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Count the number of five-digit positive integers that have exactly two digits equal to 3 and two digits equal to 5, with the remaining digit being a 2. | nvidia |
For a certain positive integer $n,$ there exist real numbers $x_1,$ $x_2,$ $\dots,$ $x_n$ such that
\begin{align*}
x_1 + x_2 + x_3 + \dots + x_n &= 1000, \\
x_1^4 + x_2^4 + x_3^4 + \dots + x_n^4 &= 512000.
\end{align*}Find the smallest positive integer $n$ for which this is possible. | nvidia |
Let $f(z)$ be a polynomial with complex coefficients such that $f(z^2) = f(z)^2 + 1$ for all $z$. Find all possible values of $f(1)$.
Can you help me solve this problem? | nvidia |
The function
$$t(z)=\sqrt[3]{\frac{1}{(z-4)^2}}$$
is defined on a subset of the real numbers. Determine the set of all $z$ such that $t(z)$ is defined, and express the answer in interval notation. | nvidia |
Find the sum of all numbers $x$ satisfying $x+25/x = 10.$ | nvidia |
Let $T(r)$ be the sum of the infinite series
$$2+2r+2r^2+2r^3+\cdots.$$
Let $b$ be a number between $-1$ and $1$ such that $T(b)+T(-b)=4$. Find $T(b)T(-b)$. | nvidia |
Determine the smallest positive integer $n$ such that $n^2-1$ is divisible by 121. | nvidia |
Let $a$ and $b$ be two non-negative real numbers such that $a^2 + b^2 = 4.$ Enter the ordered pair $(a,b)$ for which $ab(a+b)$ is maximized. | nvidia |
In a quadrilateral, the lengths of the sides are consecutive even integers, and the perimeter is 40. What is the length of the diagonal that separates the quadrilateral into two triangles with equal areas? | nvidia |
The function $f(x)$ is a monotonically increasing function on the entire real line. The equation $f(f(x)) = x^2$ has two real roots, $x_1$ and $x_2$, and it is known that $f(x_1) > f(x_2)$. Find the value of $f(x_1) + f(x_2)$. | nvidia |
Find all natural numbers n such that 1!+2!+...+n! is a perfect square. | nvidia |
A set of positive integers contains exactly one number that is a perfect square and exactly two numbers that are perfect cubes. If the sum of all the numbers in the set is 317, what is the largest possible value of the perfect square number? | nvidia |
Let $a$ and $b$ be two complex numbers such that $a^2+b^2=-4$. Find the sum of all real numbers $c$ that satisfy the equation
\[a\left(c+bi\right)+b\left(c-bi\right)=12.\] | nvidia |
Two numbers are relatively prime if their greatest common divisor is 1. Prove that if the product of two relatively prime positive integers is a perfect square, then both integers are perfect squares themselves. | nvidia |
Find the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients.
Another problem inspired by this one:
Determine the number of polynomials of the... | nvidia |
Find the smallest number that is a multiple of the sum of any two distinct integers from 1 to 9. | nvidia |
Find the smallest positive integer $n$ such that $n$ is divisible by 5 and the sum of its digits is a multiple of 7. | 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 |
When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.
When the sum of the three integers is divided by $12$, what is the remainder? | nvidia |
Find all real functions $f(x)$ that satisfy the equation
$f(x-f(y))=f(x)+f(f(y)-y)-x$
for all real numbers $x$ and $y$, given that $f(7)=0$. | 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 $a$.
The positive four-digit integers that use each of the four dig... | nvidia |
In the triangle $ABC$, the points $D$ and $E$ are such that $BD$ is perpendicular to $AC$ and $CE$ is perpendicular to $AB$. If $BD = 4, CE = 7,$ and the area of the triangle $ABC$ is $18,$ find the area of the quadrilateral $ADEC$. | nvidia |
Given the cubic equation $x^3 + ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are integers, find the possible values of $c$ such that all three roots of the equation are distinct perfect squares. | nvidia |
Given a square with side length $s$, one of its vertices is located at the origin $(0,0)$ and another at $(s,0)$. A third vertex is located at $(a,b)$. Find the coordinates of the fourth vertex, given that the square's sides are parallel to the axes and that the length of the side between the third and fourth vertices ... | nvidia |
The coefficients of the quadratic equation $ax^2+bx+c=0$ are such that the sum of the fourth powers of its roots is equal to 41 and the sum of the squares of its roots is equal to 5. Determine $a$ assuming that $b+c=2$. | nvidia |
Determine the value of the three-digit integer $\overline{aba}$ that is divisible by 9, where $a$ and $b$ are distinct digits. | nvidia |
The numbers from 1000 to 1004 are written on five cards. On each move, we simultaneously flip over any two cards and write the sum of the numbers on the faces down on each card. After a sequence of moves, it is found that all five cards have the same number on both sides. What is the smallest number that could be on th... | nvidia |
Let $f(x)$ be a polynomial of degree $5$ with real coefficients. The polynomial $f(x)$ has roots $x_1$ and $x_2$ such that $x_1^2+x_2^2=10$ and $x_1^3+x_2^3=-20$. If the coefficient of the $x^3$ term is $3$, find the coefficient of the $x^4$ term. | nvidia |
Find the smallest positive integer $a$ such that the equation $ax\equiv 2023\pmod {2024}$ has a solution. | nvidia |
Let $a$, $b$, $c$ be positive numbers such that $a+b+c=4$ and $ab+bc+ca=5$. Find the largest possible value of $a^3+b^3+c^3$. | nvidia |
The product of the squares of three consecutive positive integers is 9455. What is the sum of the fourth powers of the three original integers? | nvidia |
Find the smallest four-digit palindrome number that is divisible by 7 and has exactly two digits that are perfect squares. | nvidia |
Let $p(x)$ be a polynomial with integer coefficients such that
$$p(1)=p(3)=p(5)=p(7)=p(9)=2.$$
Find $p(0)$ modulo $5$. | nvidia |
The region $R$ is defined by the set of all points $(x,y)$ such that $|x+y| \le 1$ and $|xy| \le 1$. Find the area of $R$. | nvidia |
Find the sum of all integer divisors of 70 that also divide 21. (Recall that the divisors of an integer may be positive or negative.) | nvidia |
Let $n$ be a positive number such that when $237$ is divided by $n$, the remainder is $7$. Compute the sum of all possible two-digit values of $n$. | nvidia |
Let $a$ and $d$ be real numbers. If the polynomial $x^2+ax+d$ has exactly one real root and $a=2d-3$, find the value of the sum of all possible values of $d$. | nvidia |
The graph of the function
\[y=\frac{x^2+1}{x^2-1}\]
consists of two separate pieces, as it is undefined at x=1 and x=-1.
Find all points on the graph that are an equal distance from (1,2) and (-1,2). | nvidia |
Given functions $f(x) = ax+b$ and $g(x) = x^2 + c$, where $a$, $b$, and $c$ are real constants, it is known that $g(f(x))$ is a quadratic function in $x$. Determine $a$ such that $g(f(x))$ can be factored into a perfect square trinomial for all $x$. | nvidia |
Find the largest value of $m$ such that $3x^2+mx+24$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
What is the positive difference between the sum of the first 25 positive odd integers and the sum of the first 18 positive even integers? | nvidia |
Find the positive value of $m$ such that the equation $4x^2 + mx + 2 = 0$ has a solution that is a positive integer. | nvidia |
Let $S$ be the set of all integer sequences $(a_n)_{n=1}^{\infty}$ satisfying the following properties:
(i) There exists a positive integer $N$ such that for all $n>N$, we have $a_{n+2}=a_{n+1}+a_n$.
(ii) For every positive integer $m$, there exists a positive integer $n$ such that $a_n$ is divisible by $m$.
Find th... | nvidia |
What is the largest positive integer $n$ such that $3^n \div 27^2$ is an integer? | nvidia |
Let $T$ be the set of all positive real numbers. Let $g : T \to T$ be a function such that
\[g(x) + g(y) = g(x^y + y^x)\]for all $x,$ $y \in T$. Find the largest possible value of $g(3)$. | nvidia |
Find the largest negative integer in base 5 that can be expressed as the difference of two positive integers in base 5. | nvidia |
Consider a function $g : \mathbb{Z}^+ \to \mathbb{Z}^+$ such that for any positive integers $m$, $n$, and $p$,
\[g(m \times n + p) = g(m) + g(p)\times g(n).\]
Given that there exists a unique positive integer $a$ such that $g(a) = a$, determine the possible values of $a$. | nvidia |
A set of five distinct positive integers has the following properties:
- The sum of the numbers in the set is 20.
- When any two numbers are selected from the set and added together, the result is a different positive integer each time.
- For every three numbers selected from the set, there is at least one pair of n... | nvidia |
Find the smallest four-digit integer that is divisible by the sum of its distinct, non-zero digits. | nvidia |
Let $f(n)$ be the sum of the positive integer divisors of $n$. For how many values of $n$, where $1 \le n \le 25$, is $f(n)$ prime? | nvidia |
The graph of $y=f(x)$ is shown below, with $1$ unit between grid lines. Assume $f(x)$ is defined only on the domain shown.
What is the sum of all integers $c$ for which the equation $f(x)=c$ has exactly $6$ solutions?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm... | nvidia |
What is the positive integer $M$ for which $$ 33^2 \times 77^2 = 11^2 \times M^2 \? $$ | nvidia |
The function $f(x)$ satisfies
\[f(x+y) = f(x) + f(y)\]for all real numbers $x$ and $y.$ Find the value of $f(-1)$ given that $f(3) = 5.$ | nvidia |
Find all complex numbers $z$ such that $|z^2+1|=|z|^2+1$. | nvidia |
A subset of the integers $1,2,\cdots,100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?
$\text{(A) } 50\quad \text{(B) } 66\quad \text{(C) } 67\quad \text{(D) } 76\quad \text{(E) } 78$
| nvidia |
Find a three-digit number that is divisible by 15 and also satisfies the condition that its hundreds and units digits are even, while its tens digit is odd. | nvidia |
Find the number of positive integers less than or equal to 100 that have exactly three divisors. | nvidia |
Let $A$ be the set of all positive integers less than $50$ that are not divisible by $4$, and let $B$ be the set of all positive integers less than $50$ that are not divisible by $3$. Determine the number of elements in the set $A \cap B$ that are one more than a multiple of $5$. | nvidia |
In a certain triangle, the sum of the squares of the medians is equal to 54. If the length of the longest side of the triangle is $a$, find the area of the triangle. | nvidia |
The decimal representation of $\frac{5}{7}$ is $0.\overline{a_1a_2a_3a_4a_5a_6}$. Find the value of the sum $a_1+a_3+a_5$. | nvidia |
Determine all possible values of $\theta$ such that the matrix
$\begin{pmatrix} \cos \theta & \sin \theta \\ \cos \theta & \sin \theta \end{pmatrix}$
has no inverse. | nvidia |
If $a$, $b$, and $c$ are positive integers such that $a+b+c=15$, find the maximum value of $abc$. | nvidia |
The average of the numbers $1, 2, 3,\dots, 98, 99,$ and $x$ is $100x$. What is $x$?
A new problem:
A set of 5 positive integers has an average of 10. If the set contains the numbers 2, 5, and 12, and the remaining two numbers are consecutive integers, find the sum of the two consecutive integers. | nvidia |
Find the cubic polynomial with real coefficients that has $2 + \sqrt{3}i$ and $-1$ as roots. | nvidia |
Find all real numbers $x$ such that \[3 \le \frac{x}{2x-5} < 8.\](Give your answer in interval notation.) | nvidia |
Let $a_1, a_2, \ldots$ be a geometric sequence, and let $S_n$ denote the sum of the first $n$ terms of the sequence. If the ratio $\frac{S_{2n}}{S_{2n+1}}$ is a constant that does not depend on $n$, for all positive integers $n$, find the common ratio. | nvidia |
Let $x$ and $y$ be two positive real numbers such that their sum is 35. If the sum of $x^2$ and $xy$ is 525, what is the product of $x$ and $y$? | nvidia |
Let $a, b,$ and $c$ be positive real numbers such that $a + b + c = 3.$ Suppose that these numbers are written on three different cards, one number per card, and given to three students, Adriana, Bruna, and Carla, one card to each. After sharing the values on their cards with each other, only Adriana was able to surely... | nvidia |
Let's consider the function $f(x)=\frac{x}{3x^2+5}$.
Determine $c$ such that the function $f(x)+c$ is equal to its inverse. | nvidia |
The sum of the cubes of two positive integers is 91, and the sum of their squares is 29. What is the difference between the integers? | nvidia |
A square and a circle intersect in such a way that the diameter of the circle passes through the center of the square and the circle intersects the square at the midpoint of two of its sides. If the area of the circle is 25π square centimeters, what is the area of the square? | nvidia |
In a circle with radius 10 units, an isosceles triangle is inscribed such that the two equal sides are tangent to a smaller circle with radius 5 units. What is the area of the shaded region between the two circles and outside the triangle? | nvidia |
The two graphs $y = \log_2 x$ and $y = \log_{1/2} x$ intersect at a single point $(a,b)$. Find $b$.
[asy]
import graph; 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 tickdownlength=-0.15inch;... | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of
\[\frac{x + y}{xyz}.\] | nvidia |
A positive integer is called a circular divisor number if it is divisible by the sum of any of its adjacent pairs of digits (including the case where the first and last digits are considered adjacent). How many circular divisor numbers are there from 10000 through 99999, inclusive? | nvidia |
Find the number of 1's in the binary representation of the product of two consecutive prime numbers whose sum is 60. | nvidia |
For a positive integer n, find the minimum value of $a_1 + a_2 +... + a_n$ given that $a_1, a_2,..., a_n$ are non-negative real numbers such that $\sqrt{a_1} + \sqrt{a_2} +... + \sqrt{a_n} = 4$ and $\sqrt{a_1}^2 + \sqrt{a_2}^2 +... + \sqrt{a_n}^2 = 6$. | nvidia |
Let $T$ be the sum of all positive integers $m$ such that $m^3+25m^2-2400m-2000$ can be expressed as the product of two consecutive integers. Find the remainder when $T$ is divided by $125$. | nvidia |
Let $f(x)=x^3+ax^2+bx+c$ be a polynomial with rational coefficients and real zeros, $r$, $s$, and $t$. The zeros are such that the squares of two of the zeros sum up to the cube of the third zero. How many values of $a$ are possible? | nvidia |
Let $f(x)$ be a function that maps each positive integer $x$ to the smallest integer $y$ such that $x^2+y^2$ is divisible by 4. Determine the sum of all possible values of $f(4)+f(5)+f(6)+f(7)$. | nvidia |
Find all real numbers $x$ which satisfy \[\frac{x-1}{x-3} \ge 2.\](Give your answer in interval notation.) | nvidia |
Let $n$ be a positive integer such that $2n + 1$ and $3n + 1$ are both perfect squares. What is the smallest possible value of $n$? | nvidia |
Find the smallest positive integer $n$ such that $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}$ is equal to a fraction with a denominator less than or equal to 10. | nvidia |
Find the smallest positive integer whose square has exactly 5 positive integer factors, not counting 1 and itself. | nvidia |
Let $g$ be a function taking the positive integers to the positive integers such that
\[g(m + n) \cdot g(mn) = g(m)^2 + g(n)^2\]for all positive integers $m$ and $n.$
Let $p$ be the number of possible values of $g(3),$ and let $t$ be the product of all possible values of $g(3).$ Find $p + t.$ | nvidia |
Let \[f(x) = \left\{
\begin{array}{cl}
-3x & \text{if } x < 2, \\
x^2-7x+10 & \text{if } x \ge 2.
\end{array}
\right.\]Find the sum of all values of $x$ such that $f(x) = 0$. | nvidia |
Determine the value of $b$ such that the graph of the function
$$f(x)=\frac{x^3-2x^2+bx-1}{x^2+x-2}$$
has a horizontal asymptote at $y=2$. | nvidia |
Find the maximum value of
\[x^3 + x^2y + xy^2 + y^3\]over all real numbers $x$ and $y$ such that $x + y = 1$. | nvidia |
Find the sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$. | 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 |
A positive number $m$ can be expressed as the product of the cubes of two distinct primes. If the sum of these two primes is 13, determine $m$. | nvidia |
What is the smallest positive integer that has exactly eight distinct positive factors? | nvidia |
Find all real values of $x$ that satisfy \[\frac{1}{x^2+1} > \frac{3}{x} + \frac{17}{10}.\](Give your answer in interval notation.) | nvidia |
Ava writes down the smallest positive multiple of 24 that is a perfect square, the smallest positive multiple of 24 that is a perfect cube, and all the multiples of 24 between them. How many integers are in Ava's list? | nvidia |
A portion of the graph of a quadratic function $f(x)$ is shown below.
Let $g(x)=-f(x)$ and $h(x)=f(-x)$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=g(x)$ intersect, and $b$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, then what is $10a+b$?
[asy]
size(150);
real ... | nvidia |
Let $S$ be a set of all real numbers $x$ for which there exists a real number $y$ such that $\tan (x+y)=y$. Prove that $S$ contains a non-degenerate closed interval.
\begin{align*}
\end{align*} | nvidia |
Determine the largest value of $n$ for which $13^n$ has exactly three digits. | nvidia |
Find the number of positive, three-digit integers that have exactly one 7 as a digit and do not contain a 4 as a digit.
(Hint: Instead of counting the numbers with at least one 7, count the numbers with no 7's and subtract from the total count of three-digit numbers without 4's.) | nvidia |
We define a function $f(x)$ such that $f(11)=34$, and if there exists an integer $a$ such that $f(a)=b$, then $f(b)$ is defined and
$f(b)=3b+1$ if $b$ is odd
$f(b)=\frac{b}{2}$ if $b$ is even.
What is the smallest possible number of integers in the domain of $f$? | nvidia |
Find the number of distinct arrangements of the digits 1, 2, 3, and 4 in a 4-digit positive integer such that the sum of the digits in the even positions is equal to the product of the digits in the odd positions. | nvidia |
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