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Consider the equation $x^3 - 2x^2 + mx + n = 0$ with real coefficients. It is known that the sum of the squares of the roots is equal to 7. Determine the value of m. | nvidia |
Let $a_1, a_2, \dots$ be a sequence for which $a_1=2$, $a_2=3$, and $a_n=\frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$?
Another problem:
Consider a sequence of integers $a_1, a_2, a_3, \dots$ such that $a_1 = 1$ and for all $n \ge 2$, $a_n$ is the smallest positive integer that ha... | nvidia |
Consider the inequality
\[a^3 + b^3 + c^3 \ge k(a + b + c)(ab + bc + ca),\]
where $a,$ $b,$ $c$ are nonnegative real numbers and $k$ is a real number.
Find the smallest real number $k$ such that the inequality holds for all possible choices of $a,$ $b,$ $c.$ | nvidia |
Find the sum of the solutions of: $\sqrt{x^2-2x+5}=2(\sqrt{x^2-2x+5}-1)$ | nvidia |
Select $n$ pairs $\{x_i,y_i\}$ with $x_i<y_i$ from the set of integers $\{1,2,3,\dots,2019\}$, ensuring that no two pairs share an element and that each sum $x_i+y_i$ is distinct and does not exceed 2019. What is the largest possible value of $n$? | nvidia |
Find the sum of all real values of $x$ that satisfy
\[x = \frac{1}{1-x} + \frac{1}{(1-x)^2} + \frac{1}{(1-x)^3} + \dotsb.\] | nvidia |
Find the smallest positive integer $k$ such that the product of the first $k$ positive integers is divisible by $13^3$. | nvidia |
Let $p$ be a prime number. There are $p$ different complex numbers $w$ such that $w^p=1$. For how many of these is $w^{\frac{p-1}{2}}$ a real number? | nvidia |
Let $P$ be a set of 10 distinct positive integers. Suppose that each element of $P$ is a divisor of the sum of the other 9 elements. Find the smallest possible value of the sum of the 10 elements in $P$. | nvidia |
$n$ dice are rolled simultaneously. The probability that at most two of them show a 6 is $\frac{3}{8}$. Find $n$. | nvidia |
Find the maximum value of the sum of the lengths of all the edges of a rectangular solid with a given volume, if its three dimensions form an arithmetic progression. | nvidia |
Find the sum of all positive integral values of $n$ for which $\frac{n+6}{n}$ is an integer. | nvidia |
In a regular hexagon, three semicircles are drawn on the sides. The radii of the first two semicircles are 5 units and 3 units, and the radius of the third semicircle is a positive integer number of units. The semicircles are placed such that the diameter of the third semicircle lies on the line that passes through the... | nvidia |
A box contains twelve marbles, some of which are blue and the rest of which are green. When two marbles are drawn at random at the same time, the probability that both marbles are blue is $\frac{1}{22}$. How many marbles in the box are blue? | nvidia |
Determine $k$ such that the difference of the roots of the equation
$$x^3-4x^2+kx-7=0$$
be equal to $3$.
Can you help me solve it? | nvidia |
The sum of the cubes of the integers from 1 to $n$ is given by the formula $\frac{n^2(n+1)^2}{4}$. Find the smallest positive integer $n$ such that the sum of the cubes of the integers from 1 to $n$ is divisible by 12, but not by 24. | nvidia |
Find the smallest positive integer that is both an integer power of a prime number and has exactly 3 distinct digits when written in base 10. | nvidia |
Consider a function $f(x)$ that satisfies the following property: for every real number $x$ in its domain, the number $1 - x$ is also in its domain and $f(x) + f(1 - x) = x^2$. Determine the largest possible set of real numbers that can be in the domain of $f$. | nvidia |
Find the smallest positive integer $c$ for which $x^2 + cx + 2019$ can be expressed as the product of two binomials with integer coefficients. | nvidia |
The sum of the squares of two consecutive positive integers is 365. What is the sum of the reciprocals of these integers? | nvidia |
Determine the smallest positive integer $m$ such that
$$\left(\frac{x+y+z}{3}\right)^m\ge \frac{x^m+y^m+z^m}{3}$$ for any nonnegative real numbers $x$, $y$ and $z$ | nvidia |
Given that the polynomial $x^3 + ax^2 + bx - 5$ has three distinct rational roots, find the sum of all possible values of $b$. | nvidia |
The vertices of a triangle are $(x_1, y_1), (x_2, y_2),$ and $(x_3, y_3)$. The midpoints of the sides of the triangle are $(0,0), (1,2),$ and $(3,1)$. Find the sum of the coordinates of the vertex that is opposite to the side with the midpoint at $(0,0)$. | nvidia |
Given that $f(x) = x^3$, find $x$ such that $f(x) = f(f^{-1}(6)) + 1$. | nvidia |
Let $f(x)$ be a cubic polynomial such that $f(x) + f(2x) + f(3x) = 2x^2 - 1$ for all real numbers $x$. Find $f(5)$. | nvidia |
Find the number of pairs of non-congruent rectangles with integer side lengths that have an area of 25 square units. | nvidia |
Find the smallest positive integer n such that the product of its digits is equal to 5!. | nvidia |
The function $f(x)$ satisfies
\[f(x) + f(y) = f(x+y) + f(x-y)\]for all real numbers $x$ and $y.$ Find all possible values of $f(x).$ | 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 $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d$ be two distinct polynomials with real coefficients such that the product $f(x)g(x)$ is a perfect cube. Determine $a+b+c+d$ if $f(1)=g(1)=2$. | nvidia |
The ratio of two positive integers is 3:4 and their sum is 105. What is the value of the smaller of the two integers? | nvidia |
In a regular hexagon, how many quadrilaterals can be formed whose vertices are the vertices of the hexagon and have at least one pair of opposite sides that are parallel? | nvidia |
A hotel has 78 rooms, each with a different number of guests. The total number of guests in all the rooms is 178. If the number of guests in each room is a single-digit number (in base 9), and the sum of the digits of the room numbers (also in base 9) is 7, find the total number of rooms that have exactly 3 guests. | nvidia |
Let $x$ be a positive real number. Find the minimum value of $4x^5 + 5x^{-4}.$ | nvidia |
In the equation
$$\frac{1}{x^2+1}+\frac{1}{(x-1)^2+1}+\frac{1}{(x-2)^2+1}=\frac{3}{2}$$
determine $x$ such that the difference between the largest and smallest roots is equal to 2. | nvidia |
Determine the smallest integer $n$ such that $\lfloor\sqrt{n}\rfloor+\lfloor\sqrt{n+1}\rfloor+\lfloor\sqrt{n+2}\rfloor$ is equal to 12. | nvidia |
Find the smallest whole number greater than 200 that has exactly 4 positive factors. | nvidia |
What is the greatest integer less than $\sqrt{300}$ that can be expressed as the sum of two distinct perfect squares? | nvidia |
If $n$ is a positive integer such that
$\lfloor \sqrt{2} n\rfloor -\lfloor n/\sqrt{2}\rfloor = 15$,
find $n$. | nvidia |
Find the value of $r$ such that the sphere with equation $x^2+y^2+z^2-4x-8y+14z-75=0$ intersects the plane $2x-3y+5z=10$ in a circle of radius $r$ centered at the point $(3,2,1)$. | nvidia |
Let $ABCD$ be a convex quadrilateral. The midpoints of its sides are connected to form a smaller quadrilateral $PQRS$. It is known that the diagonals of quadrilateral $ABCD$ intersect at a point lying inside the quadrilateral. Prove that the diagonals of the quadrilateral $PQRS$ are also intersecting and find the ratio... | nvidia |
The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the the smallest possible value of $m+n.$ | nvidia |
Determine the number of positive integers $n$ such that the equation
\[\lceil x \rceil + \lceil 2x \rceil + \lceil 3x \rceil = n\]has exactly two solutions for $x \in (0, 1).$ | nvidia |
Find the radius of the largest circle that can be inscribed in a right-angled triangle with legs of length 5 and 12, and hypotenuse of length 13. The center of the circle must lie inside the triangle. | nvidia |
In the equation $x^2+4x-12=0$, the roots $x_1$ and $x_2$ are such that $x_1^2+x_2^2=16$. Determine the value of $x_1+x_2$. | nvidia |
Find the number of ordered pairs $(x,y)$ of complex numbers such that
\[x^4 y^3 = x^2 y^7 = 1.\] | nvidia |
Given that $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ are unit vectors such that $\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0},$ find all possible values of $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}).$ | nvidia |
Find the sum of the reciprocals of the divisors of $120$. | nvidia |
Given the equation
$$5x^2-kx+1=0$$
determine $k$ such that the difference of the roots be equal to unity.
Write another problem inspired by this one:
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 |
Let $f(x)$ be a function such that
\[f(x) + f(y) = f(x+y) + 2009\]for all real numbers $x$ and $y.$ If $f(5) = 2014,$ find $f(15).$ | nvidia |
The sum of the areas of two circles is equal to the area of a circle with a diameter of 30 inches. One of the circles has a radius of 5 inches, while the other circle's radius is an integer number of inches. Determine the radius of the second circle. | nvidia |
Let $S$ be the set of all positive integer divisors of $10!$ that are not divisible by 3. For how many elements $n$ of $S$ does the decimal representation of $\frac{10!}{n}$ terminate? | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-px+q=0$$
are such that
$(x_1)^2$ and $(x_2)^2$ are the roots of another quadratic equation
$$x^2-rx+s=0$$
If $q$ and $s$ are positive integers with the same prime factorization, determine $p$. | nvidia |
Find the number of real solutions of the equation
\[\frac{x}{100} = \sin x.\] | nvidia |
Find the sum of all integer divisors of 120 that are also factors of 20, and determine the number of such divisors. | nvidia |
Let $a$, $b$, and $c$ be real numbers such that $a+b+c=10$ and $ab+ac+bc=25$. Compute the maximum integer value of
\[\frac{a^3+b^3+c^3}{abc}.\] | nvidia |
Find the number of different remainders that $2^n$ can leave upon division by $12$ as $n$ ranges over the positive integers. | nvidia |
Find the number of positive integers less than 100 that are multiples of both 3 and 5, but not of 7. | nvidia |
Let $n$ be a positive integer and $x_1, x_2, \ldots, x_n$ be $n$ real numbers. Find all possible values of $n$ for which there exist $x_1, x_2, \ldots, x_n$ satisfying
$$\sum_{i=1}^n x_i = \sum_{i=1}^n x_i^2 = \sum_{i=1}^n x_i^3 = 1.$$ | nvidia |
How many different 3-digit positive integers exist that have at least one 5 in them? (Note that we don't allow "3-digit" integers that start with 0, such as 012.) | nvidia |
Let $a$, $b$, $c$, and $d$ be real numbers such that the equation
\[x^4+ax^3+bx^2+cx+d=0\]
has roots $-1$, $-1$, $2$, and $2$. Find the sum of the roots of the equation
\[x^4+2ax^3+(a^2+b)x^2+(ab+c)x+abc+d=0.\] | nvidia |
Find the product of all real values of $x$ such that the function $$g(x) = \frac{x}{x-\frac{x^2}{x-\frac{x^3}{x}}}$$ is undefined. | nvidia |
Given that $p$ and $q$ are prime numbers and $pq<100$, find the number of distinct prime factors of $N=p^{10}+q^7$. | nvidia |
Problem:
In a regular polyhedron, the number of edges is equal to the number of faces. Determine the number of vertices of this polyhedron. | nvidia |
Let $\alpha$ be a real number such that $0<\alpha<1$. Let $S$ be the sum of $\lfloor n\alpha \rfloor$ for all positive integers $n$ such that $1\leq n \leq \frac{1}{\alpha}$. Find $\alpha$ if $S=\frac{1}{2}(\frac{1}{\alpha}-1)(\frac{1}{\alpha}+1)$. | nvidia |
If $m$ and $n$ are positive 3-digit integers such that $\gcd(m,n)=5$, what is the smallest possible value for $\mathop{\text{lcm}}[m,n]$? | nvidia |
What is the largest positive integer that has exactly six distinct positive factors and can be expressed as the sum of two perfect squares? | nvidia |
Let $a$, $b$ and $c$ be the positive integer roots of the equation $x^3-17x^2+51x-13=0$.
What is the least possible value of the sum $|a-b|+|b-c|+|c-a|$? | nvidia |
In the 3-dimensional space consider the set $C$ of points $(x,y,z)$ such that $x$, $y$, $z$ are non-negative real numbers and the following conditions hold:
\begin{align*}
&2x^2+3y^2+z^2=1\\
&2x^2+6y^2+3z^2=1
\end{align*}
For every natural $n$, denote by $t_n$ the minimum number such that for every point $M\in... | nvidia |
Let $m$ and $n$ be two positive integers such that $m+n=48$. Given that the product of $m$ and $n$ is equal to three times the sum of their squares, find the value of $m^2+n^2$. | nvidia |
Find the number of positive integers that divide both 1200 and 1800, and also divide their difference. | nvidia |
Let $a$ and $b$ be positive real numbers. Find the minimum value of $\frac{a^2+b^2}{ab}$, given that $a + b = 1$. | nvidia |
A complex number $a+bi$ is chosen randomly from the set of all complex numbers with $-1\le a\le 1$ and $-1\le b\le 1.$ Find the probability that
\[\left( \frac34 + \frac34i \right) z\]
is also in the set of all complex numbers with $-1\le a\le 1$ and $-1\le b\le 1.$ | nvidia |
Find a positive integer that is divisible by 14 and has a square root between 25 and 25.3. | nvidia |
Determine all positive integers $n$ such that the decimal representation of $\frac{1}{n}$ contains a block of 5 zeros in a row. | nvidia |
Find the median of the set of positive divisors of 72. | nvidia |
An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the square of the length of each side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| 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 sum of all positive integer solutions to the equation $xy=77$ that satisfy the condition $x \leq y$. | nvidia |
Let $f$ be a function defined on the positive integers, such that
\[f(x+y) = f(x) \cdot f(y)\]for all positive integers $x$ and $y.$ Given $f(5) = 3$ and $f(7) = 5,$ find $f(12).$ | nvidia |
Find the smallest positive integer $n$ such that any positive integer greater than $n$ can be expressed as the sum of a prime number and a positive integer that is not coprime to $n$. | nvidia |
Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ$, find $z^{2000}+\frac 1{z^{2000}}$. | nvidia |
Find the largest positive integer $m$ such that $12m \equiv 15m \pmod{36}$ and $m$ is a multiple of 5. | nvidia |
Find all positive real numbers $x$ that satisfy
\[x \sqrt{12 - x} + \sqrt{12x - x^3} \ge 12.\]
Another problem:
Find all positive real numbers $x$ and $y$ that satisfy the inequality
\[x^2 \sqrt{y} + y^2 \sqrt{x} \ge x^3 + y^3.\] | nvidia |
In a quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at a point $O$ such that $AO/OC = BO/OD = 3/2$. The diagonals form four triangles, each with area 12. Find the area of the quadrilateral. | nvidia |
Find the positive integer $n$ such that the number $a = \sqrt{n!}$ satisfies the property that for every positive integer $m \geq n$, $\left\lfloor m^2/a\right\rfloor + \left\lfloor (m+1)^2/a\right\rfloor + \left\lfloor (m+2)^2/a\right\rfloor = \left\lfloor (m+3)^2/a\right\rfloor$. | nvidia |
Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$. | nvidia |
For how many values of $c$ is it true that:
(1) $c$ is a positive integer such that $c \le 30$.
(2) the quadratic equation $x^2 + (3c+2)x + 2c^2 = 0$ has two integer solutions? | nvidia |
An octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 108 centimeters. What is the number of square centimeters in the area of the octagon? | nvidia |
Find the sum of the first 100 positive integer multiples of 7 that do not contain the digit 2. | nvidia |
Find the product of all the solutions of the equation $\frac{x^2}{2x+1} = \frac{3}{x-1}$. | nvidia |
Let $T$ be the set of real numbers of the form $2\cos\theta$, where $\theta$ is a real number, such that
\[\frac{1}{2} \le 2\cos\theta \le 1.\]
Find the smallest positive integer $k$ such that for all positive integers $j \ge k,$ there exists a real number $t \in T$ such that $t^j = 1.$ | nvidia |
Find the sum of the squares of two consecutive integers, given that the difference of their cubes is 214. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 2.$ Find the maximum value of $a^2 b^4 c^3.$ | nvidia |
Determine the maximum number of boxes of pens that can be packed into a large box, given that each box of pens must contain a number of pens that is a positive multiple of the number of boxes of pens packed, and that there are 48 pens in total. | nvidia |
The sum of the squares of two consecutive integers is 365. What are these integers? | nvidia |
Find all values of $x$ in the interval $[0, \pi]$ such that the sequence $\sin x, \sin 2x, \sin 3x$ is geometric. Express your answer(s) in terms of $\pi$. | nvidia |
Given the system of equations
$$x+y+z=6,$$
$$x^2+y^2+z^2=14,$$
find the largest whole number $N$ satisfying the inequality $|xy+yz+zx|<N$. | nvidia |
If $n$ is the smallest positive integer for which there exist positive real numbers $a$ and $b$ such that
\[(a + bi)^n = (a - bi)^n,\]compute $\frac{b}{a}.$ | nvidia |
In a right triangle, the length of the hypotenuse is 10 cm, and one leg is 6 cm. Find the length of the other leg, knowing that the area of the triangle is 24 square centimeters. | nvidia |
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