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The ratio of the areas of two similar triangles is 4:9, and the length of the corresponding side of the larger triangle is 6 cm. Find the length of the corresponding side of the smaller triangle. | nvidia |
Consider a polynomial of the form $P(x) = (x^2 + ax + b)^m$, where $a$ and $b$ are constants and $m$ is a positive integer. It's given that the coefficients of the polynomial $P(x)$ form a geometric sequence with a common ratio of $2$. Find the sum of all possible values of $m$ and the degree of the polynomial $P(x)$. | nvidia |
Find the smallest positive integer m such that
$\begin{pmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix}^m = \mathbf{I}.$ | nvidia |
Let f be a function from the set of positive integers to the set of nonnegative integers. Assume that f satisfies the two properties below:
• Property 1. If x is a positive integer, f(x)=0 if and only if x is a power of 2.
• Property 2. If x is a positive integer and y is a positive integer such that 2^(r+1)>y, then f... | nvidia |
In the complex plane, two complex numbers $a+bi$ and $c+di$ are reflected across the line $y=x$. The reflected complex numbers are $b+ai$ and $d+ci$. Given that $a+b=3$, $c+d=5$, and $a+c=7$, find the value of $b+d$. | nvidia |
Find all positive values of $c$ such that the inequality $|x^2 + c(x-3)| < x^2$ has real solutions for $x$. Express your answer in interval notation. | nvidia |
Find the number of non-negative integers $n$ that satisfy
\[\sqrt{99} < 10^n < \sqrt{9999}.\] | nvidia |
A planar graph $G$ has $18$ vertices, $40$ edges, and $22$ faces, where each face is a triangle or a pentagon. A chord of $G$ is a line segment connecting two non-adjacent vertices that lie on the same face. Determine the maximum number of chords that $G$ can have. | nvidia |
Let $a$ and $b$ be real numbers such that $a^2 + 2b^2 \le 3$ and $2a^2 + b^2 \le 4.$ Find the largest possible value of $ab$. | nvidia |
Determine a positive real number $a$ such that
\[\frac{a}{\sqrt[3]{2}} + \frac{a^2}{\sqrt[3]{2^2}} + \frac{a^3}{\sqrt[3]{2^3}} + \dotsb = 16.\] | nvidia |
Determine the number of possible values of $n$ for which the base-$n$ number $123456_n$ is a perfect square, where $n$ ranges from 2 to 100. | nvidia |
Let $g(x)$ be a function such that $g(2x+1) = 4x^2+2x+1$ for all real numbers $x$. Find $g(x^2+2)$. | nvidia |
Find the number of positive five-digit integers of the form $abcba$ that are divisible by 55. | nvidia |
Find the smallest positive integer $m$ such that
\[\sum_{k=1}^{m} \sin(k^3 \theta) \cos(k^2 \theta)\]
is an integer, where $\theta = \frac{\sqrt{2}}{2013}$. | nvidia |
Given a set of nonnegative real numbers $x_1,x_2,\ldots,x_n$, determine the minimum value of $\frac{x_1^2+x_2^2+\cdots+x_n^2}{x_1x_2+x_2x_3+\cdots+x_{n-1}x_n+x_nx_1}$ for all possible values of $n$. | nvidia |
In the set of all positive integers from 1 to 150, find the largest subset such that for any two distinct elements, a and b, in the subset, the absolute difference between a and b is greater than the square root of either a or b. | nvidia |
Find all values of $k$ for which the expression $k^2 - 4ik + 3$ can be written in the form $a+bi$, where $a$ and $b$ are integers and $i^2 = -1.$ | nvidia |
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 |
Let $x$ and $y$ be two positive integers such that $x\neq y$. If the harmonic mean of $x$ and $y$ is $h$, find the minimum value of $xy$ if $x+y=4h$. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a + 2b + 3c = 1$. Find the minimum value of $\dfrac{1}{a} + \dfrac{2}{b} + \dfrac{3}{c}$. | nvidia |
Find the product of all values of $t$ such that $t^2 = 36$. | nvidia |
Consider the sphere defined by $x^2-4x+y^2-6y+z^2-8z+33=0$ and the plane defined by $x+2y+3z=15$. Find the sum of the squares of all the coordinates of all the points of intersection of the sphere and the plane. | nvidia |
Count the number of parallelograms formed by connecting points on a 4x4 grid such that no two points share the same x or y coordinate, and exactly one pair of points lies on a diagonal of the grid. | nvidia |
A function $f:\{1,2,3,4,5,6,7\}\rightarrow \{1,2,3,4,5,6,7\}$ has the property that for all $n\in \{1,2,3,4,5,6,7\}$,
$$f(f(n))+f(n+1)+1\equiv 0\pmod 7.$$
Determine the smallest possible value of
$$f(1)+f(2)+f(3)+f(4)+f(5)+f(6)+f(7).$$ | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a^2+b^2+c^2=1$ and $(a+b+c)^2=2$. Find the minimum possible value of $\frac{ab+bc+ca}{abc}$. | nvidia |
Find the smallest positive integer $c$ such that the equation $x^3 - cx - 2 = 0$ has exactly one real solution. | nvidia |
What is the smallest number n>2 such that the fraction (2^n + 1)/(2^(n+1)) can be written as the sum of two distinct unit fractions, i.e., fractions of the form 1/x, where x is a positive integer? | nvidia |
Determine the base $b$ such that $\log_{b}{0.5}$ is equal to $-2$. | nvidia |
In a rectangular prism, each edge is labeled with a positive integer. The product of the numbers along each of the diagonals is the same. What is the largest possible product of the numbers along the diagonals if the sum of the numbers along the edges is 50? | nvidia |
Find all values of $x$ that satisfy the equation $|x-3|=2x+4$. Express your answers in simplest fractional form. | nvidia |
Find the sum of all integers from 45 through 60, inclusive. | nvidia |
Let $n$ be a positive integer such that $8n+1$ is a perfect square. Find the greatest common divisor of $n^2+3n+1$ and $n+1$. | nvidia |
Suppose that there exist two polynomials f(x) and g(x) of degree 4 with nonzero complex coefficients a, b, c, d, e and f such that the roots of f(x) are exactly the reciprocals of the roots of g(x). If the product of the roots of f(x) is 2, determine the possible values of the coefficient a. | nvidia |
Determine the number of positive integers less than 1000 that are not divisible by 2, 3, or 5. | nvidia |
Find all real numbers $x$ such that $[x] + x = 10$, where $[x]$ is the greatest integer less than or equal to $x$. Express your answer as a decimal. | nvidia |
Let $f(x)$ be a function defined on the interval $[0,1]$ such that $f(0)=f(1)=0$ and $f(x)$ is an increasing function for $x\in[0,1]$. Determine the largest possible value of the definite integral $\int_0^1 f(x)dx$ given that $f(x) + f(1-x) = 1$ for all $x \in [0,1]$. | nvidia |
Determine the number of different remainders that $p^2$ can leave upon division by $120$, as $p$ ranges over the integers $n$ such that $n>5$ and $\gcd(n, 120) = 1$. | nvidia |
Let $A$ and $B$ be the points in 3-dimensional space with coordinates $(1,2,3)$ and $(4,5,6)$, respectively. Let $S$ be the sphere with center $O$ and radius 1, and let $T$ be the set of all points that are 3 times as far from $A$ as from $B$. Find the sum of the distances from the origin to the two points in $S\cap T$... | nvidia |
Let $g$ be a function defined as follows: for any real number $x$, if $x$ is positive then $g(x)=x^2+1$, otherwise $g(x)=-x$. Given that the function $h$ is the composition of $g$ with itself, evaluate $h(h(2)-h(-1))$. | nvidia |
Let $a$, $b$, $c$, $d$ be positive real numbers such that $abcd=1$. Find the minimum value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}$. | nvidia |
When three numbers are added two at a time the sums are 29, 46, and 53. What is the sum of all three numbers? | nvidia |
Consider a sequence of positive real numbers $a_1, a_2, a_3, \dots$ such that every term after the first is the sum of the squares of its two immediate neighbors. For how many different values of $a_1$ does the term $a_1 + 1$ appear somewhere in the sequence? | nvidia |
In 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 |
Determine the number of distinct ordered-pair solutions $(x, y)$ of the equation $xy = 36$, where $x$ and $y$ are positive integers. | nvidia |
Find the value of $x$ such that $x \cdot (x+2) \cdot (x+4) + \frac{48}{x} = 100.$ | nvidia |
A three-digit number $\underline{xyz}$ is divisible by 11 if the sum of the odd-position digits $x$ and $z$ is equal to the even-position digit $y$. Determine the smallest three-digit number $\underline{xyz}$ that is divisible by 11 and has no repeated digits. | nvidia |
Let $c$ be a real number randomly selected from the interval $[-5,5]$. Then, $p$ and $q$ are two relatively prime positive integers such that $p/q$ is the probability that the equation $x^4+9c^2=(2c^2-6c)x^2$ has $\textit{at least}$ two distinct real solutions. Find the value of $p+q$. | nvidia |
Let $x$ be a point on the unit circle in the second quadrant such that the angle between the radius vector and the $x$-axis is twice the angle between the radius vector and the $y$-axis. Determine the $y$-coordinate of point $x$. | nvidia |
Find $x$ such that $\log_x 4=\log_{27} 3$. | nvidia |
Find the smallest positive integer $N$ such that $N$, $N+1$, and $N+2$ are consecutive terms in an arithmetic progression, and $N$ is divisible by $2^3$, $N+1$ is divisible by $3^2$, and $N+2$ is divisible by $5$. | nvidia |
A square grid is divided into smaller squares, with each side of the grid containing $n$ squares. Three points are chosen uniformly at random on the grid, with no point lying on the boundary or intersection of two squares. What is the probability that the triangle formed by these three points is acute? | nvidia |
Let $a$ and $b$ be real numbers. If the polynomial $x^2+ax+b$ has two distinct real roots and $a=2b-3$, find the value of the sum of all possible values of $b$. | nvidia |
If $n$ is a positive integer, for how many values of $n$ is $12^{12}$ the least common multiple of the positive integers $6^6$, $8^8$, and $n$? | nvidia |
Find the sum of the values of $x$ such that $\tan 2x \sin 3x = \sin 5x$, where $x$ is measured in degrees and $0 < x < 90$. | nvidia |
The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | nvidia |
If $x$ and $y$ are positive integers such that $4x+2y=120$, what is the greatest possible value of $xy$? | nvidia |
If $a$ and $b$ are positive integers such that $(a+bi)^n = 1+2i$, what is $a+bi$? | nvidia |
In a square, a smaller square is inscribed such that its vertices are on the midpoints of the sides of the larger square. If the area of the smaller square is 64 square units, what is the area of the region between the two squares? | nvidia |
If $a$ and $b$ are positive integers such that $\gcd(a,b) = 18$, then what is the smallest possible value of $\gcd(12a,20b)$? | nvidia |
Suppose there exist non-zero real numbers $a, b, c,$ and $d$ such that the following system of equations has a non-zero solution:
$$ax^2+by^2+cz^2+dw^2=0$$
$$bx^2+cy^2+dz^2+aw^2=0$$
$$cx^2+dy^2+az^2+bw^2=0$$
$$dx^2+ay^2+bz^2+cw^2=0$$
Find all possible values of the sum $a+b+c+d$. | nvidia |
Two positive integers $x$ and $y$ are chosen such that $x$ is the largest prime number less than $50$ and $y$ is the smallest positive integer with exactly four positive divisors. What is $x+y$? | nvidia |
In the system of equations
\begin{align*}
x+ay&=2,\\
ax+y&=3,
\end{align*}
$a$ is a constant such that when $x = y$, the system has a solution.
Determine $a$. | nvidia |
Find all real $x$ such that \[\left\lfloor x \lfloor x \rfloor\right \rfloor = 29.\] | nvidia |
Find the number of ways to arrange the digits of the number 12345 in a way that the resulting 5-digit number is divisible by 11. | nvidia |
In a group of 50 people, each person has a unique favorite number from 1 to 50. For each pair of people, the sum of their favorite numbers is distinct. Find the maximum number of people who can be paired such that no two pairs have the same sum of favorite numbers. | nvidia |
A cone has a height of 5 units and a base radius of 4 units. A sphere is cut from the cone such that its volume is equal to that of the remaining cone. Find the radius of the sphere. | nvidia |
Find the number of divisors of $2007$ that are divisible by $3$ but not by $9$, given that its prime factorization is $3^2 \cdot 13 \cdot 17$. | nvidia |
Given that $\lfloor\log_{2}x\rfloor=5$, how many possible integer values of $x$ are there? | nvidia |
There are three equal circles arranged such that their centers are the vertices of an equilateral triangle. The radius of the circles is 20 cm. The perimeter of the shape formed by the arcs inside the triangle is painted blue. The area of the blue-painted shape is 570 cm^2. Find the length of one side of the equilatera... | nvidia |
A polynomial $P(x)$ is called self-centered if it has integer coefficients and $P(100) = 100.$ Find the sum of all numbers $c$ such that $P(x) = x^3 + cx^2 + 2017x + 1$ is self-centered. | nvidia |
Given the expression $\frac{1}{3}-\frac{1}{9}+\frac{1}{27}-\frac{1}{81}+\frac{1}{243}$, determine a new expression with common differences between the denominators that sums up to $\frac{5}{7}$. The denominators are consecutive powers of a number and there should be 6 terms in the expression.
Consider an expression in... | nvidia |
Find the sum of all integer solutions $x$ such that $x^2$ divides 120, and $x$ is not divisible by 3.
Can you solve it? | nvidia |
Let $r$ be a real number, $|r| < 2,$ and let $z$ be a complex number such that
\[z + \frac{1}{z} = r.\]Find $|z|.$ | 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 $A$ and $B$ be $n\times n$ and $m\times m$ matrices of zeroes and ones, respectively, such that each row and each column of $A$ and $B$ contains exactly one 1. Let $C$ be the $n\times m$ matrix such that $C_{ij}$ is the number of 1s in the $i$th row of $A$ and the $j$th column of $B$. Prove that $\det C=0$. | nvidia |
Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$. | nvidia |
In a regular octahedron, the midpoints of the 12 edges are the vertices of a smaller octahedron. The ratio of the surface area of the smaller octahedron to that of the larger is $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | nvidia |
Problem:
Consider a polygon in the complex plane formed by the points 0, $z$, $2z$, $3z$, $\ldots$, $(n-1)z$, where $n$ is a positive integer greater than or equal to 3. The area of the polygon is known to be $\frac{n(n-1)}{2} |z|^2 \sin \theta$, where $\theta$ is the angle between the positive real axis and the line s... | nvidia |
Let $p(x)$ be a polynomial with positive integer coefficients such that $p(1)=21$ and $p(21)=201$. What is the greatest possible value of $p(13)$? | nvidia |
Find the number of integer values of x that satisfy the condition:
$$2x < 5$$
$$3^x < 10$$ | nvidia |
What is the smallest positive integer that can be expressed as a sum of two distinct perfect squares in exactly three different ways? | nvidia |
Circle $C$ has its center at $(3, -8)$ and a radius of $15$ units. What is the sum of the $x$-coordinates of the two points on circle $C$ that are also on the $x$-axis? | nvidia |
How many distinct four-digit positive integers have digits that are either all prime numbers or all composite numbers? | nvidia |
Find the smallest positive integer $m$ such that:
(i) $m^3$ can be expressed as the sum of two consecutive squares;
(ii) $3m - 11$ is a perfect cube.
Determine the value of $m$. | nvidia |
Find the product of the solutions to the equation $6x^2-11x+3=0$, given that one of the solutions is the reciprocal of the other. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R} $ be defined by $ f(x) = x^3 + x + \sqrt{2} $. How many real numbers $x $ are there such that $ x < 0 $ and both the integer part and the fractional part of $f(x)$ are rational numbers with denominators at most $ 5 $? | nvidia |
Find the number of three-digit numbers that can be formed using the digits 2, 3, 5, and 7, with the condition that the sum of the digits in the hundreds and tens places is equal to the digit in the units place. | nvidia |
Let $x$, $y$, and $z$ be distinct real numbers that sum to $0$. Compute \[ \dfrac {xy+yz+zx}{x^2+y^2+z^2}. \] | nvidia |
Find a positive integer that is divisible by 24 and whose square root is a number between 9 and 9.2. | nvidia |
Let $f(n)$ be the number of factors of $n$ that are not factors of $n+1$. Determine the smallest positive integer $m$ for which there exists a positive integer $n$ such that $f(n) = m$. | nvidia |
Find all possible values of the parameter $b$ such that the equation $\sin^2x + \sin x \cos x + b=0$ has real solutions and find the sum of all such values of $b$. | nvidia |
Find the maximum value of
\[\frac{1}{1-a_1} + \frac{1}{1-a_2} + \dots + \frac{1}{1-a_{12}},\]
given that $a_1,$ $a_2,$ $\dots,$ $a_{12}$ are positive real numbers such that $a_1 + a_2 + \dots + a_{12} = 1.$ | nvidia |
The function $g(x)$ satisfies
\[g(x+y) = \frac{g(x)g(y)}{x+y}\]for all positive real numbers $x$ and $y.$ If $g(2) = 4,$ find $g(4).$ | nvidia |
Find the smallest four-digit integer that is divisible by the sum of its distinct, non-zero digits. | nvidia |
There are 3 complex numbers $a+bi$, $c+di$, and $e+fi$. If $b=1$, $e=-a-c$, and the sum of the numbers is $-i$, find $d+f$. | nvidia |
Find all real values of $x$ that satisfy the equation
$$\left(\frac{x^3+2x-1}{x^2-1}\right)^{x^2-4}=1.$$
Can you solve this problem using algebraic manipulations and the properties of exponents? | nvidia |
Find the positive value of $x$ for which the area of the triangle formed by the points $(2, 3)$, $(7, x)$, and $(5, -2)$ is 8 square units. | nvidia |
For positive integers $m$, define $T_m$ to be the minimum value of the sum
\[\sum_{j=1}^m \sqrt{(3j-2)^2+b_j^2},\]where $b_1,b_2,\ldots,b_m$ are positive real numbers whose sum is $25$. Find the unique positive integer $m$ for which $T_m$ is also an integer. | nvidia |
Determine the last two digits of the number 555! | nvidia |
In the equation $x^2-4x-3=0$, the sum of the roots is $4$ and the product of the roots is $-3$. The roots $x_1$ and $x_2$ of the equation $x^2-kx+k^2=0$ are such that $x_1^2+x_2^2=40$. Determine $k$. | nvidia |
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