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Determine the value of $x$ such that $\sinh(\log_{5}(x+1)) = \frac{3}{2}$. | nvidia |
Find the maximum value of the expression $|a^2-b^2+2abi|$, where $a$ and $b$ are positive real numbers, $i$ is the imaginary unit, and $a^2+b^2=1$. | nvidia |
Find the number of positive integers less than or equal to 1000 whose digits are all distinct and either 1 or 2. | nvidia |
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that
\[\begin{aligned} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7 &= 1 \\
4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7 &= 12 \\
9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7 &= 123. \end{aligned}\]Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$. | nvidia |
Find the greatest integer value of $b$ for which the expression $\frac{9x^3+4x^2+11x+7}{x^2+bx+8}$ has a domain of all real numbers. | nvidia |
Given that $\left\lfloor \sqrt{x}\right\rfloor=5$, find the value of $x$ such that $\left\lceil \frac{x}{2}\right\rceil$ is also an integer. | nvidia |
Find all real solutions to $x^4+(2-x)^4=34$. Enter all the solutions, separated by commas. | nvidia |
Determine all real numbers $x$ such that $(2-3i)x$ is equal to its complex conjugate. | nvidia |
Compute the number of distinct subsets of size 5 that can be formed from a set of 11 elements, where exactly 2 of the elements are identical. | nvidia |
In a triangle, the sum of the lengths of the two shorter sides is 15 cm, and the difference between the lengths of the longer side and the shorter side is 3 cm. Find the length of the shorter side. | nvidia |
Determine $k$ such that the system of equations $y=x^3$ and $y=x^2+k$ has exactly three solutions. | nvidia |
A particle starts at the point $(6,8)$ and moves randomly to one of the neighboring points, either $(x-1,y)$, $(x,y-1)$, or $(x-1,y-1)$, each with a probability of $\frac{1}{3}$. The movement continues until the particle hits one of the coordinate axes for the first time. Determine the probability that the particle hit... | nvidia |
Let $P(x)=x^3-2022x+m$ be a polynomial with integer coefficients, and let $a$, $b$, and $c$ be the three roots of $P(x)$. If the sum of all the possible products of any two of these roots is equal to 250, find the product $abc$. | nvidia |
In a right triangle, a circle is inscribed such that it touches the three sides of the triangle. If the area of the triangle is $K$, determine the radius of the inscribed circle in terms of $K$ and the lengths of the sides of the triangle. | nvidia |
Find the number of positive integers $n$ such that $S(n)+S(S(n))+...+S^{2019}(n)$ (where $S^{2019}(n)$ denotes the composition of $S(n)$ with itself 2019 times) is equal to 2022. | nvidia |
The quadratic equation $x^2 + (a+b)x + ab = 0$ has two solutions for $x$ when $a \neq b$. However, when $a=b$, the equation has only one solution. What is the sum of the values of $a$ for which the equation $x^2 + (a+b)x + ab = 0$ has only one solution, given that the sum of the solutions of the equation is equal to 10... | nvidia |
Find the smallest positive integer $a$ such that the equation $$ax \equiv 1 \pmod{55}$$ has a solution, given that $5x\equiv 1 \pmod{11}$ and $11x\equiv 1 \pmod{5}$. | nvidia |
Let $S$ be the set of all $3$-digit positive integers that have no zeros and whose digits are all distinct. If the $30^\text{th}$ smallest integer in $S$ is $a,$ find the remainder when $a$ is divided by $4.$ | nvidia |
Let set $\mathcal{B}$ be a 60-element subset of $\{1,2,3,\ldots,80\},$ and let $T$ be the sum of the elements of $\mathcal{B}.$ Find the number of possible values of $T$. | nvidia |
What is the smallest base $b$ such that there exist more than $b^2$ different $b$-digit numbers where all digits are distinct and the sum of digits is equal to $b+2$? | nvidia |
Find the set of points $(x,y)$ that satisfy the equation
$$|x| + |y| + |x+y| = 4.$$
Determine the geometric shapes that can be used to describe this set. | nvidia |
The vertices of a regular polygon with n sides are colored with red and blue colors such that any two adjacent vertices have different colors. The number of blue vertices is equal to the number of red vertices. What is the number of possible colorings of the polygon? | nvidia |
The equation $2x^2+ax+b=0$ has real roots. Given that $\log_2{a}$ is between the consecutive integers $m$ and $n$, and $\log_3{b}$ is between the consecutive integers $p$ and $q$, determine $m+n+p+q$. | 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 |
What is the smallest positive integer $n$ such that $2^n$ is a factor of $1512$? | nvidia |
Consider the set of all positive integers that have exactly three digits in base $6$, and exactly $4$ digits in base $5$. What is the sum of the smallest and largest numbers in this set? | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x) g(y) - g(xy) = x^2 + y^2\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(3),$ and let $t$ be the sum of all possible values of $g(3).$ Find $m \times t.$ | nvidia |
Find the smallest positive integer $n$ such that $2^n$ is the sum of two distinct prime numbers. | nvidia |
Find the greatest value of $y$ such that the graph of the line $y=3x-5$ lies below the graph of the parabola $y=x^2$ for all $x$ in the interval $[0,y]$. | nvidia |
Let $\mathbb{P}$ be the set of positive integers, and let $g: \mathbb{P} \rightarrow \mathbb{R}$ be a function satisfying the following properties:
(1) $g(p) = 0$ for any prime number $p$.
(2) For any integer $n \ge 2$, there exist integers $a$ and $b$ such that $g(n) = \frac{g(a) + g(b)}{2}$ and $ab = n$.
Find the ... | nvidia |
Count the number of four-digit integers that have exactly one digit appearing more than once, with the condition that no digit is zero and the repeating digit appears consecutively. | nvidia |
What is the largest five-digit palindrome number that is divisible by 11 and has exactly three digits that are multiples of 3? | nvidia |
Let $x$ and $y$ be positive real numbers such that $xy = 4.$ Find the set of all possible values of $x^2 + y^2.$ | 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 |
Determine the largest composite number that can be expressed as the sum of two prime numbers, where the larger prime is less than 20. | nvidia |
Find the value of $a$ so that the lines with the given equations are perpendicular. \begin{align*}
y &= 2x+13 \\
6y+ax &= 6.
\end{align*} | nvidia |
Let $n$ be a positive integer such that the prime factorization of $n$ is $p_{1}^{2} \times p_{2}^{3} \times p_{3}$, where $p_{1}, p_{2}, p_{3}$ are distinct prime numbers. How many ordered triples of positive integers $(x, y, z)$ satisfy the equation $xyz = n$? | nvidia |
Find the largest value of $y$ that satisfies the inequality $|2y-1|\leq5$, given that $y$ is a non-negative integer. | nvidia |
Find the smallest positive integer $n$ such that $n$, $n^2$, $n^3$, and $n^4$ are all square-free. | nvidia |
Consider a rational number $\frac{m}{n}$, where $m$ and $n$ are positive integers. When expressed as a decimal, the decimal representation either terminates or repeats. Determine the smallest possible value of $n$ such that the decimal representation of $\frac{m}{n}$ repeats for any positive integer $m$ that is relativ... | nvidia |
Find the product of four real numbers $x$ that are not in the domain of $$f(x) = \frac{1}{1+\frac{1}{1+\frac{1}{1+\frac 1x}}}.$$ | nvidia |
Count the number of positive integers less than 100 that have only prime digits. | nvidia |
Let $$A=(a_{ij})_{1\le i,j\le n}$$ be an integer matrix where the sum of all the elements in the matrix is 100. Find the maximum value of $$\sum_{i,j}a^2_{ij}.$$ | nvidia |
The roots of two quadratic equations $x^2+px+q=0$ and $x^2+rx+s=0$ are all distinct and form a geometric sequence. Determine the value of $\frac{p^2}{q}-\frac{r^2}{s}$. | nvidia |
Find the smallest positive integer $m$ such that $2x^2 + mx + 24$ can be expressed as the sum of three squares of linear factors with integer coefficients. | nvidia |
Find the value of $y$ such that the equation
$\log_2(y^2+3y+1) = \log_2(y^2-5y+7)$ holds true. | nvidia |
The numbers $a$, $b$, and $c$ are positive integers such that $a \cdot b \cdot c = 360$. The number $360$ is written in base $d$, where $d$ is a prime number such that $d \ge 7$. What is the remainder when the product of the digits of the base-$d$ representation of $360$ is divided by $d$? | nvidia |
In a regular polygon, the sum of all exterior angles is equal to the sum of all interior angles. Find the number of sides of this polygon. | nvidia |
A convex hexagon $P_1P_2P_3P_4P_5P_6$ is said to be equi-areal if there exists a point $Q$ in its interior such that the area of each triangle $QP_iP_{i+1}$, $i=1,2,3,4,5$, is the same, and the area of triangle $QP_6P_1$ is $2$ times the area of each triangle $QP_iP_{i+1}$, $i=1,2,3,4,5$. Determine the maximum number o... | nvidia |
A square with side length $x$ cm is inscribed in a circle with radius $r$ cm. The area of the circle is $πr^2$ square cm. To find the area of the square, Sarah rounded the value of $π$ to 3.14 and the value of $r$ to 5 cm. She then used the rounded values to calculate the area of the circle and subsequently the area of... | nvidia |
A bookstore has three shelves, each with room for exactly one book. There are four different books that need to be placed on the shelves, but only three of them can be placed at a time. In how many different ways can the books be arranged on the shelves if the order of the books matters? | nvidia |
Find the number of distinct points $P$ in the plane such that the triangle formed by the points $(0,0)$, $(4,0)$, and $P$ has area 4 and the triangle formed by the points $(0,0)$, $(4,3)$, and $P$ has area 3. | nvidia |
Let $g(x)$ be a function defined for all real numbers, and consider the equation
$$g(x) - 3g(-x) = x^2 + 1.$$
Find all real values of $x$ for which $g(x) = g(2x)$. | nvidia |
Given that the polynomial $p(x)=(x-1)(x^2-1)(x^4-1)\cdots (x^{2^n}-1)$ has the sum of its coefficients equal to zero, determine the largest possible value of $n$. | nvidia |
In a quadrilateral, let P be the centroid of the triangle formed by the vertices A, B, and C, and let Q be the centroid of the triangle formed by the vertices A, C, and D. If the area of the triangle APQ is 3/5 of the area of the triangle ABC, find the ratio of the area of the triangle ACD to the area of the triangle A... | nvidia |
Problem:
Let $f(x)$ be a polynomial with integer coefficients such that the number of divisors of $f(x)$ is equal to the number of divisors of $x$ for all positive integers $x$. What is the degree of $f(x)$? | nvidia |
Find the sum of the digits of the smallest positive integer that satisfies the following conditions:
I. The product of its digits is $24$, and
II. Each digit is smaller than the one on its right. | nvidia |
Consider a set of real numbers $\{x_1, x_2, \dots, x_n\}$. Define the function $f(x_i)$ as the smallest integer that is greater than or equal to $x_i$, for each $i$. Given that $f(x_1) + f(x_2) + \dots + f(x_n) = k$, where $k$ is an integer, find the largest possible value of $n$ if $|x_i| < 1$ for all $i$. | nvidia |
What is the largest integer value of n such that the sum of the cubes of the first n positive integers is less than 10000? | nvidia |
What is the smallest positive integer $n$ such that the sum of the divisors of $n$ is a perfect square? | nvidia |
Consider the function $f(x)=\left\lfloor{x}\right\rfloor+\left\lceil{x}\right\rceil.$ Find the sum of the values of $x$ in the interval $0<x<5$ at which $f(x)$ takes an even value. | nvidia |
In the expression $a^2\cdot a^5$, $a$ is a positive number such that $a^2=5$. Evaluate $a^5$. | nvidia |
Let $p$ be a prime number such that when $2p^2$ is divided by a positive integer $m$, the remainder is 3. Determine the largest possible value of $m$. | nvidia |
A square and a regular hexagon are inscribed in the same circle. The vertices of the square are the midpoints of the sides of the hexagon. Find the ratio of the area of the square to the area of the hexagon. | nvidia |
The tens digit of a two-digit number can be any of the digits $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, or $9$. Consider a set of $15$ two-digit numbers such that each tens digit is represented at least once. What is the smallest possible number of two-digit numbers from this set such that the sum of the tens digits is $... | nvidia |
Find all integer values $k$ such that $\binom{20}{10}+\binom{20}{k}=\binom{21}{11}$. | nvidia |
What is the largest integer that can be expressed as the sum of three distinct perfect squares, with each square being no greater than 100? | nvidia |
Find a quartic polynomial with integer coefficients such that the complex number $1 + i \sqrt{7}$ is a root of the polynomial, and the constant term of the polynomial is equal to 100. | nvidia |
Let $a_n$ be a sequence of positive integers such that $\log_{10} a_{n+1} - \log_{10} a_n$ is an arithmetic sequence. Given that $\log_{10} a_1 = 1$ and $\log_{10} a_2 = 2$, find the least positive integer $n$ such that $\log_{10} a_n$ is an integer. | nvidia |
A polynomial has integer coefficients, and its roots are all integers except for one root which is of the form $p/q$, where $p$ and $q$ are relatively prime positive integers. What can you say about the factorization of the constant coefficient of the polynomial? | nvidia |
A square grid is drawn on the Cartesian plane, with each side length being a positive integer. A triangle is formed by choosing three vertices from the grid points. What is the maximum possible area of such a triangle if the square grid has side length 10? | nvidia |
A right triangle has a leg of length $x$ and a hypotenuse of length $2x$. The area of the triangle is equal to the sum of the lengths of its sides. Find the value of $x$. | nvidia |
Let $g(x)$ be a function such that $g(0) = 1$ and
\[g(x+y+z) = g(xy+yz+zx) + xyz\]for all real numbers $x$, $y$, and $z.$ Find $g(x).$ | nvidia |
Consider the set of all points $(x,y)$ in the coordinate plane that satisfy the equation $|x| + |y| = 1$. If we were to plot all such points, what would be the maximum number of regions into which the resulting graph could split the plane if we also draw all the lines that pass through the origin and are perpendicular ... | nvidia |
In the polygon shown, each side is perpendicular to its adjacent sides, and all 24 of the sides are congruent. The perimeter of the polygon is 48. Find the area of the polygon.
[asy]
unitsize(0.5 cm);
draw((3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(6,2)--(6,3)--(7,3)--(7,4)--(6,4)--(6,5)--(5,5)--(5,6)--(4,6)--(4,7)--(3,7)... | nvidia |
Let $f(x)$ be a function such that
\[f(x) f(y) = f(xy) + f(x) + f(y) - 2\]for all real numbers $x$ and $y.$ Find the largest possible value of $f(1).$ | nvidia |
What are the last two digits in the sum of factorials of the first 100 positive integers? | nvidia |
Let $x$ and $y$ be two real numbers such that $0<x<1$ and $0<y<1$. Let $F(n)$ denote the base-$n$ number $0.\overline{xyxy}_n$, where $n\geq 2$ is an integer. Determine the smallest positive integer $k$ such that $\prod_{n=2}^k F(n)$ can be written as a sum of integer fractions $\sum_{i=1}^m \frac{a_i}{b_i!}$ where $... | nvidia |
Let $x$ and $y$ be nonzero real numbers such that
\[(3 + 5i)(x + yi)\]is a real number. Find $\frac{x}{y}.$ | nvidia |
Find the point in the $xz$-plane that is equidistant from the points $(1,-1,0),$ $(2,1,2),$ and $(3,2,-1).$
Problem:
A cylindrical tank with a diameter of 10 meters and a height of 15 meters is placed on a horizontal plane. Three points $A,$ $B,$ and $C$ are marked on the tank such that $A$ is at the center of the cir... | nvidia |
Given that $a$, $b$, and $c$ are positive integers such that $a^2+b^2=c^2$, solve for $d$: $d=\displaystyle\frac{\sqrt{a^2+b^2}}{a+b}$. Express as a common fraction. | nvidia |
Find the largest possible area of a triangle inscribed in an ellipse with its foci on the $y$-axis, if the center of the ellipse lies on the line $x = 4$ and the length of the semiminor axis is 5. | nvidia |
Find the sum of all constants $t$ such that the cubic $x^3 + tx^2 + 1$ can be factored in the form $(x+a)(x^2+bx+c)$, where $a$, $b$, and $c$ are integers. | nvidia |
A sphere with center $(1,0,0)$ and radius $3$ intersects a cone with vertex at the origin and axis along the positive $z$-axis. The angle between the $z$-axis and the slant height of the cone is $30^\circ$. Find the coordinates of the point of intersection. | nvidia |
Find all $a$ such that $a^{\lfloor a \rfloor} = 33$.
Express your answer using only integers and the operations $+$, $-$, $\cdot$, $/$, and exponentiation with integer exponents. | nvidia |
Inside an equilateral triangle $\triangle ABC$, there is a smaller equilateral triangle $\triangle DEF$. The points $E$ and $F$ divide the sides $AB$ and $AC$ into the segments $AE,EB$ and $AF,FC$, respectively. If $AE=x$, $EB=y$ and $AF=z$, determine the values of $x,y,z$ such that the areas of the regions $\triangle ... | nvidia |
Determine $n$ such that $i^{2n+1}=i^n$ | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $a^2b^3c^2 = 1.$ Find the minimum value of
\[2a + 5b + 3c.\] | nvidia |
Find the number of positive integers less than or equal to 10000 that have distinct digits, all of which are prime. | nvidia |
Let $f(x)$ be a function defined for all positive integers $x$. For all positive integers $x$ and $y$ such that $x+y$ is a perfect square, we have
\[f(x) + f(y) = x^2 + 3xy + y^2\]
Let $T$ denote the sum of all the values of $x$ for which $f(x) = 2025$. Compute the remainder when $T$ is divided by $1000$. | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x \cdot f(y)) = x + y\]for all real numbers $x$ and $y$. | nvidia |
Find the largest three-digit palindrome number that is divisible by 7 and has exactly one prime digit. | nvidia |
The sum of the squares of three consecutive positive integers is 7805. What is the sum of the cubes of the three original integers?
A circle is inscribed in a triangle with sides 8, 15, and 17. Find the radius of the circle. | nvidia |
Find the smallest positive integer $n$ such that the least common multiple of $n$ and $48$ is $192$. | nvidia |
What is the smallest positive integer $n$ such that $6^n$ has more digits than $2^{10}$? | nvidia |
Find the equation of a hyperbola with center at the origin, transverse axis along the x-axis, and conjugate axis along the y-axis, such that the sum of the distances from any point on the hyperbola to its two foci is equal to 8. | nvidia |
Given a cube with side length $x$ and a sphere with radius $y$, both made of the same material, find the maximum volume of the sphere that can be inscribed within the cube, subject to the constraint that the surface area of the cube is equal to the surface area of the sphere. | nvidia |
The equation $y=ax^2+bx+c$ represents a parabola with vertex $\left(\frac{1}{4},-\frac{9}{8}\right)$. The problem is to find the smallest possible value of $a$ given that $a+b+c$ is an integer and $a>0$.
Here is another problem inspired by this one:
Find the largest possible value of $k$ such that the ellipse $x^2+ky... | nvidia |
Find the number of ways to color a cube with 3 different colors, such that no two adjacent faces have the same color. | nvidia |
Let $f(n)$ be the base-10 logarithm of the sum of the elements of the $n$th row in Pascal's triangle. Express $\frac{f(n)}{\log_{10} 2}$ in terms of $n$. Recall that Pascal's triangle begins
\begin{tabular}{rccccccccc}
$n=0$:& & & & & 1\\\noalign{\smallskip\smallskip}
$n=1$:& & & & 1 & & 1\\\noalign{\smallskip\smallsk... | nvidia |
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