instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find the value of x such that the expression \[\dfrac{x-2}{3x^2-16x-12}\] is undefined. | nvidia |
Let $a_n=\ds\sum_{k=1}^n k!$.
Find the smallest positive integer $n$ such that
$\ds\frac{a_{n+1}}{a_n}>2015$ | nvidia |
Point $P$ lies on the circle $(x-2)^2+(y-3)^2=9$ and is 5 units from the line $y=6$. Find the sum of all possible $x$-coordinates that satisfy the given conditions. | nvidia |
Determine the largest value of $m$ such that the quadratic expression $2x^2 + mx + 50$ can be expressed as the product of two linear factors with integer coefficients. | nvidia |
What is the largest possible value of $n$ for which the $n$ consecutive positive integers $2n+1, 2n+2, \ldots, 3n$ all have exactly one positive divisor other than 1 and themselves? | nvidia |
Suppose $x$ and $y$ are integers such that $xy+5x+4y=-5$. Find the greatest possible value of $y$. | nvidia |
How many 4-digit numbers are there such that the sum of the first two digits is 5, the sum of the last two digits is 3, and no digit appears more than once in the number? | nvidia |
Find the number of ways to arrange the letters of the word "referee" so that the vowels (E and E) are always together and the consonants (R, F, and R) are always together. | nvidia |
Let $x$ be the smallest positive real number such that $y = x + \frac{1}{x}$ is equal to $z + \frac{1}{z}$ where $z = \frac{x}{2}$. Find $x$. | nvidia |
Determine the value of $x$ in the expression $\left(\dfrac{a}{b}\right)^x$ such that the result is always 1, regardless of the values of $a$ and $b$, assuming that $a$ and $b$ are non-zero integers. | nvidia |
Let $a_1,a_2,\ldots$ be a sequence determined by the rule $a_n= \frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n=3a_{n-1}+1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 2008$ is it true that $a_1$ is less than each of $a_2$, $a_3$, and $a_4$?
Consider a sequence $b_1, b_2, \ldots$ where $b_n$ is defin... | nvidia |
Find the minimum value of
\[\frac{(x-1)^2}{y^2} + \frac{(y-1)^2}{x^2}\]
for positive real numbers $x$ and $y$ such that $xy = 2$. | nvidia |
At the 2007 Math Olympics, Team Canada won $17$ out of a possible $100$ medals. Which one of the following is closest to the fraction of medals that they won? $$
\frac{1}{4} \qquad \frac{1}{5} \qquad \frac{1}{6} \qquad \frac{1}{7} \qquad \frac{1}{8}
$$ | nvidia |
The line $y=-2x+c$ intersects the parabola $y=x^2+1$ at two points. Determine the value of $c$ such that the distance between these two points is equal to $\sqrt{20}$. | nvidia |
The greatest common divisor of two positive integers $m$ and $n$ is 17. Determine the smallest positive integer $a$ so that $4m + 7n + a$ is divisible by 17. | nvidia |
Find the smallest value of $x$ such that $\frac{x^2+2x+1}{x^2-4x+3} \ge 0$. | nvidia |
Let $x$ and $y$ be two positive real numbers such that $x + y = 35.$ Enter the ordered pair $(x,y)$ for which $x^5 y^2$ is maximized. | nvidia |
What is the smallest positive integer $m$ such that $\frac{m}{m+67}$ is equal to a terminating decimal? | nvidia |
In $\triangle ABC$, $\measuredangle BAC=65^{\circ}$ and $\measuredangle ACB=35^{\circ}$. $AB=1$. Let $N$ be the midpoint of segment $\overline{AB}$. Point $D$ lies on side $\overline{BC}$ such that $\overline{AD}\perp\overline{CN}$. Extend segment $\overline{BC}$ through $C$ to point $F$ such that $DF=FC$. Then $BF=\fr... | nvidia |
Find the least positive integer $x$ that satisfies $x+4609 \equiv 2104 \pmod{12}$. | nvidia |
Let $x$ and $y$ be two real numbers selected at random from the interval $[0,1]$. What is the probability that the sum of the roots of the quadratic equation $z^2+xz+y=0$ is positive? | nvidia |
Determine the value of $x$ such that the equation $|x^2-4|=|x-2|$ has exactly one solution. | nvidia |
How many three-digit positive integers have at least two digits that are the same? | nvidia |
Find the minimum number of integers Alice needs to choose so that their product is guaranteed to be even, regardless of the specific integers chosen. | nvidia |
Find a positive integer $x$ such that $x$, $x^2$, and $x^3$ have a total of exactly 11 distinct positive factors. | nvidia |
The polynomial $x^2+bx+2008b$ can be factored over the integers if and only if its discriminant is a perfect square. Thus, we have $b^2-4\cdot 1 \cdot 2008b = k^2$ for some integer $k$. Rearranging the equation, we get $(b-2008)^2-k^2=2008^2$. Thus, $|S|$ is the sum of all possible values of $|b-2008|$ such that $2008^... | nvidia |
A trapezoid $ABCD$ is inscribed in a circle with center $O$. The lengths of the sides of the trapezoid are $AB = x$, $BC = y$, $CD = x + a$, and $DA = y + b$, where $a$ and $b$ are positive constants. In terms of the length of the perpendicular bisector of the height of the trapezoid, $\omega$, the area of the trapezoi... | nvidia |
A cube has a total of eight vertices, each of which is assigned a non-negative integer. The sum of the integers on the vertices of each face of the cube is the same. If the sum of the integers on the vertices of one face is 12, what is the sum of the integers on all eight vertices? | nvidia |
Find all real numbers that are tripled when added to their cubes. | nvidia |
Let $A$ and $B$ be two points on the hyperbola $y = \frac{1}{x},$ such that when the tangents at $A$ and $B$ are drawn, they are parallel. Then for any such pair of tangents, the $x$-coordinate of their point of intersection $P$ is always the same. Find this $x$-coordinate. | nvidia |
Find the value of $m+n$ given that the quadratic expression $4x^2+12x+9$ can be expressed as $(2x+m)^2+n$, where $m$ and $n$ are constants. | nvidia |
Let $x$ and $y$ be real numbers such that $|x+y|=0$ and $|x-y|=2$. Determine $x^2+y^2$. | nvidia |
Find the largest possible value of $x$ such that $1<\left|x-\frac{1}{x}\right|<2$ for $x>0$. | nvidia |
A positive integer $n$ is a "lucky number" if every even factor of $n$ is also a factor of $120.$ Find the largest lucky number. | nvidia |
A portion of the graph of $f(x)=ax^2+bx+c$ is shown below. The distance between grid lines on the graph is $1$ unit.
What is the value of $a+b+2c$?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-... | nvidia |
Find the largest value of $n$ such that $3x^2 +nx + 72$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
Let p(x) be a polynomial of degree 4 with integer coefficients.
Assume that there exists a positive integer k such that p(3k)=5 for any integer k.
What is the smallest possible value of p(0)? | nvidia |
Let \(a_{1}\), \(a_{2}\),..., \(a_{n}\), \(a_{n+1}\) be positive integers such that
\[a_{1}<a_{2}<\cdots<a_{n}<a_{n+1}.\]
Find the smallest positive integer \(n\) for which
\[\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n}}{a_{n+1}}<2.\] | nvidia |
Given positive integers $x$, $y$, and $z$ such that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$, what is the least possible value of $xyz$ such that $x$, $y$, and $z$ are consecutive integers? | nvidia |
Find the smallest positive integer with exactly 12 positive integer divisors. | nvidia |
If $a$, $b$, and $c$ are digits and $0.ab0c$ can be expressed as $\frac{1}{y}$ where $y$ is an integer such that $0<y\le9$, then what is the largest possible value of $a+b+c$? | nvidia |
What is the smallest positive integer that can be written in the form $2002m + 44444n$, where $m$ and $n$ are integers? | nvidia |
Find the coordinates of the point that divides the line segment joining the points $(3,7)$ and $(5,1)$ into the ratio $2:3$. | nvidia |
Find the product of the magnitudes of the two complex numbers $2+i$ and $1-i$. | nvidia |
Let $m$ be the product of all positive integers less than $4!$ which are invertible modulo $4!$. Find the remainder when $m$ is divided by $4!$.
(Here $n!$ denotes $1\times\cdots\times n$ for each positive integer $n$.) | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $a+b+c$ is even and $a<b<c$. What is the largest integer that always divides $(b-a)(c-b)$? | nvidia |
What is the largest positive integer $n$ for which $n^2+5n+1$ is divisible by $n^2-5n+1$? | nvidia |
Find the largest constant $C$ so that
\[x^2 + y^2 + 4 \ge C(x + 2y)\]for all real numbers $x$ and $y.$ | nvidia |
Given that the graph of $y = f(x)$ is an upside-down parabola with vertex $(2,3)$ and $x$-intercept at $(4,0)$. Find the value of $k$ for which the graph of $y=f(x)+k$ intersects the $x$-axis at exactly one point. | nvidia |
Find the point in the $xy$-plane that is equidistant from the points $(0,3,4),$ $(1,1,2),$ and $(2,0,3).$ | nvidia |
Nine tiles are numbered $1, 2, 3, \cdots, 9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
| nvidia |
A positive integer $n$ can be expressed as $3^m \cdot k$, where $k$ is an integer and $k$ is not divisible by $3$. If $n$ is written in base $3$, what is the minimum value of $m$ such that the last digit of $n$ in base $3$ is $2$? | nvidia |
Find the smallest positive integer $n$ such that $\left(\frac{1}{2}\right)^{n}$ can be written as a finite decimal. | nvidia |
Let $\mathbf{A}$ be a point on the circle $x^2 + y^2 = 1$ that is $240^\circ$ counterclockwise from the point $(1,0)$. Find the $y$-coordinate of $\mathbf{A}$. | nvidia |
Let $S$ be the set of all positive integers that are multiples of 100 and less than 1000. Let $A$ and $B$ be two subsets of $S$ such that $A$ consists of all positive integers in $S$ that are multiples of 30, and $B$ consists of all positive integers in $S$ that are multiples of 50. Let $x$ be an element selected rando... | nvidia |
Find the value of $x$ that satisfies the equation $x^{x^{x^{x^{\cdots}}}} = 4$. | nvidia |
Let $g$ be a function defined on the positive integers, such that
\[g(xy) = g(x) + g(y)\]for all positive integers $x$ and $y.$ Given $g(15) = 21$ and $g(75) = 27,$ find $g(450).$ | nvidia |
Determine the number of complex numbers $z$ that satisfy the conditions $|z|=2$ and $|z^2-\overline{z}^2|=4$. | nvidia |
Let's $a$ be the smaller and $b$ be the larger roots of the quadratic equation $x^2-cx+d=0$, such that $(a-1)(b-1)=2$. Given that $a+b+c=7$, $d=5$, find $c$. | nvidia |
Let $a,$ $b,$ $c$ be real numbers such that $\{a,b,c\}=\{\tan x,\tan y,\tan z\},$ where $x,$ $y,$ $z$ are acute angles and $x+y+z=\frac{\pi}{2}.$
Find the maximum value of the expression
\[|\cos (a-b)|\cdot |\cos (b-c)|\cdot |\cos (c-a)|.\] | nvidia |
There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? | nvidia |
If $m$ is a positive integer, then let $g(m)$ be the product of the digits of $\frac{1}{3^{{}^m}}$ that are to the right of the decimal point. What is the smallest positive integer $m$ such that $g(m) > 50$? | nvidia |
Find the sum of the cubes of the solutions of $x^2+7x-3=0$. | nvidia |
Given a set of nonnegative numbers $\{x_1, x_2, x_3\}$, determine the number of distinct ways to construct a $3\times3$ matrix where each row and each column sum up to a given positive integer $n$, such that the element in the $i$th row and $j$th column is either $x_i$ or $x_j$. | nvidia |
The sum of the squares of two positive integers is 225, and the difference of their squares is 135. What are the two integers? | nvidia |
Find the smallest positive integer that can be expressed as a product of two different prime numbers and is also a multiple of 11. | nvidia |
What is the maximum number of non-overlapping circles that can be packed around a central circle such that each non-overlapping circle touches the central circle and two of its neighbors? | nvidia |
Given the points $A(2,-5)$, $B(-1,7)$, $C(-2,-3)$, find the point $D$ such that the quadrilateral $ABDC$ is a parallelogram. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) - y) = g(y - x^2) + 2g(x) y\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the product of all possible values of $g(2).$ Find $m \times t.$ | nvidia |
Let $p$ and $q$ be two distinct prime numbers. Find the number of positive integers $n$ such that $p^p$ and $q^q$ are both factors of $n!$, and $n$ is the smallest positive integer with this property. | nvidia |
Find the sum of all the positive divisors of 162. | nvidia |
Find the largest value of $x$ that satisfies the inequality $\sqrt{x^2-4x+4} \leq 3$. | nvidia |
The sum of the squares of two positive integers is 28125. What is the greater of the two integers? | nvidia |
Let $F(x)$ be a polynomial such that $F(6) = 15$ and\[\frac{F(3x)}{F(x+3)} = 9-\frac{48x+54}{x^2+5x+6}\]for $x \in \mathbb{R}$ such that both sides are defined. Find $F(12)$.
| nvidia |
Point Q lies on the line $y= 2$ and is 5 units from the point $(1,-1)$. Find the sum of the squares of all possible $x$-coordinates that satisfy the given conditions. | nvidia |
Find the maximum number of distinct integer values of $c$ such that the quadratic expression $x^2 - 25x + c$ can be factored into the product of two linear binomials with integer coefficients, where the sum of the constants in the binomials is positive. | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that $xyz=1$. Determine the minimum possible value of
\[x^2y+y^2z+z^2x.\] | nvidia |
Find the sum of all integers $m$ for which there exists an integer $n$ such that $m + \frac{n}{m} = n$. | nvidia |
Let $x,$ $y,$ and $z$ be three positive real numbers whose sum is 1. If no one of these numbers is more than twice any other, then find the minimum value of the product $xyz.$ | nvidia |
Given that $x$ is a positive integer, $\sqrt[3]{x}$ is greater than 7 and less than 8. How many integer values of $x$ satisfy this condition? | nvidia |
Find the area of the triangle with vertices $(3,-5),$ $(-2,0),$ and $(1,-6).$ | nvidia |
In the equation $|x^2-3x-1| = x+2$, find the sum of all possible values of $x$ that satisfy the equation. | nvidia |
A triangle $ABC$ has vertices at $(0,0,0),$ $(1,0,0),$ and $(0,1,t),$ where $t$ is a real number. Find the maximum possible area of triangle $ABC$. | nvidia |
Let $A$ be a set of points in the plane such that the distance from any point in $A$ to the point $(0,0)$ is equal to the distance from that point to the line $x+y=1$ and the line $x-y=1$. Determine the number of intersection points between the line $x=0$ and the set of points $A$. | nvidia |
A fair coin is flipped five times. What is the probability that the sequence of heads and tails will be such that no two adjacent flips will be the same? | nvidia |
A sequence of complex numbers is defined by $a_0 = 1$ and $a_{n+1} = \frac{1}{2}(a_n + \frac{1}{a_n})$ for $n \ge 0$. Find the smallest positive integer $n$ such that $a_n$ is a real number. | nvidia |
If $a$, $b$, $c$ are nonzero real numbers such that $a^2+b^2+c^2=2ab+2bc+2ca$, find the value of $\left|\frac{a+b+c}{a-b+c}\right|$. | nvidia |
What is the smallest positive integer $n$ such that the product of the smallest and largest factors of $n$ is 27? | nvidia |
The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real who... | nvidia |
Find the minimum possible value of the expression $\sqrt{\frac{x+y}{x-y}} + \sqrt{\frac{x-y}{x+y}}$, where $x$ and $y$ are positive real numbers with $x > y$. | nvidia |
Find the number of positive integers $n$ that satisfy the inequality $125 < n^3 < 512$. | nvidia |
Find the number of positive integers less than 10000 whose digits sum up to a multiple of 6 and the number formed by the last two digits is divisible by 4. | nvidia |
How many positive five-digit integers have the property that the sum of their digits is the same when read forwards and backwards? | nvidia |
Consider a convex polygon with $n$ sides. Let $S$ be the set of all segments connecting pairs of vertices that do not share a common edge. The segment $s \in S$ is said to be "proportional" if the ratio of the lengths of the two parts into which it divides the polygon's perimeter is a rational number.
Find the maximum... | nvidia |
The diagonals of a regular hexagon intersect at a point O. Prove that triangle AOC is equilateral and find the measure of angle AOB. | nvidia |
Consider the function $g(x)$ that maps positive real numbers to real numbers, such that
\[x^2g(y) - y^2g(x) = g \left( \frac{x^2}{y^2} \right)\]for all positive real numbers $x$ and $y.$ Determine all possible values of $g(81).$ | nvidia |
A rectangle has a length of x inches and a width of (20-2x) inches. The area of the rectangle is 40 square inches. Determine the sum of all possible lengths of the rectangle. | nvidia |
Count the number of four-digit positive integers that have exactly two even digits. The even digits can be in any position, and the remaining two digits can be any odd number. | nvidia |
Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| nvidia |
Find the largest three-digit number whose digits are distinct and form a geometric sequence, such that when the digits are rearranged in descending order, the resulting number is divisible by 11. | nvidia |
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