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The sum of the areas of two squares with sides of consecutive integers is 253. Find the side length of the larger square. | nvidia |
Let $S$ be the set of all positive integers whose digits are all $1$ or $2$. Let $T$ be the set of all positive integers whose digits are all $3$ or $4$. Find the smallest integer $n$ such that there exists a one-to-one function $f: S \to T$ with the property that for any $x, y \in S$, the number of digits in the pro... | nvidia |
The function $g(x)$ satisfies
\[g(xy) = g(x)y^2\]for all positive real numbers $x$ and $y.$ If $g(2) = 8,$ find $g(3).$ | nvidia |
How many different convex quadrilaterals with integer side lengths are there such that the sum of the squares of their sides equals 800? | nvidia |
A number in the form of $a^b$ where $a$ and $b$ are positive integers and $b > 1$ is called a perfect power. Find the perfect power that is closest to the cube root of 1999, and determine the sum of its base and exponent. | nvidia |
What is the smallest whole number that must be a factor of the sum of any two consecutive positive integers? | nvidia |
A cuboid with a volume of 240 cubic units has a height equal to half of the sum of its base edges. A larger cuboid has the same proportions between its base edges and height but has a volume that is 1.5 times the volume of the smaller one. Find the ratio of the sum of the base edges of the larger cuboid to the sum of t... | nvidia |
Given $a$ and $b$ are real numbers such that $|a+b|=2$ and $|ab|=3$, find the smallest possible value of $|a^2+b^2|$. | nvidia |
Let $g(x)$ be a function such that $g(0) = 2$ and
\[g(xy) = g \left( \frac{x^2 + y^2}{3} \right) + (x + y)^2\]for all real numbers $x$ and $y.$ Find $g(x).$ | nvidia |
In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. A circle is drawn inside the triangle so that it is tangent to all three sides. The minimum possible radius of such a circle can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x+y))=g(x)g(y)+x+y\]for all $x,y$. Find the sum of all possible values of $g(2)$ | nvidia |
Find the sum of the infinite series $$\frac{2}{5^{1}}+\frac{2}{5^{2}}+\frac{2}{5^{3}}+\frac{2}{5^{4}}+\frac{2}{5^{5}}+...$$ | nvidia |
Let a and b be non-zero complex numbers such that
|a^2/b|+|b^2/a|=1.
Find the sum of all possible values of
|b^2/a|+|a^2/b|. | nvidia |
A circle is inscribed in a square and a regular hexagon is inscribed in the circle. Determine the ratio of the area of the circle to the area of the hexagon. | nvidia |
Let $a$ and $b$ be positive integers. Given that $a^{2b} = 2^{15} \cdot 4^{13} \cdot 8^{11}$, find the least possible value of $ab$. | nvidia |
A triangle is randomly chosen from all possible triangles with vertices on the points $(0,0)$, $(0,3)$, $(4,0)$, and $(4,3)$. What is the probability that the chosen triangle has a right angle? | nvidia |
The function $g(x)$ satisfies
\[g(x + y) = g(x) + g(y)\]for all real numbers $x$ and $y.$ Find all possible values of $g(0).$ | nvidia |
Determine all real $m$ and $n$ such that for all real $x$, $x^2+mx+n=0$ has the same roots as $x^2+mx+n(x^2+mx+n)^2=0$. | nvidia |
There exists a positive real number $y$ such that $ \sin (\arccot (y)) = y $. Find the value of $y^2$. | nvidia |
The sum of the cubes of two consecutive integers is 693. What is the value of the product of the two integers? | nvidia |
Two pairs $(x,y)$ of real numbers satisfy the equation $x+y=3xy=5$. Given that the solutions $x$ are in the form $x=\frac{a\pm b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, what is the value of $a+b+c+d$? | nvidia |
Find the monic cubic polynomial, in $x$, with real coefficients, which has $2 + i$ as a root and satisfies the condition that the sum of the coefficients of the polynomial is 10. | nvidia |
Find the sum of the squares of all positive integers from 1 to 25. | nvidia |
Find the smallest digit $d$ such that the number $23d456$ is divisible by 11. | nvidia |
Find the sum of all real solutions to the equation $x^2+9=6x+\frac{9}{x}$. | nvidia |
In the first row of Pascal's Triangle, all the numbers are equal to 1. In the second row, the numbers are 1, 2, and 1, where the middle number is the sum of the two numbers just above it. In general, each number in Pascal's Triangle is the sum of the two numbers just above it and to the left and the right.
In the rows... | nvidia |
Simplify the expression $\sqrt{21+12\sqrt{3}}$ into the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$. | nvidia |
In the graph of $\frac{x^2+3x+2}{x^3+x^2-2x}$, let $a$ be the number of holes in the graph, $b$ be the number of vertical asympotes, $c$ be the number of horizontal asymptotes, and $d$ be the number of oblique asymptotes. Find $a+2b+3c+4d$. | nvidia |
In $\triangle ABC$, let $D$ be the point on side $AC$ such that $AD:DC = 2:3$ and let $E$ be the point on side $AB$ such that $BE:EA = 3:2$. If $BD$ and $CE$ intersect at $F$, and $AF:FB = 5:4$, then what is $BF:FC$? | nvidia |
Let $f(x)$ be a function such that $f(0) = 1$ and
\[f(xy) = f \left( \frac{x^2 + y^2}{2} \right) + (x - y)^2\]for all real numbers $x$ and $y.$ Find $f(x).$ | nvidia |
The radius of the large circle is twice the radius of the small circle. The area of the ring between these circles is 540π square centimeters. Find the radius of the small circle. | nvidia |
Let $A_n$ denote the set of all positive integers $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point, and let $g(n)$ be the product of all elements of $A_n$. Find the smallest positive integer $m$ such that $g(m) > 10^{10}$. | nvidia |
Find all integers $a$ such that the equation $3x \equiv 2 \pmod{a}$ has a solution, and the set of solutions is equal to the set of solutions of the equation $2x \equiv 3 \pmod{a}$. | nvidia |
Let $p(x)$ be a monic polynomial of degree $n$ with real coefficients, and let $\omega$ be a complex number such that $\omega^n = 1$ and $\omega \neq 1.$ If
\[\frac{p(\omega)}{\omega^k} = \frac{p(1)}{2}\]
for some integer $k$, find $\frac{p(-1)}{(-1)^k}$. | nvidia |
A convex polyhedron has six faces, each being a regular hexagon. Determine the number of edges of the polyhedron if its vertices can be colored with three colors such that any face has all three colors. | nvidia |
Let $m$ and $n$ be positive integers such that $2^m \cdot 5^n = 10^6$. Determine all possible values of $m+n$. | nvidia |
Find the largest value of $x$ that satisfies the equation $|x-5|=12$. | nvidia |
In a regular octahedron, the midpoints of the twelve edges are the vertices of a smaller octahedron. The ratio of the surface area of the smaller octahedron to that of the larger is $a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. | nvidia |
A space curve is parameterized by the equations
$$\begin{cases}
x=a \cos t\\
y=b \sin t\\
z=c \sqrt{1-\sin^2t}
\end{cases}$$
where $a,$ $b,$ and $c$ are positive constants. The points $(3,0,4)$ and $(0,1,-3)$ lie on the curve. Find the point on the curve that is farthest from the origin. | nvidia |
Find the sum of the odd, positive integers less than 89. | nvidia |
Find the number of real values of $x$ such that the decimal representation of $\frac{x}{120}$ terminates and $x^2 + x - 6 = 0$. | nvidia |
Let \(a\) and \(b\) be two positive integers such that \(a<b\). Consider a rectangle with dimensions \(a\) and \(b\). It is known that the perimeter of the rectangle is a divisor of its area, and that the area of the rectangle is a divisor of its perimeter. How many different values can the perimeter of this rectangle ... | nvidia |
A right pyramid with an octagonal base is placed on a coordinate plane so that its vertices are located at $(0,0)$ and $(4\sqrt{2},4\sqrt{2})$. A point $P$ is located on the edge of the pyramid so that it is 8 units from each of the two mentioned vertices. What is the length of the line segment from $P$ to the center o... | nvidia |
Find the harmonic mean of the squares of the first three positive integers. | nvidia |
Find the largest integer value of $m$ that satisfies the following inequality:
$$\frac{m}{2\sqrt{m+1}} < 1$$ | nvidia |
Find the minimum value of
\[x^2 - 2xy + 3y^2\]over all real numbers $x$ and $y.$ | nvidia |
In triangle $ABC$, $\angle A$ is a right angle and $AB = 25$. Find the largest possible value of $\tan C$ such that the area of the triangle is 150 square units.
Note: This problem requires a different approach than the original one, as it involves finding the maximum value of a trigonometric function while satisfying... | nvidia |
Find the largest three-digit number $N$ such that the value of its hundreds digit plus the absolute difference between its units and tens digits equals $P(N)$. | nvidia |
Let $a$ and $b$ be two real numbers such that $a^3 - b^3 = 133$.
Determine the smallest possible value of $a^2 + b^2$. | nvidia |
Find a positive integer $x$ such that $x^2$ is a multiple of 12, and $x$ is between 12 and 20. | nvidia |
How many different convex quadrilaterals can be formed by connecting four of the vertices of a cube, such that all four vertices are not coplanar? (Two quadrilaterals are different if they do not share all four vertices.) | nvidia |
Find the sum of the coefficients of the polynomial $P(x)$, given that its roots are the squares of the roots of the equation $x^3 - 2x^2 - 5x + 1 = 0$. | nvidia |
Find the number of positive real numbers $x$ that satisfy the inequality
$$\frac{\sin(x-1)\sin(x-3)\sin(x-5)}{\sin(x+1)\sin(x+3)\sin(x+5)} > 0.$$ | nvidia |
Find $x$ such that $\log_x 81=\log_2 16$. | nvidia |
Let $P(x)$ be a polynomial such that $P(P(x))$ is equivalent to $P(x^2)$ for infinitely many $x$. Find the polynomial $P(x)$ that has the least degree. | nvidia |
The base of a pyramid is a square. The volume of the pyramid is equal to twice its surface area (the area of the four lateral faces and the base). Find the length of the pyramid's side, assuming that its slant height equals the diameter of its base. | nvidia |
How many non-negative integers can be formed using all the digits of the number 123456789 exactly once, such that the resulting number is divisible by 27? | 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 |
The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $a+b+c$.
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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;
re... | nvidia |
What is the smallest positive integer that is guaranteed to be a factor of the sum of any three consecutive positive integers? | nvidia |
In the quadratic equation $ax^2 + bx + c = 0$, the coefficients $a$, $b$, and $c$ are all integers. If the sum of the roots is $1$ and the product of the roots is $6$, what is the sum of all possible values of $a$? | nvidia |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three points in 3-dimensional space. Determine the largest possible value of the sum of the squares of the areas of the three triangles formed by connecting any two of the three points with the origin. | nvidia |
$\triangle ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:
$\textbf{(A)}\ s^2\le8r^2\qquad \textbf{(B)}\ s^2=8r^2 \qquad \textbf{(C)}\ s^2 \ge 8r^2 \qquad\\ ... | nvidia |
The function $f(x)$ satisfies
\[f(xy) = \frac{f(x)}{y}\]for all positive real numbers $x$ and $y.$ If $f(30) = 20,$ find $f(40).$ | nvidia |
Find the sum of all possible positive integer values of $c$ such that the quadratic equation $3x^2 + 4x + c = 0$ has rational roots. | nvidia |
Solve the equation $\sqrt{2x+1}=\sqrt{x+2}$, where $x$ is a real number. | nvidia |
Given that m is an integer and 0 < 3m < 24, what is the sum of all possible integer values of m? | nvidia |
The sum of the squares of two consecutive integers is 85. Determine the absolute value of their difference. | nvidia |
In a rectangular coordinate system, the vertices of a triangle are located at (0, 0), (a, 0), and (0, b), where a and b are positive real numbers. The area of the triangle is equal to 16 square units. Find the value of a + b. | nvidia |
Let $a$ and $b$ be two positive integers such that their sum and product are given. Determine the value of $a^2 + b^2$ if the product of $a$ and $b$ is 16900 and their sum is 260. | nvidia |
In the diagram below, $P(x,y)$ is a point on the graph of the function $f(x)$ such that $f(x) = x^2-2x+1$ and $f^{-1}(x)$ exists. $Q$ is the point obtained by reflecting $P$ about the line $y=x$. $R$ is the point obtained by reflecting $P$ about the $x$-axis. What is the ratio of the distance between $P$ and $R$ to the... | nvidia |
Compute the largest integer $n$ such that $2401^n$ divides $2401!$. | nvidia |
How many different positive integers can be represented as the sum of two distinct members of the set $\{1, 2, 3, 4, 5, 6\}$? | nvidia |
A square pyramid has a base with a side length of $s$ units and four triangular faces that meet at the apex. The sum of the $x$-coordinates of the vertices of the base is $s+1$. Find the $x$-coordinate of the centroid of the pyramid. | nvidia |
In a regular dodecagon, a diagonal is drawn, dividing the dodecagon into two regions. What is the smallest number of diagonals that can be drawn in the dodecagon such that each of the resulting regions has an equal number of vertices? | nvidia |
Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of
\[\frac{1}{a} + \frac{1}{b}.\] | nvidia |
Let $q(x)$ be a polynomial with integer coefficients such that $q(50) = 50.$ Find the maximum number of integer roots $r$ of the equation $q(r) = r^2.$ | nvidia |
What is the probability that the sum of the numbers on two six-sided dice is not less than 11? | nvidia |
Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$?
$\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text... | nvidia |
The point with coordinates $(3,4)$ is the midpoint of the segment with one endpoint at $(-2,1)$. Find the difference of the coordinates of the other endpoint. | nvidia |
In the base-3 system, let $S$ be the set of all positive integers whose cube, when written in base 3, contains exactly three digits.
Determine the smallest integer $n$ such that all integers less than $n$ are in $S$. | nvidia |
Problem:
Let $S$ be the set of all complex numbers of the form $r+si$, where $r$ and $s$ are integers, that can be expressed as a finite sum of terms of the form $a_n(2+i)^n$, where $a_n$ is a non-negative integer less than or equal to $2^n$. Find the maximum value of $|z|$ for all $z \in S$ that can be expressed as a... | nvidia |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 312^\circ.$ | nvidia |
Given three non-overlapping cubes of unit length, with centers at $(3, 4, 1)$, $(1, 2, 3)$, and $(5, 6, 7)$, find the equation of the plane passing through $(1, 2, 3)$ such that the volume of the parts of the cubes on one side of the plane is equal to the volume of the parts of the cubes on the other side. | nvidia |
Determine the value of $x$ such that $x\ast x = x\ast(x+1)$. | nvidia |
Three positive numbers are in an arithmetic progression. If the sum of the squares of the two smaller numbers is 65, and the product of the two larger numbers is 105, find the largest number. | nvidia |
Determine the monic polynomial in $x$ with real coefficients that has the following roots: $2$, $1-i$, and $3+\sqrt{2}$. | nvidia |
Determine the number of divisors of 180 whose sum of digits is a perfect square. | nvidia |
Let's assume that the angle measures are $4x$ and $5x$. Since the two angles are complementary, $4x+5x=90$.
Therefore, $9x=90$, and the measures are $\ds\frac{4}{9}\cdot 90=40^\circ$ and $\ds\frac{5}{9}\cdot 90=50^\circ$. Therefore, the ratio of the measures is $\ds\frac{40}{50}=\frac{4}{5}$.
The smaller angle's measur... | nvidia |
Find the sum of all numbers $x$ satisfying $x^2+10/x = 20$. | nvidia |
A circle is inscribed in a right triangle with legs 8 and 15. The sum of the areas of the two smaller triangles formed by the circle and the legs of the triangle is equal to half the area of the triangle. Find the radius of the inscribed circle. | nvidia |
How many four-digit whole numbers have at least two digits that are perfect squares? | nvidia |
Find the number of integer values of x that satisfy the condition:
$$2x < 5$$
$$3^x < 10$$ | nvidia |
Find the smallest positive integer $n$ such that the sum of its prime factors is equal to the number of its divisors. | nvidia |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | nvidia |
Let $f(x)$ be a function defined for all real numbers $x$, such that the set of all points $(x,f(x))$ forms a line parallel to the $y$-axis. Suppose this line intersects the graph of $y=x^2$ at precisely two points. Let the function $g(x)=2f(x)+1$. How many intersection points can the graph of $y=g(x)$ have with the gr... | nvidia |
Find the value of $a$ for which the equation $ax^2+3x-2=0$ has a solution $x=1$ and the sum of the roots is equal to $-\frac{7}{2}$. | nvidia |
Given that $n$ is an integer and $0 < 4n <30$, what is the sum of all possible integer values of $n$? | nvidia |
Given that $a$ and $b$ are positive integers such that $\gcd(a,b) = 18$, find the smallest possible value of $\gcd(a^2,b^3)$. | nvidia |
A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 3-primable positive integers are there that are less than 1000? | nvidia |
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