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Let $\cal A$ be the set of all nonnegative integers, and let $T$ be the set of all pairs of real functions defined on $\cal A$ such that for every function $f$ in $T$, there exist a function $g$ in $T$ and a positive integer $N$ such that for all integers $n>N$, $f(n)=|g(n)-f(n-1)|$ and $f(n)\ne g(n)$. Determine the ma... | nvidia |
Let $a$ be a positive real number. Determine the smallest value of $a$ for which the expression $a^{\sqrt{a}}$ is an integer. | nvidia |
A certain positive integer $n$ has the property that $2n+1$ is a perfect square. What is the smallest possible value of $n$ such that $n$ itself is also a perfect square? | nvidia |
Let $a,$ $b,$ $c,$ and $d$ be positive real numbers such that $abcd=1$. Determine the largest possible value of
$\frac{a}{1+a} + \frac{b}{1+b} + \frac{c}{1+c} + \frac{d}{1+d}$ | nvidia |
The digits of a four-digit number $ABCD$ are such that the product $AB \cdot CD$ is divisible by 12. What is the largest possible value of the digit $A$? | nvidia |
A circle with center $C$ has a radius of length $r$. A right triangle $ABC$ is inscribed in the circle, with $A$ and $B$ on the circle and $C$ as the right angle. A square is inscribed in the triangle, with one side on the hypotenuse and the other sides on the legs of the triangle. If the area of the square is $\frac{1... | nvidia |
What is the minimum number of lines that can be drawn through three given points on a plane such that each point is touched by exactly one line, and no two lines intersect at any of the points? | 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 |
Let A be the event that the number on the first die is greater than the number on the second die, let B be the event that the number on the second die is greater than the number on the third die, and let C be the event that the number on the third die is greater than the number on the fourth die.
(a) Evaluate P ( A an... | nvidia |
Given that $x^2 - 3mx + 2m^2 + 1 = 0$ has two distinct real roots, find the smallest possible value of $m$ so that the roots differ by exactly 3. | nvidia |
Find the smallest number whose sum of the squares of its proper divisors is greater than the number itself. | nvidia |
The areas of three concentric circles are in the ratio 1:4:9. The sum of their perimeters is 48π cm. Find the area of the largest circle. | nvidia |
If $a$ and $b$ are positive integers such that $(a+bi)^2(a-bi)^2 = 10$, what is $a+bi$? | nvidia |
If the point $(4,8)$ is on the graph of $y=f(x)$, and $k(x)=(f(x))^3$ for all $x$, then there is one point that must be on the graph of $y=k(x)$. What is the sum of the coordinates of that point? | nvidia |
Given the points $A(0,0)$, $B(2,3)$, and $C(4,6)$, find the point $D(x,y)$ such that the sum of the squares of the distances from $D$ to $A$, $B$, and $C$ is minimized. | nvidia |
Compute the number of nondecreasing sequences of positive integers $a_1 \leq a_2 \leq a_3 \leq a_4 \leq a_5 \leq a_6 \leq a_7$ such that $1 \leq a_i \leq 4$ for $1 \leq i \leq 7$. | nvidia |
In $\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:
$\textbf{(A)}\ 1:3 \qquad \textbf{(B)}\ 3:4 \qquad \textbf{(C)}\ 4:3 \qquad \textbf{(D)}\ 3:1 \qquad \textbf{(E)}\ 7:1$
A line intersects two circ... | nvidia |
The function $f(x)$ takes positive real numbers to real numbers, such that
\[f(x) + f(y) = f \left( \frac{x+y}{2} \right)\]for all positive real numbers $x$ and $y.$ Find all possible values of $f(20).$ Enter all possible values, separated by commas. | nvidia |
Determine the value of $b$ such that the polynomial $x^3 - 2x^2 - bx + 1$ can be factored as the product of two binomials and a monomial. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a^2+b^2=2$, $b^2+c^2=3$, and $c^2+a^2=5$. What is the value of the product $abc$? | nvidia |
A pentagon has three sides of length 2, 3, and 4 units, and the lengths of the other two sides are x and y units, where x and y are integers. What is the largest possible perimeter of the pentagon such that the sum of the lengths of any three sides is greater than the sum of the lengths of the remaining two sides? | nvidia |
Let $\tau_n(x)$ be defined as follows:
$\tau_1(x)=x^2$
For $k\ge2$,
$\tau_k(x)=\tau_1(\tau_{k-1}(x))=\tau_{k-1}^2(x)$
Find all $z$ such that $\tau_4(z)=z+168960$. | nvidia |
Let $x$ be a real number in the interval $(0,1)$ such that the decimal representation of the number $x^2$ ends in $0.\overline{2}$. Find the sum of all possible values of $x$. | nvidia |
Suppose that $(a_n)$ is a sequence of real numbers satisfying
\[a_{n+1}=a_n^2+a_{n-1}\]and that $a_1=1$ and $a_3=5$. What is $a_2$? | nvidia |
What is the sum of all the positive integer divisors of 24 that are also perfect squares? | nvidia |
Find all possible values of $b$ such that the graph of the exponential function $y=b^x$ intersects the graph of the circle $x^2 + (y - 1)^2 = 1$ at exactly one point. | nvidia |
The sum of the squares of the roots of the equation $x^3+ax^2+bx+c=0$ is $25$. Find all possible values of $b$ if $a$ and $c$ are integers. | nvidia |
Find the value of $x$ in the expression $2^{4x} \cdot 3^{-2x}$ when it is known that $\left(\frac{2}{3}\right)^{6x} = \frac{1}{64}$. | nvidia |
Circle $B$ has its center at $(-6, 2)$ and a radius of $10$ units. What is the sum of the $y$-coordinates of the two points on circle $B$ that are also on the $y$-axis? | nvidia |
Let $z$ be a complex number such that
$$z^3-12z^2+47z-60=0.$$
Determine the least possible value of $|z-1|$. | nvidia |
Consider the matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, where $a, b, c,$ and $d$ are real numbers. Suppose that the matrix $A$ satisfies the equation $A^2 = kI$, where $I$ is the identity matrix and $k$ is a positive integer. Determine the value of $k$ if it is known that the trace of the matrix $A$ (th... | nvidia |
Consider a quadrilateral with vertices at $(0,0)$, $(6,5)$, $(9,2)$, and $(3,-3)$.
Prove that this quadrilateral is a parallelogram and find the point of intersection of its diagonals. | nvidia |
Find the sum of the values of $y$ that satisfy the equation $|2y| + 5 = 23$. | nvidia |
In a triangle with side lengths $a$, $b$, and $c$, where $c$ is the longest side, the cosine of the largest angle is given by $\frac{a^2+b^2-c^2}{2ab}$. Determine the number of triangles with integer side lengths between 1 and 100 for which the cosine of the largest angle is a rational number. | nvidia |
Determine the sum $a + b$ of real numbers $a$ and $b$ if the polynomial $x^5+ax+b$ is divisible by $(x-1)^2(x^2+1)$. | nvidia |
[asy] draw(circle((0,6sqrt(2)),2sqrt(2)),black+linewidth(.75)); draw(circle((0,3sqrt(2)),sqrt(2)),black+linewidth(.75)); draw((-8/3,16sqrt(2)/3)--(-4/3,8sqrt(2)/3)--(0,0)--(4/3,8sqrt(2)/3)--(8/3,16sqrt(2)/3),dot); MP("B",(-8/3,16*sqrt(2)/3),W);MP("B'",(8/3,16*sqrt(2)/3),E); MP("A",(-4/3,8*sqrt(2)/3),W);MP("A'",(4/3,8*s... | nvidia |
Find the sum of the squares of the roots of the equation $x^6-3x^4+2x^2-1=0$ that satisfy $|x|>1$. | nvidia |
Find the product of all positive integers less than 100 that are cubes of perfect cubes. | nvidia |
Find the number of positive integers that are both a perfect square and a triangular number, where a triangular number is a number that can be represented as the sum of consecutive integers, starting from 1. | nvidia |
Given that the product of two positive integers' greatest common divisor and least common multiple is 900, if one integer is 60, find the smallest possible value of the other integer. | nvidia |
A set of positive integers contains exactly $20$ integers that are divisible by $3$. If there are no consecutive integers in the set that are both divisible by $3$, what is the maximum number of integers in the set that can be less than or equal to $100$? | nvidia |
Consider a circle with equation $x^2+y^2=r^2$, where $r$ is a positive integer. For how many values of $r$ does the line $y=2x$ intersect the circle at exactly two points, and the line $y=x+1$ intersects the circle at exactly one point? | nvidia |
The sum of the squares of two consecutive positive integers can be represented as $(n+1)^2+n^2=2n^2+2n+1$. Show that this expression can also be written as $N^2+(N+1)^2$ for some integer $N$. What is the smallest value of $N$? | nvidia |
Determine all possible real values of $a$ for which the expression $\frac{1}{\sqrt{a - x}} + \frac{1}{\sqrt{x - a}}$ can be simplified into a single function for all real values of $x$. | nvidia |
Let's denote by $R_n$ the decimal number containing $n$ consecutive $7$'s. For example:
$R_2=77, \qquad R_3=777, \qquad R_5=77777.$
Determine $\gcd(R_{2001},R_{2000})$. | nvidia |
Find the number of ordered pairs of integers $(a,b)$ with $2\le a<b\le 200$ such that $a^2 - b^2$ is divisible by 10. | nvidia |
Find the largest positive real number $k$ such that for all nonnegative real numbers $a$ and $b$, the inequality
\[\frac{a+b}{3} \ge k\sqrt{ab} + |a-b|\]
holds. | nvidia |
Find the maximum value of $\sqrt{x} - x\sqrt{x},$ over all real numbers $x > 0.$ | nvidia |
Let $a$ and $b$ be two positive real numbers such that $a + b = 50.$ Find the ordered pair $(a,b)$ for which $a^3 b^4$ is maximized. | nvidia |
Let $A$ be a subset of $\{1, 2, 3, \ldots, 2004\}$ such that $|A \cap \{1, 2, 3, \ldots, k\}| \geq \frac{1}{2}k$ for all $k$. What is the minimum value of $n$ such that every $n$-element subset of $A$ contains at least two elements that are relatively prime? | nvidia |
Let $S$ be the set of integers between $1$ and $2^{40}$ whose decimal expansions have exactly two digits with a value greater than 5. If a number is chosen at random from $S,$ the probability that it is divisible by 7 is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | nvidia |
Find the sum of two positive integers $x$ and $y$, where $x$ is the smallest positive integer such that $x^2 + 1$ has only two positive divisors, and $y$ is the largest positive integer less than 100 such that $y^3 + 1$ has exactly three positive divisors. | nvidia |
What is the largest positive integer that is not the sum of a positive integral multiple of $10$ and a perfect square greater than 1? | nvidia |
Determine the value of $a$ such that the polynomial $ax^3 + (a^2 - 2)x^2 - 5ax + 1$ has exactly two distinct real roots. | nvidia |
Simplify the expression $|a^2+b| - |a^2-b|$ if $a$ is a positive real number and $b$ is a negative real number. | nvidia |
Let $a$ and $b$ be positive real numbers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{ab} = 2$ and $a^2 + b^2 = 10$. Find the value of $a+b$. | nvidia |
In $\triangle PQR$, point $S$ is the midpoint of side $QR$. Point $T$ is on $PQ$ such that $PT:PQ =2:5$. Point $U$ is on $PS$ such that $PU:US=2:1$. If the area of $\triangle STU$ is 25, determine the area of $\triangle PQR$. | nvidia |
Let $x$ be a positive real number such that $x^3=5000+27x$. Express $x$ as a decimal. | nvidia |
Find the length of the segment whose endpoints are (-3, 4) and (5, -2) if it is known that the midpoint of the segment lies on the line y = 2x + 1. | nvidia |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}^n = \mathbf{I}.\] | nvidia |
Let $\{a_n\}$ be a sequence such that $a_1=2$ and for all $n\geq 2$, $a_n=\frac{a_{n-1}}{a_{n-1}+n}$. Compute $\sum_{n=1}^{\infty}a_n$. | nvidia |
Find the sum of the distinct divisors of $n^4 - n^2$, where $n$ is a positive integer such that $n^2$ is a perfect square.
(Note: This problem requires a different approach than the original problem, as it involves finding the divisors of a given expression rather than prime factors.) | nvidia |
Let $x, y,$ and $z$ be positive real numbers such that $xyz = 32$. Find the minimum value of
\[x^2 + 4xy + 4y^2 + 2z^2.\] | nvidia |
Let $f(x)=3x-2$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f(x^{2})$. | nvidia |
Compute the number of ways to arrange the letters in the word "COMPUTER" if all the vowels must be together and all the consonants must be together. | nvidia |
The function g(t) satisfies
\[g(t \cdot s) = g(t) + g(s)\] for all real numbers t and s. Find all possible values of g(1). | nvidia |
In the set of real numbers $\{0,1,c,\frac{1}{c}, d\}$, it is known that the arithmetic mean of the five numbers equals $c$, and the arithmetic mean of the first four numbers equals $\frac{1}{d}$. Determine the geometric mean of the set $\{1,c,\frac{1}{c}\}$. | nvidia |
The product of two positive integers plus their sum is 95. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | nvidia |
Consider a function $f(x)$ defined on the set of non-negative integers, such that for every positive integer $k$, the equation $\sum_{i=0}^n f(i) = kn$ has a unique solution $n$.
Find all possible values of $f(100)$. | nvidia |
How many positive three-digit integers less than 500 have at least two digits that are the same? | nvidia |
The vertices $A$, $B$, $C$, and $D$ of a tetrahedron are located on the axes of a Cartesian coordinate system and have integer coordinates. Given that the vertex $D$ is located at $(0,0,d)$ and the faces $ABC$ and $ABD$ have the same area, determine the minimum value of the volume of the tetrahedron. | nvidia |
In the rectangle $ABCD$, the point $E$ is chosen on the side $AB$ such that the area of the triangle $ADE$ is equal to the area of the triangle $BCE$. Find the length of $AE$, given that $AB = 20$ and $AD = 12$. | nvidia |
In the triangle $ABC$ with angle $A=30^{\circ}$, the product of the lengths of the segments that the angle bisector of $A$ divides the side $BC$ into equals the square of the length of the segment that the angle bisector of $C$ divides the side $AB$ into. Find the measure of angle $B$. | nvidia |
The hypotenuse of a right triangle has a length of 10 units, and one of the legs has a length of 6 units. What is the maximum value of the area of the triangle that can be achieved by adjusting the length of the other leg? | nvidia |
In a convex quadrilateral, the sum of the squares of the diagonals is twice the sum of the squares of the sides. Find the measure of the angle formed by the diagonals, in degrees. | nvidia |
Find all possible values of the real number $c$ such that the function $g : \mathbb{R} \to \mathbb{R},$ defined by $g(x) = c - |x|,$ satisfies the functional equation $$g(g(x)) = g(2x)$$for all $x.$ | nvidia |
A collection of coins consists of pennies, nickels, dimes, and quarters. The total value of the coins is $25, and the total number of coins is 75. If the number of pennies is greater than the number of nickels, the number of nickels is greater than the number of dimes, and the number of dimes is greater than the number... | nvidia |
Find the smallest positive whole number $n$ such that the inequality $2n^2 + 5n - 3 > 0$ is satisfied for all values of $n$. | nvidia |
The sum of three consecutive positive odd integers is twice their product. What is the sum of the reciprocals of the three integers? | nvidia |
Find all real solutions to $x^3+(x+1)^3+(x+2)^3=(x+3)^3$. Enter all the solutions, separated by commas. | nvidia |
Find the value of $k$ such that the sum of the squares of the roots of the cubic equation $x^3 - 2x^2 + kx + 1 = 0$ is equal to 15. | nvidia |
If $A$, $B$, $C$, and $D$ are four complex numbers such that $A+B+C+D=i$, $A-B=C-D=2i$, and $A^2+C^2=10$, find $B^2+D^2$. | nvidia |
Find the smallest positive integer $k$ for which $k^2$ is divisible by 24 and $k^3$ is divisible by 1080. | nvidia |
An infinite array is constructed as follows:
\begin{align*} 2 \qquad 4\,\ \qquad 6\,\ \qquad 8 \qquad &\cdots\\ 1 \qquad 2\,\ \qquad 3 \qquad &\cdots\\ \frac 12 \qquad 1\qquad &\cdots\\ \frac 14 \qquad &\cdots\\ &\ddots \end{align*}
In general, the first entry of each row is $\frac{1}{2}$ times the first entry of the p... | nvidia |
Find the number of positive integers $n$ such that $n^2 + 2n + 1$ is a perfect square and $50 < n < 250$. | nvidia |
Let $x$ and $y$ be two positive real numbers such that their sum is 10 and the sum of their squares is 68. What is the product $xy$? | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xy) + f(xz) - f(x) f(yz) \ge 1\]for all real numbers $x,$ $y,$ and $z.$ | nvidia |
Consider a square grid of numbers where each number is the sum of the two numbers above it and the number to its left. The top row consists of the sequence of natural numbers starting from 1. The leftmost column also consists of the sequence of natural numbers starting from 1. Find the sum of the numbers in the 50th ro... | nvidia |
Let $a,b,c$ be positive integers satisfying the system of equations:
\[\begin{aligned} a+b+c&=26 \\ ab+bc+ac &= 165 \\ abc &= 900.\end{aligned}\]
Find the maximum value of the sum $|a-b|+|b-c|+|c-a|$ among all solutions $(a,b,c)$ of the system. | nvidia |
Find the smallest positive integer x that satisfies the congruence relation:
4x ≡ 11 (mod 23). | nvidia |
Find the least possible value of $|a-b|$ where $a$ and $b$ are distinct positive integers such that $a^b+b^a$ is a perfect square and $\gcd(a+b,27)=1$. | nvidia |
Given the function $f(x)=3x^3+2$, find the value of $x$ so that $f^{-1}(x)=4$. | nvidia |
Find the sum of all values of $x$ such that the expression $\frac{x^3-3x^2-4x+12}{x^2-3x-4}$ is equal to zero. | nvidia |
The values $p$ and $q$ are chosen so that the quadratic equations $x^2+px+5=0$ and $x^2+qx+6=0$ have the same discriminant. Determine the number of possible integer values for $p+q$. | nvidia |
Let $S$ be the set of positive integers $n$ such that $\sqrt{n}$ can be represented as the sum of two polynomials of non-negative integer coefficients with the same variable $x$, evaluated at $x = i$. That is, there exist polynomials $f(x)$ and $g(x)$, with only non-negative integer coefficients, such that $\sqrt{n} = ... | nvidia |
Find the sum of the squares of the coefficients in the expansion of $(x+y)^n$ if the sum of the coefficients is 1024 and the sum of the coefficients in the expansion of $(x-y)^n$ is 32. | nvidia |
Given the equation $x^3 + ax^2 + bx + c = 0$, determine the value of $a$ such that the sum of the squares of the roots is equal to the product of the roots plus 5. | nvidia |
The three roots of the cubic $ 30 x^3 - 50x^2 + 22x - 1$ are distinct real numbers strictly between $ 0$ and $ 1$. If the roots are $p$, $q$, and $r$, what is the sum
\[ \frac{1}{1-p} + \frac{1}{1-q} +\frac{1}{1-r} ?\] | nvidia |
In triangle $ABC$, the area of the largest side is $x\%$ more than the area of the smallest side, while the area of the middle side is $y\%$ less than the area of the smallest side. If the perimeter of the triangle is 60 cm, find the ratio of the areas of the largest and smallest sides, given that $x = 20$ and $y = 10$... | nvidia |
Two concentric circles with radii $r_1$ and $r_2$, where $r_1 > r_2$, are given. A line passing through the origin intersects the larger circle at $A$ and $B$, and the smaller circle at $C$ and $D$, such that $AB = 2CD$. Find the smallest positive value of the ratio $\frac{r_1}{r_2}$ for which this condition holds. | nvidia |
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