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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 number of ordered pairs of integers $(x,y)$ such that when $x$ is divided by $y$, the quotient and remainder are 2-digit numbers whose digits are the same but in reverse order. | nvidia |
A portion of the graph of $y = G(x)$ is shown in red below. The distance between grid lines is $1$ unit.
Compute $G(G(G(G(G(1)))))$.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tick... | nvidia |
A goat is tied to a point on the circumference of a circle with radius $r$, and the length of the leash is $L$. Determine the area within the circle that the goat cannot access if the length of the leash is less than the radius of the circle. [asy]
draw(Circle((0,0),5),black+linewidth(1));
draw((0,0)--(5,0)--(4,3),blac... | nvidia |
The number $n$ is a three-digit positive integer and is the sum of the squares of three distinct integers $a$, $b$, and $c$, where $a$, $b$, and $c$ are each less than 10. What is the largest possible value of $n$ such that $a$, $b$, and $c$ are in an arithmetic progression? | nvidia |
Determine all values of $x$ that satisfy the equation $|x + 2| = \sqrt{3x + 5}$. | nvidia |
Let $g(x)$ be a function defined for all positive real numbers satisfying the conditions $g(x) > 0$ for all $x > 0$ and
\[g(x + y) = \sqrt{g(xy) - 1}\]for all $x > y > 0.$ Determine $g(2010).$ | nvidia |
Find the largest positive integer $k$ such that there exists a positive integer $n$ satisfying the equation $5n^2+12n+k=0$. | nvidia |
Find the number of distinct positive integers that can be represented as the sum of the digits of a number in the form $2^n$ where $n$ is a positive integer less than or equal to $12$. | nvidia |
A positive integer $n$ has the property that its square has exactly 5 divisors. What can be said about the number of divisors of $n^3$? | nvidia |
Let's define a sequence of positive integers $a_n$ such that
$$a_{n+1} = a_{n-1} \cdot a_n$$
for all $n \geq 2$. Given that $a_1 = 2$ and $a_2 = 3$, compute the sum of the exponents of the prime factorization of $a_{10}$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $\sqrt{x} + \sqrt{y} = 5$ and $x + y = 23$. Find the value of $\sqrt{x \cdot y}$. | nvidia |
Given a triangle $ABC$ with vertices $A(-3,1)$, $B(7,1)$, and $C(5,-3)$, find the length of the longest median of the triangle. | nvidia |
Three consecutive integers are such that the sum of the integer and the square of the next integer is 46 less than twice the square of the third integer. What is the value of the smallest integer? | nvidia |
Find the value of $x$ such that the product of $3^x$ and $2^x$ is equal to the square root of $2^{24} \cdot 3^{24}$. | nvidia |
Find the smallest five-digit number that is divisible by 21. | nvidia |
Let $Q$ be the product of the first $50$ positive even integers. Find the largest integer $m$ such that $Q$ is divisible by $2^m$. | nvidia |
A cube has six faces, each of which is a different color. The colors are blue, green, red, yellow, white, and black. Each face shares an edge with four other faces. The probability that no two adjacent faces have colors that are next to each other in the rainbow (ROYGBIV) is $p/q$, where $p$ and $q$ are relatively prim... | nvidia |
Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12},$ find all possible values for $x.$
(Enter your answer as a comma-separated list.) | nvidia |
Let x and y represent two natural numbers such that the sum of the reciprocals of x and y is equal to 1/10. Determine the values of x and y for which the quantity \(\sqrt{x^{2}+y^{2}}\) is minimized. | nvidia |
Let $x$ be the greatest integer such that $x^{2}$ is less than $t^{2}+t$ and $x^{2}$ is a multiple of $t$. Find the least value of $t$ that satisfies the equation $x^{2}=2\left(t^{2}+t\right)-3$. | nvidia |
What is the minimum number of distinct integers that must be selected from the set {1, 2,..., 100} to ensure that there are at least two integers whose difference is a multiple of 7? | nvidia |
Let $ABCD$ be a cyclic quadrilateral inscribed into a circle with center $O$, and $EFGH$ be the quadrilateral formed by the midpoints of the sides of $ABCD$. Prove that the angle between the diagonals of $EFGH$ equals the angle between the diagonals of $ABCD$. | nvidia |
Let $x, y, z$ be positive real numbers such that $xyz = 1$. Find the minimum value of
\[x^2\left(y+\frac{1}{y}\right)+y^2\left(z+\frac{1}{z}\right)+z^2\left(x+\frac{1}{x}\right).\] | nvidia |
A square consists of red and green stones arranged in a certain pattern. Each row and column contains an odd number of stones, and the total number of stones in the square is divisible by 6. Prove that the number of red stones and the number of green stones are congruent modulo 6. | nvidia |
A polynomial with rational coefficients has all the numbers \[1+\sqrt{2}, \; 2+\sqrt{3}, \; 3+\sqrt{4},\; \dots, \; 1000+\sqrt{1001}\]as roots, but with the additional condition that the coefficients of the polynomial must be integers. What is the smallest possible degree of such a polynomial? | nvidia |
Given that $x$ is a positive real number and $f(x) = \sin^4 x + \cos^4 x + 1$, find the value of $x$ for which $\sqrt{f(x)} = \frac{\sqrt{6}}{2}$. | nvidia |
Let $p$ be a prime number and $a$ be an integer such that $p$ does not divide $a$. For any positive integer $n$, let $S(n)$ denote the sum of the digits of $n$ when written in base $p$. Let $m$ be a positive integer such that $a^{m-1} + a^{m-2} + \cdots + a + 1$ is divisible by $p$. Prove that $S(a^m) = S(a)$. | nvidia |
Find the sum of all numbers from $1$ to $6$ using only the digits $1,2,3,4,5,6$ once. | nvidia |
Determine the number of divisors of 180 whose sum of digits is a perfect square. | nvidia |
Given the point $F=(f,0)$ and a line $y=x-3$, find a point $P$ on the line such that the distance between $F$ and $P$ is minimal. | nvidia |
Let $g(x)=\sqrt{x+1}$. Find the product of all $x$ that satisfy the equation $g(x^2)=g^{-1}(x)$. | nvidia |
Let $a$ and $b$ be positive real numbers such that $2a + 7b = 35.$ Find the maximum value of $ab.$ | nvidia |
Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\ldots+k^2$ is a multiple of $225$. | nvidia |
In triangle $ABC,$ if the angle bisector of $\angle BAC$ intersects side $\overline{BC}$ at point $D$ and the circumcircle of triangle $ABC$ at point $E$, then find the value of $\frac{BD}{DC}\cdot\frac{CE}{EB}$. | nvidia |
Consider the polynomial $Q(x)=x^4+dx^3+ex^2+fx+g$ such that the sum of the squares of its zeros is equal to the difference between the product of its zeros taken two at a time and the mean of its zeros.
If the leading coefficient of $Q(x)$ is 1 and the $y$-intercept is $g=12$, determine $f$. | nvidia |
If $x$ is a real number such that $3\log_{2}{x}=15$, find the value of $2x$. | 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 |
Let $f(x)$ be the number of divisors of a positive integer $x$. $\mathcal{T}$ is the set of positive integers such that for all elements $n$ in $\mathcal{T}$, we have that $f(n)=6$ and $1\le n\le 1000$. If $m$ is the number of elements in $\mathcal{T}$, compute $f(m)$. | nvidia |
The moon has a surface area that is 1/5 that of Earth. The surface area of the Earth is 200 square acres. The land on the moon is worth 6 times that of the land on the Earth. If the total value of all the land on the earth is 80 billion dollars, what is the total value in billions of all the land on the moon? | nvidia |
A polynomial $P(x)$ of degree $n$ is defined as the product of $n$ linear factors. The coefficients of the polynomial are all integers, and each factor is of the form $(x+a_i)$, where $a_i$ is a positive integer. Given that the polynomial has exactly $n$ positive integer roots, and that the sum of the coefficients of t... | nvidia |
Find the length of the shortest path from the origin to a parabola that passes through the points $(2,4),$ $(6,12),$ and $(4,16),$ assuming the path can only consist of line segments that are either horizontal or vertical. | nvidia |
In a triangle, the sum of the lengths of the two legs is 18 cm, and the difference between the lengths of the legs is 4 cm. Find the length of the hypotenuse. | nvidia |
The polynomial $P(x)$ satisfies $P(1) = 1$ and $P(\alpha)P(\alpha^{-1}) = 1$ for all complex numbers $\alpha \neq 0$. Prove that $P(x) = x^n$ for some integer $n$. | nvidia |
The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $c$. (The grid lines are spaced one unit apart.)
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownle... | nvidia |
Determine all positive integers $n$ such that
\[\prod_{k = 0}^n \left( 1 + \frac{1}{k!} \right) \le \frac{8}{7}.\] | 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 |
What is the largest integer that can be added to $\frac{1}{7}$ so that the resulting sum is less than $\frac{3}{4}$? | nvidia |
Find the largest possible value of the sum $x^2 + y^2$ given that $x$ and $y$ are integers with a product of 1000. | nvidia |
Find the degree of the polynomial with rational coefficients that has the roots \(\sqrt[3]{2}+2\), \(\sqrt[3]{2}\omega+2\) and \(\sqrt[3]{2}\omega^2+2\), where \(\omega\) is a primitive third root of unity, and all of its coefficients are integers. | nvidia |
Find the smallest positive integer that is a multiple of the product of three distinct composite numbers. | nvidia |
Given that $a$ and $b$ are nonzero real numbers such that $a+\frac{1}{b}=20$ and $b+\frac{1}{a}=\frac{3}{7},$ find all possible values for $a.$ | nvidia |
Five cards are numbered $1, 2, 3, \cdots, 5,$ respectively. Each of two players randomly selects and keeps two of the cards, and sums those two values. The probability that both players obtain an even sum is $a/b,$ where $a$ and $b$ are relatively prime positive integers. Find $a+b.$ | nvidia |
Find a positive integer $k$ such that the numbers $\sqrt{12+k}$, $\sqrt{96+k}$, and $\sqrt{216+k}$ are three consecutive terms of a geometric progression. | nvidia |
Find the smallest positive integer that is divisible by all the prime numbers less than 20 and leaves a remainder of 1 when divided by 30. | nvidia |
In a convex polygon, the sum of the interior angles is 5400 degrees. The polygon has one interior angle of 180 degrees, and the rest of the interior angles are equal. Find the number of sides of the polygon. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) - y) = g(x^2 + y) + 2g(x) y\]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 |
Let $f(n)$ be a function defined as follows:
$f(n) = \left\{ \begin{aligned} &n^2+1& \text{ if } n \text{ is a prime number} \\ &f(a) + f(b)& \text{ if } n \text{ is a composite number and } n = a \times b \text{ is a prime factorization of } n. \end{aligned} \right.$
Find the smallest positive integer $n$ such that ... | nvidia |
Determine the sum of all single-digit replacements for $x$ such that the number ${45{,}x29}$ is divisible by 9. | nvidia |
Let p be the product of all the positive integers less than or equal to 13. Determine n if n is the smallest integer such that pn is divisible by 19. | nvidia |
Let $A$ be a set of $8$ people, such that there are $4$ pairs of friends among them. Let $B$ be the set of all possible subsets of $4$ people from $A$ that contain exactly one friend from each pair. If all $4$-person subsets from $A$ are chosen at random and equally likely, what is the probability that the chosen subse... | nvidia |
How many 4-digit positive integers exist that satisfy the following conditions: (A) Each of the first two digits must be 1, 4, or 5, and (B) the last two digits cannot be the same digit, and (C) each of the last two digits must be 5, 7, or 8? | nvidia |
Suppose that $a$ and $b$ are positive integers for which $a$ has $3$ factors and $b$ has $a$ factors. If $b$ is divisible by $a$, then what is the least possible value of $b?$
Another problem:
Find the least positive integer $n$ such that it has exactly $6$ divisors, and its largest divisor is $3$ times its smallest ... | nvidia |
Find the number of all non-congruent squares modulo 12 among the first 500 positive integers. | nvidia |
Find the number of real solutions to the equation $\sqrt{x^2+1} = \sqrt{x}+1$ where $x$ is a positive number. | nvidia |
What is the minimum number of integer-coordinate points that can be graphed in the coordinate plane such that there are four collinear points and no three points form an equilateral triangle? | nvidia |
The function $g(x)$ satisfies
\[g(xy) = g(x) + g(y)\]for all positive real numbers $x$ and $y.$ Given that $g(2) = 1,$ find $g(8).$ | nvidia |
Let $a$ and $b$ be positive real numbers such that the equations $x^2 + ax + 2b = 0$ and $x^2 + 2bx + a = 0$ have roots that satisfy the following condition: the sum of the squares of the roots of the first equation is equal to the product of the roots of the second equation. Find the largest possible value of $a - b$. | nvidia |
Find the largest integer $n$ such that $6n + 1$ is a multiple of $13$ and $8n - 1$ is a multiple of $11$. | nvidia |
Find all values of $b$ such that $b+\lceil b\rceil=17.8$. | nvidia |
A bag contains 4 coins, each of which is either fair or has both sides tails. Two of the coins are chosen at random and flipped. Given that both coins land tails, the probability that the other two coins in the bag are also both tails is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n... | nvidia |
Find the value of $x$ such that $\log_2(x+2) + \log_2(x-2) = 5$. | nvidia |
The sum of the squares of two consecutive numbers is 12345. What are the two numbers? | nvidia |
Find the range of all possible values of $m$ such that the graphs of the functions $f(x)=mx-2$ and $g(x)=m^2x+m-1$ intersect on the x-axis. | nvidia |
Given a square $ABCD$ of side length 20 cm, a smaller square $XYZW$ is drawn inside it such that the vertices $X$, $Y$, $Z$, and $W$ lie on the sides $AB$, $BC$, $CD$, and $DA$ of the larger square, respectively. If the area of the shaded region is three times the area of the smaller square, find the length of the side... | nvidia |
The product of two positive integers minus their difference is 45. The integers are relatively prime, and each is less than 30. What is the sum of the two integers? | nvidia |
Find all real numbers $a$, $b$, and $c$ so that the expression $(a + bi)^2 - (a - bi)(c + bi)$ is a real number. Enter all the solutions, separated by commas. | nvidia |
The area of a square equals the square of a length of the side of the square. The perimeter of a square equals the sum of the lengths of all four sides. The sum of the areas of two squares is 65, while the difference in their areas is 33. Find the sum of their perimeters. | nvidia |
In the system of equations
\begin{align*}
5x-2y&=5,\\
-3x+ay&=-2,
\end{align*}
the value of $a$ allows the system to have exactly one solution $(x,y)$ with integer components.
Determine $a$. | nvidia |
Find the number of positive integers less than 100 that are divisible by 3 but not by 7. | nvidia |
Suppose that $x$ and $y$ are positive integers such that $(x+yi)^2 = 15+20i$. What is $x+yi$? | nvidia |
Determine the number of ways to arrange the letters of the word COMBINATION such that the vowels appear in alphabetical order. | nvidia |
Find the sum of all positive integer values of $k$ such that the equation $3x^3-2x^2+kx-1=0$ has at least one real root in the interval $[-1,1]$. | nvidia |
Determine all possible values of $t$ such that the expression $\sqrt{\frac{t^2+4}{t^2-16}}$ is defined and nonzero. | nvidia |
Consider the sequence of numbers $\{x^2\},$ $\lfloor x^2 \rfloor,$ and $x^2,$ where $\{x^2\} = x^2 - \lfloor x^2 \rfloor.$ Given that $x$ is nonzero, determine the value of $x$ such that this sequence is an arithmetic progression. | nvidia |
Determine $a$ such that the equation $$ax^3-6x^2+5x+a=0$$ has two roots satisfying $x_1x_2=2$ and $x_1+x_2=5$. | nvidia |
A square $ABCD$ is inscribed in a semicircle with diameter $d$ so that vertex $A$ lies on the diameter and vertices $B$ and $C$ lie on the semicircle. It is given that the perimeter of the square is equal to $d$. Find the ratio of the area of the square to the area of the semicircle. | nvidia |
Consider the set of all complex numbers of the form $a+bi$, where $a$ and $b$ are integers, that can be expressed in base $2+i$ using the digits $0,1,2,\ldots,4$. That is, the equation
\[a+bi=d_n(2+i)^n+d_{n-1}(2+i)^{n-1}+\cdots+d_1(2+i)+d_0\]is true for a unique choice of non-negative integer $n$ and digits $d_0,d_1,\... | nvidia |
Find the sum of all $x$ such that the expression $\frac{x^2-3x+2}{x^2-3x-4}$ is undefined. | nvidia |
Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $3(6x+1)\equiv 4\pmod p$.
New Problem:
Find the smallest positive integer $n$ such that there exists a polynomial $f(x)$ with integer coefficients, satisfying the condition that $f(x) \equiv 0 \pmod{x^2-1}$ ... | nvidia |
Let $a$ be the number of positive multiples of $9$ that are less than $45$. Let $b$ be the number of positive integers that are less than $45$, and a multiple of $3$ and a multiple of $5$. Compute $(a - b)^2$. | nvidia |
Given that $a$ and $b$ are two real numbers such that $ab>0$ and $a\neq b$, and that the complex numbers $1$, $a+bi$, and $a-bi$ form an isosceles triangle in the complex plane, find the relationship between $a$ and $b$. | nvidia |
Suppose that $(u_n)$ is a sequence of real numbers satisfying
\[u_{n+2}=2u_{n+1}+u_n\]and that $u_3=9$ and $u_6=128$. What is $u_5$? | nvidia |
A museum has a collection of artifacts, some of which are ancient and some of which are rare. If there are three times as many artifacts that are ancient as there are artifacts that are rare, 15 artifacts that are both ancient and rare, and 10 artifacts that are neither ancient nor rare, how many artifacts are in the c... | nvidia |
A point $P$ lies on the circle centered at $(4,0)$ with radius $5$. Another point $Q$ lies inside this circle. Find the maximum value of the sum of the distances from $P$ to $(0,0)$ and from $Q$ to $(2,0)$, given that $P$ and $Q$ lie on the same line passing through the center of the circle. | nvidia |
A sequence is defined recursively by
\[b_0=1-\cos\left(\frac{3\pi}{14}\right)\]and
\[b_{n+1}=\frac{1}{4}\left(1+\sqrt{2b_n-1}\right)^2\]for $n\ge0$. Find the smallest positive integer $n$ such that $b_n=b_0$. | nvidia |
Determine the number of solutions of the equation
$$\cos x = \frac{1}{x}$$
on the interval $(0, 2\pi)$ that are in the form of $\pi/2 + n\pi$, where $n$ is an integer. | nvidia |
In the quadratic equation $ax^2 + bx + c = 0$, the coefficients $a$, $b$, and $c$ are positive integers such that the product of the roots is equal to the sum of the roots plus 3. Determine the value of $b$ when $a$ is equal to 1 and $c$ is equal to 5. | nvidia |
Suppose $x$ and $y$ are positive integers such that $\gcd(x,y)$ is divisible by exactly $5$ distinct primes and $\mathop{\text{lcm}}[x,y]$ is divisible by exactly $20$ distinct primes.
If $x$ has fewer distinct prime factors than $y$, then $x$ has at most how many distinct prime factors? | nvidia |
Find the number of positive integers less than 720 that are either a perfect square or a perfect fifth power. | nvidia |
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