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Let $a,$ $b,$ $c,$ and $d$ be positive real numbers. Find the number of real roots of the equation
\[\begin{vmatrix} x^2 & a & -b \\ -a & x^2 & c \\ b & -c & x^2 \end{vmatrix} = \begin{vmatrix} d^2 & a & -b \\ -a & d^2 & c \\ b & -c & d^2 \end{vmatrix}.\] | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 32.$ Find the minimum value of
\[x^2 + 4xy + 4y^2 + 2z^2.\] | nvidia |
Determine the range of values for $x$ such that both $4x$ and $5x$ are in the interval $(2, 3)$. | nvidia |
Consider the functions $f(x) = x^2 - 2x$, $g(x) = -\arctan(x)$, and $h(x) = \frac{4}{x}$. These functions have domains of all real numbers, all real numbers, and all real numbers except 0, respectively. Determine the product of the numbers in the set $\{2, 3, 4\}$ such that the corresponding function is invertible on i... | nvidia |
Let $a_n$ be the sum of all the residues of the polynomial $$P(x)=(x-1)(x-2)(x-3)\cdots(x-n)\left(x-\dfrac{1}{2}\right)\left(x-\dfrac{3}{2}\right)\left(x-\dfrac{5}{2}\right)\cdots\left(x-\dfrac{2n-1}{2}\right),$$where $n$ is a positive integer.
Determine the largest integer $n$ such that $a_n<0$. | nvidia |
Given that the area and perimeter of a rectangle with length $x$ inches and width $2x$ inches have the same numerical value as the area and perimeter of a square with side length $x+3$ inches, find $x$. | nvidia |
The volume of a rectangular box is equal to the surface area of the box. If the length, width, and height of the box are all different positive integers, find the smallest possible length of the longest edge of the box. | nvidia |
Determine the smallest positive integer $m$ such that the number of divisors of $m$ is equal to the number of digits in $m$, and $m$ has exactly three distinct prime factors. | nvidia |
Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}.$ Find the number of possible values of $S.$
| nvidia |
Find the smallest positive integer $n$ such that $n!$ is divisible by $n^2$. | nvidia |
The point $(5,3)$ lies on the graph of the function $y=g(x)$. Find the sum of the coordinates of the point that must be on the graph of the function $y=g\left(\frac{x}{2}-1\right)+2$. | nvidia |
Find the radius of the circle that passes through the points $(1,3)$, $(2,4)$, and $(3,5)$, and has its center on the line $y = x + 2$. | nvidia |
In a convex quadrilateral ABCD, the lengths of the diagonals are 16 and 20 units. The diagonals intersect at point E. How many units are in the length of the smallest line segment that can be drawn from E to one of the vertices so that the remaining three segments (AE, BE, and CE, or AE, DE, and CE) can no longer form ... | nvidia |
In a triangle, the sum of the squares of two sides is 200, and the product of these sides is 150. Find the length of the third side. | nvidia |
In a parallelogram, the length of one side is 18 units and the angle between that side and one of the diagonals is 30 degrees. Find the area of the parallelogram. | nvidia |
Find the smallest positive integer $n$ such that the sum of the digits of $2^n$ is equal to the sum of the digits of $n^2$. | nvidia |
Find the product of all numbers $x$ satisfying $\frac{x^2+1}{x} = 7.$ | nvidia |
A sequence of consecutive even numbers starts with -4 and ends with 6. If the sum of all the numbers in the sequence is -30, what is the difference between the number of positive and negative numbers in the sequence? | nvidia |
Three concentric circles are given such that the radius of the outer circle is twice the radius of the middle circle, and the radius of the middle circle is thrice the radius of the inner circle. Find the ratio of the areas between the middle and outer circles to the area between the middle and inner circles. Express y... | nvidia |
Find all real numbers $a$ such that the roots of the equation
$x^2-2ax+1=0$
satisfy the condition $x_1x_2+x_1+x_2=0$. | nvidia |
Determine the number of ways to arrange the letters of the word "PINBALL" such that no two identical letters are adjacent. | nvidia |
Let $a$ and $b$ be the two smallest positive integers such that $a^2-b^2=48$. Determine the value of $ab$. | nvidia |
Let $p(x)$ be a polynomial with integer coefficients such that $p(17) = 34$ and $p(23) = 46$. What is the smallest positive integer $n$ such that $p(x+n) = p(x)$ for all $x$? | nvidia |
Find a positive integer that is divisible by 25 and whose square root is a number between 10 and 10.2. | nvidia |
Find the largest power of $x$ that divides the determinant of the matrix
$$\begin{bmatrix} 1 & x & x^2 \\ 1 & x+1 & (x+1)^2 \\ 1 & x+2 & (x+2)^2 \end{bmatrix}.$$ | nvidia |
Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than $30$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$. | nvidia |
A regular polygon $P_1$ with 80 sides is drawn in the Cartesian plane. The sum of the y-coordinates of the vertices of $P_1$ is 2011. The midpoints of the sides of $P_1$ form a second polygon, $P_2$, also with 80 sides. Finally, the midpoints of the sides of $P_2$ form a third polygon, $P_3$, with the same number of si... | nvidia |
A square with a side length of $s$ units and a circle with a radius of $r$ units are drawn on a coordinate plane, as shown below.
[asy]
size(5cm,5cm);
draw((-2,0)--(7,0));
draw((0,-1)--(0,7));
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw(circle((0,0),1));
label("$(s,s)$",(3,3),NE);
label("$(r,0)$",(1,0),S);
[/asy]
T... | nvidia |
A hexagon is inscribed in a circle such that each vertex of the hexagon lies on the circle and the sides of the hexagon are equal in length. The radius of the circle is 12 cm. What is the area of the hexagon? | nvidia |
Let $d$ be a positive number such that when $109$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. | nvidia |
The degree measure of the sum of the interior angles of a convex polygon with $n$ sides is 1800. What is the degree measure of the sum of the interior angles of a convex polygon with $n + 2$ sides? | nvidia |
How many distinct ordered pairs of positive integers $(x,y)$ are there so that the sum of the reciprocals of $x$ and $y$ is $\frac{1}{6}$? | nvidia |
Let $a$, $b$, $c$, $d$, and $e$ be positive integers such that
\[a+b+c+d+e=2010\]and
\[\frac{a}{b+c+d+e}+\frac{b}{a+c+d+e}+\frac{c}{a+b+d+e}+\frac{d}{a+b+c+e}+\frac{e}{a+b+c+d}=1.\]Determine $a^2+b^2+c^2+d^2+e^2$. | nvidia |
Let $z$ be a complex number such that $z^2 + z + 1$ is a multiple of $5$. Determine the smallest positive integer $m$ such that $z^3 + m$ is also a multiple of $5$. | nvidia |
Let $p(x)$ be a polynomial such that $p(0)=0$, and for any positive integer $n$, the value of $p(n)$ is a prime number. For how many positive integer values of $n$ can the value of $p(n)$ be a two-digit prime number? | nvidia |
Two circles with centers $A$ and $B$ intersect at the points $C$ and $D$. The equation of the circle centered at $A$ is $x^2+y^2-4x-6y=0$, and the equation of the circle centered at $B$ is $x^2+y^2-2x-ky=0$. If the points $A$ and $B$ have the same $y$-coordinate, what is the value of $k$? | nvidia |
Let $f(x)=16x+3$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f((2x)^{-1})$. | nvidia |
A quadratic equation $ax^2+bx+c=0$ has two roots, $x_1$ and $x_2$, such that $x_1+x_2=12$ and $x_1^2+x_2^2=120$. Find the product of the roots. | nvidia |
Let $x$ be the number of positive integers less than $50$ that are divisible by both $3$ and $5$. Let $y$ be the number of positive integers less than $50$ that are divisible by $3$, $5$, or both. Compute $(x - y)^2$. | nvidia |
Find the product of the roots of the equation $\sqrt{x-2}+\sqrt{x+3}=3$. | nvidia |
Determine the largest three-digit positive integer that is a multiple of 49. | nvidia |
Determine the largest possible degree of $p(x)$ such that the graph of the rational function $\frac{2x^3+5x^2-x+1}{p(x)}$ has a horizontal asymptote and a slant asymptote. | nvidia |
What is the largest positive integer that divides the product of every set of five consecutive positive integers? | 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 the product of all values of $t$ such that $t^2 = 36$. | nvidia |
How many different triangles can be formed by connecting three of the dots in a $5\times 5$ square array of dots, such that the triangle has at least one side that is parallel to the grid? | nvidia |
Find the positive integer n such that 2n + 1 is a perfect square and 4 ≤ n ≤ 10. | nvidia |
Find the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A( -3, 1)$ and $B(2, 0)$. | nvidia |
Find the equation of the circle that passes through the points (-2,4), (3,-8), and (1,2), and has its center on the x-axis. | nvidia |
Find the largest value of $n$ such that $5x^2+nx+48$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
Find the product of two different prime numbers $p$ and $q$ ($p,q\geq3$) such that when $q$ is subtracted from $p$, the difference is the least possible positive prime number. | nvidia |
Let $a$ be a real number. Solve the following equation:
$$x + \frac{1}{x+a} = 2$$
Find the value of $a$ such that the equation has exactly three solutions. | nvidia |
Let $f(n)$ be the sum of the squares of the digits of a positive integer $n$. A number $n$ is called great if it has $d\geq 3$ digits, and if $f(n) = \frac{f(f(n))}{d}$. What is the maximum number of digits that a great number can have? | nvidia |
Determine the largest possible number of elements that can be removed from the set {1, 2, 3,..., 100} such that none of the sums of three distinct remaining elements are multiples of 7. | nvidia |
Let $a$, $b$, and $c$ be real numbers that satisfy the equation
$$\frac{1}{a+1}+\frac{1}{b+2}+\frac{1}{c+3}=1.$$
Suppose that $a$, $b$, and $c$ are all positive. Find the value of
$$\frac{a}{a+1}+\frac{b}{b+2}+\frac{c}{c+3}.$$ | nvidia |
The graph of $y=f(x)$ is shown below, with $1$ unit between grid lines. Assume $f(x)$ is defined only on the domain shown.
What is the sum of all integers $c$ for which the equation $f(x)=c$ has exactly $6$ solutions?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm... | nvidia |
Find the largest possible degree of the polynomial $p(x)$ such that the graph of the rational function $\frac{p(x)}{x^3-2x^2+3x+1}$ has a slant asymptote. | nvidia |
Find the maximum value of $\frac{x}{\sqrt{x^2+4}}$, where $x$ is a real number. | nvidia |
The circle with center $C$ and radius $4$ units is tangent to two other circles of radii $2$ and $3$ units. The two smaller circles are externally tangent to each other at the point $P$. Find the area of the shaded region. | nvidia |
Find the positive value of $n$ such that the equation $9x^2+nx+1=0$ has exactly one solution in $x$. | nvidia |
Given that $\sum_{i=1}^{n}x_i^{y_i} = z$, where $x_i$, $y_i$, and $z$ are positive integers, find the smallest $n$ such that for any positive integer $m$, there exists a sequence of positive integers $a_1, a_2, ..., a_m$ such that $a_1^{a_2} + a_2^{a_3} + ... + a_m^{a_1} = \sum_{i=1}^{n}x_i^{y_i}$. | nvidia |
In a triangle $ABC$, the angle bisectors of angles $A$ and $B$ intersect at point $I$. The side lengths $AB$, $AC$, and $BC$ are positive integers such that $AB < AC < BC$. If $AB=12$, what is the smallest possible value of $BC$ such that no circle can be inscribed in triangle $ABC$? | nvidia |
A circle intersects the $x$-axis at two points $A$ and $B$, and a parabola with the equation $y^2 = 4x$ intersects the circle at two points, one of which is the center of the circle. If the coordinates of the center of the circle are $(x, 0)$, and the distance between $A$ and $B$ is $6$, find the radius of the circle. | nvidia |
A rectangular prism has a height of $10$ inches and a base that is a square with side length $6$ inches. A pyramid is cut from the prism by making two planar cuts that intersect at the center of the base. The first cut is perpendicular to the base, and the plane of the second cut forms a $60^\circ$ angle with the plane... | nvidia |
The square root of a positive integer is to be increased by 15. What is the largest value of the integer such that the result is a perfect square? | nvidia |
How many positive integers $n$ have the property that both $n$ and $2n$ are factors of the product $\left(2^{10}\right)\left(3^{12}\right)\left(5^{15}\right)$? | nvidia |
Determine the largest value of $n$ for which the expression $n^2-4n+5$ is a perfect square. | nvidia |
Let $n$ be a positive integer. The sequence $a_1, a_2, \ldots, a_n$ is defined as follows:
for each $i$ such that $1\le i\le n$, $a_i$ is the smallest integer that is not the sum of any nonempty subset of $a_1,a_2,\ldots,a_{i-1}$.
Determine $a_{100}$. | nvidia |
Given that $$(a+b+c)(ab+ac+bc)=36$$and that $$a^2(b+c)+b^2(a+c)+c^2(a+b)=12$$for real numbers $a$, $b$, and $c$, what is the value of $abc$? | nvidia |
Find the value of $y$ such that the equation $\displaystyle\sqrt[4]{y - \frac{x^2}{4}} = 3$ has exactly one solution. | nvidia |
The sum of the squares of three consecutive positive integers is equal to the sum of the squares of four consecutive positive integers. What is the smallest possible value of the largest integer among these seven integers? | nvidia |
If the sum of the three digits of a number is 12, and the sum of the square of its digits is 98, what is the value of the smallest digit? | nvidia |
Find the sum of the reciprocals of the positive divisors of 24, except 1 and 24 themselves, if you are given that the sum of the reciprocals of the positive divisors of 6 (again excluding 1 and 6) is $\frac{1}{2}$. | nvidia |
A circle is inscribed in a square so that the circle's diameter coincides with one of the sides of the square. If the coordinates of the circle's center are (3, 5), and the length of the square's side is 8 units, find the distance between the circle's center and the point where the circle touches the square (not on the... | nvidia |
Let $a_1, a_2, \dots, a_n$ be positive real numbers such that $\sum_{i=1}^n a_i^p = 1$, where $p > 1$. Find the maximum value of $\sum_{i=1}^n a_i \sqrt{1 - a_i^p}$. | nvidia |
If $b$, $x$, and $y$ are positive integers such that $x^{\log_b{y}}=3$, $y^{\log_b{x}}=4$, and $b>3$, find $b$. | nvidia |
Find all values of x between 0° and 360° such that cos(2x) = -0.5. | nvidia |
Let $x,$ $y,$ and $z$ be real numbers such that $x + y + z = 1.$ Find, in terms of $p$ and $q,$ the maximum value of $px^2 + qy^2 + z^2,$ where $p$ and $q$ are fixed positive real numbers. | nvidia |
Find the value of $b$ such that the equation
$$(x^2+2x+3)^2-6(x^2+2x+3)+b=0$$
has exactly two distinct real solutions. | nvidia |
Given that $a$, $b$, and $c$ are positive integers, find the smallest positive integer that can be expressed as $ac$ or $bc$ where the greatest common divisor of $ac$ and $bc$ is $c$. | nvidia |
Find the smallest positive integer $n$ such that the equation $\frac{n}{120+2x} = \frac{3}{4}$ has a solution in positive integers for $x$. | nvidia |
The equation
$$x^3-x^2-15x-7=0$$
has a root $x = p/q$ such that $p, q \in \Z$ and $\gcd(p,q)=1$.
Prove that $pq$ is divisible by 7. | nvidia |
In a regular polygon, the number of diagonals is equal to four times the number of sides. Determine the number of sides of the polygon. | nvidia |
Let $n$ be a positive integer greater than $5$. Consider the set $A$ of all positive integers less than $n$ which are not divisible by any prime divisor of $n$. Find the remainder when the product of all elements of $A$ is divided by $n$. | nvidia |
Find the number of integers in the set $\{1, 2, 3, \ldots, 100\}$ that can be written as the sum of a perfect square and a perfect cube. | nvidia |
Find the largest integer $n$ such that there are at least $5$ primes whose squares end in the same $n$ digits. | nvidia |
Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2-n^2$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$
| nvidia |
Solve for all real numbers $x$ that satisfy the inequality
\[\frac{2x-1}{x+2} < \sqrt{x+1} < \frac{3x+1}{x+1}.\] | 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 |
Let $h(x)$ be a function that satisfies $h(h(x)) = 2x + 1$ for all real numbers $x$. If $h(5) = 3$, then find $h(2)$. | nvidia |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | nvidia |
Let $g$ be a non-constant polynomial such that
\[g(x) - g(2x) + g(3x) = \frac{x[g(x)]^2}{2014}\]for all real numbers $x.$ Find the largest possible value of $g(-1).$ | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-(2a+1)x+2a^2+a+4=0$$
are such that
$x_1^3+x_2^3=5$.
Determine $a$. | nvidia |
Find the smallest positive integer $n$ such that the equation $x^3 + nx^2 + 2019x + 1 = 0$ has at least one rational root. | nvidia |
In a regular octahedron, the midpoints of the twelve edges are the vertices of a smaller polyhedron. The ratio of the surface area of the smaller polyhedron to that of the larger is $a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. | nvidia |
Let $a$, $b$, $c$ be positive integers such that $a^2+b^2=c^2$.
Determine the number of possible ordered triples $(a,b,c)$ with $\ds\f{a}{c}\in \left(0.249,0.251\right)$ and $\ds\f{b}{c}\in \left(0.949,0.951\right)$. | nvidia |
Find the number of positive whole number divisors of $2^{10} \cdot 3^5 \cdot 5^2$ that are not multiples of 6. | nvidia |
Find the sum of all values of $m$ such that the quadratic $x^2 + mx - 20$ can be factored in the form $(x+c)(x+d)$, where $c$ and $d$ are integers. | 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 |
A polynomial with integer coefficients has a root of the form $2+\sqrt{3}$. Given that the polynomial is monic and has degree 4, determine the coefficient of its cubic term. | nvidia |
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