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A certain number has the property that the product of its digits is equal to the sum of its digits. What is the smallest three-digit number with this property? | nvidia |
Find all values of $k$ so that the domain of
\[b(x) = \frac{kx^2 + 2x - 5}{-5x^2 + 2x + k}\]is the set of all real numbers. | nvidia |
There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? | nvidia |
If the ratio of the sum of the squares of two numbers to the square of their difference is equal to the ratio of the sum of the numbers to their difference, and the sum of the numbers is 20, what is the value of the smaller number? | nvidia |
What is the smallest positive integer that has exactly 8 positive divisors and is divisible by 12? | nvidia |
Let $a$ and $b$ be two positive real numbers such that $b$ varies inversely with the square root of $a$. If $b=4$ when $a=9$, find the value of $b$ when $a=16$. | nvidia |
Find the smallest positive integer that has four distinct prime divisors and can be expressed as a difference of two perfect squares in exactly three different ways. | nvidia |
The numbers 1 through 100 are written on a blackboard and all except four are erased. The average of the four remaining numbers is 46. The average of the two smallest numbers is 27. The average of the two largest numbers is 54. Find the smallest of the four remaining numbers. | nvidia |
Let $g : \mathbb{C} \to \mathbb{C} $ be defined by $ g(w) = w^3 + 2iw - 3 $. How many complex numbers $w $ are there such that $ \text{Re}(w) < 0 $ and both the real and the imaginary parts of $g(w)$ are integers with absolute value at most $ 15 $? | nvidia |
In a triangle $ABC$, the angle bisectors of $\angle A$ and $\angle B$ intersect at point $I$. Given that $\angle AIB=100^\circ$, find the measure of $\angle C$. | nvidia |
If we let $f(n)$ denote the sum of all the positive divisors of the integer $n$, how many integers $i$ exist such that $1 \le i \le 2010$ and $f(i) = 1 + \sqrt{i} + i$? | nvidia |
Suppose $x$ and $y$ are integers such that $xy+5x+4y=-5$. Find the greatest possible value of $y$. | nvidia |
Determine the smallest positive integer $n$ such that $5^n\equiv n^5\pmod 3$. | nvidia |
Find the sum of all integer values $k$ for which $\binom{30}{15} + \binom{30}{k} = \binom{31}{16}$. | nvidia |
In a parallelogram, the point of intersection of the diagonals is called the centroid. If the coordinates of the vertices of the parallelogram are A(0, 0), B(2a, 0), C(2a + 1, 3b), and D(1, 3b), find the largest possible value of the sum of the coordinates of the centroid, assuming a and b are positive integers. | nvidia |
Given a circle with a diameter of length 10 units, centered at (3, 5), determine the sum of the coordinates of all points (x, y) on the circle such that the line tangent to the circle at point (x, y) is perpendicular to the line through points (0, 0) and (3, 5). | nvidia |
Find the sum of all two-digit prime numbers that have a units digit of 3. | nvidia |
Find all pairs of positive integers $(a,b)$ such that $a^b$ and $b^a$ are each the sum of two perfect squares. | nvidia |
Let $g(x)$ be a function that takes real numbers to real numbers, such that $g(x)$ is continuous on the interval $[0, 2\pi]$, $g(0) = 0,$ and
\[g(x+y) + g(x-y) = g(2x)g(2y)\]for all real numbers $x$ and $y$ on the interval $[0, 2\pi]$.
Find the largest possible value of $g(\pi)$. | nvidia |
Find all possible values of the parameter $a$ for which the system of equations
\[\begin{aligned} x^2+y^2+z^2&=1 \\ ax+by+cz&=1 \\ x+y+z&=1 \end{aligned}\]has a solution, and each solution is an ordered triple $(x,y,z)$ of real numbers. | nvidia |
Let $z$ be a complex number such that the absolute value of $\frac{z}{e^{iz}}$ is minimal. Determine the imaginary part of $z.$ | nvidia |
Find the value of $x^2$ if there exists a positive real number $x$ such that $\sin (\arccos (x)) = x$. | nvidia |
Let $a$ and $b$ be two relatively prime positive integers such that $\sqrt{ab}$ is an integer. Find the greatest natural number $n$ such that $(1^3+2^3+3^3+\cdots + n^3)\left[(n+1)^4+(n+2)^4+(n+3)^4+\cdots + (2n)^4\right]$ is equal to $ab$. | nvidia |
Two points $A$ and $B$ are given in the plane, and a point $P$ is chosen on the line segment $AB$ such that the ratio of the lengths of the segments $AP$ and $BP$ is $2:3$. If the coordinates of $A$ are $(a, b)$ and the coordinates of $B$ are $(c, d)$, then find the coordinates of the point $P$ in terms of $a$, $b$, $c... | nvidia |
Given a positive integer $n$, find the number of solutions $(a_1,a_2,\ldots,a_n)$ to the equation $(a_1+a_2+\cdots+a_n)^n-a_1^n-a_2^n-\cdots-a_n^n=0$, where $a_i$ are positive integers for $1\leq i\leq n$. | nvidia |
Find the sum of the roots of the equation $x^3 + ax^2 + bx + c = 0$, given that the sum of the roots is three times the product of the roots. | nvidia |
Let $g(x)$ be a real-valued function defined on an open interval $(-a,a)$, where $a > 0$, with the property that $g(g(x)) = -x$. Find the smallest possible value of $a$ such that there exists a real number $b > 0$ for which $g(x) = b\arctan (x/a)$ satisfies the condition $g(g(x)) = -x$ on the interval $(-a,a)$. | nvidia |
In a certain encoding system, each letter is replaced by a unique three-digit number in base $b$. It is given that there are exactly $100$ distinct three-digit numbers in base $b$ that do not contain the digit $0$. If the encoding system requires that the first digit of each code be non-zero, what is the minimum value ... | nvidia |
Determine the value of $a$ for which the sum of the areas of all the triangles, formed by connecting the center of a regular $n$-gon to its vertices, is equal to the area of the $n$-gon itself, where the side length of the $n$-gon is $a$. | nvidia |
Name the greatest whole number less than $100$ that has an odd number of positive factors. | nvidia |
What is the smallest positive integer $n$ such that $2n$ is a perfect square and $3n$ is a perfect cube? | nvidia |
Find the smallest positive integer $n$ such that $$3n\equiv 17\pmod {35}~.$$ | nvidia |
What is the largest three-digit integer $m$ such that the sum of its digits is equal to the sum of the digits of its square? | nvidia |
Find the number of pairs of complex numbers $(a,b)$ such that the sequence $a, b, ab, a^2b^2$ is a geometric progression. | nvidia |
Find the sum of all values of $x$ such that $3^{x^2+2x-5} = 9^{x - 2}$. | nvidia |
Find the sum of the values of $y$ that satisfy the equation $|3y| - 2 = 23$. | nvidia |
Find the value of $b$ such that the quadratic expression $bx^2 + 12x + 9$ can be written as the square of a binomial. | nvidia |
Let $x$ and $y$ be two positive numbers such that $x^2+y^2=16$.
Determine the minimum value of $\log_3 x+\log_x y+\log_y 3$. | nvidia |
Find all real numbers $\lambda$ for which there exists a nonzero, 2-dimensional vector $\mathbf{w}$ such that
\[\begin{pmatrix} 3 & 5 \\ 1 & 3 \end{pmatrix} \mathbf{w} = \lambda \mathbf{w}.\]
Enter all the solutions, separated by commas. | nvidia |
Consider a region $R$ in the plane defined by the inequality $|x|+|y| \leq 1$, where $x$ and $y$ are real numbers. A point $(x, y)$ in $R$ is said to "dominate" another point $(a, b)$ if and only if $|x| \geq |a|$ and $|y| \geq |b|$ simultaneously. Let $S$ be the set of all points in $R$ that dominate the point $(\frac... | nvidia |
The sum of the squares of four consecutive positive integers is 1000. What is the largest of these integers? | nvidia |
Find all values of $b$ such that $b+\lceil b\rceil=17.8$. | nvidia |
Find the largest integer value of $m$ such that $2m^2-16m+24$ is negative. | nvidia |
Point $Q$ lies on the parabola $y = x^2$ and is 8 units from the point $(1,4)$. Find the sum of all possible $x$-coordinates that satisfy the given conditions. | nvidia |
Let $n$ be a positive integer such that the sum of the digits of $2^n$ is equal to 10. What is the smallest possible value of $n$? | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x) f(y) - f(xy) = x^2 + y^2\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | nvidia |
Find the value of $c$ that minimizes the number of pairs of distinct integers $(a, b)$ such that $a$ and $b$ are chosen from the set $\{1, 2, \ldots, 10\}$ and $a^2 + b^2 \leq c$. | nvidia |
Let $x$ be a positive real number. Find the value of the product
$$(1+\sqrt{x})(1+\sqrt[3]{x})(1+\sqrt[4]{x})\dotsm(1+\sqrt[2001]{x}),$$
assuming that
$$(1-\sqrt{x})(1-\sqrt[3]{x})(1-\sqrt[4]{x})\dotsm(1-\sqrt[2001]{x}) = \frac{1}{2^{1001}}.$$ | nvidia |
Determine $k$ such that $\dfrac{(2k)!}{(k!)^2} = \dfrac{100!}{(50!)^2} \cdot 101$. | nvidia |
Find the sum of the squares of the roots of the polynomial $x^3 + ax^2 + bx + c = 0$, given that the sum of the roots is equal to $-a$ and the product of the roots is equal to $-c$. | nvidia |
Find all real numbers $y$ such that \[\frac{y^3-3y^2}{y^2-5y+6} - y = 4.\] | nvidia |
Let $a$ and $b$ be positive integers such that $\frac{a}{b} + \frac{b}{a} = \frac{17}{2}$. Find the least possible value of $a+b$. | nvidia |
Let's consider a sequence of consecutive odd integers from 1 to 99. Find the sum of the remainders when the squares of these integers are divided by 4. | nvidia |
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's with length $n$ such that any 3 adjacent place numbers are distinct. For example, $01010$ works, but $10011$ does not. Find the number of elements in $\mathcal{T}_{10}$. | nvidia |
Given that the polynomial $x^2-kx+16$ has only positive integer roots, find the average of all distinct possibilities for $k$. | nvidia |
Determine the number of positive three-digit integers that are multiples of 2 and end with the digit 5. | nvidia |
There are four red flags and five blue flags that can be placed on the top of a castle. At least two flags of the same color must be placed in a row. If the flags are arranged randomly, what is the probability that all flags of the same color are placed in a row? | nvidia |
Find the number of integers between 1 and 100 that can be expressed as the sum of two perfect squares. | nvidia |
Let $a$, $b$ and $c$ be positive integers less than or equal to $9$. Find all possible values of $b$ such that there exist integers $a$ and $c$ such that
\[5 \cdot 7^a + 6 \cdot 7^b + 7^c\]is divisible by 9. | nvidia |
Given a set of three positive integers less than 20, not necessarily distinct, what is the maximum number of different possible sums that can be obtained by summing all possible pairs of these three integers? | nvidia |
How many positive integers less than $100$ have the property that some positive power of it is a perfect square? | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) + y) = g(x) + g(g(y) + g(-x)) - 2x\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the sum of all possible values of $g(2).$ Find $m \times t.$ | nvidia |
Determine all real values of $x$ that satisfy
$\sqrt{x^2-2x+1}+\sqrt{x^2+4x+4}=2x+3$. | nvidia |
Let $A$ be a $2\times2$ matrix such that $A^2=\begin{pmatrix}3&2\\1&1\end{pmatrix}.$ Find the smallest possible value of $\det(A).$ | nvidia |
Let $g(x)$ be a function defined for all positive integers $x,$ such that $g(1) = 1,$ and
\[g(a + b) = g(a)g(b) + g(\gcd(a, b))\]for all positive integers $a$ and $b.$ Find $g(2007).$ | nvidia |
Find the smallest positive integer value of $n$ such that the expression $n^3 - 7n^2 + 11n - 5$ is a perfect square. | nvidia |
Let $a_1, a_2, \dots, a_m$ be a sequence of real numbers such that $\displaystyle\sum_{k=1}^{m} \cos a_k = 0$. Find the largest possible value of $m$ if the following inequality holds: $\displaystyle\sum_{k=1}^{m} \sin^2 a_k \le \frac{1}{4}$. | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $ab + bc + ca = 8.$ Find the maximum value of $abc.$ | nvidia |
The product of two positive integers is 120. Determine the least possible sum of their prime factors. | nvidia |
Find the product of the values of $x$ which satisfy $x^2 + 1992x + 1993 = 0$, given that the roots are in the form of $x = a \pm \sqrt{b}$, where $a$ and $b$ are integers. | nvidia |
The number $N$ has the property that $\lfloor\log_{10}N\rfloor=2+\lfloor\log_{5}N\rfloor.$ Determine the largest possible value of $\lfloor\log_{10}N\rfloor.$ | nvidia |
Find the number of four-digit positive integers where the sum of the digits equals 7. | nvidia |
Find the largest real number $c$ such that \[x_1^2 + x_2^2 + \dots + x_{101}^2 \geq cM^2\]whenever $x_1,x_2,\ldots,x_{101}$ are real numbers such that $x_1+x_2+\cdots+x_{101}=0$ and $M$ is the median of $x_1,x_2,\ldots,x_{101}.$ | nvidia |
Find the number of ordered pairs $(x,y)$ of real numbers such that
\[8^{x^2 + y} + 8^{x + y^2} = 1.\] | nvidia |
Find the number of positive integers less than or equal to 100 that are divisible by exactly two different prime numbers. | nvidia |
Find the greatest whole number $n$ such that $111,111\cdot n$ is divisible by $222$. | nvidia |
Find the longest possible length of the remaining side of a triangle with sides of length 6 cm, 8 cm and a right angle, such that the sum of the squares of the two shorter sides is equal to 100 square cm. | nvidia |
Find the positive value of $n$ such that the equation $9x^2+nx+1=0$ has exactly one solution in $x$. | nvidia |
Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.
| nvidia |
The diagonals of a parallelogram intersect at the point $(2, 3)$. One vertex of the parallelogram is located at the point $(5, 7)$. Find the equation of the circle passing through all four vertices of the parallelogram. | nvidia |
In the triangle $ABC$, $D$ is a point on $AB$ such that $\angle ADC=90^{\circ}$. $E$ is a point on $AC$ such that $DE$ is perpendicular to $AC$. $AE$ intersects $CB$ at $F$. In $\triangle AED$, $AE=20$, $ED=15$ and $DA=25$. Find the lengths of $DE$ and $EF$. | nvidia |
Find the smallest integer value of $a$ for which the expression $\frac{2x^2+3x-5}{x^2+ax-3}$ is undefined for exactly two real numbers. | nvidia |
Given that $a$, $b$, and $c$ are positive constants, find the largest possible value of the expression $\frac{xyz}{ax+by+cz}$, where $x$, $y$, and $z$ are non-negative real numbers. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(0) = 0$ and
\[f(x+y+f(x+y)) = f(x) + f(y) + xy\]for all real numbers $x$ and $y.$
Determine the number of possible values of $f(1)$. | nvidia |
Let $a$, $b$, and $c$ be integers such that $\frac{b+c-a}{a},$ $\frac{a+c-b}{b},$ and $\frac{a+b-c}{c}$ are all integers. If $a+b+c$ is divisible by 9, must $abc$ be divisible by 9? | nvidia |
A convex polygon is randomly selected from the set of all convex polygons with 10 vertices such that all the vertices are on the circumference of a given circle. What is the probability that the polygon contains the center of the circle? | nvidia |
Determine the sum of all single-digit replacements for $y$ such that the number ${12{,}y54}$ is divisible by 9. | nvidia |
Find the number of positive divisors of $360$ that are not multiples of $6$ but are multiples of $4$. | nvidia |
Let $f(x)$ be a polynomial of degree $n$ with real coefficients, and let its roots be $r_1,$ $r_2,$ $\dots,$ $r_{n}.$ There are exactly $k$ distinct values among
\[|r_1|, |r_2|, \dots, |r_{n}|.\] Determine the minimum number of real roots that $f(x)$ can have if it has exactly $m$ coefficients that are equal to zero. | nvidia |
Let $f(n)$ be the sum of the proper positive integer divisors of $n$. For how many values of $n$ does $f(n)$ equal $n$, given that $2 \le n \le 50$? | nvidia |
Find the largest positive integer $n$ for which $n!$ can be expressed as the sum of $n-3$ consecutive positive integers, and the smallest positive integer in this sum is greater than $n$. | nvidia |
Let $a,$ $b,$ $c$ be a three-term arithmetic series where all the terms are positive, such that $abc = 64.$ Find the smallest possible value of $b.$ | nvidia |
Given the polynomial expression $P(x) = ax^4 + bx^3 + cx^2 + dx + e$, if the sum of the squares of all coefficients equals the value of $P(1)$, and the sum of the absolute values of all coefficients equals $|P(-1)|$, determine the coefficient $c$.
(Note: Assume all coefficients are integers.) | nvidia |
Find the value of $b$ such that the equation $\ln{(x^2+bx+c)} + \ln{(x^2-bx+c)} = \ln{(x^4+1)}$ has at least one real solution. | nvidia |
Determine the number of positive integer divisors of $127^3$ that are not divisible by any prime number greater than 7. | nvidia |
Determine all real numbers $ a$ such that the inequality $ |x^2 + 2ax + 3a|\le2$ has exactly one solution in $ x$. | nvidia |
Let $a$, $b$ and $c$ be positive integers such that $a$ and $b$ are relatively prime and $ab=c^2$. Prove that the number of positive divisors of $a$ is less than or equal to the number of positive divisors of $b$. | 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 |
Find the greatest common divisor of all positive integers that can be expressed as the product of two consecutive terms of a geometric sequence with integer terms. | nvidia |
Given a rectangle with a fixed perimeter of 20 cm, where one side is twice as long as the other, determine the dimensions of the rectangle such that its area is equal to the area of the given isosceles right triangle. | nvidia |
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