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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.
New problem:
Consider a quadratic polynomial of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are integers. Suppose that the polynomial can be factored as the product of two line... | nvidia |
Find the smallest positive integer $n$ such that there exists an integer $x$ satisfying the congruence $3x \equiv 1 \pmod{2^n}$. | nvidia |
Let $k$ be a positive number such that when $97$ is divided by $k$, the remainder is $3.$ Compute the sum of all possible two-digit values of $k$. | nvidia |
Find the product of the roots of the equation \[x^2 \sqrt{x} - 3x^2 + 2\sqrt{x} + 5 = 0,\]given that all of the roots are real and nonnegative. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that
$$x^2+\frac{1}{y}=2,\qquad y^2+\frac{1}{z}=2,\qquad z^2+\frac{1}{x}=2.$$
Determine the value of $x+y+z$. | nvidia |
A positive integer $m$ is said to be cube-perfect if $\frac m2$ is a perfect square and $\frac m6$ is a perfect cube.
Determine the smallest positive integer $m$ for which $\frac{m^2}{54}$ is also a perfect cube. | nvidia |
Find the number of positive integers $k$ such that $k$ is the sum of the digits of its own square. | nvidia |
Find the largest integer value of $m$ such that the product of the roots of the equation $2x^2-5mx+3=0$ is positive and the sum of the roots is less than 5. | nvidia |
In triangle $DEF$, $DE = 19$, $EF = 22$, and $FD = 25$. Points $P$ and $Q$ are on $\overline{DE}$ with $P$ on $\overline{DQ}$, points $R$ and $S$ are on $\overline{EF}$ with $R$ on $\overline{ES}$, and points $T$ and $U$ are on $\overline{FD}$ with $T$ on $\overline{FU}$. In addition, the points are positioned so that ... | nvidia |
Problem:
The graph of the function $f(x) = \sqrt{x+1} + \sqrt{x-1}$ intersects the line $y = kx$ at two points, where $k$ is a positive constant. Find the range of values of $k$ such that the distance between the two points of intersection is greater than 2.
(No further comments or explanations, the problem is complet... | nvidia |
Compute the value of $a$ such that the equation $(a+2)^2+(a^2+4)=2a^2+8a+8$ holds true. | nvidia |
Determine the maximum number of Mondays that can occur in a 30-day month.
(Note: In a 30-day month, the maximum number of Mondays can occur if the month starts on a Monday.) | nvidia |
The length of the hypotenuse of a right triangle is $h$ , and the radius of the inscribed circle is $r$. The ratio of the area of the circle to the area of the triangle is
$\textbf{(A) }\frac{\pi r}{h+2r}\qquad \textbf{(B) }\frac{\pi r}{h+r}\qquad \textbf{(C) }\frac{\pi}{2h+r}\qquad \textbf{(D) }\frac{\pi r^2}{r^2+h^2}... | nvidia |
Count the number of three-digit integers that have exactly one digit equal to 5, and the remaining two digits are distinct and add up to 10. | nvidia |
There are four positive integers that are divisors of each number in the list $$36, 72, -12, 114, 96.$$Find the sum of these four positive integers.
Write another problem inspired by this one:
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 |
Determine the value of $k$ such that the product of the roots of the polynomial $(3x^2 +k)^{12}$ is equal to 64. | nvidia |
Find the maximum value of $b$ such that the quadratic $x^2+bx+2000$ has two real roots with a difference of less than 20. | nvidia |
Given that $x$ is a positive integer and $x^2 + x - 12 < 0$, what is the sum of all possible integer values of $x$? | nvidia |
Given that $a$ and $b$ are positive integers and that $a+b=24$, what is the value of $ab$ if $2ab + 10a = 3b + 222$? | nvidia |
A circle with center (p, q) passes through the points (-2, 7) and (6, -3).
Find the point equidistant from these two points that lies on the line pq. | nvidia |
Let $a$ and $b$ be real numbers such that
\[e^{a+b} = e^a + e^b + e^{ab}.\]
Find the minimum value of $e^a + e^{-a}.$ | nvidia |
Find the value of the sum of the coefficients of the polynomial
$$p(x) = (x-1)^2(x-2)^3(x-3)^4$$
which can be obtained by expanding the product
$$(x-1)(x-1)(x-2)(x-2)(x-2)(x-3)(x-3)(x-3)(x-3).$$ | nvidia |
Consider two real numbers $x$ and $y$ such that $-3 < x - y < 7$. If $\frac{x+y}{y}$ is an integer, determine its value. | nvidia |
Let $x$ be a positive real number such that $x + \frac{1}{x} = 98.$ Find\[\sqrt{x} + \frac{1}{\sqrt{x}}.\] | nvidia |
A right triangle with legs of length $a$ and $b$ is inscribed in a circle with center $O$. When the triangle is rotated about the leg of length $a$, the volume of the cone produced is $V_1$. When the triangle is rotated about the leg of length $b$, the volume of the cone produced is $V_2$. If the ratio of the areas of ... | nvidia |
Let \(a_1<a_2<\cdots <a_n\) be an arithmetic sequence such that the sum of the squares of the first two terms is 85, and the sum of the squares of the last two terms is 765. Find the common difference of the sequence. | nvidia |
Let $x$ be a positive real number such that $\lfloor x \rfloor$ and $\lfloor x^2 \rfloor$ have the same parity. Determine the largest integer $n$ such that $\left( \left\lfloor \frac{x}{n} \right\rfloor - \left\lfloor \frac{x^2}{n} \right\rfloor \right)$ has the same parity as $\lfloor x \rfloor$. | nvidia |
Among all pairs of real numbers $(a, b)$ such that $\tan (\tan^{-1} a) = \tan (\tan^{-1} b)$ with $a,b \in (-\frac{\pi}{2},\frac{\pi}{2})$, a number is randomly selected from the set of all possible values of $|a-b|$. Compute the probability that the number selected is equal to 0. | nvidia |
Count the number of positive integers less than 100 that are multiples of 7 and have a tens digit of 3. | nvidia |
Find the sum of the first eight positive multiples of 17. | 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 $\frac{b+c}{a}.$
A new problem:
The polynomial $p(x) = ax^3 + bx^2 + cx + d$ has three roots, one of which is $x = 2$. If the sum of the other two roots is $-3$, and the product of the other two roots is $-4$, find $... | nvidia |
Find the value(s) of $z$ such that $4zy-6y+3z-9=0$ is true for all values of $y$. | nvidia |
Let $\mathbf{B}$ be a $3 \times 3$ matrix, with real entries, such that $\mathbf{B}^4 = \mathbf{I}$, where $\mathbf{I}$ is the identity matrix. Find the maximum possible value of the determinant of $\mathbf{B}$. | nvidia |
The sum of the digits of a three-digit number $xyz$ is 12. If the sum of the digits of the number $yzx$ is 9, find the number of possible values of $x$. | nvidia |
Let $a$ and $b$ be nonzero real numbers such that
\[(3 + 4i)(a + bi)\]is a real number. Find $\frac{a}{b}.$ | nvidia |
The points $A = (0,0)$ and $B = (1,0)$ lie on a circle centered at the origin. Determine the radius of the circle such that the point $C = \left( \frac{3}{4}, k \right)$ also lies on the circle. | nvidia |
Find all real numbers $x$ such that $|x^3| = 10$ and $\lfloor x \rfloor = -3$. | nvidia |
The sum of two perfect squares is equal to the sum of two perfect cubes. If one of the squares is equal to 4 and one of the cubes is equal to 8, find the sum of the two unknown numbers. | nvidia |
Find the cubic polynomial with real coefficients such that its roots are $2, 1 + \sqrt{3}i,$ and $1 - \sqrt{3}i$. | nvidia |
Find the values of $x$, $y$, and $z$ such that the matrix
$$\begin{pmatrix} x^2 & y^2 & z^2 \\ xy & yz & zx \\ x & y & z \end{pmatrix}$$ is singular.
Determine the possible values of the sum $$\frac{x}{x+y+z}+\frac{y}{x+y+z}+\frac{z}{x+y+z}.$$ | nvidia |
Let $x$ and $y$ be real numbers such that $3x + 2y \le 7$ and $2x + 4y \le 8.$ Find the largest possible value of $x + y.$ | nvidia |
Find the sum of the squares of the roots of the equation $x^{21} + 1 = 0$. | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=7$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $a+b+c$.
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re... | nvidia |
Find the smallest value of $a$ such that the function $f(x) = |x-a| + |x+2|$ takes on the value $10$ for some real number $x$. | nvidia |
Let
$a_k=\ds\sum_{i=0}^k(-1)^i\cdot {k-i \choose i}\cdot 3^{k-2i}$.
For which $n$ does there exist a natural number $p$ such that
$\ds\sum_{k=0}^na_k={p \choose n}$? | nvidia |
Find the number of solutions in integers of the equation
$1729x+1768y=1$ | nvidia |
Suppose that $a$ and $b$ are nonzero real numbers, and that the equation $x^2+ax+b=0$ has solutions $a$ and $b$. Find the ordered pair $(a,b).$ | nvidia |
Given the parabola $y=x^2+5x-6$ and the line $y=-x$, find all $x$ such that the area between the parabola and the line is positive. Express your answer in interval notation. | nvidia |
Find the largest two-digit positive integer that is divisible by all the prime numbers between 3 and 11, inclusive. | nvidia |
Find the 25th term in the sequence of all positive integers that can be formed by rearranging the digits of the number 12546, where each digit is used exactly once. | nvidia |
A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $24$ cm across as the top and $8$ cm deep. What was the radius of the ball (in centimeters)?
$\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 8\sqrt{3} \qquad \textbf{(... | nvidia |
In the sequence of positive integers $a_1$, $a_2$, $a_3$, $\dots$, let $S_n$ be the sum of the first $n$ terms and $P_n$ be the product of the first $n$ terms. If you are given the value of $P_{2023}$, then you can uniquely determine the value of $S_n$ for some integer $n$. What is this integer $n$? | nvidia |
A rectangle is inscribed inside a circle in such a way that the two vertices of the rectangle lie on the diameter of the circle. The perimeter of the rectangle is equal to 20 cm. Find the radius of the circle. | 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 |
In a pair of similar isosceles triangles, the ratio of the area of the smaller triangle to the area of the larger triangle is 1:4. If the length of one leg of the smaller triangle is 6 cm, find the perimeter of the larger triangle. | nvidia |
In a triangle, the length of the altitude is 25 units less than the length of the median to the same side. The segment that joins the centroid and the midpoint of the side divides the triangle into two regions whose areas are in the ratio 3:2. Let x be the length of the segment joining the midpoint of the side and the ... | nvidia |
How many positive integer divisors of $2004^{2004}$ are divisible by exactly 2004 positive integers?
| nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 2.$ Find the maximum value of $\frac{abc}{a^3 + b^3 + c^3}.$ | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of
\[\frac{x + y}{xyz}.\] | nvidia |
Find all real numbers $k$ for which there exists a nonzero, 2-dimensional vector $\mathbf{v}$ such that
\[\begin{pmatrix} 1 & 8 \\ 2 & 1 \end{pmatrix} \mathbf{v} = k \mathbf{v}.\]Enter all the solutions, separated by commas. | nvidia |
Below is a portion of the graph of a quadratic function, $y=f(x)=ax^2+bx+c$:
It is known that $f(-3)=f(2)=0$ and that the equation $f(x)=-1$ has exactly one real solution. What is the value of $f(0)$? | nvidia |
What is the smallest positive integer n such that there are exactly 3 integers between 1 and n (inclusive) that are congruent to 2 (mod 7)? | nvidia |
Find the product of the solutions of the equation $x^2 + 2x + 1 = 0$, given that one of the solutions is a complex number. | nvidia |
Let $a$ and $b$ be real numbers such that $a^2 = 7-4\sqrt{2}$ and $b^2 = 7+4\sqrt{2}$. Find $|a-bi|$ | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that $xyz = 8.$ Find the minimum value of $x + 2y + 4z.$ | nvidia |
Find the coordinates of the point that divides the line segment joining the points $(2,4)$ and $(6,8)$ into three equal parts. | nvidia |
Find the product of all the solutions of the equation $\frac{3y}{15}=\frac{2}{y}$. | nvidia |
The arithmetic mean of four numbers is 15. Two of the numbers are 10 and 18 and the other two are equal. What is the product of the two equal numbers? | nvidia |
Find all real values of $x$ that satisfy $\frac{x(x+1)}{(x-4)^2} \ge 12.$ (Give your answer in interval notation.)
New Problem:
Find all real values of $x$ that satisfy $\frac{(x-1)(x+2)}{x^2+1} \le 0.$ (Give your answer in interval notation.) | nvidia |
Find the sum of the exponents of the variables after simplifying the expression $\sqrt[4]{(12x^3y^8z)^3}$. | nvidia |
Let $S$ be the number of sets $\{a_1, a_2, \dots, a_n\}$ that contain only nonnegative integers and satisfy the following conditions:
$a_{i+1}-a_i \geq 3$
$a_{n}-a_1=100$
Find the largest value of $n$ for which $S>0$. | nvidia |
A box contains 12 white balls and 8 red balls. Three balls are drawn from the box at random. What is the sum of all possible values of the number of white balls drawn, such that the probability of drawing those white balls is equal to the probability of drawing the same number of red balls? | nvidia |
Find all values of $a$ and $b$ such that the inequality $x^4-ax^2+b>0$ has no real solutions for $x$. Express your answer as a set of constraints on $a$ and $b$. | nvidia |
Given that the complex numbers $x$ and $y$ satisfy the equation $x + yi = (1 + 2i)^2$, find the product $xy$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x) f(y) - f(xy) = x^2 + y^2\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | nvidia |
If $a$,$b$, and $c$ are positive real numbers such that $a(b+c) = 152$, $b(c+a) = 162$, and $c(a+b) = 170$, then find $abc.$ | nvidia |
Let $a$ and $b$ be two real numbers such that $a+b=6$ and $a^2+b^2=12$. Find $ab(a+b)$. | nvidia |
Two real numbers $x$ and $y$ satisfy $x-y=4$ and $x^3-y^3=28$. Compute $xy$. | nvidia |
Find the number of distinct ordered pairs of positive integers $(x,y)$ that satisfy the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{3}$. | nvidia |
Find the number of positive integers $n$ such that $n^2 + 2n + 1$ is a perfect square and $20 < n < 150$. | nvidia |
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum? | nvidia |
Find the largest possible sum of three distinct prime numbers, where the sum is less than 100. | nvidia |
Find the smallest positive integer $k$ for which $k^2$ is divisible by 24 and $k^3$ is divisible by 1080. | nvidia |
In triangle $DEF$, point $G$ lies on side $EF$ such that the length of $DG$ is twice the length of $GF$. Given the coordinates of $D(0, 0)$, $E(6, 0)$, and $F(0, 9)$, find the sum of the coordinates of point $G$. | nvidia |
What is the smallest positive integer that can be expressed as a sum of two positive integers, one with exactly 3 distinct positive factors and the other with exactly 4 distinct positive factors? | nvidia |
Find the maximum possible value of the product
\[(x+y+z)(x^2+y^2+z^2),\]
where $x,$ $y,$ and $z$ are real numbers satisfying $x^2+y^2+z^2=1$. | nvidia |
The sum of the positive integer divisors of a positive integer $m$ is 56. Find $m$. | nvidia |
If the product of the digits of a three-digit number $abc$ is $a^x \cdot b^y \cdot c^z$, where $x, y, z$ are positive integers, and the sum of the digits is $a + b + c = 2^3$, find the maximum value of $x + y + z$. | nvidia |
Using only the digits 2, 3, and 4, how many positive five-digit integers can be made that are palindromes? | nvidia |
In the row of Pascal's triangle that begins with 1 and then 15, find the fifth number. | nvidia |
Find the positive real number $x$ such that $x^{\lfloor x \rfloor} = 2000$. Express $x$ as a decimal. | nvidia |
Find the number of real solutions $(x,y)$ to the system of equations:
$$\begin{cases} x^2+y^2=4 \\ x^3-y^3=8 \end{cases}$$ | nvidia |
Determine the largest integer $n$ such that the binomial expansion of $(x+1)^n$ contains no term with a coefficient ending in 0. | nvidia |
The graph of the polynomial
$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$
has five distinct $x$-intercepts, one of which is at $(0,0)$. Which of the following coefficients cannot be zero?
$\textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$ | nvidia |
What is the sum of three consecutive even integers such that the middle integer is equal to the sum of the squares of the smallest and largest integers? | nvidia |
What is the smallest positive integer $n$ such that the sum of the digits of $2^n$ is equal to 10? | nvidia |
Let $\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a, b, c$ are distinct digits. Find the sum of the elements of $\mathcal{S}.$
| nvidia |
Find a positive integer $N$ with four or fewer digits such that $N$ is the multiplicative inverse of $2^{1024}+1$ modulo 10,000. | nvidia |
Find the difference between the product of all the numbers from $-10$ to $10$ and the product of all the even numbers from $-10$ to $10$. | nvidia |
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