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If $x$ and $y$ are positive real numbers such that $5x^2 + 10xy = x^3 + 2x^2 y,$ what is the value of $x$? | nvidia |
Determine the number of distinct subsets of the set {A, L, B, M, O, R, T} such that each subset contains exactly three elements. | nvidia |
What is the smallest positive integer $n$ such that $2^n-1$ is divisible by $5^3$? | nvidia |
The sequence $a, ar, ar^2, \ldots$ is a geometric sequence, and the sum of all the terms is $18$. The sum of the terms at even places is $3$ times the sum of the terms at odd places. Find the common ratio $r$. | nvidia |
The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Determine the value of $b$. | nvidia |
Find the smallest positive integer $k$ such that $
z^{10} + z^9 + z^6+z^5+z^4+z+1
$ divides $z^k-1$. | nvidia |
Find the common ratio of the infinite geometric series:
$$\frac{3}{4}+\frac{3}{4}x+\frac{3}{4}x^2+\dots$$
where $x$ is a real number such that $|x|<1$.
Given that the sum of the series is equal to $2$, find the value of $x$. | nvidia |
At the 2007 Math Olympics, Team Canada won $17$ out of a possible $100$ medals. Which one of the following is closest to the fraction of medals that they won? $$
\frac{1}{4} \qquad \frac{1}{5} \qquad \frac{1}{6} \qquad \frac{1}{7} \qquad \frac{1}{8}
$$ | nvidia |
Find the number of 5-digit numbers that have exactly three identical digits and start with a digit other than the repeated one. | nvidia |
Find the sum of the elements in the 10th row of Pascal's triangle that are not divisible by 3. | nvidia |
Count the number of positive integers $m$ such that there exists an integer $n$, for which $m$ is the smallest positive integer satisfying $2^n + m = 3^k$, where $k$ is a positive integer. | nvidia |
A sequence of integers $b_1, b_2, b_3, \ldots$ is chosen so that $b_n = b_{n - 1} + b_{n - 2}$ for each $n \ge 3.$ The sum of the first $m$ terms of this sequence is $m$ for every positive integer $m.$ Determine the sum of the squares of the first $m$ terms of this sequence for every positive integer $m.$ | nvidia |
A palindrome is a number that reads the same forwards and backwards, such as 3003. How many positive four-digit integers are palindromes? | nvidia |
Find the area of the triangle with vertices at (-3, 4), (0, 0), and (5, -2). | nvidia |
Find the smallest four-digit integer that is divisible by the sum of its distinct, non-zero digits. | nvidia |
How many 5-digit numbers can be formed using the digits 0-9, such that the sum of the digits is even and the number contains at least one 3 or 5? | nvidia |
Let $a,$ $b,$ and $c$ be three positive integers whose sum is 120. If $a,$ $b,$ and $c$ are consecutive terms of an arithmetic progression, then find the maximum value of the sum of their squares $a^2+b^2+c^2.$ | nvidia |
Let A denote a set of positive integers such that any element in A is divisible by at least one element less than or equal to its square root. Determine the smallest possible value of the largest element in A. | nvidia |
Given the coordinates of $A=(1,0)$, $B=(0,1)$, $C=(-1,0)$, and $D=(0,-1)$ on the unit circle centered at the origin, calculate the sum of the dot products of the vectors $\overrightarrow{OA}$, $\overrightarrow{OB}$, $\overrightarrow{OC}$, and $\overrightarrow{OD}$ with the vector $\overrightarrow{OE}=(1,\sqrt{3})$. | nvidia |
The line $y = 3$ intersects the graph of $y = 4x^2 + x - 1$ at the points $A$ and $B$. The distance between $A$ and $B$ can be written as $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are positive integers that do not share any factors other than one. Find the value of $m - n$. | nvidia |
Determine the number of positive integers $b$ less than $20$ such that the congruence $bx \equiv 2 \pmod{20}$ has a solution in $x$. | 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 |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(0) = 0$ and
\[g(x^2 + y^2) = (x + y) (g(x) - g(y))\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(1),$ and let $t$ be the sum of all possible values of $g(1).$ Find $m \times t.$ | nvidia |
A math book has a large number of pages, each having a unique number from 1 to n. However, one page is missing. Given that the sum of all the page numbers is 141, what is the number of the missing page? | nvidia |
Let $a_1,a_2,\ldots,a_n$ be real numbers greater than 1 such that $a_1\cdot a_2\cdots a_n>2^n$. Find the minimum value of
\[\frac{a_1+1}{2}+\frac{a_2+1}{2}+\cdots+\frac{a_n+1}{2}.\] | nvidia |
There are values $C$ and $D$ such that
\[\frac{Dx+15}{x^2+8x+12}=\frac{2}{x+2}+\frac{C}{x+6}.\]
Find $C+D$. | nvidia |
Find the sum of all positive divisors of $70$ that are also divisors of $21$. | nvidia |
Find the number of permutations of the letters in the word "LISTEN" such that no two vowels are adjacent to each other. | nvidia |
The product of three consecutive positive integers is equal to the sum of the cubes of the same integers. What is the largest integer? | nvidia |
The equation $x^3-4x^2+x+6=0$ has two roots whose sum is $-1$.
Find the product of the three roots of this equation. | nvidia |
Find the integer $n$, $0 \le n \le 7$, such that \[n \equiv -3737 \pmod{8}.\] | nvidia |
Let $f(x)$ be a real-valued function such that $f(xy)=f(x)+f(y)$ and $f(x^2)+f(x)=0$ for all $x\neq 0$. Determine the value of $f(123)$. | nvidia |
Find the sum of the first 10 positive integers that are one more than a multiple of 5. | nvidia |
Given that $\gcd(2a,3b) = 56$ and $\gcd(3a,4c) = 168$, where $a, b, c$ are positive integers, determine the smallest possible value of $\gcd(2b,3c)$ | nvidia |
Find the number of distinct positive integers that divide 360 and are multiples of 3. | nvidia |
There are three consecutive numbers $n$, $n+1$, and $n+2$. The square of the largest number is equal to the sum of the squares of the other two numbers. Determine the middle number. | nvidia |
How many positive integers less than 1000 have the property that the sum of their digits is 9 and the product of their digits is divisible by 7? | nvidia |
Consider a sequence of real numbers $(v_n)$ that satisfies the recursive relation:
\[v_{n+2} = 3v_{n+1} - 2v_n\]
Given that $v_4 = 24$ and $v_7 = 365$, find the value of $v_6$. | nvidia |
Determine the sum of the two smallest two-digit primes and the smallest three-digit prime. | nvidia |
Determine the value of $y$ that satisfies $\sqrt[4]{y\sqrt{y^2}}=5$. | nvidia |
The domain of the function $f(x) = \arccos(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2024}$, where $m$ and $n$ are positive integers and $m>1$. Find the the smallest possible value of $m+n.$ | nvidia |
In the equation $5x^2-kx+1=0$ determine $k$ such that the difference of the roots be equal to unity.
New Problem:
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 smallest positive integer that is a multiple of 7 and has exactly three digits that are the same when its digits are reversed. | nvidia |
Given that $\gcd(3m+1,13m-4)=1$, find the largest possible value of $\gcd(7m-11,2m+27)$. | nvidia |
A positive integer $k$ can be represented as the product of three distinct factors $a$, $b$, and $a^2+b^2$, where $a$ and $b$ are each one-digit numbers. What is the largest possible value of $k$ that is a three-digit number? | nvidia |
Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability t... | nvidia |
Find the maximum value of $\sqrt{x} - x\sqrt{x},$ over all real numbers $x > 0.$ | nvidia |
A circle has three chords of the following lengths: $11$, $15$, and $x^3$. What are all of the positive integer values of $x$ such that the circle exists and the three chords can form a triangle inside the circle? Separate your answers using commas and express them in increasing order. | nvidia |
Compute the value of $p$ such that the sum of the squares of the roots of the equation $x^3 - 12x^2 + px - 45 = 0$ is equal to 116. | nvidia |
What is the least positive integer that can be multiplied by 27 to yield a number that ends with the digit 9? | nvidia |
Let $p$ be a prime number such that $p^2-p+1$ is divisible by a perfect square greater than 1. Prove that $p^3+1$ is divisible by the same perfect square. | nvidia |
A fair six-sided die is rolled twice, and the numbers shown on the top face of the die are recorded each time. What is the probability that the sum of the two numbers is a prime number? | nvidia |
Find the value of $x$ such that $\log_2(x + 3) = 5$. | nvidia |
Consider a sequence $b_n$ defined as follows: $b_1=2$ and for each positive integer $n>1$, $b_n=2^nb_{n-1}+n^2$. Determine the smallest positive integer $n$ such that $b_n$ is divisible by $3$. | nvidia |
Find the number of four-digit numbers such that the sum of the first and last digits is equal to the sum of the second and third digits, and the sum of the first and second digits is equal to the sum of the third and last digits. | nvidia |
Let $n$ be the number of positive divisors of $x$ and $m$ be the number of positive divisors of $y$. If $\lceil{\frac{n}{m}}\rceil=2$, how many possible ordered pairs of integers $(x,y)$ are there? | nvidia |
How many six-digit positive integers have at least two consecutive digits that are the same? | nvidia |
Find the minimum value of
\[x^2 + xy + y^2\]over all real numbers $x$ and $y.$ | nvidia |
Let $x$ be a three-digit number. Define a function $f(x)$ such that $f(x)$ is the integer obtained by rotating the digits of $x$ one position to the left (e.g., $f(123)=231$). Find the least positive integer $n$ such that for any three-digit number $x$, if $x+f(x)+f(f(x))$ is divisible by $n$, then so is $f(f(f(x)))$. | nvidia |
Let $x$ be a real number such that the expression $x + \frac{x}{\sqrt{1+x^2}} + \frac{x}{\sqrt{1+\frac{x^2}{1+x^2}}} + ...$ is finite. Find all possible values of x. | nvidia |
The parabola $y=ax^2+bx+c$ has vertex $(2, -1)$. Given that $a$ and $c$ are integers and that the parabola passes through the point $(0, 5)$, find the value of $a$. | nvidia |
In triangle $ABC$, the bisectors of $\angle A$ and $\angle C$ intersect at $I$, and $AB = 24$. Extend $BI$ to intersect the circumcircle of $ABC$ at $D$. The length of arc $BCD$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$. | nvidia |
Let $x$ and $y$ be the side lengths of two rectangles with integral dimensions. Given that the ratio of their areas is $\frac{441}{400}$ and that the ratio of their perimeters is $\frac{21}{20}$, find the value of $xy$. | nvidia |
A circle with a radius of 5 units is centered at the point (3,4). Find the length of the segment that connects the point (2,3) with the center of the circle. | nvidia |
Simplify the expression $\frac{1}{1+\frac{1}{1-\frac{1}{1+\frac{1}{1-x}}}}$ and find the value of $x$ for which the expression is equal to $\frac{3}{5}$. | nvidia |
Determine the number of distinct, natural-number factors of $n$ that are also multiples of $k$, given that $n=2^3 \cdot 3^2 \cdot 5$ and $k=2^2 \cdot 3$. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 6.$ Find the minimum value of
\[\frac{1}{a} + \frac{4}{b} + \frac{9}{c}.\] | nvidia |
Let $\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a, b, c$ are distinct digits. Find the sum of the elements of $\mathcal{S}.$
| nvidia |
Determine the value of $k$ such that the sum of the digits of the number $4k27$ is a multiple of $3$.
Let $S$ denote the sum of the digits of the number $4k27$, then
$S=4+k+2+7=k+13$. In order for $S$ to be a multiple of $3$, the digit $k$ should be $2$, $5$ or $8$. | nvidia |
In the $xyz$-coordinate system, there are eight points with coordinates $(0,0,0), (3,0,0), (0,3,0), (0,0,3), (3,3,0), (3,0,3), (0,3,3)$, and $(3,3,3)$. Find the point that can be removed so that the remaining points have a greater average distance between them. | nvidia |
Find the largest three-digit positive integer that is divisible by 53. | nvidia |
Find $w$, such that $5^65^w=25$. | nvidia |
Find all real numbers $a$ such that the roots of the polynomial
$$x^3 - 6x^2 + 21x + a$$form an arithmetic progression and are not all real. | nvidia |
Let $m$ and $n$ be nonzero real numbers such that
\[(3 + 2i)(m - ni)\]is pure real. Find $\frac{m}{n}.$ | nvidia |
Four standard six-sided dice are rolled. We are told there is no three-of-a-kind, but there is a pair of dice that show the same number. These two dice are set aside, and the other two dice are re-rolled. What is the probability that after re-rolling these two dice, at least three of the four dice show the same value? | nvidia |
Let $f(x)$ be a monic polynomial of degree $n$ such that $f(x_k) = x_k^2$ for $n+1$ distinct integers $x_1, x_2, \ldots, x_{n+1}$. Find the value of $f(x_1+x_2+\cdots+x_{n+1})$. | nvidia |
A circle of radius $r$ is inscribed in the triangle with the sides $4$, $5$, and $6$. What is the area of the largest regular hexagon that can be inscribed in the circle? | nvidia |
Let $d$ be a positive number such that when $109$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. | nvidia |
Let $x$ be a positive real number such that $x + \frac{1}{x} = 41.$ Find $\sqrt{x} + \frac{1}{\sqrt{x}}.$ | nvidia |
Let $S$ be the set of all positive divisors of the number $100$. What is the smallest subset $T$ of $S$ such that any element of $S$ can be expressed as a product of elements of $T$? | nvidia |
Let $w$ be a complex number such that $|w-3|=5$. Find $|w|$. | nvidia |
Let $f(x)$ be a polynomial of degree $n$ such that for all $x$, $|f(x^n)| = |f(x)|^n$. Find all possible polynomials $f(x)$. | nvidia |
Find all values of $x$ such that $\frac{x-3}{4x}$ is undefined. | nvidia |
How many 5-digit numbers with at least one even digit can be constructed from the digits 1, 2, 3, 4, and 5, with the condition that each digit can be used only once? | nvidia |
Let $x_1, x_2, \ldots, x_n$ be positive real numbers such that $x_1^2 + x_2^2 + \cdots + x_n^2 = 1$. Find the maximum value of
\[x_1x_2 + x_2x_3 + \cdots + x_nx_1.\] | nvidia |
In triangle ABC, D is a point on side BC such that AD is the largest possible altitude. Find the length of AD, given that the area of triangle ABC is 24 square units. | nvidia |
Find the largest prime factor of $n$, where $n$ is the product of all the positive integers from $1$ to $10$. | nvidia |
The function $g(x)$ satisfies $g(x+y)=g(x)g(y)$ for all real numbers $x$ and $y$, and $g(2)=3$. Find $g(3)$. | nvidia |
Find the sum of the infinite series $1+2\left(\dfrac{1}{1998}\right)+3\left(\dfrac{1}{1998}\right)^2+4\left(\dfrac{1}{1998}\right)^3+\cdots$. | nvidia |
Given that $\alpha$ and $\beta$ are the roots of the polynomial $x^2-x-1=0$, find the quadratic polynomial $f(x)$ such that $f(f(x))=xf(x)+\alpha+\beta$. | nvidia |
Let $f(x)$ be a cubic function, and let $g(x) = f^{-1}(x)$ be the inverse of $f(x)$. The graph of $g(x)$ is shown below:
[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 |
A sphere is inscribed in a right tetrahedron with three mutually perpendicular edges that lie along the positive x, y, and z axes. The volume of the sphere is 32π cubic units. Compute the area of the tetrahedron's face that lies in the first octant. | nvidia |
Let $a$ and $b$ be two complex numbers such that $a+b=10$ and $ab=-40$. Find $a^3+b^3$. | nvidia |
Given that the quadratic equation $ax^2 + bx + c$ has no real roots, determine the possible values of $a$ such that $a^2 + b^2 + c^2 = 100$, where $b$ and $c$ are fixed integers. | nvidia |
Compute
\[\sum_{1 \le i < j < k} \frac{1}{7^i 11^j 13^k}.\](The sum is taken over all triples $(i,j,k)$ of positive integers such that $1 \le i < j < k.$) | nvidia |
Find the number of positive integers less than or equal to 100 that are divisible by 3 but not by 5. | nvidia |
In $\triangle ABC$, the sides have integer lengths and $AB=AC$. Circle $\omega$ has its center at the circumcenter of $\triangle ABC$. An incircle of $\triangle ABC$ is a circle in the interior of $\triangle ABC$ that is tangent to all three sides of the triangle. Suppose that the incircle is externally tangent to $\om... | nvidia |
The vertices of a tetrahedron in 3D space have $y$-coordinates that sum to 15. Find the sum of the $y$-coordinates of the centroids of the faces of the tetrahedron. | nvidia |
Let the two integers be x and y. The sum of the squares of the two integers is 193, and the product of the two integers is 84.
The sum of the squares of two positive integers is given as x^2 + y^2 = 193. The product of the two integers is given as x*y = 84.
We need to find the sum of the two integers, i.e., x + y.
... | nvidia |
Let $p$ and $q$ be positive integers such that\[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\]and $q$ is as small as possible. What is $q-p$?
$\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$
| nvidia |
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