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We have a deck of $100$ cards, with each card having a unique number from $1$ to $100$. We want to draw two cards from the deck, but with a special condition: the number on the second card should be the sum of the number on the first card and a specific constant $k$, where $k$ is a positive integer. In how many ways ca... | nvidia |
Given the equation of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, determine the $x$-coordinate of the focus that lies on the $x$-axis, assuming that the distance between the vertices is $\frac{2}{3}$ of the distance between the foci. | nvidia |
The polynomial $p(x) = x^6 + Ax^4 + Bx^2 + C$ has exactly three real roots. Determine $C$ if it is known that all the real roots of $p(x)$ are integers. | nvidia |
An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free? | nvidia |
Let $a$ and $b$ be real numbers, and let $x_n$ be a sequence of real numbers such that
$$a^n + b^n = x_{2n-1} + x_{2n}$$for all positive integers $n$. Define
$$C = \sum_{n=1}^{\infty} \frac{1}{x_{2n-1}^2} - \frac{1}{x_{2n}^2}$$and
$$D = \sum_{n=1}^{\infty} \frac{1}{a^{2n}} - \frac{1}{b^{2n}}.$$Determine $\frac{C}{D}... | nvidia |
Find the smallest positive value of $k$ such that the equation $14x^2 - kx + 504 = 0$ has integral solutions. | nvidia |
Find the smallest positive integer $b$ for which $x^2 + bx + 2008$ factors into a product of two polynomials, each having integer coefficients, and the sum of the coefficients of the two polynomials is equal to the product of the coefficients. | nvidia |
Find the integer $n$, $0 \le n \le 7$, such that \[n \equiv -3737 \pmod{8}.\] | nvidia |
Let $g(x)$ be a function such that
\[g(x+y)+x^2=g(x)+g(y)\]for all real numbers $x$ and $y.$ If $g(2) = 3$ and $g(4) = 5$, then compute $g(8)$. | nvidia |
Find the largest possible number $b$ such that for any triangle with sides of length $3$, $4$, and $5$, it is always possible to place a circle of radius $b$ inside the triangle such that the circle touches at least two sides of the triangle. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2y + xy^2 = x + y.$ Find the minimum value of $\frac{x}{y} + \frac{y}{x}.$ | nvidia |
Find the equation of the circle that passes through the points $(0,0)$, $(3,4)$, and $(6,8)$. | nvidia |
Find all values of $b$ such that $b+\lceil b\rceil=17.8$. | nvidia |
The integers $G$ and $H$ are chosen such that
\[\frac{G}{x+5}+\frac{H}{x^2-4x}=\frac{x^2-2x+10}{x^3+x^2-20x}\]for all real values of $x$ except $-5$, $0$, and $4$. Find $H/G$. | nvidia |
Compute the smallest positive integer $n$ such that
\[\sum_{k = 0}^n \log_2 \left( 1 + \frac{1}{2^{2^k}} \right) \ge 1 + \log_2 \frac{2014}{2015}.\] | nvidia |
Let $x$ be a real number such that $|x-5| \leq x^2$. Find the range of values of $x$ that satisfy this inequality. | nvidia |
Calculate the number of times 172 can be subtracted from 9975 such that the result is still a positive integer. | nvidia |
Find the number of distinct diagonals that can be drawn in a convex decagon (10-sided polygon). | 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 |
Let $\phi (x)$ be a function defined for all real $x$ and such that its inverse $\phi^{-1}(x)$ exists for all $x$ in its range.
If the graphs of $y=\phi(x)$ and $y=\phi(x-4)$ are drawn, at how many points do they intersect? | nvidia |
Let $g:\mathbb{R}\to\mathbb{R}$ be a function such that $g(1)=1$ and
$$g(x^3-y^3)=(x-y)(g(x)^2+g(x)g(y)+g(y)^2)$$
for all real numbers $x$ and $y$. Find the number of possible values of $g(2)$. | nvidia |
Compute the smallest positive integer $n$ such that
\[\sum_{k = 0}^n \log_2 \left( 1 + \frac{1}{2^{2^k}} \right) \ge 1 + \log_2 \frac{2014}{2015}.\] | nvidia |
Let $g(x)$ be a function that, given a positive integer $x$, returns the largest possible integer $m$ such that $m$ is a divisor of $x^3$ and $m$ is a perfect square. Given that $x$ is a multiple of 12, what is the smallest value of $x$ such that $g(x) > 144$? | nvidia |
Let $a$ and $b$ be two positive integers such that their sum is a multiple of their greatest common divisor $d$. Determine the least possible value of $a+b$ if $a$ and $b$ share exactly two distinct prime factors, and $\frac{a}{d}-\frac{b}{d}=1$. | nvidia |
Find the area of the triangle with vertices $(2,-4),$ $(-1,2),$ and $(0,-7).$ | nvidia |
Let $a$ and $b$ be real numbers, and let $n$ be a positive integer. Suppose that the complex number $z = a + bi$ satisfies the equation $z^n = 1$. Determine the largest possible value of $a + b$, given that $a$ and $b$ are not both equal to zero. | nvidia |
The area of a circle inscribed in a square is 14\pi cm^2 . Find the perimeter of this square in the simplified radical form a\sqrt{b}+c where a , b , and c are integers. What is the value of the sum a+b+c ? | nvidia |
Consider three complex numbers $m+ni$, $p+qi$, and $r+si$. If $n=2$, $r=m-p$, and the sum of the numbers is $4i$, find $q+s$. | nvidia |
Find the smallest positive integer that has exactly 12 factors, including 1 and itself, such that the sum of its distinct prime factors is a perfect square. | nvidia |
Let $m$ be the smallest positive value of $c$ for which $\left|\frac12-ci\right| = \frac34$ and let $n$ be the largest positive value of $c$ for which this equation is satisfied. Compute $m+n$. | nvidia |
For any positive integer $n$, the number of perfect squares greater than $n^3+1$ and less than $n^3+n^2$ is:
$\text{(A) } 0\quad\qquad \text{(B) } 1\quad\\ \text{(C) } \frac{n-1}{2} \text{ for n odd, } \frac{n}{2} \text{ for n even}\quad\\ \text{(D) } n-1\quad \text{(E) } n$ | nvidia |
Find the sum of all real values of $s$ for which the equation $\frac{1}{\sin x}=\frac{s-\cos x}{4}$ has exactly two real solutions in the interval $[0,2\pi)$. | nvidia |
Find the number of ways to choose 5 books out of 10, such that no two books are written by the same author. | nvidia |
Find the smallest possible value of $c$ given that $a,$ $b,$ $c$ form a geometric sequence with $a+b+c=12$ and all terms are positive. | nvidia |
Let $b$ and $c$ be real numbers. If the polynomial $x^2+bx+c$ has exactly one real root and $b=c+1$, find the value of the product of all possible values of $c$. | nvidia |
Given a positive integer $n$, let $S(n)$ denote the sum of the digits of $n$. For how many values of $n$ is $n+S(n)+S(S(n))=2007$? | nvidia |
What is the sum of all the integers from 1 to 100 that are not divisible by 5? | nvidia |
How many of the 200 smallest positive integers are divisible by 3 or 5?
A positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3, and a positive integer is divisible by 5 if and only if its last digit is 0 or 5. Therefore, the number of positive integers less than or equal to 200 tha... | nvidia |
Find the number of triplets of positive integers $(a, b, c)$ such that $a^2 + b^2 + c < 50$ and $a + b + c$ is a multiple of 3. | nvidia |
Compute the number of distinct subsets of size $3$ that can be formed from a set containing $10$ red balls and $15$ blue balls. | nvidia |
The arithmetic mean of five numbers is 21. Three of the numbers are 17, 24, and 20, and the remaining two numbers are equal. What is the sum of the two equal numbers? | nvidia |
Find the number of possible real roots of the polynomial
\[x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0,\]
where $a_0, a_1, a_2, a_3, a_4$ are real numbers such that $a_0 > 0$ and
\[\frac{a_1}{a_0} > \frac{a_2}{a_1} > \frac{a_3}{a_2} > \frac{a_4}{a_3}.\] | nvidia |
Let $T$ be the set of real numbers of the form $\cos\theta$, where $\theta$ is a real number, such that
\[\frac{1}{2} \le \cos\theta \le \frac{\sqrt{2}}{2}.\]Find the smallest positive integer $k$ such that for all positive integers $n \ge k,$ there exists a real number $t \in T$ such that $t^n = 1.$ | nvidia |
A regular hexagon is inscribed in a circle of radius 10. The sum of the lengths of all the segments formed by the vertices of the hexagon and the points of tangency of the circle with the sides of the hexagon can be written in the form
\[a + b \sqrt{3},\]
where $a$ and $b$ are positive integers. Find $a+b$. | nvidia |
Let $f(x)$ be a polynomial of degree $2$ satisfying
\[f(x+1) = 3f(x)-f(x-1)\]for all real numbers $x$ and $f(0)=1$, $f(1)=10$.
Find $f(2)$. | nvidia |
Find the number of perfect squares that can be represented as the sum of two perfect cubes. | nvidia |
Find $x$ such that $\log_{12}3x=2$. | nvidia |
Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $3(6x+1)\equiv 4\pmod p$. | 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 |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x^3 + y^2g(z)) = x^2g(x) + zg(y^3)\]for all real numbers $x,$ $y,$ and $z.$
Let $m$ be the number of possible values of $g(0),$ and let $t$ be the product of all possible values of $g(1).$ Find $m + t.$ | nvidia |
A sequence of real numbers $a_1, a_2, \ldots, a_n$ has the property that for any $1\leq k \leq n$, the sum of the first $k$ terms is equal to the sum of the squares of the first $k$ terms minus the sum of the cubes of the first $k$ terms. Find the sum of the squares of the first $n$ terms. | nvidia |
Compute the ordered pair of integers $(a,b)$ such that $a>1$ and \[\log_a(a+1) \cdot \log_{a+1} (a+2) \dotsm \log_{b-2} (b-1) \cdot\log_{b-1} b=2,\] where the product has exactly 2190 terms. | nvidia |
The digits of a four-digit number are arranged in a circular pattern, such that the sum of the digits adjacent to each other is a perfect square. If the digits 3 and 4 are adjacent to each other, what is the largest possible sum of the other two digits? | nvidia |
Let $m$ be a positive integer. Let $a$ and $b$ be two distinct positive integer divisors of $m$ such that the greatest common divisor of $a$ and $b$ is equal to $1$. If $a$ is the smallest divisor of $m$ greater than $1$, find the smallest possible value of $b$ such that the least common multiple of $a$ and $b$ is equa... | nvidia |
In a three-dimensional space, three points $P,$ $Q,$ and $R$ have coordinates $(x_P, y_P, z_P),$ $(x_Q, y_Q, z_Q),$ and $(x_R, y_R, z_R)$ respectively. The points $P,$ $Q,$ and $R$ are equidistant from each other, and each of the distances is equal to $\sqrt {32}.$
Determine the number of possible values of the area o... | nvidia |
Find the number of triples $(a,b,c)$ of positive integers such that the product $abc$ is equal to 900, and the sum $a+b+c$ is divisible by 5. | nvidia |
Find the largest constant $C$ so that
\[x^2 + y^2 + 4 \ge C(x + 2y)\]for all real numbers $x$ and $y.$ | 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 smallest positive integer $n$ such that the system of congruences
$$\begin{cases}
x^2 \equiv 2 \pmod n \\
x^3 \equiv 3 \pmod n
\end{cases}$$
has no solution. | nvidia |
Problem:
A polynomial $P(x)$ has three distinct positive roots $x_1$, $x_2$, and $x_3$. The product of the roots is $64$, and the sum of the reciprocals of the roots is $1$. Find the value of $P(0)$.
Note: This problem requires a different approach to solve, as it involves using the product of the roots and the sum of... | nvidia |
What is the smallest positive integer that can be written in the form $2002m + 44444n$, where $m$ and $n$ are integers? | nvidia |
Given the Pythagorean triple (3, 4, 5) representing the lengths of the sides of a right triangle, find the minimum possible area of a rectangle inscribed within this triangle if one corner of the rectangle coincides with the right angle of the triangle and the sides of the rectangle are parallel to the legs of the tria... | nvidia |
Consider the set of all triangles $ABC$ such that $AB=10$, $BC=11$, and the area of $\triangle ABC$ is 20 square units. What is the maximum number of obtuse angles that any triangle in this set can have? | nvidia |
Let $x$ be the number of positive multiples of $9$ that are less than $45$. Let $y$ be the number of positive integers that are less than $45$, and a multiple of $3$ and a multiple of $3$. Compute $(x - y)^2$. | nvidia |
Prove that the expression \[\frac{\cos 3\theta}{\cos\theta}+\frac{\sin 3\theta}{\sin\theta}\] is always equal to 4 for all values of $\theta$ where it is defined. | nvidia |
In the diagram, $\triangle ABC$ is a right triangle with $\angle B = 90^\circ$. Points $D$ and $E$ are chosen on sides $AB$ and $BC$ such that $\angle ADB = \angle BEC = 45^\circ$. Find the ratio of the area of quadrilateral $ADEC$ to the area of triangle $ABC$. | nvidia |
In triangle $APB$, $A$ and $B$ are fixed points, and $P$ is a variable point. What is the locus of all possible points $P$ such that the sum of the distances from $P$ to $A$ and $B$ is constant? | nvidia |
Simplify $((5p+1)-2p\cdot4)(3)+(4-1\div3)(6p-9)$ to a much simpler expression of the form $ap-b$ , where $a$ and $b$ are positive integers. | nvidia |
Find all $x$ such that $x^2+5x<6$. Express your answer in interval notation. | nvidia |
If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? | nvidia |
Find the sum of the values of $x$ which satisfy $x^2 +1992x = 1993$. | nvidia |
Let the complex number $z$ represent the point on the real axis. Since the distances from this point to $(2,0)$ and $(0,-4)$ are equal, it follows that $$|z-2|=|z+4i|.$$ Find $z$. | nvidia |
Let $p$ be a prime number and let $a$ be an integer such that $p \nmid a$. Prove that the equation $ax \equiv 1 \pmod{p^2}$ has a solution if and only if $a \equiv 1 \pmod{p}$. If this is the case, prove that the solution is unique modulo $p^2$. | nvidia |
Find all positive integral values of $b$ for which the only solution of the system of equations
$$
\begin{cases}
2y+1=3z\\
bz+2y=9
\end{cases}
$$
is $y=1$, $z=1$. | nvidia |
A cube has one vertex at the origin, and the three vertices adjacent to it are at the points $(2,0,0)$, $(0,3,0)$, and $(0,0,4)$. Find the coordinates of the vertex opposite the origin, given that they are all integers. | nvidia |
Determine the sum of all real numbers $x$ satisfying
\[(x^2-4x+2)^{x^2-5x+2} = 1.\] | nvidia |
Find the largest positive integer that divides $\sqrt{432} \cdot \sqrt[3]{1024} \cdot \sqrt[4]{625}$ exactly. | nvidia |
For how many positive integer values of $n$ does the equation $\sqrt{n+\sqrt[3]{n}}=m$ have a solution, where $m$ is also a positive integer? | nvidia |
Find all real numbers $x$ which satisfy \[\frac{x-1}{x-3} \ge 2.\](Give your answer in interval notation.) | nvidia |
Find the smallest possible value of the sum of two distinct prime numbers, each greater than 1000, that are also coprime to each other. | nvidia |
Consider the set of all positive integers that have exactly three digits in base $6$, and exactly $4$ digits in base $5$. What is the sum of the smallest and largest numbers in this set? | nvidia |
A set of 7 cards contains 2 cards with the number 0, 3 cards with the number 1, and 2 cards with the number 2. Two cards are drawn from the set (without replacement). What is the probability that the sum of the numbers on the two cards is less than 3? | nvidia |
How many positive integers less than $1000$ can be expressed as the sum of three distinct positive perfect squares? | nvidia |
Let $x$ be a real number such that $[x] + [2x] + [3x] = 15.$ Find $x.$ | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c=1$. Find the maximum value of $\frac{ab}{a+c} + \frac{bc}{b+a} + \frac{ca}{c+b}$. | nvidia |
Find the integer $n$, $0 \le n \le 11$, such that \[n \equiv 10389 \pmod{12}.\] | nvidia |
Find the positive difference between the two values of $x$ that satisfy the equation $\displaystyle\sqrt[3]{\frac{x^2+1}{2}}=x-1$. | nvidia |
What is the sum of all integer values $n$ for which $\binom{20}{n}+\binom{20}{10}=\binom{21}{11}$? | nvidia |
Eight positive integers are written on the faces of a regular octahedron. Each vertex is labeled with the product of the four numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $1024$, then what is the sum of the numbers written on the faces? | nvidia |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | 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 |
Find all values of $x$ such that $x + \lfloor x \rfloor = 12.5$. | nvidia |
Let $f(x)$ be a function defined for positive integers $x$, such that
$f(1)=1$ and
$f(x) = |f(x-1)-x|$, if $x>1$.
Determine $f(2007)$. | nvidia |
On square $PQRS$, point $K$ lies on side $PQ$ and point $L$ lies on side $RS$, so that $PK=KL=LS=20$. Find the area of the square $PQRS$. | nvidia |
What is the maximum value of the product $xy$ if $x$ and $y$ are positive real numbers such that $x^2 + y^2 = 5$? | nvidia |
How many three-digit positive integers exist, all of whose digits are 2's and/or 5's? | nvidia |
A perfect square is a number that can be expressed as the square of an integer. How many four-digit perfect squares are there such that the product of their digits is also a perfect square? | nvidia |
Find the smallest positive integer that has exactly four distinct positive factors and is not a perfect square. | nvidia |
Find the largest number of non-overlapping regular hexagons that can be inscribed in a circle with a radius of 8 units, such that each hexagon shares at least one vertex with the circle. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of
\[\frac{1}{a} + \frac{1}{b}.\] | nvidia |
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