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Let $n$ be a positive integer greater than 2.
As $n$ ranges over all positive integers greater than 2, how many different remainders can $2^n$ leave upon division by 99? | nvidia |
Find the sum of all the numbers from 1 to 100 that are neither divisible by 3 nor by 5. | nvidia |
Find the product of the values of $y$ that satisfy the equation $y^2+3y-2=11$. | nvidia |
Find the number of four-digit numbers that have a symmetric structure, meaning the first and fourth digits are the same, and the second and third digits are the same. | nvidia |
The integer $n$ is the largest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\frac{n}{15}$.
| nvidia |
Given that $1 + i$ is a root of the cubic polynomial $ax^3+bx^2+cx+d$, find the value of $a+b+c+d$, given that the coefficient of $x^3$ is $2$ and the constant term is $-4$. | nvidia |
A square is drawn such that one of its sides coincides with the line $y = 5$, and so that the endpoints of this side lie on the parabola $y = x^2 + 3x + 2$. What is the area of the square? | nvidia |
A box contains 30 balls, each labeled with a different integer from 1 to 30. How many ways can the balls be divided into two groups such that the sum of the numbers on the balls in one group is equal to the sum of the numbers on the balls in the other group? | nvidia |
Let $a$ and $b$ be two real numbers such that $a^2+b^2=8$ and $ab=3$. Compute $a^6+b^6$. | nvidia |
Find the sum of the coefficients of the polynomial
$$P(x) = (x-1)(x-2)(x-3)(x-4)(x-5)(x-6).$$ | nvidia |
Find all natural numbers $n$ such that $n$, $n^2+1$ and $n^2+2$ are all prime. | nvidia |
A binary sequence is a sequence of digits that consists only of 0s and 1s. Count the number of 6-digit binary sequences that contain exactly three 1s. | nvidia |
The geometric mean of an even number of consecutive positive integers is $x$. Find the product of the smallest and largest of the integers in terms of $x$. | nvidia |
Find the smallest positive integer $n$ for which $f(n) = 4$, where $f(n)$ denotes the number of distinct ordered pairs of positive integers $(a, b)$ such that for each ordered pair, $a^2 + b^2 = n$. | nvidia |
Find all 4-digit palindromes that are divisible by 7 and have the form $abba$, where $a$ and $b$ are digits. What is the minimum value of $a$? | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that $x + y + z = 6.$ Find the maximum value of $xy^2z^3.$ | nvidia |
Find the number of positive integers $n$ such that $\sqrt{n}$ is an integer and the sum of the digits of $n$ is a perfect square. | nvidia |
Determine $k$ such that the sum of the roots of the quadratic equation $$x^2 - 5x + k = 0$$ is equal to the sum of the squares of the roots. | nvidia |
Find the smallest integer $a$ such that $3a \equiv 1 \pmod{103}$. | nvidia |
The sum of the squares of three consecutive positive integers is equal to the sum of the cubes of their reciprocals. What is the smallest possible value of the middle integer? | nvidia |
Find all tuples (a, b, c) of positive integers satisfying the condition:
cos(aπ) + cos(bπ) = cos(cπ)
such that a, b, and c are distinct and form an arithmetic progression. | nvidia |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | nvidia |
Find the largest $n$ such that $a_n$ can be expressed as the sum of two perfect squares, where $a_n$ is the number obtained by concatenating the integers from $1$ to $n$ from left to right. | nvidia |
Carolyn and Paul are playing a game starting with a list of the integers $1$ to $8.$ The rules of the game are:
$\bullet$ Carolyn always has the first turn.
$\bullet$ Carolyn and Paul alternate turns.
$\bullet$ On each of her turns, Carolyn must remove one number from the list such that this number has at least o... | nvidia |
Determine the value of $a$ such that the equation $\sin (x+a) = \cos (x-a)$ has solutions for any value of $x$. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $x^2+y^2+z^2=12xy+4yz+2zx$. Compute $x^2y+y^2z+z^2x$. | nvidia |
The reality game show Survivor is played with 16 people divided into two tribes of 8. In the first episode, two people get homesick and quit. If every person has an equal chance of being one of the two quitters, and the probability that one person quits is independent of the probability that any other person quits, wha... | nvidia |
Suppose that we have a right triangle $ABC$ with the right angle at $B$ such that $AC = \sqrt{61}$ and $AB = 5.$ A circle is drawn with its center on $AB$ such that the circle is tangent to $AC$ and $BC.$ If $P$ is the point where the circle and side $AC$ meet, then what is $CP$? | nvidia |
A rectangle has a length that is a common multiple of 12, 18, and 24, and a width that is a common divisor of these three numbers. If the area of the rectangle is 144, find the sum of the length and the width. | nvidia |
Determine the largest integer $n$ such that the sum of the series
$$20+40+60+\cdots +20n$$
does not exceed 1000. | nvidia |
Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}.$ Find the number of possible values of $S.$
| nvidia |
A positive integer $n$ can be expressed in two different ways as a sum of two perfect squares: $n=a^2+b^2=c^2+d^2$, where $0<c<a$ and $0<b<d$. Find the smallest possible value of $n$ if each of the four numbers $a, b, c, d$ is divisible by 3. | nvidia |
Given that $a, b,$ and $c$ are real numbers such that the matrix
\[\begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix}\]is not invertible, find all possible values of
\[\frac{a^2}{b^2 + c^2} + \frac{b^2}{a^2 + c^2} + \frac{c^2}{a^2 + b^2}.\] | nvidia |
How many three-digit integers are there such that their digits are not necessarily in increasing order, but the sum of the digits is exactly 12? | nvidia |
Let $g(x)$ be a function that satisfies the following equation for all real numbers $x$ and $y$:
$$g(x + y) = 3^xg(y) + 5^y g(x).$$
Given that $g(2) = 1$ and $g(3) = 5$, find the value of $g(5)$. | nvidia |
Consider the parallelogram with vertices $(12,36)$, $(12,75)$, $(30,102)$, and $(30,63)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | nvidia |
Find the smallest positive integer $n$ such that $73^n \equiv 1 \pmod{74}$. | nvidia |
A 3x3x3 cube is composed of 27 smaller cubes, each with a different integer value. The values are arranged such that the sum of the values on each face of the larger cube is the same. Determine the value of the central cube. | nvidia |
A polynomial $P(x)$ with integer coefficients is said to be "reversible" if there exists an integer $k$ such that $P(k) = P(-k)$. For example, the polynomial $x^3 + 3x^2 - x + 1$ is reversible because $P(2) = P(-2)$. Find the sum of the coefficients of all reversible polynomials of degree 2 with integer coefficients th... | nvidia |
What is the largest positive integer $m$ for which there is a unique integer $k$ such that $\frac{5}{12} < \frac{m}{m + k} < \frac{3}{7}$? | 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 |
How many four-digit numbers have the property that the sum of their digits is equal to 12 and no digit is greater than 5? | nvidia |
What is the smallest positive integer $n$ such that $\frac{n}{n+101}$ is equal to a terminating decimal? | nvidia |
The sum of the digits of a three-digit number is 12. What is the largest possible value of the sum of the digits of its square? | nvidia |
If $\alpha$, $\beta$ and $\gamma$ are positive integers such that $\sqrt[3]{\alpha x} \cdot \sqrt[3]{\beta x} \cdot \sqrt[3]{\gamma x}=6x$, where $x$ is a positive real number, determine $\alpha \beta \gamma$. | nvidia |
Find the smallest value of $d$ such that $3$ is in the range of $f(x)=x^2-2x+d$. | nvidia |
Given that the polynomial $x^2 + mx - 20$ has only positive integer roots, find the average of all distinct possibilities for $m$. | nvidia |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 1000^\circ.$ | nvidia |
The sum of the cubes of two positive integers is 300, and the product of the two integers is 15. What is the sum of the two integers? | nvidia |
Find the smallest positive integer $x$ such that the equation $5x \equiv 3 \pmod{12}$ has a solution, and determine the number of solutions less than or equal to $30$. | nvidia |
Given that $(\sin t+i\cos t)^n=\sin nt+i\cos nt$ is true for all real $t$, how many positive integers $n$ less than or equal to 1000 are there that also satisfy the equation $(\cos t+i\sin t)^n=\cos nt+i\sin nt$ for all real $t$? | nvidia |
Using only the digits 1, 2, and 3, how many positive five-digit integers can be made that are palindromes? | nvidia |
Given a rectangle with a length of 20 units and a width of 12 units, if the length is decreased by $a$ units and the width is increased by $a$ units, determine the value of $a$ such that the area of the new rectangle is 10% larger than the area of the original rectangle. | nvidia |
For how many four-digit positive integers is the sum of the digits equal to 9? | nvidia |
Given that $\sqrt{3} - 1$ is a root of the equation $x^4 + px^2 + q = 0$, where $p$ and $q$ are rational numbers, compute the value of $p$ such that the equation has exactly two real roots. | nvidia |
Consider the expression $\cos(4x) + \sin(4x)$ where $x$ is a real number. Determine the largest possible value of the expression for $x$ in the interval $[0, \frac{\pi}{2})$. | nvidia |
In triangle $ABC$, $M$ is a point on side $AB$ such that $\frac{AM}{MB} = \frac{1}{3}$. If $N$ is a point on side $BC$ such that $BC = 4MN$, and if $AN = CN$, find the ratio of the areas of triangles $AMN$ and $ABC$. | 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 |
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum? | nvidia |
Compute the determinant of the matrix
\[\begin{pmatrix} 2x-3y & 1 \\ 2x^2+y & x-2 \end{pmatrix},\]
where $x$ and $y$ are real numbers. | nvidia |
The perimeter of a square is 96 cm. Find the number of square centimeters in the area of the square. | nvidia |
Find the largest $2$-digit number such that its square has the same digits when read forward and backward (i.e., the square is a palindrome).
There are no 2-digit palindromes. | nvidia |
Find all values of $y$ such that $\frac{2}{\sqrt{2y}}-\frac{1}{\sqrt{2y-1}}=1$. | nvidia |
Find the number of positive integers less than 500 that are not divisible by 2, 3, or 5. | nvidia |
Find the smallest positive integer m such that √m is an irrational number and m contains the digit 5 exactly twice. | nvidia |
Find the number of distinct ordered-pair solutions (x, y) to the equation x + 2y = 20, where x and y are positive integers and x < y. | nvidia |
Let $f$ be a function from the set of positive real numbers to the set of real numbers such that
(i) $f(1) = 2,$
(ii) $f(x^2) = f(x) + f(1/x)$ for all $x > 0,$ and
(iii) $f(x) + f(y) = f(xy) + f(x/y)$ for all $x,y > 0.$
Determine the number of possible functions $f(x)$. | nvidia |
Determine the complex number $w$ satisfying the equation $3w-2\bar{w}=4+20i$. Note that $\bar{w}$ denotes the conjugate of $w$. | nvidia |
Find all positive values of $x$ that satisfy the equation:
$$\sqrt{6x} \cdot \sqrt{12x} \cdot \sqrt{2x} \cdot \sqrt{8x} = 48$$ | nvidia |
Find the integer n such that the following product is less than 0.05:
$$\left(1-\frac12\right)\cdot\left(1-\frac13\right)\cdot\left(1-\frac14\right) \dotsm \left(1-\frac1{n+1}\right)$$ | nvidia |
Given a cube with sides of length 6, what is the maximum number of unit cubes (cubes with sides of length 1) that can be carved out of it such that no two of the carved unit cubes share a common face? | nvidia |
Find the greatest common divisor of all positive integers that can be expressed as the sum of a positive integer and its reverse. For example, 11 (1+11), 26 (15+51), and 37 (18+81) are integers of this form. | nvidia |
Find the smallest positive integer $n$ such that $5n$ leaves a remainder of $2$ when divided by $7$. | nvidia |
A positive integer $n$ can be expressed as the difference of two positive integers $a$ and $b$ such that $\gcd(a,b) = d$. What is the smallest possible value of $n$ for a given value of $d$? | nvidia |
Suppose that $a$, $b$, $c$ are positive integers such that $a!+b!+c!=50$. Determine the largest possible value of $a+b+c$. | nvidia |
A palindrome is a number that reads the same forwards and backwards. Find the largest base 10 palindrome that can be represented as the sum of two distinct base 7 numbers, each with exactly 3 digits. | nvidia |
Given that a real number $x$ satisfies \[\sqrt{64-x^2}-\sqrt{36-x^2}=4.\]Find the value of $\sqrt{64-x^2}+\sqrt{36-x^2}$. | nvidia |
Find the surface area of the solid obtained by rotating the set of points in three-dimensional space that satisfy the inequality $(x-1)^2 + y^2 + (z-2)^2 \leq 1$ about the $z$-axis. | nvidia |
In a set of five consecutive integers, the sum of the smallest and largest numbers is equal to twice the middle number. If the second smallest number is 17, find the product of the three largest numbers. | nvidia |
Find the smallest possible four-digit palindrome number $abba$ such that $a$ and $b$ are digits and $a$ is even, and $abba$ is divisible by 9. | nvidia |
A polynomial $P(x)$ is defined for all real numbers and satisfies $P(x+1)=P(x-1)$ for all $x$. If $P(0) = 0$, determine the smallest possible degree of $P(x)$ such that it has exactly 10 distinct roots in the interval $-10\leq x \leq 10$. | nvidia |
Let $p$ be a prime number and $a$ be an integer such that $1 \leq a \leq p-1$. Given that $a^2 \equiv 1 \pmod{p}$ and $a \not\equiv 1 \pmod{p}$, find $a^{(p-1)/2} \pmod{p}$, as a residue modulo $p$. | nvidia |
Let the integers $m$ and $n$ be such that the roots of the quadratic equation
$$x^2-mx+n=0$$
be equal to the solutions to the system
$$x+y=1$$
$$xy=m.$$
Determine $n$. | nvidia |
Find the largest value of $x$ that satisfies the equation $|x-5|=12$. | nvidia |
Find the number of distinct four-digit numbers that can be formed using the digits 1, 2, 3, and 4, such that the product of the digits is 12 and the number is divisible by 3. | nvidia |
Find the sum of all values of $x$ that are not in the range of the function $f(x) = \frac{1}{x^2 - 4x + 4}$, where $x$ is a real number. | nvidia |
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that
$$f(x)-f(y)=f(x-y)(x^2-xy+y^2)$$for all $x$ and $y$. | nvidia |
Determine the maximum number of real solutions to the equation $\ds\sum\limits_{i=1}^{n} \arctan(x^i) = 0$, where $n$ is a positive integer. | nvidia |
The sum of the squares of the digits of a three-digit number is equal to the square of the sum of its digits. If the number is divisible by 3, what is the sum of its digits? | nvidia |
Find the real number $a$ such that $\sum_{n=1}^{\infty} \frac{a^n}{n^2 + 1} = \frac{\pi^2}{12}$. | nvidia |
What is the sum of all integer values $n$ for which $\binom{30}{15}+\binom{30}{n}=\binom{31}{16}$? | nvidia |
What is the smallest integer value of $x$ such that the number $x^2$ has the same digits when rotated 90 degrees clockwise? | nvidia |
Find the value of $x$ if the area of the triangle with vertices $(0,0)$, $(x,2x)$, and $(0,2x)$ is 2 times the area of the triangle with vertices $(0,0)$, $(x,0)$, and $(0,2x)$. | nvidia |
Find all solutions to the equation
\[ \sqrt{x^2-2x+5}+\frac{4}{\sqrt{x^2-2x+5}}=6.\] | nvidia |
Given two positive integers $a$ and $b$, find the minimum value of the expression $\frac{a+b}{ab}$, where $\frac{a}{b}$ is an irreducible fraction and $b > a$. | nvidia |
In the Cartesian plane, a square with side length $5$ units has two opposite corners on the x-axis. A larger square with side length $8$ units has two opposite corners on the y-axis. The two squares share a common corner with coordinates $(x, y)$. Find the product of the distances from the origin to the remaining two c... | nvidia |
What is the sum of the values of $x$ that satisfy the equation $x^2-5x+5=9$? | nvidia |
Let $a$ and $b$ be positive real numbers such that each of the equations $x^2 + ax + 2b = 0$ and $x^2 + 2bx + a = 0$ has real roots. Find the smallest possible value of $a + b.$ | nvidia |
Find the value of $x+y$ for real numbers $x$ and $y$ that satisfy the equation $x^2 + y^2 = 2(x-y) + 1$. | nvidia |
Let $\mathbf{P}$ be a $3 \times 3$ matrix, with real entries, such that $\mathbf{P}^4 = \mathbf{I}$ and the trace of $\mathbf{P}$ is 0. Find the number of different possible determinants that $\mathbf{P}^2$ can have. | nvidia |
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