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Let $f(x)$ be a function defined for all real numbers $x$ with the following properties:
(i) $f(0) = 0.$
(ii) If $x < y,$ then $f(x) < f(y).$
(iii) For any real number $t,$ $f(x + t) = f(x) + f(t).$
(iv) For any real number $r > 0,$ $f(rx) = rf(x).$
Let $S$ denote the set of real numbers $x$ such that $f(x)$ is ratio... | nvidia |
Determine the number of integers n for which the cube of the sum of the digits of n is equal to the sum of the cubes of the digits of n^2. | nvidia |
If five points are selected on a circle, how many different chords can be drawn between the points such that no three points are collinear? | nvidia |
Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1.$ Point $P$ lies inside the circle so that the region bounded by $\overline{PA_1},\overline{PA_2},$ and the minor arc $\widehat{A_1A_2}$ of the circle has area $\tfrac{1}{7},$ while the region bounded by $\overline{PA_3},\overline{PA_4},$ an... | nvidia |
Let $S$ be a sphere with radius $r$, and let $T$ be a tetrahedron inscribed within $S$ such that the lengths of all four of its faces are equal. If a plane intersects the sphere in a circle with radius $c$, what is the radius of the largest such circle that intersects each of the tetrahedron's edges at a single point? | nvidia |
Find the length of the segment of the number line whose endpoints satisfy $|x-\sqrt[7]{32}|=2$. | nvidia |
What is the sum of all positive integers less than 100 that are cubes of perfect cubes? | nvidia |
How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999?$ For example, both $121$ and $211$ have this property.
$\mathrm{\textbf{(A)} \ }226\qquad \mathrm{\textbf{(B)} \ } 243 \qquad \mathrm{\textbf{(C)} \ } 270 \qquad \mat... | nvidia |
Among all pairs of real numbers $(x, y)$ such that $\cos \cos x = \cos \cos y$ with $-15 \pi \le x, y \le 15 \pi$, Lisa randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$. | nvidia |
A rectangular garden measures $x$ meters by $y$ meters, where $x$ and $y$ are positive integers. The area of the garden is strictly between 48 and 50 square meters. What are the possible values of $x$ and $y$? | nvidia |
In a sequence of tennis matches, the probability that the server wins a point is 3/5, and the probability that the receiver wins a point is 2/5. If the server wins a point, the probability that they will win the next point is 2/3, and if the receiver wins a point, the probability that they will win the next point is 1/... | nvidia |
In the equation
$$y = x^2 + ax + b$$
the sum of the roots is $-2$ and the product of the roots is $-7$. Find the vertex of the parabola. | nvidia |
If $x$ and $y$ are two real numbers such that $x^2+y^2=4$ and $xy=-\frac{1}{2}$, find the sum of the possible values of $x+y$. | nvidia |
The sum of the digits of a positive integer is equal to the sum of the digits of its square. The integer has only two distinct digits and is less than 100. What is the integer? | nvidia |
Let $f(x)$ be a polynomial of degree 2006 with real coefficients, and let its roots be $r_1,$ $r_2,$ $\dots,$ $r_{2006}.$ There are exactly 1006 distinct values among
\[|r_1|, |r_2|, \dots, |r_{2006}|.\]What is the minimum number of real roots that $f(x)$ can have? | nvidia |
A number is divisible by $9$ if the sum of its digits is divisible by $9.$ For example, the number $19\,836$ is divisible by $9$ but $19\,825$ is not.
If $D\,767\,E89$ is divisible by $9,$ where $D$ and $E$ each represent a single digit, what is the sum of all possible values of the sum $D+E?$ | nvidia |
A circle with center $(2,9)$ passes through the point $(4,3)$. Find the sum of the coordinates of the point where the circle intersects the line $y=x$. | nvidia |
Find the sum of the positive divisors of 18 that are also prime numbers. | nvidia |
Determine the value of $a$ such that the quadratic equation $ax^2 - 6x + 2 = 0$ has a discriminant that is twice the value of its constant term. | nvidia |
A line with slope 3 intersects a line with slope 5 at the point $(10,15)$. The $x$-intercepts of these two lines are $a$ and $b$ units away from the $y$-axis, respectively. Find the value of $ab$. | nvidia |
What is the least positive integer divisible by each of the first six positive integers that are multiples of 3? | nvidia |
Find the number of positive integral values of $a$ for which the system of inequalities
$$
\begin{cases}
2x^2-5x+a>0\\
x^2+a-4<0
\end{cases}
$$
has exactly two distinct integer solutions. | nvidia |
Let $x$ and $y$ be positive numbers such that $x^{\sqrt y}=y^{\sqrt x}$. Find the maximum value of $x+y$. | nvidia |
Find the number of positive four-digit integers whose square root is an integer, that is, perfect squares. | nvidia |
Let $S$ denote the set of all two-digit numbers that are divisible by 5 and have digits adding up to 10. Compute the product of all the numbers in $S$. | nvidia |
The sum of the lengths of the edges of a cuboid (rectangular prism) is 84 cm.
What are the dimensions of the cuboid with the maximum volume that satisfies this condition? | nvidia |
Find the smallest positive integer $k$ such that there exist at least three distinct positive integers $n$ for which $n!$ ends with exactly $k$ zeros. | nvidia |
Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 2$ consecutive positive integers. | nvidia |
Find the smallest possible four-digit palindrome number $abba$ such that $a$ and $b$ are digits and $a$ is even, and $abba$ is divisible by 9. | nvidia |
Find the set of all real numbers $x$ such that both $x^2$ and $x^3$ are within a distance of 1 from the number 5. | nvidia |
An integer between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?
| nvidia |
Let $x$ be a real number such that $\lfloor x \rfloor = -3.$ Find the number of possible values of $\lfloor x^3 \rfloor$. | nvidia |
Find the value of $k$ such that the difference of the roots of the equation
$$3x^2-kx-1=0$$
is equal to twice the sum of the roots.
To solve this problem, we need to use the Vieta's formulas, which relate the roots of a polynomial to its coefficients. Specifically, we can use the fact that the sum of the roots of a qu... | nvidia |
Find the sum of the exponents of the variables after simplifying the expression $\sqrt[4]{(12x^3y^8z)^3}$. | nvidia |
Find the number of positive integers less than or equal to 100 that have a multiplicative inverse modulo 101. | nvidia |
An octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 108 centimeters. What is the number of square centimeters in the area of the octagon? | nvidia |
A cubic equation is given in the form $$2x^3+3x^2+kx+1=0,$$ where $k$ is a real number. Solve the equation for $x$ such that the product of its roots is equal to $-2$. | nvidia |
Let $x$ be a number such that
\[
\sqrt{x}+\frac{1}{\sqrt{x}}=3
\]
Determine $\ds \f{1}{x}+x$. | nvidia |
Let $f(x)$ be a function such that
\[f(x) f(y) - f(xy) = x + y\]for all real numbers $x$ and $y.$ Find $f(2).$ | nvidia |
A polynomial $P(x)$ with real coefficients is such that $P(x^2) = (x^2 - 1)(x^4 + ax^2 + b)$, where $a$ and $b$ are real numbers. It is given that the roots of the equation $P(x) = 0$ are either imaginary or complex, and the sum of the imaginary parts of all the roots of $P(x) = 0$ is equal to $4$.
Find the value of $... | nvidia |
Find the product of all numbers $x$ satisfying $\frac{x^2+1}{x} = 7.$ | nvidia |
Let z be a complex number such that \(\ds |z|=1\) and \(Im(z)+\ds\frac{z}{|z|}\cdot Re(z)=0.\) If \(z\) lies on the positive half-plane, determine the measure of the angle \(\varphi\) which \(z\) makes with the ray \([0,1)\). | nvidia |
Find the sum of the cubes of the solutions to
\[|x^3 - x + \frac{1}{n}| = \frac{1}{n},\]
where n is a positive integer. | nvidia |
Find all values of $b$ such that $b+\lceil b\rceil=17.8$. | nvidia |
Given that $x$ is real and $x^3+\frac{1}{x^3}=52$, find $x+\frac{1}{x}$. | nvidia |
In a large city, there are five major highways that intersect each other at different points. No two highways are parallel to each other. The city has 15 police stations, and each police station is located at an intersection of two or more highways. What is the minimum number of police stations that can cover all inter... | nvidia |
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 5. If $\frac{S_{2n}}{S_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term. | 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 |
Find the smallest possible value of the sum of the positive divisors of a positive integer greater than 25 that is also a multiple of 6. | nvidia |
Let $x$ and $y$ be positive real numbers such that $2x + 5y < 120.$ Find the maximum value of
\[xy (120 - 2x - 5y).\] | nvidia |
Consider the inequality $$3|2x-a|\le 2|x+b|$$ Find the set of all values of $a$ and $b$ such that the solution set of the inequality consists of three distinct intervals on the number line. | nvidia |
A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$? | nvidia |
Determine the maximum number of regions a plane can be divided into by drawing 12 lines, such that no three lines intersect at the same point. | nvidia |
Consider a triangle formed by connecting the midpoints of the sides of a larger triangle. Each vertex of the smaller triangle has a number associated with it, which is the sum of the numbers at the two vertices of the larger triangle that lie on the line segment connecting the midpoint to the opposite vertex. The numbe... | nvidia |
The sum of the positive integers $a$ and $b$ is 48. Given that the product of $a$ and $b$ is minimized, find the product. | nvidia |
How many positive whole numbers have cube roots that are less than $10$? For example, $20$ would count since $\sqrt[3]{20}<10.$
Write another problem inspired by this one:
Determine the largest integer $n$ such that the sum of the squares of its digits is less than or equal to $n$. For example, for $n=12$, we have $1... | nvidia |
Two random points are chosen uniformly in the unit square. What is the probability that the sum of the coordinates of one point is greater than the product of the coordinates of the other point? | nvidia |
Let $x$, $y$, and $z$ be positive integers such that $x^3+y^3+z^3 = 6xyz$. Determine the largest possible value of $x+y+z$. | nvidia |
Let $S$ be the sum of the squares of the first 100 prime numbers. Find the largest integer $m$ such that $S$ is divisible by $2^m$. | nvidia |
Determine the smallest value of $n$ for which the expression $2^n+5^n$ is divisible by 7. | nvidia |
What is the value of the expression $\frac{1}{3}+\frac{1}{2}+\frac{1}{6}$, given that the sum of the two smaller fractions is equal to the larger fraction? | nvidia |
The sum of the $y$-coordinates of the vertices of a quadrilateral in the Cartesian plane equals 24. Find the sum of the $y$-coordinates of the midpoints of the diagonals of the quadrilateral. | nvidia |
If $z_1$ and $z_2$ are complex numbers such that $|z_1|=3$ and $|z_2|=5$, and if the real part of $z_1 \overline{z_2}$ is 8, find the imaginary part of $\overline{z_1} z_2$. | nvidia |
Find the smallest positive integer $n$ such that $3^{12} \cdot 9^n$ is divisible by $27^{10}$. | nvidia |
The product of two positive integers is 24. The positive difference of these two integers is 2. What is the sum of the two integers? | nvidia |
Let $\mathcal{S}$ be the set of all possible values of the greatest common divisor of $3n+4$ and $n$, where $n$ ranges over the positive integers. Find the product of all the elements of $\mathcal{S}$ that are not divisible by 4. | nvidia |
Given a polynomial with integer coefficients and roots $\frac{2}{3}$, $\frac{3}{4}$, $-\frac{5}{6}$, and $k$, where $k$ is an integer, find the smallest positive value of the constant term $e$ such that the polynomial is monic, i.e., its leading coefficient is 1. | nvidia |
Given that $a+b+c=12$, $ab+ac+bc=31$, and $abc=60$, find the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$. | nvidia |
The vertices of a triangle are located at $(0, 0)$, $(b, 0)$, and $(0, c)$, where $b > 0$ and $c > 0$. Given that the area of the triangle is $450$ square units, find the ordered pair $(b, c)$. | nvidia |
Find the number of positive integers less than or equal to 1000 that end in 3 and are divisible by 7. | nvidia |
Evaluate the expression \[ \frac{a+2}{a+1} \cdot \frac{b-1}{b-2} \cdot \frac{c + 8}{c+6} , \] given that $c = b-10$, $b = a+2$, $a = 4$, and none of the denominators are zero. | nvidia |
Find the value of $a$ such that the polynomial $x^3-ax^2+7x-12$ is divisible by $(x-2)^2$. | nvidia |
The sum of the squares of three consecutive positive integers is 135. The product of the first and last integers is 21. What is the sum of the three integers? | nvidia |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}^n = \mathbf{I}.\] | nvidia |
Find the number of positive divisors of 540 that are also multiples of 3. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 6.$ Find the maximum value of $(x^2+1)(y^2+1)(z^2+1).$ | nvidia |
Given that the polynomial $x^2 + px + 25$ has only positive integer roots, find the average of all distinct possibilities for $p$. | nvidia |
Let $x$ be a positive real number such that
\[3(x-2)^2 + 4\left( \frac{x}{x+1} \right)^2 = 18.\]Find all possible values of $y = \frac{(x+1)^2 (x - 2)}{2x + 1}.$ Enter all possible values, separated by commas. | nvidia |
In a certain triangle, the sum of the length of the hypotenuse and one leg is 36 units, and the product of the lengths of the legs is 420 square units. Find the area of the triangle in square units. | nvidia |
Find the number of perfect cubes less than 1000 that can be expressed as the sum of three consecutive integers. | nvidia |
A five-digit number is divisible by 7 if the sum of its digits is divisible by 7. What is the largest digit A such that the number 14523A is divisible by 7? | nvidia |
What is the smallest positive integer $n$ such that $6n$ has exactly four positive integer divisors? | nvidia |
Let $x$ and $y$ be two distinct positive integers such that $\frac{1}{x}+\frac{1}{y}=\frac{1}{7}$. Determine the pair $(x, y)$ that maximizes the quantity $xy(x+y)$. | nvidia |
Find the largest integer $n$ such that $\sqrt{n}$ is less than or equal to the sum of the reciprocals of the first three positive integers. | nvidia |
Find the product of the roots of the equation $x^4+2x^3-7x^2-12x+18=0$, given that two of the roots are complex conjugates. | nvidia |
The polynomial $Q(x)=x^4+ax^3+bx^2+cx+d$ is defined by its roots $\pm 2i$ and the root $x=3$. Given that the sum of the real roots of the polynomial is $4$, determine the sum of the product of the real roots of $Q$ and the product of the non-real roots of $Q$. | nvidia |
Find the smallest number that can be expressed as the sum of four different cubes of prime numbers. | nvidia |
In a given list of numbers, the arithmetic mean of the perfect squares is equal to 15, and the arithmetic mean of the perfect cubes is equal to 80. Determine the arithmetic mean of the numbers in the list that are both perfect squares and perfect cubes. | nvidia |
Find the number of positive integers $n \le 1000$ such that $n$ can be expressed as the difference of two perfect squares. | nvidia |
Let $x$ and $y$ be any two prime numbers, with $x$ greater than $y$. The smallest integer which is a multiple of all possible numbers of the form $x^2+y^2$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 10$ | nvidia |
Find the smallest positive real number $k$ such that
\[\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \ge k \sqrt{\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{d} \cdot \frac{d}{a}}\]
for all positive real numbers $a,$ $b,$ $c,$ $d.$ | nvidia |
Find the sum of the squares of the divisors of 36. | nvidia |
Let the operation $\R$ be defined as follows: $a\,\R\, b=a^b\cdot b^a$. Prove that there is only one solution to the equation $a\,\R\, x=x\,\R\, a$ for any positive integer $a$, and find the solution. | nvidia |
Find the sum of all integers m such that $\binom{28}{7} + \binom{28}{8} = \binom{29}{m}$. | nvidia |
Solve the equation $|x|+|x+1|+|x+2|=6$ and find the sum of the solutions. | nvidia |
The points $A$ and $B$ are located in a coordinate system at the points $(6,0)$ and $(0,8)$ respectively. Find the area of the rectangle with the sides parallel to the axes of the coordinate system and with the vertices at the points $A$ and $B$. | nvidia |
Let $x$ and $y$ be real numbers such that $x^6-y^4=y^2-x$, $x^3+y^3 = 20$, and $x > y$. Determine the value of $x^3-y^2$. | nvidia |
Find the positive difference between the sum of the first 30 positive multiples of 3 and the sum of the first 20 positive multiples of 5. | nvidia |
How many odd four-digit numbers are there such that the sum of the hundreds and thousands digits is 9? | nvidia |
Find the units digit in the sum of all natural numbers from 1 to 99, inclusive. | nvidia |
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