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Two positive integers $p$ and $q$ are chosen such that $p$ is the smallest positive integer with exactly four positive divisors and $q$ is the largest integer less than $500$ with only two positive divisors. What is $p+q$? | nvidia |
Determine all possible pairs of positive integers $(a,b)$ such that $x^3+ax+b$ is divisible by $x^3+2x+1$. | nvidia |
Let $A$ be a set of $n$ elements. A collection of subsets of $A$ is called a "closed family" if it contains the union and intersection of any two of its members. Let $F$ be a closed family of subsets of $A$, and suppose that $F$ contains all the singletons (subsets of size 1) and the empty set. Let $S$ be a subset of $... | nvidia |
Find the positive real number x such that [ x ] ⋅ x = 30 . Express x as a decimal. (Here [ x ] denotes the greatest integer less than or equal to x .) | nvidia |
Find $a^3+b^3$ if $a$ and $b$ are positive integers such that \[\begin{aligned} a+b&=23 \\ a^2-ab+b^2 &= 325.\end{aligned}\] | nvidia |
Find all real values of $x$ that satisfy $\frac{x^2+x^3-2x^4}{x+x^2-2x^3} \ge -1.$ (Give your answer in interval notation.) | nvidia |
Let's have two triangular prisms with integer side lengths, such that the base of the larger prism is twice the area of the base of the smaller prism and the height of the larger prism is twice the height of the smaller prism.
What is the maximum possible volume of the smaller prism such that the total surface area of ... | nvidia |
Let $a$, $b$, $c$ be positive integers such that $\frac{ax+b}{cx+d}$ is an integer, where $d$ is a divisor of $a$. Find the greatest integer value of $a$ such that $a^2 + bc = 2019$. | nvidia |
Find the sum of the two smallest one-digit composite numbers and the smallest two-digit composite number. | nvidia |
A polynomial of degree 4 is said to be "nice" if its coefficients are all integers and it has at least two distinct integer roots. Let's define the "niceness" of a polynomial as the number of ways to factor it into two quadratic factors with integer coefficients, where each factor has at least one integer root.
Find t... | nvidia |
Let $f$ be a function defined on the set of all real numbers such that $f(x+y)=f(x)f(y)$ for all $x$ and $y$. Suppose also that $f(x) \neq 0$ for all $x \neq 0$ and that $f(1)=1$. Let $a$ be a real number such that $a \neq 0, 1, -1$ and let $b=f(a)$. Determine the range of values of $b$. | nvidia |
A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order. | nvidia |
Find the number of lattice points on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ and $b$ are relatively prime positive integers. | nvidia |
Find the coordinates of the point that divides the line segment joining the points $(2,6)$ and $(7,-3)$ into the ratio $2:3$ internally. | nvidia |
Let $x$ and $y$ be two positive real numbers such that $x + y \le 100$ and $xy \ge 100$. Determine the minimum value of the expression $\frac{x}{y} + \frac{y}{x}$. | nvidia |
Find the smallest positive integer $n$ such that $|n + \sqrt{n}|$ is an integer. | nvidia |
For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 50$.
(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? | nvidia |
Consider a function $g$ defined on the positive integers, satisfying the equation
\[g(x + y) = g(x) \cdot g(y)\]for all positive integers $x$ and $y.$ We know that $g(3) = 13$ and $g(7) = 53.$ Find $g(17).$ | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
Find the smallest positive integer $n$ such that $n^2$ is divisible by 4, 8, 16, and 32. | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
The degree measure of the sum of the interior angles of a convex polygon with $n$ sides is 1800. What is the degree measure of the sum of the interior angles of a convex polygon with $n + 2$ sides? | nvidia |
A hexagon is inscribed in a circle of radius $r$, such that one vertex is at the center of the circle and the other vertices are on the circle. The length of each side of the hexagon is equal to $\sqrt{r^2-n^2}$, where $n$ is a positive integer. If the area of the hexagon is $\frac{A}{B}\pi r^2$, where $A$ and $B$ are ... | nvidia |
Find the largest three-digit positive integer that is a multiple of 7 and has exactly 4 distinct prime factors. | nvidia |
Let $p(x)=ax^3+bx^2+cx+d$ be a cubic polynomial such that $p(1)=p(2)=p(3)=0$. Find the sum of all possible values of $\frac{p(4)}{p(5)}$. | nvidia |
What is the number of positive integers less than $10!$ that are divisible by neither $7!$ nor $8!$? | nvidia |
The square of the sum of two numbers is 288, while the sum of their squares is 338. What is the product of the two numbers? | 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 $a$ and $b$ be positive real numbers such that $a + b = 10$ and $ab = 4$. Find the value of $a^3 + b^3$. | nvidia |
Find the sum of the positive whole number divisors of 945 that are not multiples of 3. | nvidia |
Determine the largest real number $\alpha$ such that the inequality
\[\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{cd}{c+d}+\frac{da}{d+a}\le \alpha(a+b+c+d)\]
holds for all positive real numbers $a,$ $b,$ $c,$ $d.$ | nvidia |
How many integers $N$ less than $1000$ can be written as the sum of $j$ consecutive positive odd integers from exactly 5 values of $j\ge 1$?
| nvidia |
Find all real values of $x$ that satisfy $\frac{x(x+2)}{(x-3)^2} \ge 8.$ (Give your answer in interval notation.) | nvidia |
For what values of $k$ does the equation $A_k = A_{k+1}$ have solutions, where $A_k = \frac{(2n)!}{k!(2n-k)!}x^k$, where $n$ is a positive integer and $x$ is a real number? | nvidia |
Find the sum of all positive integral values of $k$ for which $\frac{k^2+2k-1}{k+1}$ is a perfect square. | nvidia |
How many four-digit numbers are there whose digits satisfy the condition that the sum of the squares of the first and third digits is greater than the sum of the squares of the second and fourth digits? | nvidia |
Find the value of $x$ such that $\log_{\sqrt{x}} (3x+1) = 3$. | nvidia |
Six congruent balls are colored with three different colors: two blue, three red, and one white. The balls are then placed into six indistinguishable boxes so that no box contains more than one ball. Determine the number of possible arrangements of the balls in the boxes. | 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 |
The polynomial $g(x)=x^4-2ax^2-8a^2$ has four distinct real roots $x_1,\ldots,x_4$. A polynomial $Q$ of degree 4 has the property that
\[Q\left(\frac{x_j^2}{x_j+a}\right)=0\]for $j=1,\ldots,4$. Determine the value of $a$ such that $Q(1)=Q(-1)$. | nvidia |
Let $f(x) = x^4 - 32x^2 + m$, where $m$ is an integer. Suppose that the roots of $f(x)$ are $\pm \sqrt{a}$ and $\pm \sqrt{b}$, where $a$ and $b$ are positive integers. Find the smallest possible value of $a + b$ such that $f(x)$ has four distinct integer roots. | nvidia |
What is the smallest integer $n$ such that $n$ increased by $25\%$ is a perfect square? | nvidia |
Find the number of integer solutions to the equation $x^2 + y^2 = 85$, where $x$ and $y$ are integers. | nvidia |
A square is drawn such that one of its sides coincides with the line $y = 5$, and so that the endpoints of this side lie on the parabola $y = x^2 + 3x + 2$. What is the area of the square? | nvidia |
When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.
When the sum of the three integers is divided by $12$, what is the remainder? | nvidia |
A quadrilateral has vertices at coordinates (0,0), (1,1), (3,0), and (2,a). Find the smallest value of a such that the quadrilateral is a trapezoid. | nvidia |
What is the sum of the values of $x$ that satisfy the equation $x^2-5x+5=9$? | nvidia |
Find the number of positive integer factors of $3^5 \cdot 5^4$ that are divisible by 25 but not by 27. | nvidia |
Determine the smallest positive integer n such that the two integers 8n+3 and 9n+4 share the same units digit. | nvidia |
Find the number of ordered triples of non-negative integers (a, b, c) that satisfy the following conditions:
a + b + c = 12
a is even
b and c are odd
0 < a < b < c
Determine the number of such triples. | nvidia |
Find the area of the triangle formed by the two endpoints of the major axis and one of the endpoints of the minor axis of the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$. | nvidia |
Given that $\sum_{k=1}^{25}\cos 3k=\csc \frac ab,$ where angles are measured in degrees, and $a$ and $b$ are relatively prime positive integers that satisfy $\frac ab<90,$ find $a+b.$ | nvidia |
The sum of the squares of four consecutive natural numbers is divisible by a positive integer $n$. Find the remainder when the product of these four numbers is divided by $n$. | nvidia |
For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 50$.
(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? | nvidia |
Consider the sequence $a_n$ where $a_1=1$ and $a_{n+1}=\frac{a_n+1}{2}$ for $n\geq 1$. Prove that the sequence converges and find its limit. | nvidia |
Compute the value of $y$ such that
$\left(1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}\cdots\right)\left(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots\right)=1-\frac{1}{y}+\frac{1}{y^2}-\frac{1}{y^3}+\cdots$. | nvidia |
Triangle $ABC$ is an equilateral triangle with $AB=25$, and point $M$ is the midpoint of $AB$. $D$ is a point on line $BC$ such that $\angle AMD=120^\circ$. If the area of triangle $AMD$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ i... | nvidia |
How many positive integers $n$ satisfy the inequality $\frac{1}{n} \leq \frac{1}{12}$? | nvidia |
Let $X$ be the number of 5-digit numbers that are palindromes, and $Y$ be the number of 6-digit numbers that are multiples of 3. Find $X+Y$. | nvidia |
Find the smallest positive integer $m$ such that all the roots of the equation $z^6 + z^3 + 1 = 0$ are $m^{\text{th}}$ roots of unity. | nvidia |
Let $s_n$ denote the sum of the first $n$ positive integers. How many of the first 100 values of $n$ are such that $s_n$ is a perfect square? | nvidia |
Find the sum of the coefficients in the expansion of the product $\left(1+x\right)^6$. | nvidia |
Find the largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$. | nvidia |
A sequence of seven distinct integers is constructed such that the difference between any two consecutive terms is either 1 or 2, and the sum of the seven terms is 77. How many such sequences are possible? | nvidia |
Find a positive integer that is divisible by 24 and whose square root is a number between 15 and 15.2. | nvidia |
Find the largest constant $C$ so that
\[x^2 + y^2 + 1 \ge C(x + y)\]for all real numbers $x$ and $y.$ | nvidia |
Find the number of positive three-digit integers that leave a remainder of 5 when divided by 7 and are multiples of 3. | nvidia |
$ABC$ is a triangle: $A=(0,0), B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\triangle ABC$ can have?
$\textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ \frac{3}{2} \qquad \textbf{(D)}\ \frac{13}{2}\qquad \textbf{(E)}\ \text{there is no minimum}$
| nvidia |
Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a+b+c = 28.$ Find the largest possible value of $b.$ | nvidia |
Let $N$ be the sum of all positive integers less than $6!$ that are not divisible by $2, 3,$ or $5$. Find the remainder when $N$ is divided by $6!$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that for any real numbers $x$ and $y$,
$$f(x+y)f(x-y)=f(x)f(y)+f(xy)(x^2-y^2).$$
Determine all possible values of $f(1)$. | nvidia |
Let $S$ be the set of complex numbers of the form $x + yi,$ where $x$ and $y$ are real numbers, such that
\[\frac{\sqrt{2}}{2} \le x \le \frac{\sqrt{3}}{2}.\]Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a complex number $z \in S$ such that $z^n = 1.$ | nvidia |
In the Cartesian plane, the graph of the hyperbola $\frac{x^2}{4}-y^2=1$ intersects the graph of the ellipse $x^2+4y^2=k$ in exactly one point. Find all possible values of $k.$ | nvidia |
Determine $b$ such that the polynomial $x^3 + bx^2 + 2x + 1$ can be factored into the product of two polynomials, one of which is linear and the other quadratic. | nvidia |
Let $z$ be a complex number such that
\[|z-2|+|z+2i|=3.\]
Find the largest possible value of $|z|$. | nvidia |
A cube is divided into unit cubes by dividing each edge of the cube into n equal parts, where n is a positive integer. A subset of the unit cubes is colored such that for each unit cube in the subset, the average number of colored unit cubes in each of the adjacent unit cubes is an integer. Given that one of the unit c... | nvidia |
Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$. | nvidia |
Find all real numbers $a$ such that $\lceil \lceil a \rceil + 2 \rceil = \lceil 3a - 1 \rceil.$ | nvidia |
Find the number of ordered pairs $(x,y)$ of real numbers such that
\[x^2 y^3 = x^4 y = 2.\] | nvidia |
When two fair 6-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that the two digit number $ab$ (where $a$ and $b$ are digits) and $a$ and $b$ are all divisible by 3? | nvidia |
Find the number of positive integers less than or equal to 1000 that are not divisible by any of the prime numbers 2, 3, or 5. | nvidia |
In a right-angled triangle $ABC$, with $\angle B = 90^\circ$, point $D$ lies on side $AB$ and point $E$ lies on side $BC$, so that $AD = DE = EC$. Given that the area of triangle $ABC$ is $24$ square units, find the length of side $AB$. | nvidia |
Consider a sequence of positive numbers $a_1, a_2, \ldots, a_n$ such that each term is greater than 1. Find the maximum possible value of the expression $\log_{a_1}(a_1/a_2) + \log_{a_2}(a_2/a_3) + \cdots + \log_{a_{n-1}}(a_{n-1}/a_n) + \log_{a_n}(a_n/a_1)$. | nvidia |
A rectangular room has two square holes cut out from the opposite corners. The area of one hole is four times the area of the other. If the remaining area of the room is 120 square meters, find the smallest possible length of the side of the larger square hole. | nvidia |
Let $p$ and $q$ be two positive integers such that $\gcd(p,q)=1$. The smallest integer which can be expressed in the form $p^3+q^3$, where $p$ and $q$ are relatively prime, is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 18$ | nvidia |
For positive integers $n$, let $f(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point. How many positive integer divisors does $f(2010)$ have? | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 1.$ Find the maximum value of
\[x^2y^2z^2.\] | nvidia |
The function $f(x) = \frac{x^2-7x+12}{x^2-x-2}$ has vertical asymptotes at two values of $x$. Find the sum of the values of $f(x)$ at the points where the graph of $y=f(x)$ intersects its horizontal asymptote. | nvidia |
Find the real number $x$ such that $$\sqrt{x}+\sqrt{\frac{1}{x}}=\sqrt[3]{3+\sqrt{x}}+\sqrt[3]{\frac{3}{x}+\frac{1}{\sqrt{x}}}$$ | nvidia |
Let $S$ be the set of all positive integers that are divisors of $120$ and less than $20$. Let $a$ and $b$ be two distinct elements of $S$ such that $a+b=20$. Determine the largest possible value of the least common multiple of $a$ and $b$. | nvidia |
Consider the triangular array of numbers with 0, 1, 2, 3, $\dots$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.
\[
\begin{array}{ccccccccccc}
& & & & & 0 & & & & & \\
& & & & 1 & & 1 & & & & \\
& & & 2 & & 2 & & 2 & & & \\
& & 3 & & 4... | nvidia |
A $2\times2$ matrix $\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ is called *quasimagic* if the sum of the entries in each row and column is equal to $20$. Determine all values of $t$ such that the matrix $\begin{pmatrix} t & 7 \\ 3t & 5 \end{pmatrix}$ is quasimagic. | nvidia |
Let $x$, $y$, and $z$ be distinct real numbers that sum to $0$. Compute \[ \dfrac {xy+yz+zx}{x^2+y^2+z^2}. \] | nvidia |
Find the product of all composite numbers between 1 and 15 that are not divisible by 3. | nvidia |
Find the number of divisors of $2^{12} \cdot 3^7 \cdot 5^3$ that are divisible by $2^4 \cdot 3^2$ but not by $5^2$. | nvidia |
Given that the point $(4,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=3f(4x)+5$. What is the sum of the coordinates of that point? | nvidia |
Find the smallest positive integer n such that the product of 3.2 and n is greater than or equal to 12. | nvidia |
What is the least positive integer with exactly $10$ positive factors? | nvidia |
Let $g(x)$ be a function such that
$$g(x) =
\begin{cases}
|x|+2 &\text{if }x<0, \\
x^2+1&\text{if }x\ge 0.
\end{cases}
$$
Determine the value of $a$ such that $g(a) = g(-a)$. | nvidia |
Find the smallest positive integer $n$ such that $2^n-1$ is relatively prime to $168$ and is greater than $15$. | nvidia |
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