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Given a positive integer n>2 determine the largest integer less than
log_{n}[(n^2+1)/(n^2-1)] | nvidia |
In a large office building, there are 23 elevators, each with a different capacity. The building manager wants to assign a specific number of people to each elevator such that the total number of people is maximized, but no elevator is overloaded. Given that the capacities of the elevators are consecutive integers from... | nvidia |
Let \(x\) and \(y\) be two positive real numbers such that \(x+y=\frac{1}{2}\). Determine the product \(xy\) if the following equation holds:
\[\frac{1}{x}-\frac{1}{y}+3\log_2(x)-7\log_3(y) = \frac{5}{2}.\] | nvidia |
The sum of the cubes of two positive integers is 100. The difference of the two integers is 5. What is the product of the two integers? | nvidia |
The vertices of a triangle are reflected across a line, resulting in a new triangle that is congruent to the original one. The coordinates of the vertices of the original triangle are $(2,3)$, $(4,5)$, and $(6,7)$, while the coordinates of the corresponding vertices of the reflected triangle are $(8,9)$, $(10,11)$, and... | nvidia |
A biased coin is flipped $n$ times, where $n$ is a positive integer. The probability of getting heads on any given flip is $p$, where $0 < p < 1$. It is observed that the probability of getting an even number of heads is exactly $\frac{1}{2}$. What is the largest possible value of $n$ for which this condition holds? | nvidia |
The equation $x^3+2x^2+bx+c=0$ has two real solutions that satisfy $x_1^2+2x_1+x_2^2+2x_2=-1$. Find $c$. | nvidia |
Determine the number of possible integer side lengths that can form a right triangle with legs of 8 units and 15 units. | nvidia |
The sum of the digits of a positive integer is equal to 24. What is the smallest number that can be added to this integer to make it a palindrome? | nvidia |
Let set $\mathcal{B}$ be a 20-element subset of $\{1,2,3,\ldots,100\},$ and let $T$ be the product of the elements of $\mathcal{B}.$ Find the number of possible values of $T$ that are perfect squares. | nvidia |
Determine the sum of the cubes of the exponents of the prime factors of the number $2^3 \cdot 3^2 \cdot 5$. | nvidia |
The sum of the squares of the roots of the equation
$$x^3-3x^2+2x-1=0$$
is equal to 3.
Determine the product of the roots. | nvidia |
Given that $a$, $b$, and $c$ are positive integers, find the smallest integer value for $b$ such that $(ab+c)^2 \ge 123456$. | nvidia |
There is a unique two-digit positive integer $t$ for which the last two digits of $11\cdot t$ are $36$.
What is $t$? | nvidia |
Three fair coins are tossed, and the numbers $x,y,z$ of heads obtained are recorded. Find the probability that $x+y+z$ is a prime number. | nvidia |
Let $a,$ $b,$ $c,$ and $d$ be positive real numbers such that $a+b+c+d=1$. Find the minimum value of
$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}.$$ | nvidia |
Let $a$, $b$, $c$ be real numbers such that
$$(a+b+c)(ab+bc+ca)=3,$$
$$a^3(b+c)+b^3(a+c)+c^3(a+b)=3.$$
Find $a+b+c$.
Can you solve it? | nvidia |
What is the smallest positive integer $m$ such that $m^2$ has exactly $5$ distinct positive divisors? | nvidia |
A circle $C$ with center $(2,3)$ and radius $\sqrt{5}$ is given. There are real numbers $p$ and $q$ such that the line with slope $n$ passing through the origin does not intersect $C$ if and only if $p<n<q$. Find $p+q$. | nvidia |
A sequence of squares is inscribed in a circle with radius 10. The area of the first square is 36 square units, and the area of each subsequent square is 75% of the area of the previous square. What is the total area of the infinite sequence of squares? | nvidia |
What is the maximum number of points that can be chosen in the interior of a unit cube such that the distance between any two of them is at least 1/3? | nvidia |
A square and a circle are drawn inside a rectangle. The area of the square is 49 square meters, and the diameter of the circle is equal to the length of the rectangle. If the width of the rectangle is 3 meters less than the length of the rectangle, find the area of the circle in terms of π. | nvidia |
Let $m$ and $n$ be positive integers such that $5m+7n$ is a multiple of $12$. Find the largest value of $5m+7n$ that is less than $425$. | nvidia |
A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ participants, of the numbe... | nvidia |
If $\alpha$ and $\beta$ are acute angles such that
$\sin{\alpha}=\cos{\beta},$
$\cos{\alpha}=1-\sin{\beta},$
and $\sin{\alpha}<\cos{\alpha},$ find the value of $\tan{\frac{\alpha}{2}}.$ | nvidia |
Find the product of all positive odd factors of 729 that are not perfect squares. | nvidia |
If $a$, $b$, and $c$ are positive integers such that $a^4 \cdot b^2 \cdot c^3$ can be simplified to $m^3 \cdot n^2$ for some integers $m$ and $n$, what is the minimum possible value of $a+b+c+m+n$? | nvidia |
The vertices of a parallelogram are $A(0,0)$, $B(t,3)$, $C(t^2,6)$, and $D(t^3,9)$, where $t$ is a real number. Find the smallest possible area of this parallelogram. | nvidia |
Let $P$ be the product of the first $100$ positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k .$
| 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 |
Find the sum of the squares of the reciprocals of the positive divisors of 120. | nvidia |
Find the number of ordered pairs $(x,y)$ of complex numbers such that
\[x^4 y^3 = x^2 y^7 = -1.\] | nvidia |
A function $f$ defined on the positive integers is said to be scattered if $f(x) + f(y) > \lceil \log_2 (xy) \rceil$ for all positive integers $x$ and $y.$ Let $h$ be a scattered function such that $h(1) + h(2) + \dots + h(10)$ is as small as possible. Compute the minimum possible value for $h(8).$ | nvidia |
Given the function $g(x) = \sin(x^n)$, where $n$ is a positive integer, determine the set of values of $n$ such that the range of $g(x)$ on the interval $[0, 2\pi)$ is the entire interval $[-1, 1]$. | nvidia |
Find the smallest number whose digits, when multiplied together, produce a product of 1440. The digits of the number must all be different, and the number cannot begin with 0. | nvidia |
What is the smallest positive integer $n$ such that the product of $n$ and a given integer $m=2345$ is a number whose digits are in increasing order? | nvidia |
Find the smallest possible value of a positive integer $m$ such that $m$ is the product of three distinct prime factors $p$, $q$, and $r$, where $p$, $q$, and $r$ are consecutive prime numbers and $m$ is a four-digit number. | nvidia |
The polynomial $P(x)=ax^3+bx^2+cx+d$ has the property that $P(x)P(-x)=P(x^2)$. If $a$ and $d$ are positive integers, what is the minimum possible value of $a+d$? | nvidia |
Find the number of non-congruent convex pentagons with all sides being straight line segments and all vertices lying on the grid of squares of the Cartesian coordinate system, whose perimeter is equal to $2010$ units and area is a positive integer. | nvidia |
In the Cartesian plane, a point $(x,y)$ is called a "center of symmetry" if there exist two distinct points $(a,b)$ and $(c,d)$ such that the midpoint of the segment connecting $(a,b)$ and $(c,d)$ is $(x,y)$, and the slopes of the lines passing through $(x,y)$ and $(a,b)$, and $(x,y)$ and $(c,d)$, respectively, are neg... | nvidia |
Find the largest number $k$ such that the equation
$$x^2-kxy+y^2=1$$
has no integer solutions for $x$ and $y$. | nvidia |
Let $a$ and $b$ be two real numbers such that $|a|=|b|=1$ and $a+b=-1.$ Find the exact value of the expression $|ab-2|.$ | nvidia |
Find the smallest positive four-digit number that is a perfect square and has exactly three distinct prime factors. | nvidia |
Determine the smallest positive integer whose digits sum to a prime number and is divisible by 11. | nvidia |
Let's find all three-digit positive integers whose cubes begin with the digits 729. | nvidia |
Determine $x$ such that $5^{x-1}+5^{x+1}=132$. | nvidia |
Find the sum of all constants $k$ such that the quadratic $x^2 + kx + 9$ can be expressed as $(x+a)^2 + b$, where $a$ and $b$ are integers and $a$ is non-zero. | nvidia |
The sum of the cubes of two positive integers is 306. The difference of the squares of the two integers is 63. What is the sum of the two integers? | nvidia |
Find the number of positive integers $b$ less than $16$ such that the equation $by \equiv 3\pmod{16}$ has no solutions in $y$. | nvidia |
What is the sum of the squares of the three consecutive integers starting from the smallest positive integer that is not a factor of the number 30? | nvidia |
Two coins are tossed simultaneously, and the probability that at least one of them shows a head is equal to the probability that they both show tails plus the probability that only one of them shows a head. What is the probability that each of the coins shows a head? | nvidia |
Find the sum of all integer solutions to $|n-1| < |n-5| < 12$. | nvidia |
Find the value of $a$ such that the equation $x^2 + 2ax + a^2 = 0$ has only one solution. | nvidia |
How many different $6\times 6$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0, the sum of the entries in each column is 0, and the sum of the entries in the main diagonal is 0? | nvidia |
Consider a square matrix $M$ whose entries are integers, such that its determinant is $0$.
Let $N$ be a matrix obtained by interchanging two rows of $M$.
Determine all possible values of the determinant of $N$. | nvidia |
The product of two prime numbers is 391. Find the sum of these two prime numbers. | nvidia |
What is the smallest number of digits $n$ such that there exists a number in base-$n$ representation that contains at least one digit greater than 1 and can be read the same forwards and backwards? | nvidia |
Determine $a$, given that the polynomial $(x-a)^2(2x+a)$ has exactly three distinct real roots and that the sum of the roots is equal to 0.5. | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that $x+y+z = 6$. Compute the maximum value of
\[\frac{xy+yz+zx}{xyz}.\] | nvidia |
Consider the cubic equation $$2x^3 - 9x^2 + 15x - 8 = 0.$$ Find the sum of all the positive integer roots of this equation that are less than 4. | nvidia |
Given a sequence of numbers defined by $a_n = 2^{n-1} + 1$, find the value of $n$ such that the geometric mean of the first $n$ terms is equal to the arithmetic mean of the first $n-1$ terms. | nvidia |
How many four-digit numbers can be formed using the digits $1, 2, 3, 4$ such that the product of the first and last digits is equal to the product of the middle two digits? | nvidia |
Consider a cubic polynomial $Q(x)$ with real coefficients such that
\[Q(x^2+x)\leq Q(x^3+1)\]
for all real numbers $x$. Find the product of the roots of $Q(x)$. | nvidia |
Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\sin x + \cos x = \sqrt{2}.$ Enter all the solutions, separated by commas. | nvidia |
When three positive integers are divided by $47$, the remainders are $25$, $20$, and $3$, respectively.
When the sum of the three integers is divided by $47$, what is the remainder? | nvidia |
Let $P$ be a point inside triangle $ABC$ such that
$\overrightarrow{PA}=\frac23\overrightarrow{PB}$ and
$\overrightarrow{PC}=\frac23\overrightarrow{PD}$,
where $D$ is the intersection point of lines $BP$ and $AC$.
Find the ratio of the area of triangle $ABC$ to the area of triangle $APC$. | nvidia |
Find the smallest positive integer $c$ for which $x^2+cx+2009$ factors into a product of two binomials, each having integer coefficients. | nvidia |
Find the largest positive integer $n$ such that
\[\sin^n x + \cos^n x \ge \frac{1}{n}\]for all real numbers $x.$
Determine the value of $a$ for which the inequality
\[\frac{1}{\sin x + \cos x} + \frac{1}{\sin x - \cos x} \ge a\]holds true for all real numbers $x$. | nvidia |
Find the sum of all $x$ that satisfy the equation $\frac{-9x}{x^2-1} = \frac{2x}{x+1} - \frac{6}{x-1}.$ | nvidia |
Given that the parabola $y=ax^2+bx+c$ passes through the points $(-1,7)$, $(5,7)$, and $(6,10)$, determine the value of $a+b+c$. | nvidia |
What is the largest number that divides all numbers of the form $3n^2+1$, where $n$ is a positive integer? | nvidia |
The sum of the reciprocals of two consecutive positive integers is 7/40. What is the product of the two integers? | nvidia |
Let $f(n)$ be the product of all the positive integers less than or equal to $n$. If $f(6) + 2 = 720$, calculate the sum of all integers $n$ such that $f(n)$ is a multiple of $11$. | nvidia |
Let $n$ be a positive integer, and let $b_0, b_1, \dots, b_n$ be a sequence of real numbers such that $b_0 = 25$, $b_1 = 40$, $b_n=0$, and $$ b_{k+1} = b_{k-1} - \frac{5}{b_k} $$for $k = 1,
2, \dots, n-1$. Find $n$. | nvidia |
Let $n$ be a positive integer such that all the digits in its decimal representation are prime digits. Let $P(n)$ be the product of these digits. Find all $n$ such that $P(n)$ itself is a prime number. | nvidia |
Find the sum of the square roots of all the perfect squares between 10 and 100. | nvidia |
What is the smallest positive integer $n$ such that the sum of the first $n$ even positive integers is greater than 1000? | nvidia |
Find the sum of all possible positive integer values of $b$ such that the quadratic equation $2x^2 + 5x + b = 0$ has rational roots. | nvidia |
Let $f(x)=\frac{x+1}{x^2+2}.$ Prove that $|f(x)|<\frac{\sqrt{6}}{2}$ for any real number $x$, and show that $\sqrt{6}/2$ is the best upper bound by giving an example of a sequence $(x_n)$ of real numbers such that $\lim_{n\to\infty}|f(x_n)|=\frac{\sqrt{6}}{2}$. | nvidia |
Find the smallest positive integer $m$ such that
\[\begin{pmatrix} -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{pmatrix}^m = \mathbf{I}.\] | nvidia |
The expression $\sqrt[3]{2700}$ can be simplified as $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ is as small as possible. Now, consider the equation $x^3 = 2700$. How many solutions for $x$ are there such that $x$ is a positive integer? | nvidia |
In $\triangle PQR$, point $S$ is the midpoint of side $QR$. Point $T$ is on $PR$ such that $PT:TR = 2:3$. Point $U$ is on $PS$ such that $PU:US = 5:2$. If the area of $\triangle STU$ is 24, determine the area of $\triangle PQR$. | nvidia |
The function $f(x)$ satisfies
\[f(x - y) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(x) \neq 0$ for all real numbers $x.$ Find $f(3).$ | nvidia |
Let $g(x)$ be a function defined for all positive integers $x$, satisfying the conditions
$$g(x) g(y) = g(x + y + xy)$$
for all positive integers $x$ and $y$, and
$$g(x) \neq g(y)$$
whenever $x \neq y.$ Find $g(7).$ | nvidia |
Find the largest value of $k$ such that the inequality $\frac{x^2}{11}+\frac{y^2}{13}+\frac{z^2}{17}\geq k(x^2+y^2+z^2)$ holds for all real numbers $x$, $y$, and $z$. | nvidia |
Let $A$ and $B$ be two points on the positive part of the coordinate plane such that $A$ has integer coordinates, $B$ is located to the right of the vertical line $x=2012$, and $\angle AOB = 45^\circ$, where $O$ is the origin. Determine the smallest possible value of $OB$. | nvidia |
A sequence of real numbers is defined recursively as follows:
$$x_1 = 2019,$$
$$x_n = \frac{1}{\sqrt{x_{n-1}}} + 1$$for all $n \ge 2$. Compute the smallest positive integer $k$ for which $x_k$ is an integer. | nvidia |
Let $n$ be a positive integer such that the decimal expansion of $\sqrt{n}$ has only finitely many digits to the right of the decimal point.
Determine the smallest such $n$ that is not a perfect square. | nvidia |
Let $a$ and $b$ be nonnegative real numbers such that
\[\cos (ax + b) = \cos 17x\]for all integers $x.$ Find the smallest possible value of $a.$ | nvidia |
Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O,$ and that the ratio of the volume of $O$ to that of $C$ is $\frac mn,$ where $m$ and $n$ are relatively prime integers, find $m+n.$
| nvidia |
Consider the expression $\frac{(x+y)^3-(x-y)^3}{2y}$. Find the sum of the coefficients of the resulting polynomial after simplifying the expression. | nvidia |
Find the number of integers m that satisfy the inequality -12√2 ≤ m ≤ 8√2. | nvidia |
The product of the digits of a three-digit number is 5. The sum of the digits on either side of any digit is 8. Determine the number. | nvidia |
Find the constant $b$ such that the expression $\log_{z^8}{w}\cdot\log_{w^4}{z^3}\cdot\log_{z^2}{w^7}\cdot\log_{w^7}{z^2}\cdot\log_{z^3}{w^4}$ can be written as $b\log_z{w}$. | nvidia |
Find the positive integer $n$ such that
\[\sin \left( \frac{\pi}{2n} \right) + \cos \left (\frac{\pi}{2n} \right) = \frac{\sqrt{n}}{2}.\] | nvidia |
A certain operation * is defined on real numbers such that
a*b=a+b+\sqrt {a}-\sqrt {b}.
If
x*(3x+1)=x+\sqrt {x},
then compute the value of x. | nvidia |
Suppose that $a$ and $b$ are integers such that $$3b = 8 - 2a.$$How many of the first six positive integers must be divisors of $2b + 12$? | nvidia |
Find the sum of all integers $x$ such that the cube root of $3x$ is greater than 2 and less than 3. | nvidia |
Determine the smallest five-digit palindrome that is a perfect square, knowing that its third digit is a prime number. | nvidia |
Find the number of distinct solutions to the equation $\sqrt{x^2+4x+4} = \sqrt{x^2-2x+1}$, where $x$ is a real number. | nvidia |
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