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Determine the smallest positive integer whose decimal representation has six digits such that each digit appears at least twice. | nvidia |
A certain positive integer has only even digits and the sum of the digits is 30. What is the smallest possible value of the number? | nvidia |
Find the smallest positive integer $n$ such that the product of the first $n$ positive integers is divisible by 135, 468, and 1920. | nvidia |
Count the number of positive integers less than $10000$ that can be expressed as the difference between two consecutive positive integer squares. | nvidia |
Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=\frac{t_{n-1}+1}{t_{n-1}-1}$ when $n$ is even and by $t_n=\frac{t_{n-1}+t_{n-2}}{t_{n-1}-t_{n-2}}$ when $n$ is odd. Given that $t_n=\frac{1+\sqrt{5}}{2}$, find $n.$ | nvidia |
Find the 5th number in the row of Pascal's triangle that starts with 1 and then 7. | nvidia |
If $x$ is a positive number such that \[\sqrt{8x}\cdot\sqrt{10x}\cdot\sqrt{3x}\cdot\sqrt{15x}=15,\]find all possible values for $x$. | nvidia |
A sequence of real numbers $a_n$ is defined by
$a_n = 2020 - 2021 \operatorname{sign}(a_{n-1}+a_{n-2})$
for $n \geq 3,$ where $a_1$ and $a_2$ are given real numbers.
Determine the number of distinct values that $a_{10}$ can take. | nvidia |
Prove that the number $\sqrt{2+\sqrt{2+\sqrt{2+\dotsb}}}$ is an irrational number. | nvidia |
The dimensions of a rectangular solid are in the ratio $3:4:5$. If the sum of the lengths of the edges of the solid is 96 inches, find the volume of the solid. | nvidia |
Determine the smallest positive integer $k$ such that $13^k+1$ is divisible by a perfect square greater than 1. | nvidia |
A hexagon is divided into six congruent equilateral triangles. The side length of the hexagon is equal to $5n+17$, where $n$ is a positive integer. Find the number of possible values of $n$ such that the area of the hexagon is greater than 2500 square units but less than 4000 square units. | nvidia |
In the triangle $XYZ$, the coordinates of the vertices are $X(0,0)$, $Y(10,0)$, and $Z(0,8)$. The points $M$, $N$, and $P$ are on the sides $XY$, $YZ$, and $ZX$, respectively, such that the ratio of the area of triangle $MNP$ to the area of triangle $XYZ$ is $1/4$. The point $M$ is $m$ units away from $X$, and the poin... | nvidia |
What is the smallest positive multiple of 7 that is not a "lucky number", where a lucky number is defined as a positive integer that is divisible by the product of its digits? | nvidia |
Find all values of $b$ such that $b+\lceil b\rceil=17.8$. | nvidia |
Find the smallest positive integer n such that the sum of the digits of n is equal to the sum of the digits of 2n. | nvidia |
How many distinct triangles can be formed by connecting three vertices of a regular octagon, such that no two triangles share the same set of vertices? | nvidia |
Given any two positive real numbers $a$ and $b$, find the largest real number $\mu$ such that
\[a^xy + b^y \ge a^{\mu x} b^{y - \mu x}\]for all nonnegative real numbers $x$ and $y$. | nvidia |
Find the number of functions $f : \mathbb{N} \to \mathbb{N}$ such that
\[f(n m) = (f(n) + f(m))^2 - f(n) f(m)\]for all positive integers $n$ and $m$. | 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 |
Given that $a$ and $b$ are positive real numbers such that $a \ge b > 1$, find the largest possible value of $\frac{a^2}{b} + \frac{b^2}{a}$. | nvidia |
What is the sum of the reciprocals of all the positive integer divisors of 24? | nvidia |
The number 24 can be represented as the product of different factors in the form $2^a \cdot 3^b$, where $a$ and $b$ are non-negative integers. Find the representation with the smallest sum $a+b$ that is not equal to $2^0 \cdot 3^3$. | nvidia |
Find the number of distinct six-digit integers that can be formed using the digits 1, 2, 2, 3, 4, and 4, such that the integer is divisible by 3. | nvidia |
Find the number of positive integers $n$ such that $2^n$ is a divisor of $10!$ but $3^n$ is not. | nvidia |
A rectangle is inscribed in a right triangle. The sum of the lengths of the sides of the rectangle parallel to the legs of the triangle is $20$ units. Find the area of the rectangle. | nvidia |
Let $a$, $b$, $c$ be positive integers such that $a^2$, $b^2$, $c^2$ are in arithmetic progression. Determine the smallest possible value of $c$. | nvidia |
Find the smallest positive integer $m$ such that the expression $\left(\frac{m-1}{2}\right)^7 + m^2 + 3m + 9$ is a multiple of $7$. | nvidia |
Find the sum of the squares of the first seven prime numbers that have a units digit of 3. | nvidia |
Find the greatest possible value of $a$ such that the inequality $ax^2 + bx + c \leq 0$ holds true for all real values of $x$, given that $b$ and $c$ are fixed positive numbers. | nvidia |
Let $\tau (n)$ denote the number of positive integer divisors of $n$ (including $1$ and $n$). Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$.
| nvidia |
Find the largest integer that is divisible by 7 and is less than 1000. The integer should also have the property that the sum of its digits is a multiple of 5. | nvidia |
Let $f(x)$ be a function such that for all real numbers $x$, $$f(x)f(-x)=\frac{x^2}{x^2+1}.$$ If $f(3)=\frac{1}{2}$, what is $f(-3)$? | nvidia |
Find the value of $x$ such that $2^{2x+1}+1$ is divisible by $x^2+2x+1$. | nvidia |
For all positive integers $n$, let $f(n)=\log_{2002} n^2$. Find $f(11)+f(13)+f(14)$. | nvidia |
The graph of the function $y=x^2+bx+c$ is tangent to the line $y=2x+1$ at the point $(1,3)$. Determine the sum of the $x$-coordinates of the points where the graph of the function intersects the line $y=x-3$. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 6.$ Find the minimum value of
\[\frac{1}{a} + \frac{4}{b} + \frac{9}{c}.\] | nvidia |
A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles? | nvidia |
Problem:
The diagonal of a square is tangent to a circle that touches two adjacent sides of the square.
If the radius of the circle is 1 cm and the area of the circle is $\pi/16$ of the area of the square, determine the length of the side of the square in centimeters. | nvidia |
In the equation
$$3x^2+5x-2=0$$
determine $k$ such that the sum of the roots be equal to 2.
Solve for $k$. | nvidia |
Consider the sequence of numbers defined recursively by $s_1=2$ and for $n>1$ by $s_n=3s_{n-1}-s_{n-2}$ when $n$ is even and by $s_n=\frac{s_{n-1}+1}{s_{n-2}}$ when $n$ is odd. Given that $s_n=13$, find $n$. | nvidia |
Let $f(n)$ return the number of distinct ordered pairs of positive integers $(a, b)$ such that for each ordered pair, $a^2 + b^2 = n$. Note that when $a \neq b$, $(a, b)$ and $(b, a)$ are distinct. What is the smallest positive integer $n$ for which $f(n) = 3$? | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $x+y+z=1$. Find the maximum value of $xyz\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$. | nvidia |
There are 3 complex numbers $a+bi$, $c+di$, and $e+fi$. If $b=3$, $e=-a-c$, and the sum of the numbers is $2i$, find $d+f$. | nvidia |
Given a positive integer $n$, let $f(n)$ denote the smallest positive integer $k$ such that $kn$ can be represented as the sum of the squares of three distinct positive integers. What is the value of $\sum_{n=1}^{1000} f(n) \mod 1000$? | nvidia |
Find the greatest value of $a$ such that $|a^2-10a+21| \ge 10$. | nvidia |
Let $f(x)$ be a quadratic function such that the value of the expression
\[f(2x^2+3)-f(2x^2-3)\]
is $k\cdot x^2$ for some positive integer $k$. If
\[f(0)=3,\]
what is $k$? | nvidia |
Count the number of positive integers from 1 to 99999 (inclusive) that do not contain the digits 1, 3, 6, or 8. | nvidia |
Find the remainder when the product of all positive integers less than or equal to 100 that are relatively prime to 100 is divided by 100. | nvidia |
A circle with center $O$ has three tangents $\ell_1, \ell_2, \ell_3$ that form an equilateral triangle $XYZ$, with each vertex $X, Y, Z$ lying on one of the tangents. If the radius of the circle is $4$, find the length of the line segment $OX$. | nvidia |
The positive integer divisors of 175, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7? | nvidia |
Find the value of $x$ such that $\log_3(x) + \log_3(2) = 2$. | nvidia |
The sum of the squares of two consecutive positive integers is 85. What is the product of these two integers? | nvidia |
Find the number of values of $b$ such that the circle $x^2 + y^2 = b^2$ intersects the parabola $y = x^2 + b$ at exactly two points. | nvidia |
Find the product of all positive integer values of $c$ such that $8x^2+15x+c=0$ has two real roots. | nvidia |
Let $a, b, c$ be real numbers so that \begin{align*}
a^2+b^2&=20,\\
b^2+c^2&=30,\\
c^2+a^2&=40.
\end{align*}Find $\sqrt{abc(a+b+c)}$. | nvidia |
A sequence of $n$ real numbers is said to be \emph{special} if the sum of any two terms is not equal to a third term.
Let $a_n$ be the number of special sequences of length $n$ with all terms in the set $\{1,2,\dots,n\}$.
Find $a_{20}$. | nvidia |
Find the smallest positive integer such that its decimal representation contains only 1's and 0's, and the sum of its digits is divisible by 7. | nvidia |
There is a unique two-digit positive integer $t$ for which the last two digits of $11\cdot t$ are $36$.
What is $t$? | nvidia |
Let $f(x)$ be a function that satisfies the condition $f(x+y)=\left[f(x)+f(y)\right]^2$ for all real values of $x$ and $y$. If the point $(5,-2)$ is on the graph of $y=f(x)$, find the value of $f(10)$. | nvidia |
Let $\rho(n)$ denote the sum of the positive divisors of $n$, and let $\mu$ be a multiplicative function on the positive integers. If $\mu(1)=1$ and $$\rho(n) \mu(m n) = m \mu(n) + \mu(m n)$$for all positive integers $m$ and $n,$ find the sum of all possible values of $\mu(8).$ | nvidia |
Given a circle with equation $(x+2)^2+(y-3)^2=8^2$, find the point on the circle that is furthest from the line $y=x-5$. | nvidia |
Find the smallest positive integer $y$ which satisfies the congruence $23y+8\equiv 11\pmod {25}$. | nvidia |
Determine the sum of all real solutions $x$ to the equation $\sqrt{x^2+1}=\sqrt{x}+1$. | nvidia |
Find the value of $(a^2+b^2+c^2)(x^2+y^2+z^2)$ when $a, b, c$ are the roots of the equation $x^3-5x^2+7x-3=0$ and $x, y, z$ are the roots of the equation $x^3-2x^2-5x+1=0$. | nvidia |
Let $f(x)$ and $g(x)$ be two functions such that
\[f(x+a) = g(x)\]for all integers $x$, where $a$ is a positive real number. Suppose that $f(x)$ is a quadratic function and $g(x)$ is a linear function, and that $f(0) = g(0) = 0$. Find the smallest possible value of $a$ such that $f(x)$ and $g(x)$ have the same leading ... | nvidia |
The function $g(t)$ satisfies
\[g(tz) = \frac{g(t)}{z}\]for all positive real numbers $t$ and $z.$ If $g(48) = 30,$ find $g(60).$ | nvidia |
When three positive integers are divided by $24$, the remainders are $10,$ $4,$ and $12,$ respectively.
When the sum of the three integers is divided by $24$, what is the remainder? | nvidia |
How many five-digit positive integers have at least one digit that is a 5 or a 7? | nvidia |
A portion of the graph of a quadratic function $f(x)$ is shown below.
Let $g(x)=-f(x-2)$ and $h(x)=f(x+1)$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=g(x)$ intersect, and $b$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, then what is $5a+2b$?
[asy]
size(150);
re... | nvidia |
Let $P$ be a point inside triangle $ABC$ such that triangles $ABP$, $BCP$, $CAP$ have areas $A_1$, $A_2$, $A_3$ respectively.
Assume that point $P$ is chosen inside the triangle $ABC$ so that $A_1+A_2=2A_3$.
Let $M$ be the midpoint of segment $BC$.
Find $\frac{PM}{MA}$. | nvidia |
A sequence of five consecutive numbers is such that the sum of the first, third, and fifth terms is 15 times the sum of the second and fourth terms. Determine the sum of the five numbers. | nvidia |
What is the maximum number of non-overlapping circles that can be packed around a central circle such that each non-overlapping circle touches the central circle and two of its neighbors? | nvidia |
Given that the polynomial $x^2-kx+16$ has only positive integer roots, find the average of all distinct possibilities for $k$. | nvidia |
Find the number of integers $-10 \leq n \leq 10$ that satisfy the inequality $(n-3)(n+5)(n-7)<0$. | nvidia |
Find all possible values of the digit $A$ such that $42$ is divisible by $A$ and $315{,}2A8$ is divisible by $9$. | nvidia |
Let's consider a function with a similar trigonometric theme and domain restriction.
Find the smallest value of the positive integer $k$ such that the function $g(x) = \arccos(\sqrt{kx})$ has a domain that includes the interval $[0, \frac{1}{2}]$. | nvidia |
Find the maximum value of $y = \cos^2(\theta) + \sin^2(2\theta)$ over all real numbers $\theta$. | nvidia |
How many positive four-digit integers greater than 1000 have at least two consecutive digits that are the same? | nvidia |
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression with first term $a$ and common difference $d$.
Find the largest value of $n$ such that the fraction $\frac{S_n}{S_{n+1}}$ is less than or equal to $\frac{5}{6}$. | nvidia |
Find the sum of the first 50 positive integers whose square roots are integers. | nvidia |
A polynomial $P(x)$ with rational coefficients has the property that for any rational number $x$, the value of $P(x)$ is equal to either $0$ or $1$. Prove that $P(x)$ is either $0$ or $1$ for all real numbers $x$. | nvidia |
Find the largest positive integer $m$ such that when its digits are reversed, the resulting integer is $m$ times the number of digits in $m$. | nvidia |
Find the values of $x$ for which the inequality $2^x < 3^x$ holds, given that $x$ is an integer. | nvidia |
The number of diagonals in a convex polygon with $n$ sides is given by the formula $d_n = \frac{n(n-3)}{2}$. Let $S_n$ be the sum of all diagonals in a polygon with $n$ sides. For example, for a triangle $S_3=0$ and for a square $S_4=2$. Consider the set of all possible non-degenerate polygons with $n$ sides, where $n\... | nvidia |
How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\frac14$? | nvidia |
A sequence of $n$ points in the plane is drawn such that the slope of the line through each consecutive pair of points is either $1$ or $-1$. Suppose the sum of the $y$-coordinates of the points equals $k$. If the sequence is then reflected over the line $y=x$, find the sum of the $x$-coordinates of the reflected point... | nvidia |
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is m/n, where m and n are relat... | nvidia |
Let $S_n$ be the set defined as follows:
$$S_n=\{(x,y): x, y \text{ are integers, } x\geq 0, y\geq 0, x+y=n\}.$$
Thus, $S_3=\{(0,3),(1,2),(2,1),(3,0)\}$ and $S_6=\{(0,6),(1,5),(2,4),(3,3),(4,2),(5,1),(6,0)\}$.
Find the smallest $n$ such that there is a subset $T_n\subset S_n$ with 241 elements and the property that ... | nvidia |
Find all positive integers $n$ such that the number $242_n$ (the number written as $242$ in base $n$) is a palindrome in base $10$. | nvidia |
Find all real numbers $x$ which satisfy \[\frac{x^2-4x+3}{x^2+4x+5} \le 1.\](Give your answer in interval notation.) | nvidia |
Let $a,b,c$ be three real numbers such that $a$ is the $x$-coordinate of the vertex of the parabola $y=x^2+bx+c$, $b$ is the $x$-coordinate of the vertex of the parabola $y=x^2+ax+c$, and $c$ is the $x$-coordinate of the vertex of the parabola $y=x^2+ax+b$. Find $a+b+c$. | nvidia |
Three integers $x$, $y$, and $z$ are chosen uniformly at random from the set of positive integers less than or equal to $11$, such that $x\neq y$, $y\neq z$, and $z\neq x$. Let $p(x,y,z)$ be the probability that $x$, $y$, and $z$ are the lengths of the sides of a triangle. Compute $p(3,5,7)$. | nvidia |
Find the value of $a$ such that the quadratic equation $ax^2 + 2x + a = 0$ has two distinct real solutions, and the sum of the squares of the roots is equal to 10. | nvidia |
Find the value of $b$ so that the lines with the given equations are parallel. \begin{align*}
y &= 3x-5 \\
by-2x &= 7.
\end{align*} | nvidia |
What is the sum of all values of $x$ for which $\sqrt{(x+3)^{2}} = 7$?
Determine the sum of all values of $x$ for which $|x+3| = 5$. | nvidia |
Find the value(s) of $x$ such that $8xy-12y+2x-3=0$ is true for all values of $y$. | nvidia |
Find the number of positive integers $n$ such that the decimal representation of $n^2$ contains exactly two distinct digits. | nvidia |
Find the number of functions $g : \mathbb{R} \to \mathbb{R}$ such that
\[g(x + g(y)) = x - y\]for all real numbers $x$ and $y.$ | nvidia |
What is the smallest positive integer $m$ such that the product of the first $m$ positive odd integers is divisible by $3^8$? | nvidia |
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