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Let $S$ be the sum of the first $n$ terms of the sequence $\{(-1)^{a_n} \cdot n\}$, where $a_n$ is the number of divisors of $n$. Find the largest prime number $p$ such that $S$ is divisible by $p$ for all positive integers $n$. | nvidia |
Find the values of $a$ and $b$ such that the equation $z^2 + az + b = 0$ has roots that are conjugates of each other and satisfy the condition $|z_1|^2 + |z_2|^2 = 10.$ | nvidia |
Find the product of all real values of $s$ for which $\frac{1}{3x}=\frac{s-2x}{5}$ has exactly one real solution. | nvidia |
Find the sum of the solutions of the equation $|x-2|+|x+3|=5$. | nvidia |
Let's define a function $h(x)$ to be the smallest prime factor of a positive integer $x$. Now, let's consider the sequence $T_n = \sum_{k=1}^{3^{n-1}}h(3k)$. Determine the largest integer $n$ less than 500 such that $T_n$ is divisible by the sum of its digits. | nvidia |
Let $h(x)$ be a cubic polynomial and $r(x)$ be a polynomial such that the degree of $h(x) \cdot r(x)$ is 5. Determine the degree of $r(x)$. | nvidia |
Tom creates a sequence of positive integers that starts with 2, then replaces each term $n$ with $2n + 1$. Tom claims that he can keep doing this until he reaches the number $2023$. Is he correct? | nvidia |
Given two points $A(x_1,y_1)$ and $B(x_2,y_2)$ on the Cartesian plane, determine the value of $k$ such that the line passing through points $A$ and $B$ is perpendicular to the line $y=kx$. | nvidia |
Let $A$ and $B$ be two events such that $P(A)=0.3$, $P(B)=0.4$, and $P(A\cap B)=0.1$. Find the probability that $A$ occurs but $B$ does not occur. | nvidia |
Find the smallest positive integer that is both a perfect cube and has a digit sum that is a prime number. | nvidia |
Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square and $\dfrac{12}{n}$ is also an integer. | nvidia |
Let $f(n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. If $p$ is a prime number, what is $f(p^2)$? | nvidia |
In the equation $x^2 - kx + 1 = 0$, determine $k$ such that the product of the roots is equal to twice the sum of the roots. | nvidia |
Let $m$ be the smallest positive integer with the property that $2m$ is a perfect square and $3m$ is a perfect cube. What is the smallest positive integer $k$ such that $2^k \cdot m$ is a perfect sixth power? | nvidia |
Find the sum of all numbers that can be expressed as 11k+3 where k is an integer and 0 < 11k+3 < 100. | nvidia |
Let $f(x)$ be a function defined on the set of positive real numbers, such that $f(x^2) = x + 1$. Determine the value of $f(x^3)$ in terms of $x$. | nvidia |
Let $f(x)$ be a function such that for every real number $x,$ the equation
\[f(x) + f(y) = f(xy)\]holds for all real numbers $y$ that are at least 1 unit away from $x.$ If $f(1) = 3,$ what is $f(64)$? | nvidia |
Find the smallest positive five-digit number that is a palindrome and has exactly three distinct digits. | nvidia |
On the sphere $x^2+y^2+z^2=1$, there are three points $A$, $B$, and $C$, such that the plane passing through them is parallel to the plane $x+y+z=0$ and their centroid coincides with the center of the sphere.
Find the area of triangle $ABC$. | nvidia |
The following system of equations is given:
$$
\begin{align*}
|x|+|y|+|z|&=13,\\
x+2y-3z&=4.
\end{align*}
$$
Find a point that satisfies the above equations such that $x$, $y$, $z$ are all positive real numbers. | nvidia |
Let $f(x)$ be a function defined for all real numbers $x$ such that $f(x) = \lfloor x \rfloor$ if $x$ is irrational and $f(x) = \lceil x \rceil$ if $x$ is rational. Evaluate the sum
\[\sum_{n = 1}^\infty \frac{f(n \pi) + f(-n \pi)}{n^2}.\] | nvidia |
Find the value of $b$ that satisfies the equation $256_b + 325_b = 571_b$, where the base $b$ is a positive integer. | nvidia |
The sides of a rectangle are in the ratio 3:4. The length of the rectangle is increased by $20\%$. By what percent must the width be decreased so that the area of the rectangle remains unchanged? | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(x + y) = g(x) + g(y) + g(xy)$ for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the sum of all possible values of $g(2).$ Find $m \times t.$ | nvidia |
Find the sum of the positive whole number divisors of 378. | nvidia |
Let $z$ be a complex number such that $\ds z^3+z^{-3}=2i$. Determine the value of $z^6+z^{-6}$. | nvidia |
A triangle is formed by three points in the plane, each of which has a positive integer coordinate. If the sum of the absolute values of the differences of the $x$-coordinates is equal to the sum of the absolute values of the differences of the $y$-coordinates, find the smallest possible area of such a triangle. | nvidia |
Given that
$\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}$
find the greatest integer that is less than $\frac N{100}$.
| nvidia |
Find the smallest five-digit positive integer that is divisible by 42 and has the property that when its digits are reversed, the resulting number is divisible by 56. | nvidia |
For positive integers $m$, define $T_m$ to be the minimum value of the sum
\[\sum_{k=1}^m \sqrt{(3k-2)^2+b_k^2},\]where $b_1,b_2,\ldots,b_m$ are positive real numbers whose sum is $24$. Find the unique positive integer $m$ for which $T_m$ is also an integer. | nvidia |
For all positive integers $n$, what is the largest integer that always divides into the difference between $n$ and the product of $n$ and its largest prime factor? | nvidia |
The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [asy]
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;
real tickdownbase=0.3;
real who... | nvidia |
Let $\alpha$ be the angle between the positive $x$-axis and the line passing through the points $(-2, -3)$ and $(-6, 3)$. There are real numbers $p$ and $q$ such that the parabola with the equation $y = x^2 + c$ does not intersect the line through the points $(-2, -3)$ and $(-6, 3)$ if and only if $p < c < q$. Find the... | nvidia |
For a given positive integer $n$, the parabola $y^2 = 4px$ is drawn, where $p = n^2 + 1$. The area of the triangle formed by the tangent line to the parabola at the point $(p, 2p)$ and the $x$-axis is $S(n)$. Find the sum of all positive integers $n$ for which $S(n)$ is a perfect square. | nvidia |
The vectors $\mathbf{a}$ and $\mathbf{b}$ are such that the length of the projection of $\mathbf{a}$ on $\mathbf{b}$ equals $|\mathbf{a}|$. The angles between $\mathbf{a}$ and the vectors $\mathbf{e_1}=(1,0,0)$, $\mathbf{e_2}=(0,1,0)$, $\mathbf{e_3}=(0,0,1)$ are equal. Determine the angle between $\mathbf{a}$ and $\mat... | nvidia |
The function $f(x)$ satisfies
\[f(x - y) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(x) \neq 0$ for all real numbers $x.$ Find $f(3).$ | nvidia |
A right triangle has a hypotenuse of 10 cm and one of the legs has the same length as one of the sides of a square. The perimeter of the square is 10 times the length of the other leg of the triangle. Determine the ratio of the area of the triangle to the area of the square. | nvidia |
Find all real numbers $a$ such that the equation $\cos x = \frac{1}{1 + ax}$ has exactly three solutions in the interval $[0, 2\pi]$. | nvidia |
A circular Ferris wheel has 10 cars, labeled from 1 to 10 in a clockwise direction. Each car has a plaque with a number, such that the number on the plaque is the reciprocal of the car's number (i.e., car 1 has a plaque with the number 1/1, car 2 has a plaque with the number 1/2, and so on). Ten people ride the Ferris ... | nvidia |
Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{9}{10}.\] | nvidia |
A box with a fixed volume $V$ is to be designed such that the total surface area is minimized. The box has a square base with side length $x$ and height $y$. Find the value of $\frac{y}{x}$ that minimizes the total surface area. | nvidia |
The vertices of a triangle $ABC$ are located at the points $A(0, 0)$, $B(0, 12)$, and $C(15, 0)$ in the coordinate plane. A square with a side length of $x$ is inscribed in the triangle such that its vertices lie on the sides of the triangle. Express the area of the square as a function of $x$. Determine the coefficien... | nvidia |
Find the largest value of $x$ such that $\sqrt{x^2 - x + 1} = \frac{2}{x-1}$. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $xyz = 1$. Find the minimum value of
\[\frac{x^2y^2 + y^2z^2 + z^2x^2}{x + y + z}.\] | nvidia |
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ w... | nvidia |
Find the base of the logarithm such that the equation $\log_{b} (2x+3) = x-1$ has a solution $x=2$. | nvidia |
Let $x$ and $y$ be positive integers such that $x>y$ and $x$ is a multiple of $y$. If $x^2-y^2=28$, determine $x+y$. | nvidia |
Given the function
\[g(x) = \left\{
\begin{array}{cl}
x-2 &\text{ if }x>2, \\
\dfrac{x}{2} &\text{ if }x \le 2,
\end{array}
\right.\]
find the fixed points of $g(g(x))$, that is, the values of $x$ such that $g(g(x))=x$. | nvidia |
Find all positive integers $x$ and $y$ such that $x^2 + y^2 = 2^x + 2^y.$ | nvidia |
Find the value of $a$ such that the equation $$x^2-2ax+a^2-1=0$$ has two solutions and the sum of the roots is equal to $a$. | nvidia |
Find the sum of all real values of $y$ that satisfy
\[y = \sqrt{1 + \sqrt{1 + \sqrt{1 + \dotsb}}}.\] | nvidia |
A rectangle with sides of length 6 cm and 10 cm is cut into smaller squares with equal side length.
What is the largest possible number of such squares that can be obtained, given that the area of each square is at least 9 cm^2? | nvidia |
Let $P$ be a point inside a regular dodecagon $A_1A_2\dots A_{12}$, such that the areas of the triangles $PA_1A_2, PA_2A_3, \dots, PA_{12}A_1$ are all equal. Find the number of possible locations for point $P$. | nvidia |
Let $a$, $b$ and $c$ be positive integers such that $a>b>c$. Determine the largest value of the integer $m$ such that the inequality
$$\left \lfloor \frac {a!}{b!+c!}\right \rfloor >m$$
holds for all possible triples $(a,b,c)$. | nvidia |
In a three-dimensional space, a point $P$ is given by its position vector $\mathbf{p}$. Three unit vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are also given, such that the set $\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}$ is linearly independent. It is known that the point $P$ satisfies the following relation:
$$... | nvidia |
Find the largest value of $y$ that satisfies the equation $|2y-11|=22$. | nvidia |
Find the area of the triangle formed by the lines $y=2x-3$, $y=-3x+1$, and the $x$-axis. | nvidia |
What is the maximum number of consecutive positive even integers that can be multiplied together before the product exceeds 10000? | nvidia |
Consider the function
\[f(x) = ax^2 + bx + c\]
where $a$, $b$, and $c$ are integers. The equation $f(x) = 0$ has exactly two distinct roots $r_1$ and $r_2$. Suppose that the equation $f(f(x)) = 0$ has exactly four distinct roots $t_1$, $t_2$, $t_3$, and $t_4$. What are the possible values of the degree of the polynomia... | nvidia |
Let $a$ and $b$ be the roots of the equation $x^2-x-1=0$. If $n$ is a positive integer for which $a^n$ and $b^n$ are also consecutive positive integers, find the smallest possible value of $n$. | nvidia |
Determine the number of real solutions to the equation $$x^{\sqrt{2x+1}}=\sqrt{x+1}.$$ | nvidia |
Let P ( z ) and Q ( z ) be two polynomials with integer coefficients such that
$$ P(z)\cdot Q(z)=1+z+z^2+z^3+\cdots +z^{14}. $$
Determine the smallest possible value of [ P (−1)] ^{2} + [ Q (−1)] ^{2} .
Good luck! | nvidia |
Let $ABCD$ be a quadrilateral such that $P$ is the midpoint of $\overline{BC}$ and $Q$ is the midpoint of $\overline{AD}$. Given that the area of $\triangle ACD$ is $60$ and the area of $\triangle ABP$ is $25$, determine the area of $\triangle PCD$. | nvidia |
The digits of a 5-digit number are such that the last digit is twice the first digit and the average of all the digits is exactly the third digit. Determine the fifth digit if the number is divisible by 7, 11, and 13. | nvidia |
Find the value of $x$ such that $\sqrt{\frac{x^2-4}{x^2+4}}+\sqrt{\frac{x^2+4}{x^2-4}}=3$ | nvidia |
Given positive integers $m$ and $n$ such that $\frac{1}{3m} + \frac{1}{n} = \frac{1}{9}$, what is the least possible value of $mn$? | nvidia |
In a certain country, the postal system uses a unique three-letter code for each city. The code consists of three different letters, and the letters are chosen from a set of $n$ letters, where $n$ is a positive integer. Given that there are exactly 120 different codes that can be formed using these $n$ letters, find th... | nvidia |
The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [asy]
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;
real tickdownbase=0.3;
real who... | nvidia |
Find the smallest possible degree of a nonzero polynomial with integer coefficients, whose roots are all integers except $\sqrt{2014}$. | nvidia |
The largest circle that can be inscribed in the right triangle has a radius of $2$ inches. Find the area of the triangle. | nvidia |
Let $a$, $b$, and $c$ be positive integers with $a<b<c$, and consider the system of equations
\begin{align*}
2x+y&=2003,\\
y&=\lvert x-a\rvert+\lvert x-b\rvert+\lvert x-c\rvert.
\end{align*}Find the minimum value of $c$ such that the system of equations has exactly one solution. | nvidia |
A number $x$ is equal to $7\cdot24\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? | nvidia |
The function $f(x)$ is defined as the minimum value of the three numbers $|2x-1|$, $|3x-2|$, and $|x+1|$. Determine the value of $f(-0.8)$. | nvidia |
Determine all real numbers a for which there exists a positive integer n such that the number (10^n + a) is divisible by (10 + a).
Note: n can take any positive integer value. | nvidia |
The lengths of two sides of a triangle are in the ratio of 3 to 4. The length of the shorter side is increased by 15%. By what percent must the length of the longer side be decreased so that the perimeter of the triangle remains the same? | nvidia |
Let $m \ge 3$ be an integer and let $S = \{3,4,5,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.
| nvidia |
Three different numbers have the property that the sum of the two larger numbers is 25 and the difference between the two smaller numbers is 7. What is the value of the largest number if the sum of the smallest and largest numbers is 23? | nvidia |
Find the smallest positive integer $n$ such that $n!$ is divisible by $n^2$. | nvidia |
Let $a$, $b$, $c$, and $d$ be real numbers such that the equation
\[x^4+ax^3+bx^2+cx+d=0\]
has roots $-1$, $-1$, $2$, and $2$. Find the sum of the roots of the equation
\[x^4+2ax^3+(a^2+b)x^2+(ab+c)x+abc+d=0.\] | nvidia |
Find the number of positive integer solutions to the equation $x^2 + 2y^2 = 3z^2$ where $x$, $y$, and $z$ are all positive integers. | nvidia |
Find the number of 3-letter words that can be formed using the letters of the word "TUNNEL" if the word must contain at least one vowel and no letter can be repeated. | nvidia |
Find the sum of the $87^{\mathrm{th}}$ odd positive integer and its reverse number. | nvidia |
Find the value of $x$ such that $\cot x^\circ = \cot 700^\circ$. | nvidia |
Find the smallest positive integer $m$ such that $23m \equiv 567 \pmod{11}$. | nvidia |
In a conference, the number of attendees from Europe is triple the number of attendees from Asia. There are 90 attendees from Asia, and none of the attendees from Asia are also from Europe. The total number of men attending the conference is three times the number of women. How many people are attending the conference ... | nvidia |
A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls is 7 or that at least one of the rolls is a 4? | nvidia |
In a square $ABCD$ with side length 12, there are two circles located inside $\angle BAD$ which are tangent to rays $\overline{AB}$, $\overline{AD}$, and segment $\overline{BC}$. Compute the sum of the areas of these two circles. | nvidia |
Call an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n$. How many oddly powerful integers are less than $2010$? | nvidia |
In a right triangle, the ratio of the lengths of the two legs is 3:4. If the area of the triangle is 240 square units, find the length of the hypotenuse. | nvidia |
Let the sequence of positive odd integers be 1,3,5,...,19.
Find a closed form expression for the sum of the cubes of these numbers. | nvidia |
A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles? | nvidia |
Find the number of integers $n$ between 1 and 1000 (inclusive) for which the decimal representation of $\frac{1}{n}$ contains at least three consecutive zeros. | nvidia |
A sequence of positive integers $a_1, a_2, a_3,...$ is defined recursively by $a_1=1$, $a_2=1$, and $a_n = a_{n-1}a_{n-2} + 1$ for $n \geq 3$. Let $F(x)$ be a function that maps a real number $x$ to the smallest positive integer $n$ such that $a_n \geq x$. Find the sum of all values of $x$ for which $F(x) = 7$. | nvidia |
Find the number of ordered pairs of integers $(x,y)$ such that $x^2+y^2=13$ and $x \neq y \neq -x$. | nvidia |
A circle with center $(0,0)$ and radius $r$ intersects the parabola $y=x^2+1$ at two points $A$ and $B$. Find the value of $r$ such that the line passing through points $A$ and $B$ is tangent to the circle. | nvidia |
Let $n$ be a positive integer greater than $4$. A list of integers $\{a_i\}$ for $i = 1, 2, \dots, n$ contains $i$ copies of $i$. Determine the value of $n$ such that the average of all these integers is equal to $7$. | nvidia |
Find the number of positive integers less than 100 that are sums of two distinct perfect squares. | nvidia |
The sum of two positive integers is 75 and their difference is 15. What is the value of the positive difference of the cubes of the integers? | nvidia |
Let $a^2=\frac{16}{44}$ and $b^2=\frac{(2+\sqrt{5})^2}{11}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a+b)^3$ can be expressed in the simplified form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$? | nvidia |
Find the cubic polynomial, with real coefficients, whose roots are the squares of the roots of the quadratic polynomial with roots $2+i$ and $2-i$. Ensure that the coefficient of $x^2$ in the resulting polynomial is 12. | nvidia |
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