\[g(x + y) = \sqrt{g(xy) - 2}\]for all $x, y.$ Determine $g(0).$ | nvidia |
The vertices of a triangle $ABC$ are given by the complex numbers $z_A$, $z_B$, and $z_C$, where $z_B = 7+3i$ and $z_C = 12-5i$. The area of the triangle is $\frac{51}{2}$ and the centroid is $5+5i$. Find the maximum possible value of $|z_A|$.
(Note: $|z|$ represents the magnitude or modulus of the complex number $z$) | nvidia |
A sequence is defined as $y_1=a$ and $y_n=y_{n-1}+\frac{1}{y_{n-1}}$ for $n>1$, where $a$ is a positive integer. Determine the smallest value of $a$ such that the sequence contains at least two integer terms. | nvidia |
Find the number of real solutions of the equation
\[x = \cos x + \sin x.\] | nvidia |
Find all values of $b$ such that $b+\lceil b\rceil=17.8$. | nvidia |
Two circles have radii in a ratio of $2:5$. If the area of the larger circle is $3\pi$ square inches, what is the area of the smaller circle?
The ratio of the area of the smaller circle to the area of the larger circle is $4/25$. Since the area of the larger circle is $3\pi$ square inches, the area of the smaller circ... | nvidia |
Three standard, six-sided dice are rolled. The probability that the product of their values is greater than 90 is $\frac{1}{10}$. What is the expected value of the sum of the numbers shown when these dice are rolled? | nvidia |
Find the largest power of 3 that divides the sum of all the numbers from 1 to $3^{10}$. | nvidia |
The sum of the squares of two consecutive positive integers is 365. What is the larger integer? | nvidia |
Determine the number of positive integers less than or equal to 100 that are not divisible by any of 3, 5, and 7. | nvidia |
Consider the equation $y=x^3+ax^2+bx+c$. Given that $y=0$ has three distinct rational roots, find the maximum possible value of the sum of the absolute values of $a$, $b$, and $c$. | nvidia |
A triangle is inscribed in a square such that the vertices of the triangle coincide with three consecutive vertices of the square. The side length of the square is 5 inches. If the area of the triangle is 8 square inches, find the length of the segment connecting the center of the triangle and the midpoint of the side ... | nvidia |
Find the product of the largest and smallest divisors of 1024 that are powers of 2. | nvidia |
Given the quadratic equation $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are integers, find the values of $a$, $b$, and $c$ such that the equation has exactly two solutions, and the sum of the solutions is equal to $2b$. | nvidia |
Let $a$ and $b$ be two positive integers such that the number of their common divisors is 6. Determine all possible values for the greatest common divisor of $a$ and $b$. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $\log_a b = 3$ and $\log_b c = 2$. Find $\log_a \sqrt{bc}$. | nvidia |
Suppose that $a$ and $b$ are positive integers such that $(a+bi)^3 = -46+9i$. What is $a+bi$? | nvidia |
Find the product of all $y$ that satisfy the equation $\frac{3y^2-2y-1}{y^2+3y-4} = \frac{2y-1}{y-1} - \frac{y+2}{y+4}.$ | nvidia |
How many positive three-digit integers have an even number of positive factors? | nvidia |
Find the sum of all integers from 100 through 999, inclusive, that are divisible by 7. | nvidia |
If $0.8= \frac{y}{186+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers. | nvidia |
The geometric mean of an even number of consecutive even integers is z. Find the product of the two middle integers in terms of z. | nvidia |
How many integers satisfy the inequality $(x+3)^{2}\leq1$?
Problem:
Determine the values of $x$ that satisfy the inequality $2^{x+1}+2^{x-1}\leq 17$. | nvidia |
There is a unique two-digit positive integer $t$ for which the last two digits of $11\cdot t$ are $36$.
What is $t$? | nvidia |
Two cities are connected by a straight highway. The highway's width is constant and its ends are on the same latitude. Determine the angle of inclination of the highway if it is known that the difference in longitude between the two cities is $30^\circ$, and the length of the highway is 200 miles. | nvidia |
Find the value of $x$ such that the sum of the squares of three consecutive even numbers, starting from $x$, is equal to 80. | nvidia |
Let $S$ be the set of all integers $n$ such that
$$\sum_{k=0}^{n} \binom{2n}{2k} = \sum_{k=0}^{n-1} \binom{2n}{2k+1}.$$
Find the number of elements in $S$. | nvidia |
What is the area of the largest triangle that can be formed by connecting three distinct points on the graph of the equation $x^2 + 2y^2 = 4$? | nvidia |
Let $\star$ be a binary operation defined on positive integers such that $$x \star y = \sqrt{x^2+y^2}+xy.$$ Determine the value of $5 \star 3$. | nvidia |
A circle is inscribed in a triangle, such that the angle bisectors of the triangle intersect at the center of the circle. If the measure of one angle of the triangle is 60 degrees, and the measure of the other angle is 2x degrees, find the measure of the third angle in terms of x. | nvidia |
Let $S$ be the set of all nonzero real numbers. Let $f : S \to S$ be a function such that
\[f(x) + f(y) = f(xyf(x + y))\]for all $x,$ $y \in S$ such that $x + y \neq 0.$
Let $n$ be the number of possible values of $f(4),$ and let $s$ be the sum of all possible values of $f(4).$ Find $n \times s.$ | nvidia |
Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ$, find $z^{2000}+\frac 1{z^{2000}}$. | nvidia |
Let $x$ be a positive number such that $x,$ $\sqrt{x},$ and $\log_2 x$ form a geometric sequence in that order. Determine $x.$ | nvidia |
Find the largest integer x satisfying the inequality:
\[ \frac{x+1}{x^2 + 1} > \frac{1}{5}. \] | nvidia |
What is the sum of four consecutive odd integers if the sum of the first and last integers is 162? | nvidia |
Find all real numbers $x$ such that
\[\frac{\cos(x)}{2-\sin(x)}<2.\]
(Remember to express your answer in interval notation.) | nvidia |
Let $x$ and $y$ be two numbers on the number line such that the average of their absolute values equals 5, i.e. $\frac{|x|+|y|}{2}=5$, where $x$ is twice as far from 2 as $y$. Find the length of the segment joining $x$ and $y$. | nvidia |
A certain number $N$ can be written as the sum of $1000$ consecutive positive integers. If the $1000$th digit from the right of $N$ is a $7$, what is the largest possible value of the $999$th digit from the right? | nvidia |
The positive integers $a$, $b$, and $c$ are such that $ab + bc + ac$ has a common factor $m$ with $abc$.
Given that $m > 1$, find the minimum value of $m$. | nvidia |
The center of a circle lies on the line segment connecting the points $(2,3)$ and $(4,1)$. The circle passes through the point $(6,5)$. Find the sum of the squares of the coordinates of the center of the circle. | nvidia |
Find the product of the positive divisors of 315 that are not multiples of 3. | nvidia |
Given that $a$ and $b$ are positive integers, find the smallest value of $x$ such that $3x$ is divisible by 12 and $5x$ is divisible by 35. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a+b+c=1$. Find the maximum value of
\[\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}.\] | nvidia |
Let $a$, $b$, $c$, and $d$ be the roots of the equation $x^4-11x^2+13x+10=0$. The pairwise differences $|a-b|$, $|a-c|$, $|a-d|$, $|b-c|$, $|b-d|$, and $|c-d|$ are all integers and are in an arithmetic progression. Find the sum of all possible values of $a$. | nvidia |
Let $a,b,c$ be positive real numbers. Find the minimum value of
\[\sqrt{\frac{a+b}{c}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{c+a}{b}}.\] | nvidia |
Let $f(n)$ denote the number of perfect squares less than or equal to $n$. Find the sum of the smallest five positive integers $n$ such that $f(n) + f(2n) = 7$. | nvidia |
Let $p(x)$ be a monic quadratic polynomial with integer coefficients such that $p(7)=0$. If the sum of the squares of the coefficients of $p(x)$ is a perfect square, what is the smallest possible value of this sum? | nvidia |
Find the sum of the coefficients of the terms which contain a nonzero power of x in the expansion of the expression $(x^2+5y+7)^3$. | nvidia |
Find the smallest positive integer that can be represented as the sum of two perfect cubes in two different ways, with one representation having all digits odd and the other having all digits even. | nvidia |
Given a circle with center $(3, 7)$ and radius $5$, find the sum of the $x$-coordinates of the points where the circle intersects the line $y = x + 1$. | nvidia |
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x$, $y$, and $r$ with $|x|>|y|$,
\[(x+y)^r=x^r+rx^{r-1}y^1+\frac{r(r-1)}2x^{r-2}y^2+\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\cdots\]What are the first three digits to the right of the decimal point in the decimal representation... | nvidia |
Find the sum of all integer values of $m$ such that $\frac{15}{3m + 2}$ is an integer. | nvidia |
Let f(x) be a non-constant polynomial with integer coefficients such that f(1)=0 and there exists an integer n such that for all integers x with x≡1 (mod n), f(x)|f(1). Find the smallest possible degree of f(x). | nvidia |
Determine the number of ways to arrange the digits of a 6-digit number, 121456, so that no two identical digits are next to each other. | nvidia |
Let p ( n ) be the product of all positive integers less than n and relatively prime to n . Find the value of p ( 531 ) p ( 532 ) p ( 534 ) modulo 531 . | nvidia |
Let $f$ be a function defined on the positive integers, such that
\[f(xy) = f(x) + f(y)\]for all positive integers $x$ and $y.$ Given $f(10) = 14$ and $f(40) = 20,$ find $f(500).$ | nvidia |
Let $x$, $y$, and $z$ be real numbers such that $xyz=90$.
Find all possible values of $\ds x+y+z$ if \[x^2+y^2+z^2 = (x+y+z)^2.\] | nvidia |
Given a positive integer $n$, let $S_n$ be the set of all $n \times n$ real matrices whose entries in each row and each column sum to $1$. Let $T_n$ be the set of all real numbers that can be expressed in the form $\mathrm{det}(A)$, where $A \in S_n$. For each positive integer $n$, find the maximum value in $T_n$. | nvidia |
Find the sum of the divisors of 72. | nvidia |
Find the sum of the first five perfect squares that are also perfect cubes. | nvidia |
Find the number of integers that are divisors of both 6000 and 9000. | nvidia |
Given a sequence of consecutive integers, determine the largest integer that always divides the difference between the sum of the cubes of the integers and the square of the sum of the integers. | nvidia |
Find the smallest positive integer $n$ such that the equation $x^2 + 6x \equiv n \pmod{11}$ has a solution for $x$. | nvidia |
In a three-dimensional space, there are eight points that are $3$ units from the plane $z=7$ and $7$ units from the point $(2,5,7)$. What is the sum of the $x$-, $y$-, and $z$-coordinates of all eight of these points? | nvidia |
Find the largest number with distinct digits that sums up to 21. | nvidia |
Evaluate the product \[ (a-10) \cdot (a-9) \cdot \dotsm \cdot (a-1) \cdot a, \] where $a=2$. | nvidia |
What is the sum of all the four-digit positive integers that are divisible by 5? | nvidia |
Find the plane containing the points $(1,0,0)$, $(0,1,1)$, and $(-1,-1,2)$, such that the distance from the point $(0,0,0)$ to this plane is half the distance from the point $(1,1,1)$ to this plane. | nvidia |
Find all three-digit numbers that are the sum of two palindromic primes, where the digits of one prime are a permutation of the digits of the other. What is the largest such number? | nvidia |
Find the maximum value of the product of the roots of the quadratic equation \(ax^2 + bx + c = 0\), given that \(a\), \(b\), and \(c\) are positive integers, the roots are rational, and \(a + b + c = 20\). | nvidia |
Find the sum of all positive divisors of $50$ that are also divisors of $15$. | nvidia |
Find $w$, such that $5^65^w=25$. | nvidia |
Find the product of the roots of the equation
$$x^3 - x^2 \sqrt{x+1} + x \sqrt[3]{x-1} - 2 = 0,$$
given that all of the roots are real and nonnegative. | nvidia |
A point is randomly selected from the region enclosed by the parabola $y=x^2-1$ and the line $y=r$, where $r>0$. If the probability of selecting a point that lies within one unit of the parabola is $\frac{1}{4}$, find all possible values of $r$. | nvidia |
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