instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find the number of non-negative integer solutions to the system of inequalities:
\[
\begin{array}{cccc}
(1) & x+y & \geq & 2z+3 \\
(2) & y+z & \leq & 2x+1 \\
(3) & z+x & \geq & 3y-1
\end{array}
\] | nvidia |
Find the sum of all integral values of $c$ with $c\le 30$ for which the equation $y=x^2-11x-c$ has two rational roots. | nvidia |
Find the greatest $a$ such that $\frac{7\sqrt{(2a)^2+(1)^2}-4a^2-1}{\sqrt{1+4a^2}+3}=2$. | nvidia |
Let $f(x)$ be a function defined for all real numbers and satisfies
$f(x)+f(2009-x)=\ds\frac{k}{2009-x}$,
where $k$ is a constant. Determine $\ds\frac{f(x)}{2009-x}$. | nvidia |
What are the last three digits in the sum of the cubes of the first 50 positive integers? | 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 |
If $m$ is a positive integer such that $6m^2-m=393120$, find $m$. | nvidia |
Given a function $f(x) = ax^2 + bx + c$ where $a, b, c$ are positive integers. Find the number of unique tuples $(a, b, c)$ such that $f(0) = f(1) = 1$ and $f(x)$ has only one positive real root. | nvidia |
Find the sum of the digits of the product of all the prime numbers that are factors of the numbers in the sequence $1!$, $2!$, $3!$, $4!$, $5!$, $6!$, $7!$, $8!$, $9!$. | nvidia |
A positive integer is called "rounded" if it can be written in the form $aba$ where $a$ and $b$ are digits. How many positive three-digit integers are rounded and have no repeated digits? | nvidia |
Find the equation of the ellipse with foci at $(3,0)$ and $(-3,0)$ that passes through the point $(2,3)$. | nvidia |
Find the smallest positive integer $k$ such that $k \cdot 1234567$ is divisible by 11. | nvidia |
Find the integer $n$, $0 \le n \le 7$, such that \[n \equiv -3737 \pmod{8}.\] | nvidia |
A binomial $x^3+ax^2+bx+c$ is such that its square has only three terms.
Determine $b$ if $a$ and $c$ are known. | 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 |
What is the sum of all values of $z$ such that $z^2=12z-7$? | nvidia |
Let $ S $ be the set of all sides and diagonals of a regular pentagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | nvidia |
Find the sum of the positive factors of 120 that are divisible by 3. | nvidia |
A portion of the graph of $y = f(x)$ is shown in red below, where $f(x)$ is a quadratic function. The distance between grid lines is $1$ unit.
What is the sum of all distinct numbers $x$ such that $f(f(f(x)))=-3$ ?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14... | nvidia |
Let $S$ be the set of all positive integers such that every digit in the integer is 0 or 1. Find the number of elements in $S$ that are divisible by 3 and have exactly 5 digits. | nvidia |
There exists a real number $x$ such that $ \sin (\arccos (x)) = x^2 $. Find the value of $x$. | nvidia |
Given a positive integer $n$ and a real number $a$, consider the expression
$$\frac{(a-n) \cdot (a-(n-1)) \cdot \dotsm \cdot (a-2) \cdot (a-1)}{(a+(n-1)) \cdot (a+(n-2)) \cdot \dotsm \cdot (a+2) \cdot (a+1)}.$$
Determine the value of $a$ for which this expression has the maximum value, given that $n=6$. | nvidia |
Find the smallest whole number greater than 1 that has exactly four positive factors. | nvidia |
Let $x$ and $y$ be two positive real numbers such that $(x + y)^2 = 9$ and $x^2 - y^2 = 3$. Find the value of $xy$. | nvidia |
The product of two positive integers plus their sum is 95. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | nvidia |
Let $S$ be a set of all real numbers in the interval $[0, \pi]$ that satisfy the equation
$\cos^2 x + \sin^2 2x = 1$.
Find the range of values of $\sin x$ that are in $S$. | nvidia |
Find the smallest positive integer that is divisible by exactly four distinct primes and has a remainder of 1 when divided by 5. | nvidia |
Let $a$ be a positive real number such that the sum of the distances from $a$ to $x^2$ and from $a$ to $-x^2$ is equal to the distance from $a$ to $\sqrt{x}$. Find the largest possible value of $a$. | nvidia |
Find the product of all positive integers less than 100 that are cubes of perfect cubes. | nvidia |
Find all pairs of positive integers $(x,y)$ such that their sum is 30 and the greatest common divisor of $x$ and $y$ is 5. | nvidia |
The function $g(x)$ is defined for all real numbers except $x = -1$ by
\[g\left(\frac{x+1}{x}\right)=x^2.\]
Given that $u\geq0$, find the value of $g(1+e^{2u})$. | nvidia |
Let $m$ be a positive integer, and let $a_0, a_1, \dots , a_m$ be a sequence of real numbers such that $a_0 = 37$, $a_1 = 72$, $a_m=0$, and $$ a_{k+1} = a_{k-1} - \frac{3}{a_k} $$for $k = 1,
2, \dots, m-1$. Find $m$. | nvidia |
Let $\odot$ be a binary operation defined on the set of positive integers, such that $a\odot b=a^2b$ for all positive integers $a$ and $b$. Find the value of $(2\odot 3)\odot 4$. | nvidia |
Determine all positive integers $m$ such that $1_m + 1_m = 10_m$. | nvidia |
Let $m$ and $n$ be two relatively prime positive integers. The number of positive divisors of $mn$ is $32$. Given that $m$ has only two distinct prime divisors, determine the maximum number of distinct prime divisors that $n$ could have. | nvidia |
Find the smallest positive integer $n$ such that $n$ is a multiple of both $3$ and $7$, and $n$ is one more than a multiple of $5$. | nvidia |
Find the length of the segment joining the center of the circle with equation $x^2+y^2=2x+4y-1$ and the point $(13,7)$, knowing that it is a tangent to the circle. | nvidia |
Eight boys and six girls are standing in a circle, and we want to count the number of different ways that they can join hands such that no two people of the same gender hold hands. Two arrangements are considered the same if they can be obtained from each other by rotating the circle. How many such arrangements are pos... | nvidia |
Given $m$ and $n$ as positive integers, determine all values of $m$ and $n$ such that the area of each of the smaller triangles into which the parallelogram $ABCD$ is divided by a diagonal is a different perfect square.
[asy]
draw((0,0)--(18,12),Arrow);
draw((0,0)--(12,0),Arrow);
label("$O$",(0,0),W);
label("$A$",... | nvidia |
Find the value of $x$ such that $5x \equiv 1 \pmod{29}$, where $x$ is a residue modulo 29. | nvidia |
Let $P(x)$ be a polynomial with integer coefficients such that
$$P\left(\sqrt{\frac{3-\sqrt{5}}{2}}\right)=0.$$
Determine the smallest possible value of $\deg P$.
That's all! | nvidia |
The roots of the quadratic equation $x^2 + (2m - 3)x + m = 0$ are such that the sum of their reciprocals is $\frac{1}{4}$. Find the value of $m$. | nvidia |
Find the smallest positive integer $n$ such that $13\cdot n\equiv 11\pmod{17}$. | nvidia |
In the complex plane, a point $z$ lies on a circle centered at the origin with a radius of $15$. If the real part of $z$ is $\frac{3}{5}$ of its imaginary part, find the positive value of the imaginary part. | nvidia |
Let $p$ be a prime and let $a$ be an integer such that $a$ is a primitive root $\pmod p$. Define $n$ to be the smallest positive integer such that $a^n\equiv 1\pmod p$. Prove that $n$ divides $p-1$, and find the value of $n$ when $a=2$ and $p=7$. | nvidia |
Let $M$ and $N$ be two points on the legs of a right isosceles triangle, such that $\angle{MCN} = 45^{\circ}$ and $CN=CM=1$. Compute the area of the triangle $MCN$ | nvidia |
If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? | nvidia |
On a circular table there are 8 congruent cubes arranged to form an octagon shape such that each cube is connected to its two adjacent cubes with their edges touching. Determine the length of the edge of the cube such that the area of the octagon formed by the faces of the cubes towards the center of the table equals $... | nvidia |
Find the largest integer $n$ for which the equation $x^2 + 5x + n = 0$ has a real solution. | nvidia |
Find the sum of all two-digit positive integers that have exactly two non-zero digits. | nvidia |
A city has $m$ streets, where $m$ is a positive integer with two digits. If they add $3$ more streets, the total number of streets will be a multiple of $7$. If they close $2$ streets, the remaining number of streets will be a multiple of $4$. What is the largest possible value of $m$ such that the city can be divided ... | nvidia |
Let $p$ be a prime number such that $p^2$ divides $2^n + 1$ for some positive integer $n$.
Prove that $p$ is congruent to 1 modulo 4, and find all possible values of $n$ modulo $p-1$. | nvidia |
The area of the smallest square that will contain a circle of radius 4 is what? | nvidia |
Find the sum of the squares of the x-coordinates and the sum of the squares of the y-coordinates of the intersection points of the parabolas $y = (x - 2)^2$ and $x = (y + 1)^2$. | nvidia |
Find the product of the roots of the quadratic equation $x^2 + bx + c = 0$, given that the sum of the squares of the roots is 15 and the sum of the reciprocals of the roots is 1/5. | nvidia |
What is the least positive integer with exactly five distinct positive factors? | nvidia |
Let $x$, $y$, $z$ be nonzero real numbers such that $x+y+z=0$ and $x^4+y^4+z^4=x^6+y^6+z^6$. Find the value of $x^3+y^3+z^3$. | nvidia |
Find the equation of the line that is the perpendicular bisector of the line segment joining the points (2, 4) and (6, 8), given that the line passes through the point (3, 5). | nvidia |
Simplify $((5p+1)-2p\cdot4)(3)+(4-1\div3)(6p-9)$ to a much simpler expression of the form $ap-b$ , where $a$ and $b$ are positive integers. | nvidia |
Let $a$ and $b$ be real numbers such that $a^2-b^2=16$ and $a^4-b^4=272$. Determine $a^2+b^2$. | nvidia |
Find the number of positive divisors of 360 that are also multiples of 4. | nvidia |
What is the smallest positive integer that is divisible by the product of the first three positive prime numbers and has a units digit of 0? | nvidia |
The roots of the equation
$$5x^2-kx+1=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $k$. | nvidia |
The equation $(2x-a)^2 = (3x+12)^2$ has at least one solution $x$ for every value of $a$. Determine $a$ such that this equation has exactly one solution $x$. | nvidia |
Find the smallest positive integer $n$ such that $\lfloor\sqrt{n}\rfloor^3 - n$ is a perfect square. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 4.$ Find the minimum value of
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c}.\] | nvidia |
Consider the sequence of numbers defined by $a_n = n^2 + n - 1$, where $n$ is a positive integer. Determine the value of $n$ for which the sum of the first $n$ terms of the sequence is equal to $15n$. | nvidia |
How many distinct ways can the seven letters of the word "Laughed" be arranged so that no two vowels are adjacent? | nvidia |
Find the longest possible length of the remaining side of a triangle with sides of length 6 cm, 8 cm and a right angle, such that the sum of the squares of the two shorter sides is equal to 100 square cm. | nvidia |
The equation $x^2-kx-12=0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$? | nvidia |
Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$. | nvidia |
Find the smallest positive integer $n$ such that the product of its distinct prime factors is equal to $315$. | nvidia |
Let $a$, $b$, and $c$ be three positive integers such that $a$ is divisible by $b$, $b$ is divisible by $c$, and $c$ is divisible by $a$. Find the maximum possible value of $a+b+c$ if $a$, $b$, and $c$ are all different. | nvidia |
Determine the number of integers between 1 and 9 that are factors of the four-digit number 12,960. | nvidia |
Let $x$ and $y$ be any two even numbers, with $y$ less than $x$. The largest integer which divides all possible numbers of the form $x^2+y^2$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$ | nvidia |
Let $a$, $b$, $c$ be positive real numbers. Find the maximum possible value of $$\frac{a+b+c-\sqrt{a^2+b^2+c^2+2ab+2bc+2ca}}{\sqrt{abc}}.$$ | nvidia |
Find all real numbers $a$, $b$, and $c$ that satisfy the system of equations:
$a + b + c = 4$
$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 6$
$abc = 2$ | nvidia |
Let $f(x)$ be a function defined on the set of positive integers such that $f(1) = 1$ and
$$f(n) = n + \sum_{k=1}^{n-1} f(k)$$
for $n > 1$. Find $f(10)$. | nvidia |
Consider a square matrix $M$ whose entries are integers, such that its determinant is $0$.
Let $N$ be a matrix obtained by interchanging two rows of $M$.
Determine all possible values of the determinant of $N$. | nvidia |
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find the number of functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that
$$f(xf(y))=f(xy)+f\left(\frac{x}{y}\right)$$for all $x,y\in\mathbb{R}^+$. | nvidia |
Find the number of three-digit numbers whose digits are in increasing order and have a sum of 12. | nvidia |
Find the smallest positive integer n such that the sum of the first n perfect squares is divisible by n. | nvidia |
Let $S$ be the set of positive integers such that each element $x\in S$ can be expressed as the sum of two perfect squares. Find the number of $4$-digit numbers in $S$ that are in increasing order (i.e., each digit of the number is larger than the previous digit). | nvidia |
Let $m \ge 3$ be an integer and let $S = \{3,4,5,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.
| nvidia |
Find the sum of the positive divisors of 24 that are also multiples of 3. | nvidia |
Compute the value of the following expression: $\frac{14}{15} \cdot \frac{3}{11} \cdot \frac{55}{2} \cdot \frac{4}{7}$ | nvidia |
What is the greatest common divisor of all numbers that can be expressed as the sum of two squares of consecutive positive integers? | nvidia |
Count the number of positive integers less than 100 that have no repeating digits. | nvidia |
Determine if there are integers $a$, $b$, $c$, and $d$ such that $a+b+c+d=1234$, $|a-b|=|c-d|$, $abcd=567890$ and if there is such quadruple, find it (otherwise show that no such quadruple exists). | nvidia |
Find the sum of all $x$ such that the expression $\frac{x^2-3x+2}{x^2-3x-4}$ is undefined. | nvidia |
Let $\mathcal{C}$ be a circle centered at the origin with radius 257 and whose circumference has exactly 400 equally spaced markers. For each pair $(A,B)$ of consecutive markers on $\mathcal{C}$ consider the locus of points $X$ inside $\mathcal{C}$ such that there exists a point $Y$ outside $\mathcal{C}$ such that $AXB... | nvidia |
Two of the roots of the equation \[ax^3+bx^2+cx+d=0\]are $3$ and $-2.$ Given that $a \neq 0,$ compute $\frac{b+c}{a}.$ | nvidia |
Determine the largest possible value of n for which the polynomial x^n can be expressed as the sum of two or more consecutive polynomials with non-negative integer coefficients. | nvidia |
In a sphere, a cone is inscribed such that the center of the sphere is the apex of the cone and the base of the cone is a circle on the surface of the sphere. A smaller cone is also inscribed in the sphere such that its base is a circle on the surface of the sphere and its apex is on the surface of the larger cone. The... | nvidia |
Consider the polynomials \[f(x)=1-12x+3x^2-4x^3+5x^4\] and \[g(x)=3-2x-6x^3+9x^4.\] Find $c$ such that the polynomial $f(x)+cg(x)$ has degree 3. | nvidia |
A cylindrical log with radius $r$ is cut into two equal parts by a plane that passes through the axis of the cylinder and makes an acute angle $\theta$ with the base of the cylinder. What is the radius of the largest sphere that can be inscribed in one of the two parts of the log?
Express your answer in terms of $r$ a... | nvidia |
A triangle $ABC$ is inscribed in a circle and has side lengths $AB=4$, $BC=5$, and $CA=3$. The circle also contains an inscribed square that shares a side with the triangle. Find the area of the triangle. | nvidia |
For how many positive integers $n$ does $1+2+\cdots+n$ evenly divide $6n$? | nvidia |
A regular decagon can be split into ten congruent sectors of equal area. A certain linear function maps the vertices of the decagon onto the center of one of the sectors. Compute the sum of the images of all the vertices, assuming the origin lies at the center of the decagon. | nvidia |
Find the largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$. | nvidia |
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