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Let $P$ be a point on the line
$$lx+my+n=0$$
and $Q$ be a point such that the midpoint of $PQ$ is the origin.
Find the coordinates of point $Q$, if it is known that its distance from the given line is $d$.
response finished. | nvidia |
A trapezoid $ABCD$ is inscribed in a circle, where $A(-4,2)$, $B(6,0)$, $C(8,-4)$, and $D(-2,-6)$, and $AB$ is parallel to $CD$. A line with equation $y=t$ intersects $AB$ at $T$ and $CD$ at $U$, dividing the trapezoid into two smaller trapezoids. Find $t$ such that the two smaller trapezoids have equal areas. | nvidia |
Let $h(x)$ be a function on $\mathbb{R}$ such that the range of $h(x)$ is $(-\infty,0)$. Determine the largest possible value of $c$ such that the range of $f(x) \cdot h(x)$ is $(-\infty,c]$. | nvidia |
Find the greatest possible value of $t$ such that the equation $x^2+4x+t=0$ has no real solutions. | nvidia |
Determine the smallest positive integer $n$ such that $2^n+1$ is a perfect square. | nvidia |
Let $x$ and $y$ be positive real numbers such that $7x + 5y = 70.$ Find the maximum value of $xy^2.$ | nvidia |
Find the point $E$ on side $\overline{BC}$ of triangle $ABC$ such that the area of triangle $ABE$ is equal to the area of triangle $ACE$. Prove that $\overline{AE}$ bisects angle $\angle BAC$. | nvidia |
Find the smallest positive integer $a$ such that $2a^2 \equiv 337 \pmod{29}$. | nvidia |
A tower with 100 levels is being built, and each level is a rectangle with a length that is twice its width. The length of the rectangle at the first level is 10 meters, and the length of the rectangle at the second level is 12 meters. For each subsequent level, the length of the rectangle is the sum of the lengths of ... | nvidia |
Let $x,y,z$ be three positive real numbers such that $x,y,z$ is a geometric progression and $x,z,y$ is an arithmetic progression. Find the largest possible value of $\frac{x+y+z}{xyz}$. | nvidia |
Find the sum of the coordinates of the midpoint of the segment with endpoints $(4, -2)$ and $(-3, 7)$. | nvidia |
Find the sum of all positive integral values of $m$ for which $\frac{2m-3}{m}$ is an integer. | nvidia |
The sum of the first $n$ terms of a geometric sequence is denoted as $S_n$, and the sum of the first $n$ terms of the sequence formed by the partial sums of the original sequence is denoted as $T_n$. If you are given the value of $S_m$ for some integer $m$, and the common ratio $r$ of the original sequence, determine t... | nvidia |
The roots of the equation $$x^3-ax^2+bx-a=0$$ are such that the sum of the roots is equal to the sum of their reciprocals. Determine $a$. | 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 |
Let $f(x)$ and $g(x)$ be two functions such that the domain of $g$ contains the range of $f$. Suppose that for all $x$, it is true that $f(g(x))=x^2+1$. If $f(x)=x^2+kx-1$, where $k$ is a real constant, find $g(7)$. | nvidia |
We are given a set of four positive integers such that each number is a prime number, and it can be expressed as $\{A, B, A-B, A+B\}$.
The sum of these numbers is $$\sum\{A, B, A-B, A+B\} = 2A + 2B$$
There are five statements given below and one of them is correct about the sum of these numbers:
A. The sum of the n... | nvidia |
A palindrome number is a number that reads the same forwards and backwards. Find the largest palindrome number that is divisible by 7 and has exactly 3 digits. The number cannot start with 0. | nvidia |
The arithmetic mean of an odd number of consecutive odd integers is $y$. Find the sum of the smallest and largest of the integers in terms of $y$. | nvidia |
In a triangle ABC, the coordinates of the vertices A, B, and C are (x1, y1), (x2, y2), and (x3, y3) respectively. The triangle is inscribed in a circle with radius r. What is the probability that the area of the triangle is less than half the area of the circle? | nvidia |
A hexagonal prism is constructed from 12 points in 3-dimensional space. The sum of the x-coordinates of these 12 points is 36, the sum of the y-coordinates is 48, and the sum of the z-coordinates is 60. A second hexagonal prism is formed by connecting the midpoints of the edges of the first prism. Finally, a third hexa... | nvidia |
Consider the geometric sequence $a,ar,ar^2,\ldots$ where $a$ and $r$ are positive real numbers. Suppose the sum of the first four terms is equal to the product of the first two terms, and the sum of the first six terms is equal to the product of the first three terms. Find the value of $r$. | nvidia |
Let $S$ be the set of positive rational numbers less than $1$ that have a terminating decimal representation. For which values of $n$, where $n$ is a positive integer, is it possible to find $r \in S$ such that $r^n = \frac{1}{2}$? | nvidia |
Find the value of $a^3+b^3$ if $a$ and $b$ are real numbers such that \[\begin{aligned} a+b&=5 \\ a^2-ab+b^2 &= 29.\end{aligned}\] | nvidia |
Let x and y represent two natural numbers such that the sum of the reciprocals of x and y is equal to 1/10. Determine the values of x and y for which the quantity \(\sqrt{x^{2}+y^{2}}\) is minimized. | nvidia |
Let $a$ and $b$ be two distinct positive integers. Find the smallest value of $n$ such that the product of the last three digits of $a^n$ and $b^n$ is a perfect square. | nvidia |
Find the positive integer $M$ such that the difference between the cube of $M$ and $M$ itself is equal to the absolute value of the difference between the number formed by reversing the digits of $M$ and the square of the sum of its digits. | nvidia |
Find the number of real solutions of the equation
\[\frac{x}{50} = \cos x.\] | nvidia |
Let f(n) be the sum of all the divisors of a positive integer n.
If f(f(n)) = n + 2, then call n superdeficient.
How many superdeficient positive integers are there? | nvidia |
Let $S$ be the set of all positive integers $n$ such that $n$ is not a multiple of 3 and $7n+1$ is not divisible by any perfect square greater than 1. Compute the sum of all positive integers less than 500 that are not in $S$. | nvidia |
Two fair six-sided dice are rolled. What is the probability that the sum of the numbers showing on the two dice is greater than 10 and the difference between the numbers showing on the two dice is less than 4? | nvidia |
How many 4-digit positive integers exist that satisfy the following conditions: (A) Each of the first two digits must be 1, 4, or 5, and (B) the last two digits cannot be the same digit, and (C) each of the last two digits must be 5, 7, or 8? | nvidia |
In the equation $x^2+kx+5=0$, find the value of $k$ such that the sum of the squares of the roots is equal to $16$. | nvidia |
Let $S$ be the set of polynomials of the form $$ Q(z)=z^n+d_{n-1}z^{n-1}+\cdots+d_2z^2+d_1z+\frac{1}{2}, $$where $ d_1,d_2,\dots, d_{n-1} $ are rational numbers and $Q(z)$ has roots that are all located on the unit circle in the complex plane. How many polynomials in $S$ have the property that all their coefficients ar... | nvidia |
Given the function $g(x) = \left\lceil\sqrt{2x}\right\rceil$ for $x \geq 0$, and $g(x) = \left\lfloor\sqrt{-3x}\right\rfloor$ for $x < 0$, what is the smallest positive integer $n$ such that there are exactly two distinct values of $x$ that satisfy $g(x) = n$? | nvidia |
Find $m$ such that $3^4 \cdot 2^3 \cdot m = 10!$. | nvidia |
The sum of the squares of two consecutive integers is 181. What is the smaller of the two integers? | nvidia |
Find the number of integers $n$ such that $2^{n-1} < 100 < 2^n$. | nvidia |
Evaluate the expression \[ \frac{a+2}{a+1} \cdot \frac{b-1}{b-2} \cdot \frac{c + 8}{c+6} , \] given that $c = b-10$, $b = a+2$, $a = 4$, and none of the denominators are zero. | nvidia |
Determine the smallest positive integer $n$ such that the product of the first $n$ positive integers is divisible by $7^k$, where $k$ is the largest integer satisfying the condition. | nvidia |
Let $Q$ be the product of the first $200$ positive integers that are one more than a multiple of $4$. Find the largest integer $m$ such that $Q$ is divisible by $2^m$. | nvidia |
Three different positive integers are chosen. When the product of any two of them is subtracted from the product of all three, which of the following sets of numbers could be obtained?
$$
\text A. \ \ \{20, 32, 40\} \qquad \text B. \ \ \{12, 36, 72\} \qquad \text C. \ \ \{35, 50, 75\} \qquad \text D. \ \ \{21, 42, 84\}... | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
Find the number of positive integers less than or equal to 2002 that have exactly three distinct positive divisors. | nvidia |
The vertices of a triangle are $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$. Determine the coordinates of the point $(x, y)$ inside the triangle such that the sum of the distances from $(x, y)$ to the three vertices is minimized. | nvidia |
The sum of the squares of two consecutive prime numbers is between two perfect cubes. Determine the difference between these two perfect cubes. | nvidia |
The arithmetic mean of four numbers is 15. Two of the numbers are 10 and 18 and the other two are equal. What is the product of the two equal numbers? | nvidia |
Find the smallest five-digit palindrome number that is divisible by 7 and has a sum of its digits equal to 25. | nvidia |
Let $P_1$ be a regular $r~\mbox{gon}$ and $P_2$ be a regular $s~\mbox{gon}$ $(r\geq s\geq 3)$ such that the ratio of the perimeter of $P_1$ to the perimeter of $P_2$ is equal to the ratio of the area of $P_2$ to the area of $P_1$. What's the largest possible value of $r$? | nvidia |
Find the value of $x$ such that the four-digit number $\underline{x15x}$ is divisible by 18. | nvidia |
The product of the first $n$ positive integers is $n!$. The product of the first $n$ positive even integers is $2^n \cdot n!$. The product of the first $n$ positive odd integers is $\frac{(2n)!}{2^n \cdot n!}$. Find the product of the first $n$ positive integers that are multiples of 3. | nvidia |
Find the number of positive integers $a$ such that the equation $x^3 - 2ax^2 + (a^2 + 1)x - a^2 = 0$ has at least two rational roots and $a \leq 50$. | nvidia |
Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a+b+c = 12.$ Find the largest possible value of $b.$ | nvidia |
Let $n$ be a positive integer with 2015 digits. What is the smallest positive integer $k$ such that $n^3 - k$ is divisible by 8? | nvidia |
Find the smallest positive value of $k$ such that the equation $x^2 + kx - 2010 = 0$ has two integer roots. | nvidia |
A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater than the number of the green point, but les... | nvidia |
In a circle with radius 7, a chord of length $x$ is drawn. Find the maximum number of different positive integer lengths $x$ can have such that the triangle formed by the chord and the radii connecting its endpoints to the center of the circle is acute. | nvidia |
Find the smallest positive integer $n$ such that for any distinct integers $a,b,c,d$, the maximum value of $|a|+|b|+|c|+|d|$ occurs when $n|ab-cd|$. | nvidia |
Given the equations of two circles centered at (0, 0) and (7, 0) with radii r1 and r2 respectively, find the type of conic section that is formed by the set of points for which the sum of the distances from a point on the conic section to the two circles is constant. Assume that the circles do not intersect. Determine ... | nvidia |
What is the maximum number of distinct pairs of integers that can be formed from a set of 12 integers, such that the sum of each pair is a multiple of 3? | nvidia |
The sum of the squares of two consecutive even integers is 340. Find their difference. | nvidia |
Let $x$ be the number of positive multiples of $7$ that are less than $50$. Let $y$ be the number of positive integers that are less than $50$, and a multiple of $5$ and a multiple of $2$. Compute $xy(x-y)$. | nvidia |
Let $p$ and $q$ be two real numbers such that
$$p=\sqrt{q+\sqrt{q+\sqrt{q+\cdots}}}, \quad q=\sqrt{p+\sqrt{p+\sqrt{p+\cdots}}}.$$Find $p+q$ when $pq=15/4$. | nvidia |
Let $ABCD$ be a convex quadrilateral and let $E$ be the intersection of its diagonals. If $P$, $Q$, $R$, and $S$ are points on segments $AB$, $BC$, $CD$, and $DA$ such that $\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RD}=\frac{DS}{SA}$, show that the area of quadrilateral $PQRS$ is equal to half the area of quadrilateral $A... | nvidia |
In a right triangle with integer length sides, the length of the shorter leg is given as 15 units. What is the maximum possible length of the hypotenuse such that the triangle still has integer length sides? | 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 |
A moving robot starts at the point $(6,6)$ and moves until it hits one of the coordinate axes for the first time. When the robot is at the point $(x,y)$, it moves at random to one of the points $(x-1,y)$, $(x,y-1)$, or $(x-2,y-1)$, each with probability $\frac{1}{3}$, independently of its previous moves. The probabilit... | nvidia |
If $x$ and $y$ are integers such that $x+y$ is even, which of the following statements must be true?
A. $x$ and $y$ have the same parity.
B. At least one of $x$ or $y$ is even.
C. $x$ and $y$ are both even.
D. $x$ and $y$ are both odd.
Answer by listing your choices in alphabetical order, separated by commas. | nvidia |
Find the value of $c$ such that the inequality $x^2+cx+4>0$ holds true for all real numbers $x$ except when $x$ belongs to the interval $(-2, 2)$. | nvidia |
Determine the number of values of x for which the expression x^2-4/(x^2+5x+6)*(x-2) has no solution. | nvidia |
Suppose that $x$ and $y$ are positive real numbers such that $x^2-xy+2y^2=8$. Find the minimum possible value of $x^2+xy+2y^2$ such that $xy$ is a positive integer.
Note: This problem requires a different approach as we need to find the minimum value of the expression instead of the maximum value, and we also have th... | nvidia |
Given a hexagon where all six vertices are labeled with positive integers from 1 to 6, such that the sum of the numbers at adjacent vertices is divisible by 3, determine the possible labels of the vertices adjacent to 3. | nvidia |
In the decimal representation of the number $2^{99}$ there are no zeros. Prove that there exist two decimal representations of powers of 2 such that the number built up of the last n (n > 4) digits of one of these powers is equal to the number built up of the last n - 4 digits of the other power. | nvidia |
Determine all possible digits of base $b$, where $b$ is a positive integer, such that $7_{b}+2_{b}=B_{b}$, where $B_{b}$ is the largest digit of base $b$. | nvidia |
Find the value of $b$ such that the equation $(x+2)^2=b(x-1)$ has exactly two distinct real roots, given that one of the roots is $x=3$. | nvidia |
On the $x$-axis, 5 points are selected at random in the interval $[-1, 1]$. What is the probability that there will be at least one point on the left side and at least one point on the right side of the origin? | nvidia |
How many four-digit numbers have the property that the first two digits are the same and the last two digits are also the same, but the two pairs of identical digits are not the same? | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x) + g(y) + g(xy) = \frac{1}{x} + \frac{1}{y}\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(1),$ and let $t$ be the product of all possible values of $g(1).$ Find $m \times t.$ | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $\cfrac{ab}{bc}<1$ and $\cfrac{bc}{ca}<1$. Evaluate $$\left\lfloor \cfrac{\left\lceil \cfrac{ab}{bc}\right\rceil}{\left\lceil \cfrac{bc}{ca}\right\rceil}\right\rfloor +\left\lfloor \cfrac{\left\lceil \cfrac{bc}{ca}\right\rceil}{\left\lceil \cfrac{ab}{bc}\right\rceil}... | nvidia |
In a right triangle with legs of length 3 cm and 4 cm, find the maximum area of a rectangle that can be inscribed inside the triangle, with one side along the hypotenuse. | nvidia |
Consider the triangular array of numbers with 1, 2, 4, 8, $\dots$ along the sides and interior numbers obtained by multiplying the two adjacent numbers in the previous row. Rows 1 through 6 are shown.
\[
\begin{array}{ccccccccccc}
& & & & & 1 & & & & & \\
& & & & 2 & & 2 & & & & \\
& & & 4 & & 4 & & 4 & & & \\
& & 8 ... | nvidia |
The vertices of a triangle are $(2a-3, 3)$, $(4-a, -2)$, and $(3a-1, 1)$. Find the value of $a$ for which the area of the triangle is zero. | nvidia |
Determine the sum of the coefficients of the equation $4x^2+7x-2=0$, given that one of its roots is equal to the product of the roots of the equation $x^2-5x+6=0$. | nvidia |
Let $a$ and $b$ be two positive integers with different parity, such that $ab$ and $a+b$ are both divisible by 13. What is the smallest possible value of $ab$? | nvidia |
Find the least positive integer n, such that 8n+49 is divisible by 11. | nvidia |
Given a rational number $\dfrac{p}{q}$ where $p$ and $q$ are relatively prime, find the smallest number of decimal digits that repeat in the decimal expansion of $\dfrac{p}{q}$. | 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 variables $x$, $y$, $z$ satisfy the following system of equations:
\begin{align*}
x+y+z&=18\\
x^2+y^2+z^2&=122
\end{align*}Given that $z=\frac{x}{y}$, find the value of $x+y+z$. | nvidia |
Find the smallest positive integer $x$ such that the product of all digits in the number $23x54$ is divisible by 32. | nvidia |
Find the maximum value of $x$ such that the inequality $\displaystyle\frac{x^2+1}{x+1} < 5$ holds. | nvidia |
What is the least positive integer that is divisible by four distinct primes? | nvidia |
The product of the digits of a two-digit number is twice the product of the digits of another two-digit number. The units digits of both numbers are equal and not equal to zero. Determine the numbers such that their sum is minimal. | nvidia |
A cube has numbers written on each of its faces. Each edge is labeled with the sum of the numbers on the two faces adjacent to the edge. If the sum of the numbers on the edges is equal to 432, then what is the sum of the numbers written on the faces? | nvidia |
Given a square with a side length of 5 cm, inscribe a circle within it such that the distance between the center of the square and the center of the circle is minimized. What is the diameter of the inscribed circle? | nvidia |
Find the sum of the positive divisors of the positive divisors of 48. | nvidia |
What is the largest positive integer $n$ such that $2^n$ is less than $100$? | nvidia |
How many positive integers less than 100 have an odd number of positive divisors? | nvidia |
Determine the largest possible value of $n$ such that $n!$ is divisible by $2^{11}$. | nvidia |
Find the number of polynomials of the form $f(x)=ax^2+bx+c$ where $a$, $b$, and $c$ are real numbers, such that $f(f(x))=0$ for all $x$. | nvidia |
Let $P$ be a plane passing through the points $(1,2,3)$, $(4,5,6)$, and $(7,8,9)$. Find the point $(x,y,z)$ on the plane $P$ that minimizes the sum $x+y+z$. | nvidia |
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