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In a survey of 1000 people, 17% of the respondents said they preferred a certain brand of soda. Which one of the following is closest to the fraction of people who did not prefer that brand of soda?
$$
\frac{5}{6} \qquad \frac{4}{5} \qquad \frac{3}{4} \qquad \frac{2}{3} \qquad \frac{1}{2}
$$ | nvidia |
Find the area of the region enclosed by the parabola $y=x^2$, the line $y=x$, and the line $y=4$. | nvidia |
Let $g(x)=16x^4-24x^3+89x^2-194x+112$. Find the smallest positive integer $m$ such that for every real number $r$ at least one of the coefficients of the polynomial $g(x+r)$ is prime. | nvidia |
How many three-digit positive integers exist, all of whose digits are 2's and/or 5's? | nvidia |
In a regular hexagon $ABCDEF$ with side length 2, a square with a side length of $\frac{3-\sqrt{3}}{6}$ and a vertex on $\overline{AB}$ shares a common vertex $A$ with the hexagon and has sides that are parallel to those of the hexagon. Find the area of the circle inscribed in the hexagon that is tangent to all its sid... | nvidia |
Find the sum of all the values in a factor tree for the number 360 that has the fewest possible number of nodes. | nvidia |
The two-digit positive integer $n$ has the property that when divided by $4$ or $7$ its remainder is always $3$. What is the smallest value of $n$ that is a multiple of $3$? | nvidia |
What is the maximum number of consecutive positive integers that can be multiplied together before the product exceeds 1000? | nvidia |
Given a set of points $A, B, C$ in the plane, let $P$ be a point such that the angle bisector of $\angle APB$ passes through $C$. Among all such points $P$, there exists a constant $k$ such that $P$ is at a fixed distance from the point $D$, where $D$ is the intersection of the angle bisector of $\angle ACB$ with the l... | nvidia |
Let $a$, $b$, and $c$ be positive integers such that \begin{align*}
(a+b)(b+c)&=(c+a)^2,\\
(b+c)(c+a)&=(a+b)^2,\\
(c+a)(a+b)&=(b+c)^2.
\end{align*}Prove that $a=b=c$. | nvidia |
Determine the number of positive integer values of $b$ such that $x = 4$ is the only positive integer solution of the system of inequalities:
$$
\begin{cases}
3x > 2x + 2\\
4x - b > -8
\end{cases}
$$ | nvidia |
Given the function $f(x)=\frac{1}{x^2}$, if the point $(a,b)$ is on the graph of $y=f(x)$, and $g(x)=f^{-1}(x)$ for all $x$, then there is one point that must be on the graph of $y=g(x)$. What is the product of the coordinates of that point? | nvidia |
What is the smallest positive integer $x$ such that $\frac{1}{x}+\frac{1}{x+1}+\frac{1}{x+2} < 1$? | nvidia |
Find the number of positive integers less than 60 that are not perfect squares but have a divisor that is a perfect square. | nvidia |
What is the smallest positive integer $n$ such that $n^2$ is greater than $3^5$? | nvidia |
The numbers $a$, $b$, and $c$ are positive and satisfy the following equations:
\[ab-c^2=22,\]
\[bc-a^2=-13,\]
and
\[ca-b^2=3.\]
Determine the value of $abc$. | nvidia |
Find the number of different ways to arrange 12 blue and 7 red balls into three boxes, such that each box contains at least one blue ball and the number of red balls in each box is a different integer. | nvidia |
In an isosceles triangle with integer length sides, the sum of the lengths of the two equal sides is 75 units. How many units is the length of the base? | nvidia |
Find the sum of all $x$ such that the expression $\frac{x^2-3x-4}{x^2-4x+4}$ is undefined. | nvidia |
Find the smallest positive integer $M$ with the following property: of the four numbers $M$, $M+1$, $M+2$, and $M+3$, one of them is divisible by $2^3$, one of them is divisible by $3^3$, one is divisible by $5^3$, and one is divisible by $11^3$. | nvidia |
Find the smallest positive integer $n$ such that the sum of the squares of the first $n$ positive integers is greater than 1000. | nvidia |
Let $a$, $b$, $c$ be real numbers such that $a^2+2b^2+3c^2=1$ and $a+b+c=0$. Find the maximum value of $c$. | nvidia |
Find the positive difference between the solutions of the equation $x^2+5x-6=\sqrt{x^2-4x+5}$. | 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 |
Find all real numbers $a$ for which the equation $$ax^2+x+a=0$$ has real solutions. What is the product of those numbers? | nvidia |
Find the number of all integer values of $x$ such that $x^2+6x+9$ is a perfect cube. | nvidia |
Find the sum of all prime divisors of 315 that do not divide 36. | nvidia |
If $x$, $y$ are nonzero real numbers such that $x^2 + y^2 = 12xy$, find the value of $\left|\frac{x - y}{x + y}\right|$. | nvidia |
Find the sum of all even integers between 20 and 50, inclusive. | nvidia |
In the equation
$$4x^2-kx+9=0$$
determine $k$ such that the sum of the squares of the roots be equal to 4.5. | 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 |
If the point $(2,5)$ is on the graph of $y=f(x)$, and $k(x)=2f(x)+1$ for all $x$, then there is one point that must be on the graph of $y=k(x)$. What is the product of the coordinates of that point? | 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 that $x$ is a positive real number and $f(x) = \sin^4 x + \cos^4 x + 1$, find the value of $x$ for which $\sqrt{f(x)} = \frac{\sqrt{6}}{2}$. | nvidia |
The graph of the rational function $f(x) = \frac{3x^2-12x-15}{x^2-4x-5}$ has two horizontal asymptotes. Find the equation of the line that passes through the point of intersection of these two asymptotes and is perpendicular to the graph of $f(x)$ at that point. | nvidia |
Find the positive difference between the roots of the equation $x^2-12x+27=0$. | nvidia |
A quadrilateral $ABCD$ has vertices $A(1,1)$, $B(3,3)$, $C(4,0)$, and $D(0,0)$.
Prove that for any point $P$ inside the quadrilateral,
$AP^2+CP^2 \geq BP^2 + DP^2$. | nvidia |
Find the distance between the center of the circle with equation $x^2+y^2=2x-8y+1$ and the point $(-3, -2)$. | nvidia |
In isosceles triangle $ABC$ with $AB=AC$, let point $D$ be the midpoint of $\overline{BC}$ and point $E$ be the intersection of the angle bisector of $\angle BAC$ with $\overline{BC}$. Find $\cos \angle DAE.$ | nvidia |
Determine the number of integers between 1000 and 10000 that are palindromes and have exactly two distinct digits, one of which is 7. | nvidia |
Find a positive integer that is divisible by 18 and whose cube root is a number between 8 and 8.1. | nvidia |
What is the smallest positive integer $n$ such that $531n \equiv 1067n \pmod{24}?$ | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 9.$ Find the minimum value of
\[x^2 + y^3 + z^4.\] | nvidia |
Three standard dice are rolled. What is the probability that the sum of the numbers on the first two dice is equal to the number on the third die? | nvidia |
Find the sum of all real roots of the equation $\sqrt[3]{x} + \sqrt[3]{2-x} = 2.$ | nvidia |
Determine the smallest positive integer $B$ such that the three-digit number $B6B$ is divisible by $11$. | nvidia |
Consider the function:
\[g(x) = \left\{
\begin{array}{cl}
x^2+1 & \text{ if }x<0, \\
x-5 & \text{ if }x \geq 0.
\end{array}
\right.\]
Find the positive difference between the two values of $b$ that satisfy the equation $g(-1)+g(1)+g(b)=0$. | nvidia |
Find the value of k for the equation 2x^2 - 7x + k = 0, such that the sum of the squares of its roots is equal to 10.5. | nvidia |
Find all values of $x$ that satisfy the inequality
$$\frac{x^2-4}{x+2} \geq \frac{x-1}{2}.$$
Express your answer in interval notation, simplifying any fractions that occur in your answer. | nvidia |
Let $A$ and $B$ be real numbers such that $\frac{A}{x-5}+B(x+1)=\frac{-3x^2+12x+22}{x-5}$. What is $A+B$? | nvidia |
Determine the number of positive integers $a$ less than $12$ such that the equation $ax^2 + 1 = 12x$ has a solution in positive integers $x$. | nvidia |
Determine the number of distinct binary strings of length 6 that can be formed such that the number of 1's is exactly twice the number of 0's. | nvidia |
Consider the polynomials \[f(x)=x^4-3x^3+2x^2-x+1\] and \[g(x)=x^3-2x^2+3x+1.\] Find $c$ such that the polynomial $f(x)+cg(x)$ has a root at $x=2$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x) = \frac{1}{f(x + 1)} + f(f(x + 2))\]
for all real numbers $x.$
Find the number of possible values of $f(1).$ | nvidia |
Let $x$ be a positive number such that $\sqrt{x} + \frac{1}{\sqrt{x}}$ can be expressed in the form $\frac{m\sqrt{x}+n}{p}$, where $m,n$ and $p$ are positive integers with $p$ as small as possible. Determine $m+n+p$. | nvidia |
We define a function $f(x)$ such that $f(11)=34$, and if there exists an integer $a$ such that $f(a)=b$, then $f(b)$ is defined and
$f(b)=3b+1$ if $b$ is odd
$f(b)=\frac{b}{2}$ if $b$ is even.
What is the smallest possible number of integers in the domain of $f$? | nvidia |
Let A denote a set of positive integers from 1 to 30. For any integer n from set A let denote $p_i(n)$ the product of all positive integers from set A (including 1) that are less than $n$. Thus, for example, $p_i(3)=2 \cdot 1=2$. What is the largest possible integer value of $i$ for which $p_i(i) < i$? | nvidia |
What is the sum of all two-digit positive integers whose squares end with the digits 01? | nvidia |
Find the value of $x$ such that the four-digit number $\underline{x15x}$ is divisible by 18. | nvidia |
Find the number of positive integers less than or equal to 500 that are divisible by 3 but not by 4. | nvidia |
Find the product of all constants $t$ such that the quadratic $x^2 + tx - 10$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. | nvidia |
Let $S$ be a set of positive integers such that every pair of elements in $S$ is the length of two sides of a triangle whose area is not an integer. Determine the largest possible value of the maximum element of $S$. | nvidia |
The sum of the squares of two consecutive positive integers is equal to 365. What is the difference between these two integers? | nvidia |
Find the number of zeros at the end of the expansion of $256!\cdot101$. | nvidia |
The sum of the squares of the values of $x$ that satisfy the equation $\cos^2 x + 2\sin^2 x = 1$ in the interval $[0, 2\pi]$ is $k\pi^2$, where $k$ is a positive integer. Find $k$. | nvidia |
Find the number of ordered triples $(x,y,z)$ of nonnegative integers such that
\begin{align*}
x+y+z &= 12, \\
xyz &= 48.
\end{align*} | nvidia |
Determine the number of different quadrilaterals that can be formed by connecting four distinct points on a sphere, given that no three of these points are collinear. | nvidia |
In a circular segment with a central angle of 60 degrees and a radius of 10 units, two points $P$ and $Q$ are chosen on the arc such that the area of the triangle formed by the radii to $P$ and $Q$ and the chord $PQ$ is minimized. Find the maximum possible distance between $P$ and $Q$ along the arc. | nvidia |
Let $f(x)$ be a real-valued function such that for all non-zero real numbers $x,$
\[f(x) - 2f(-x) = \frac{1}{x}.\]
Find the real solutions to $f(x) = 2f(-x)$. | nvidia |
Determine the range of values for y such that the expression \[\frac{y^4-6y^2+9}{y^2-16}\] is nonnegative. | nvidia |
Find all real numbers $x$ such that
\[\frac{8^x + 27^x}{12^x + 18^x} = \frac{7}{6}.\]Enter all the solutions, separated by commas. | nvidia |
The perimeter of a right triangle with integer side lengths is equal to the perimeter of a rectangle with integer side lengths. The area of the triangle is 30 square units, and the length of the hypotenuse is 3 times the length of the shorter leg. Find the dimensions of the rectangle. | nvidia |
Find all real values of $x$ that satisfy the inequality
$$\frac{x^2-2x-1}{x^2+1} < \arctan(x) - x.$$
Give your answer in interval notation. | nvidia |
What is the sum of all positive integer divisors of the number obtained by multiplying 7 and 11? | nvidia |
Find all real numbers $a$ such that the equation $x^2-ax+\lfloor a \rfloor = 0$ has two distinct real roots. | nvidia |
Given vectors $\mathbf{a}$ and $\mathbf{b}$, the area of the parallelogram generated by the vectors $\mathbf{a}$ and $\mathbf{b}$ is $6$ and the area of the parallelogram generated by the vectors $\mathbf{b}$ and $\mathbf{c}$ is $8$, where $\mathbf{c}$ is a vector orthogonal to $\mathbf{a}$.
Find the area of the parall... | nvidia |
Determine the largest number of points that can be placed on the coordinate plane with integer coordinates such that the distances between any two of them is a prime number. | nvidia |
The prime factorization of a number is 2^3 \cdot 3 \cdot 5^2. What is the arithmetic mean of all the positive integer divisors of this number that are multiples of 15? | nvidia |
The sum of the squares of three consecutive positive integers is equal to 50. Determine the smallest of these integers. | nvidia |
Determine the number of terms in the sequence $a_1=1$, $a_n=2a_{n-1}+1$, where $1\le n\le 10$, that are divisible by 7. | nvidia |
Find the sum of all the even integers from 10 to 50 inclusive that are not divisible by 3. | nvidia |
Find the number of ways to arrange 10 distinct objects in a row such that no two consecutive objects have the same color, given that 5 objects are red and 5 objects are blue. | nvidia |
In the equation $2x^2-5x+c=0$, the sum of the solutions is $2.5$. Determine $c$ such that the product of the solutions is equal to the square of their difference. | nvidia |
Let $x$, $y$, $z$ be real numbers such that $x+y+z=5$, $xy+xz+yz=4$, and $x^2+y^2+z^2=7$. Find the value of $(x+y)(y+z)(z+x)$. | nvidia |
Given that $\frac{2x-1}{x+1}=\frac{x+5}{x-2}$, find the value of $x$ such that the reciprocals of these two expressions are also equal. | nvidia |
The function $f(x)$ satisfies
\[f(x + y) = f(x) f(y)\]for all real numbers $x$ and $y.$ Find all possible values of $f(0).$
Enter all the possible values, separated by commas. | nvidia |
Find the number of positive integers that are less than or equal to 96 and have exactly three positive divisors. | nvidia |
What is the minimum number of days that must pass from any given day of the week so that the day of the week repeats exactly three times? | nvidia |
Find the sum of all possible integer values of $a$ for which there exists an infinite geometric sequence with first term $a$ and common ratio $r$, such that the first three terms of the sequence form an arithmetic progression, and the sum of the squares of the first three terms is a perfect square. | nvidia |
Find a positive integer that is divisible by 18 and whose cube root is a number between 8 and 8.1.
Another problem:
Find a positive integer that is a perfect square and whose square root is a number between 4 and 4.1. | nvidia |
Find the value of $n$ such that $\sum_{k=0}^{n} \binom{n}{k} = \binom{2n}{n}$. | nvidia |
Find the value of $k$ such that the equation
$$\sin(x+k)=\sin(2x)$$
has exactly two solutions for $x$ in the interval $[0,2\pi)$. | nvidia |
Problem:
A quadrilateral $ABCD$ has side lengths $AB=10,BC=20,CD=30$, and $DA=40$. Lines $\ell_A,\ell_B,\ell_C$, and $\ell_D$ are drawn parallel to $\overline{BC},\overline{AD},\overline{AB}$, and $\overline{CD}$, respectively, such that the intersections of $\ell_A,\ell_B,\ell_C$, and $\ell_D$ with the interior of qua... | nvidia |
Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.
| nvidia |
Problem:
Find the greatest value of $r$ such that there exists a regular heptagon $P_1P_2P_3P_4P_5P_6P_7$ whose vertices lie on a circle of radius $r$ centered at the origin of the coordinate plane, and the y-coordinates of its vertices are distinct elements of the set $\{-3,-2,-1,0,1,2,3\}$. Each vertex $P_i$ has a no... | nvidia |
Given that $x$, $y$, and $z$ are nonzero real numbers, find all possible values of the expression
\[\frac{x^2}{|x^2|} + \frac{y^2}{|y^2|} + \frac{z^2}{|z^2|} + \frac{xyz}{|xyz|}.\]Enter all possible values, separated by commas. | nvidia |
Determine the values of $a$ and $b$ in the function $y=\frac{ax+b}{x^2-x-6}$ such that the graph of the function has vertical asymptotes at $x=3$ and $x=-2$, and a hole at $x=1$. | nvidia |
Let's consider a set $T$ of all possible subsets of the set $\{1, 2, \ldots, n\}$. Three subsets $A_1, A_2, A_3$ are chosen independently and at random with replacement from the set $T$. The probability that $A_1$ is a subset of $A_2$ and $A_2$ is a subset of $A_3$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively pr... | nvidia |
Let $a_1,a_2,\dots,a_n$ be positive real numbers such that $a_1+a_2+\dots+a_n=1$. Find the maximum possible value of the expression
\[\left\lceil{a_1}\right\rceil+\left\lceil{a_2}\right\rceil+\dots+\left\lceil{a_n}\right\rceil.\](Note that $\lceil{x}\rceil$ denotes the smallest integer greater than or equal to $x$.) | nvidia |
Find the value of $k$ such that $\log_{5}2\cdot\log_{2}3\cdot\log_{3}4\cdot\log_{4}5=k$. | nvidia |
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