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A cylindrical container is placed so that one of its bases lies on the $xy$-plane with the origin at its center. The equation of the cylinder is $x^2 + y^2 = 25$. Find the length of the segment of the $z$-axis that lies inside the cylinder. | nvidia |
The polynomial $x^4 + ax^3 + bx^2 + cx + d$ has four roots, and it is known that two of them are complex conjugates of each other. Additionally, the sum of the two complex roots is $5$, and the sum of the two real roots is $-3$. Determine the value of $a$. | nvidia |
Determine the number of real solutions of the equation \[\sin (e^x) = xe^{-x}.\] | nvidia |
The complex numbers $a+bi$ and $c+di$ are such that $\left|a+bi\right|=\left|c+di\right|$ and $\left|a+c\right|=\left|b+d\right|$.
Find the number of real values $k$ for which $\left|a+bi+k\right|=\left|c+di+k\right|$. | nvidia |
Find all values of the positive integer $n$ such that the vertices of the regular polygon with $n$ sides inscribed in the unit circle of the complex plane are the roots of a polynomial equation with integer coefficients. | nvidia |
There exists a positive real number $y$ such that $\sin (\arcsin (y)) = \frac{y^2}{2}$. Find the value of $y^3$. | nvidia |
Given that $n$ is a positive integer, and given that $\mathop{\text{lcm}}[24,n]=72$ and $\mathop{\text{lcm}}[n,27]=108$, what is $n$? | nvidia |
The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $a+b+c$.
[asy]
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real tickspace=2;
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real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
re... | nvidia |
Find the smallest positive integer $n$ such that when $\sqrt[3]{123.\overline{456}}$ is expressed as a fraction in the form $\frac{\sqrt[3]{m}}{n}$, where $m$ is an integer, the value of $m+n$ is minimized. | nvidia |
Given that $a$ and $b$ are positive integers and $x$ is a real number greater than 0, find the maximum value of $\frac{x^{\sqrt{a}}}{\sqrt[b]{2^x}}$ for $a = 6$ and $b = 3$. | nvidia |
The points $A$, $B$, $C$ are positioned on the circle with radius $10$ and center at $O$ so that the angle $\angle AOC=2\angle BOC$. Find the range of possible values of the perimeter of the triangle $\triangle ABC$. | nvidia |
Find the sum of all positive integer values of $d$ such that $3x^2 + 10x + d = 0$ has two distinct rational roots. | nvidia |
Find the smallest number of rows in a triangular lattice such that the sum of all the numbers in the lattice is greater than 1000.
Note: In a triangular lattice, each row has one more number than the previous row, starting with 1 number in the first row. | nvidia |
If $f(x)$ is a function defined for all real $x$ such that $f^{-1}(x)$ exists for all $x$ in the range of $f$, determine the number of points at which the graphs of $y=f(x)$ and $y=f(\frac{1}{x})$ intersect. | nvidia |
Find the smallest positive integer $a$ such that the system of equations:
$a \equiv 0 \pmod{11}$,
$a \equiv 2 \pmod{13}$
has a solution. | nvidia |
Find the smallest positive integer $k$ such that $k^3 \equiv 2 \pmod{15}$. | nvidia |
Find the probability that when three standard dice are tossed, the sum of the numbers obtained is greater than 15. | nvidia |
What is the largest integer $n$ such that the number $5^n$ has exactly $n$ digits in its decimal representation? | nvidia |
Find the sum of all the digits that appear before the decimal point in the decimal representation of $\frac{123456789}{999999999}$. | nvidia |
Let $x$ and $y$ be real numbers such that
$\frac{x}{y}+\frac{y}{x}=1$,
and $x^2+y^2=7$.
Find the value of $x+y$. | nvidia |
Find the sum of all the even integers between 1 and 101, excluding the multiples of 3. | nvidia |
Find the number of perfect squares that can be expressed as the sum of two consecutive prime numbers less than 20. | nvidia |
The vertices of a regular octagon are connected by a set of line segments that form a larger octagon. The transformation $T$ reflects a given line segment in one of the sides of the smaller octagon, then reflects the resulting line segment in one of the sides of the larger octagon. The transformation is applied repeate... | nvidia |
Find the smallest positive integer $n$ such that $4n$ leaves a remainder of $1$ when divided by $21$.
Express your answer as an integer from $0$ to $20$, inclusive. | nvidia |
Find all the values of x that satisfy the equation $\frac{x^2+1}{x^2-1} = \frac{x^2+5}{x^2+3}$. Express your answer as the product of all possible values. | nvidia |
Let x0,x1,x2,… be a sequence of positive integers such that
x3≡44x0(mod77)
for all non-negative integers n. What is the smallest possible value of x0? | nvidia |
Find the smallest five-digit palindrome number that is divisible by the sum of its digits. | nvidia |
Given a right triangle $ABC$ with hypotenuse $BC$ of length 25, consider a point $M$ on side $AB$ such that the circle drawn on $BM$ as a diameter intersects the segment $AC$ extended through $C$ at a point $P$, forming a right angle with side $AC$. Find the maximum possible area of triangle $BMP$. | nvidia |
Compute $j^{-100}+j^{-99}+j^{-98}+\cdots+j^{-1}+j^0+j^1+\cdots+j^{99}+j^{100}$, where $j$ is a complex number such that $j^2=-1$ and $j^3=j$. | nvidia |
Find all real numbers $x$ such that \[\frac{x^3+2x^2}{x^2+3x+2} + x = -6.\]Enter all the solutions, separated by commas. | nvidia |
Let $g$ be a function defined on the positive integers, such that
\[g(x+y) = g(x) \cdot g(y)\]for all positive integers $x$ and $y.$ Given $g(2) = 3$ and $g(6) = 27,$ find $g(12).$ | nvidia |
The polynomial $x^6 + x^3 + 1$ is written as a sum of three irreducible polynomials with real coefficients. Determine the largest possible number of summands. | nvidia |
What is the largest positive integer $n$ such that $n\cdot 0.\overline{34}$ is an integer? | nvidia |
Let $a$ and $b$ be real numbers. Determine the sum of all real solutions $x$ to the equation
$$(x^3+ax^2+bx-a)^{x^2+bx+a}=(x^2+bx+a)^{x^3+ax^2+bx-a}.$$ | nvidia |
Let $x$ be a positive real number. Find the value of $x$ that satisfies the equation:
$$\left\lceil \frac{x}{\sqrt{2}} \right\rceil + \left\lfloor \frac{\sqrt{2}}{x} \right\rfloor = 3.$$ | nvidia |
Find the number of positive perfect cube integers that are divisors of the product $\left(2^9\right)\left(3^{10}\right)\left(5^{12}\right)$. | nvidia |
Consider the following expressions:
$$\frac{2x+3}{x+2}, \ \ \frac{x+4}{2x+1}, \ \ \frac{3x-1}{x-1}, \ \ \frac{x+2}{x+3}, \ \ \frac{2x-3}{x-2}$$
Determine the value of $x$ for which the sum of the smallest two expressions is equal to the sum of the largest two expressions. | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=7$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
Find the area of the regular hexagon inscribed in the circle with the equation $x^2+y^2=4$. | nvidia |
Find the sum of all integers between 1 and 500 that are multiples of 5 but not multiples of 7. | nvidia |
The function $f(x)$ satisfies
\[f(x + y) = f(x) + f(y)\]for all real numbers $x$ and $y.$ Find all possible values of $f(0).$
Enter all the possible values, separated by commas. | nvidia |
The equation of the hyperbola is $\frac{x^2}{144} - \frac{y^2}{225} = 1.$ Find the positive value of $m$ such that the asymptotes of this hyperbola have the equations $y = \pm mx.$ | nvidia |
Find all values of $x$ so that $\arccos x > \arcsin x.$ | nvidia |
The largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$ is 890.
Let's find another problem inspired by this one.
Problem:
Find the largest positive integer $n$ such that the expression $n^2 + 100n + 2017$ is divisible by $n + 17$.
This problem requires a different approach to solve, as the expr... | nvidia |
Find the largest real number $\mu$ such that
\[p^2 + q^2 + r^2 + s^2 \ge pq + \mu qr + rs\]
for all nonnegative real numbers $p,$ $q,$ $r,$ $s.$ | nvidia |
Find the largest positive integer $a$ for which $\frac{1}{x^2 + ax + 2000}$ can be expressed as the sum of two distinct fractions with integer coefficients. | nvidia |
Quadrilateral $ABCD$ has two pairs of opposite sides that are perpendicular: $\overline{AB}\perp\overline{CD}$ and $\overline{BC}\perp\overline{AD}$. $\angle A=2\angle B$ and $\angle D=\angle C+45^\circ$. Find $\angle B$ (in degrees). | nvidia |
Let $x$ be a positive number such that $\sqrt{x} + \frac{1}{\sqrt{x}}$ can be expressed in the form $\frac{m\sqrt{x}+n}{p}$, where $m,n$ and $p$ are positive integers with $p$ as small as possible. Determine $m+n+p$. | nvidia |
Problem:
Points $A$ and $B$ are symmetric with respect to the line $y = x$. The coordinates of point $A$ are $(x, y)$. Find the coordinates of point $B$ if the area of the rectangle formed by the points $(0,0)$, $(0,y)$, $(x,y)$, and $(x,0)$ is equal to the area of the rectangle formed by the points $(0,0)$, $(0,x)$, $... | nvidia |
Consider the inequality
$$3^{x^2+y} + 3^{x+y^2} > 3^{x+y},$$
where $x$ and $y$ are real numbers.
How many ordered pairs $(x,y)$ exist such that the inequality holds? | nvidia |
Given that $p(x) = x^8 - 1$ has been factored into non-constant polynomials with real coefficients as $p(x) = p_1(x) p_2(x) \dotsm p_k(x),$ find the largest possible value of $k$ such that each $p_i(x)$ has a distinct set of roots. | nvidia |
Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{9}{10}.\] | nvidia |
In the pattern below, each number is the sum of the two numbers directly above it.
\begin{tabular}{rccccccccccc}
Row 0:& & & & & & 2\\\noalign{\smallskip\smallskip}
Row 1:& & & & & 1 & & 3\\\noalign{\smallskip\smallskip}
Row 2:& & & & 1 & & 4 & & 3\\\noalign{\smallskip\smallskip}
Row 3:& & & 1 & & 5 & & 7 & & 3\\\noal... | nvidia |
Let $p(x)$ be a monic polynomial such that its degree is greater than or equal to 1, with integer coefficients, such that it satisfies the following condition for all integers $n$ greater than 1. The expression $\frac{p(n)}{p(n-1)}$ is always an integer, multiple of $n$. Find the greatest $k$ such that $p(k) = 31$. | nvidia |
Let p be a positive integer such that p^2-8p-1 is a perfect square. Prove that (p+4)(p^2-8p-1) is also a perfect square. | nvidia |
Find the largest constant $C$ so that
\[x^2 + y^2 + 1 \ge C(x + y)\]for all real numbers $x$ and $y.$ | nvidia |
In a regular polygon with 24 sides, all the angles are equal in measure. How many degrees are in the sum of the measures of all the angles in the polygon? | nvidia |
Find the smallest positive integer $n$ such that the number of positive integers less than or equal to $n$ that are divisible by 3 is greater than the number of positive integers less than or equal to $n$ that are divisible by 5. | nvidia |
Let's consider a geometric progression whose first three terms are $a-r$, $a$ and $a+r$.
Given that $q$ is the root of the equation
$x^3-3ax^2+ (a^2+b^2-1)x-a(a^2-b^2+1)=0$,
find the least value of $|a|$ such that for any real $b$,
the expression
$q(q^2-2aq+a^2+b^2)+q(a^2-b^2-1)$
is non-negative. | nvidia |
What is the smallest positive integer $n$ such that all the roots of $z^4 - z^2 + 1 = 0$ are $n^{\text{th}}$ roots of unity? | nvidia |
A rectangle with a fixed perimeter of 24 units has two of its sides with lengths x and x + y, where y is a positive constant. Find the value of x that maximizes the area of the rectangle. | nvidia |
Determine the maximum value of the expression
\[\frac{3x^4 + 4x^2 + 1}{2x^4 + x^2 + 2},\]over all real numbers $x.$ | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x - g(y)) = g(g(x) - y) + x + g(-y)\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(0),$ and let $t$ be the sum of all possible values of $g(0).$ Find $m \times t.$ | nvidia |
Find the minimum value of
\[x^2 + xy + y^2\]over all real numbers $x$ and $y.$ | nvidia |
Given a triangle $ABC$ with sides of length 3, 4, and 5, a point $P$ is chosen inside the triangle such that the distance from $P$ to each side is 1. Find the length of the longest diagonal of the quadrilateral formed by connecting the feet of the perpendiculars from $P$ to the sides of the triangle. | nvidia |
Find the number of solutions $(x, y)$ to the system of equations:
$$x^2y^3 = x^3y^2 = -1$$
such that $x$ and $y$ are complex numbers. | nvidia |
What is the smallest positive integer $n$ such that there exist two positive integers $x$ and $y$ whose sum is $n$ and whose product is divisible by $n$? | nvidia |
Find the sum of the digits in the binary representation of the smallest positive integer that is divisible by 2^6, 3^3, and 5^2. | nvidia |
The smallest possible value of the sum of two perfect cubes that differ by a prime number. | nvidia |
Determine the number of integers $b$ greater than 4 such that the congruence $x^2 \equiv b \pmod{20}$ has a solution in $x$. | nvidia |
Find all positive real numbers $a$ that satisfy
\[a\left( \frac{1}{\sqrt{16-a^2}} - \frac{1}{\sqrt{16+a^2}} \right) + \frac{1}{\sqrt{a^2+4}} \ge 2.\]
Enter all solutions, separated by commas. | nvidia |
Find the smallest positive integer $n$ such that the sum of the digits of $n^2$ is greater than the sum of the digits of $n$. | nvidia |
Let $n \ge 5$ be an integer and let $T = \{1, 2, 3, \ldots, n\}$. Find the smallest value of $n$ such that for every partition of $T$ into three subsets, at least one of the subsets contains integers $x$, $y$, and $z$ (not necessarily distinct) such that $x + y = z$. | nvidia |
What is the maximum number of distinct triangles that can be formed by connecting 5 points on a circle, such that no three points are collinear? | nvidia |
Let $n$ be the smallest positive integer such that $[n/2]+[n/3]+[n/4]=n$. Find $n$ modulo $7$. | nvidia |
A 2-digit number is such that the ratio of the tens digit to the units digit is 3:2. If the number is squared, the resulting number is a 4-digit number that has the same tens and units digit as the original number. Find the sum of the two possible values of the original number. | nvidia |
In the diagram, $\triangle PQR$, $\triangle QRS$ and $\triangle RST$ are isosceles triangles with $PR = 12$. All the angles of each triangle are integers in degrees, and the sum of the angles in each triangle is 180 degrees. The sum of the interior angles of the quadrilateral $PQST$ is 370 degrees. Find the perimeter o... | nvidia |
Given three non-overlapping cubes of unit length, with centers at $(3, 4, 1)$, $(1, 2, 3)$, and $(5, 6, 7)$, find the equation of the plane passing through $(1, 2, 3)$ such that the volume of the parts of the cubes on one side of the plane is equal to the volume of the parts of the cubes on the other side. | nvidia |
Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12},$ find all possible values for $x.$
(Enter your answer as a comma-separated list.) | nvidia |
Find the number of distinct diagonals that can be drawn in a convex decagon (10-sided polygon). | nvidia |
Find the number of positive perfect cubes less than $10^6$ that are not multiples of 3. | nvidia |
Find $n$ such that $2^5 \cdot 3^2 \cdot n = 8!$. | nvidia |
What is the sum of all positive integer values of $n$ such that $\frac{n+18}{n}$ is an integer? | nvidia |
It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the $x$ and $y$ axes and so that the medians to the midpoints of the legs lie on the lines $y = 3x + 1$ and $y = mx + 2$. The number of different constants $m$ for which such a triangle exists is
$\textbf{(A)}\ 0\qquad ... | nvidia |
Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$. Find $g^{-1}(f(-7))$. | nvidia |
For how many ordered pairs of positive integers $(x,y),$ with $y<x\le 100,$ are both $\frac xy$ and $\frac{x+1}{y+1}$ integers?
| nvidia |
The area of a rectangle equals the product of its length and width. The perimeter of a rectangle equals twice the sum of its length and width. The sum of the areas of two rectangles is 240, while the difference in their areas is 80. Find the sum of their perimeters. | nvidia |
Count the number of positive integers from 1 to 1000 that are not multiples of 5 or 7, but are multiples of 3. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 100.$ Find the minimum value of $xy + yz + xz.$ | nvidia |
What is the sum of all of the solutions of the equation $\frac{4x}{20}=\frac{5}{x}$? | nvidia |
Let $p$ be a prime number and $m$ be a positive integer. Consider the congruence $2^m\equiv 1\pmod p$. Prove that if $p$ divides $2^m-1$, then $m$ is divisible by $p-1$. | nvidia |
Find the equation of the hyperbola with its center at the origin, whose asymptotes have the equations $3x-y=0$ and $3x+y=0$, and that passes through the point $(3,2)$. | nvidia |
Consider a polynomial $P(x) = x^3 + ax^2 + bx + c$. The sum of the roots of $P(x)$ is $-a$, the sum of the products of all pairs of roots is $b$, and the product of the roots is $-c$. Determine the value of $c$ if the roots of $P(x)$ are consecutive integers. | nvidia |
Find the smallest positive integer k such that $16^k + 21^k$ is divisible by 17. | nvidia |
Find all real numbers $y$ such that the expression $\frac{y + 2i}{y - 2i}$ is a real number. Enter all the solutions, separated by commas. | nvidia |
Let $a$ and $b$ be positive integers such that $a^2-b^2=3(a+b)$. Find the maximum possible value of $\gcd(a,b)$. | nvidia |
A sequence of five positive integers has the property that the first three terms form a geometric progression, the last three terms form an arithmetic progression, and the sum of the second and fourth terms is equal to twice the third term. Find the product of the first and last terms. | nvidia |
In an infinite geometric sequence, the sum of all terms is $42$ times the sum of the sequence that results if the first five terms of the original sequence are removed. What is the value of the sequence's common ratio? | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) - y) = g(y - x^2) + 2g(x) y\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the product of all possible values of $g(2).$ Find $m \times t.$ | nvidia |
If we let $f(n)$ denote the sum of all the positive divisors of the integer $n$, how many integers $i$ exist such that $1 \le i \le 2010$ and $f(i) = 1 + \sqrt{i} + i$? | nvidia |
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