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Let $f$ be a function that maps the set of positive integers to itself, satisfying the equation:
$f(m + n) + f(\lfloor \frac{m}{n} \rfloor) = f(m) + f(n) + 1$
for all positive integers $m$ and $n$. Find the sum of all possible values of $f(3)$. | nvidia |
Let $P_1P_2P_3P_4P_5$ be a pentagon inscribed in an ellipse. Prove that the sum of the squares of the diagonals of the pentagon is a constant, and find this constant. | nvidia |
Determine $n$ such that $2^{n-1} + 3^{n-1}$ is divisible by 5. | nvidia |
Let $p(x)=x^2+ax+b$ be a polynomial with integer coefficients and positive roots $r$ and $s$. It is known that $b=r\cdot s$ and $r+s=-a$. Assuming that the values of $a$ and $b$ are known, determine the product of all possible values of $r$ and $s$. | nvidia |
If $k$ is the smallest positive integer for which there exist positive real numbers $c$ and $d$ such that
\[(c + di)^k = (c - di)^k,\]compute $\frac{d}{c}.$ | nvidia |
Find the largest positive integer k such that every set of k consecutive squares contains at least one square that is divisible by 7. | nvidia |
Let $g(x) = \{x^2\}$. Find the smallest positive integer $m$ such that the equation $mg(xg(x)) = x^2$ has infinitely many real solutions. | nvidia |
Let $x,y,z>0$ be real numbers such that $$\cos x=0.6,\qquad \sin y=0.6,\qquad \tan z=0.6.$$ Evaluate $\cos(x+y+z)$. | nvidia |
The set $\{1, 2, 3, \ldots, n\}$ is split into two subsets, such that each number $k$ from the set is placed into one of the subsets with the probability $k/(n+1)$, and all decisions are independent. Determine the expected value of the difference of the sums of the numbers in the two subsets. | nvidia |
Let $p$ and $q$ be two complex numbers. Consider the equilateral triangle whose vertices are $p,$ $q,$ and $p+q.$
Determine all possible distances between the circumcenter of this triangle and the origin, knowing that the points $p,$ $p^2,$ $p^3,$ $q,$ $q^2,$ and $q^3$ are all vertices of a non-degenerate convex hexago... | nvidia |
Let $x$ be the largest two-digit positive integer divisible by 7 and $y$ be the largest three-digit positive integer divisible by 11. What is the value of $x+y$? | nvidia |
Find the sum of the squares of the divisors of $81$. | nvidia |
A circle with center $O$ and radius $r$ is given. We want to place $n$ congruent circles with radius $r/2$ inside the larger circle so that each of the smaller circles touches the larger circle and two of its neighbors, and the sum of the areas of the smaller circles is equal to the area of the larger circle. For how m... | nvidia |
Let $\mathbf{M}$ be a $3 \times 3$ matrix with complex entries such that $\mathbf{M}^3 = \mathbf{I}.$ If the sum of the absolute values of the diagonal entries of $\mathbf{M}$ is $4$, then find the possible values of the determinant of $\mathbf{M}$. | nvidia |
Let $x$ and $y$ be real numbers such that $|x - y| = 2$ and $x^2 + y^2 = 10$. Find the sum of all possible values of $x + y$. | nvidia |
Let $T$ be the set of $3 \times 3$ real matrices of the form
$$\begin{pmatrix}
1 & 0 & 0 \\
0 & \cos \theta & -\sin \theta \\
0 & \sin \theta & \cos \theta
\end{pmatrix}$$
such that $0 \le \theta \le \frac{\pi}{2}$.
Find the smallest positive integer $k$ such that for all positive integers $n \ge k,$ there exists a mat... | nvidia |
Let $a, b, c$ be positive integers such that $a^3 + b^3 + c^3 = n^3$ for some positive integer $n$. Find the smallest possible value of $n$ such that $(a, b, c)$ are pairwise distinct and $a + b + c$ is divisible by 4. | nvidia |
The decimal representation of the fraction $\frac{17}{27}$ is $0.\overline{629}$. What is the largest possible integer $n$ such that the digit in the $n$-th place to the right of the decimal point is 6? | nvidia |
Find the sum of all three-digit numbers that are divisible by 7 and remain divisible by 7 when their digits are reversed. | nvidia |
The number 2021 can be expressed as the sum of two positive integers in several ways, but only one of these sums has a product of its terms that is the largest possible. What is the sum of the digits of the product of the terms of this sum? | nvidia |
How many different tens digits are possible in numbers that are divisible by 6? | nvidia |
The graph of $y=f(x)$, consisting of five line segments, is shown in blue below.
Suppose that we reflect the entire graph about the line $y=x$. Find the largest value of $x$ such that $f(x)=x$.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=bla... | nvidia |
Consider the set of complex numbers of the form $z = \cos\theta + i\sin\theta$, where $\theta$ is a real number. Find the smallest value of $n$ for which there exists an integer $m$ such that $|z^n + m| \leq 1$ for every $\theta$. | nvidia |
Let $a_1 , a_2 , \dots$ be a sequence for which $a_1=2$ , $a_2=3$, and $a_n=\frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$? | nvidia |
Find the sum of all possible positive integer values of $c$ such that the cubic equation $x^3 + 2x^2 + cx + 5 = 0$ has at least one rational root. | nvidia |
If $ f(x)=\frac{x^2+bx+c}{x^2+7x+12} $ and $ f(x) $ has two vertical asymptotes at $ x=-4 $ and $ x=-3 $, find the equation of the horizontal asymptote of the graph of $ f(x) $. | 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 positive integer is called a "sum of primes" if it can be expressed as a sum of distinct prime numbers. For example, 7 is a sum of primes because it can be expressed as 2 + 3 + 2. Determine the smallest positive integer that is not a sum of primes. | nvidia |
Find the smallest positive value of $k$ such that the equation $14x^2 - kx + 504 = 0$ has integral solutions. | nvidia |
Determine the largest integer $n$ such that $n^3$ is a divisor of $20!$ | nvidia |
Find the largest value of $c$ such that the function $f(x) = x^3 - 2x^2 + c$ has exactly one real root. | nvidia |
For $a,b,c,d>0,$ the graph of the function
\[f(x)=\frac{a\sin(bx-c)}{d+\sin(bx-c)}\]intersects the line $y=a-1$ at exactly one point $(x_0,y_0)$ in the interval $-\frac{\pi}{2b}<x<\frac{\pi}{2b}.$ Find the largest possible value of $d.$ | nvidia |
Find the minimum value of $x^2+y^2$ given that $x$ and $y$ are positive integers and $xy=1998$. | nvidia |
Let $f(x)$ be a quadratic polynomial such that $f(x) + f(-x) = 12x^2$ for all real numbers $x$. If $f(0) = 5$, find the product of the roots of $f(x)$. | nvidia |
A number consists of two digits such that the digit in the tens place is greater than the digit in the units place by 3. What is the number of all such two-digit numbers if the sum of their digits is a prime number greater than 3? | nvidia |
The quadratic expression $ax^2+bx+c$ can be factored into the form $(x-p)(x-q)$, where $p$ and $q$ are rational numbers. Determine the product of all possible values of $a$, if it is known that $p$ and $q$ satisfy the condition $pq=-\frac{c}{a}$. | nvidia |
Find the value of $k$ such that the equation of the hyperbola $\frac{x^2}{k^2}-\frac{y^2}{64}=1$ has a pair of conjugate diameters whose slopes are $m$ and $\frac{-1}{m}$, where $m$ is positive. | nvidia |
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \neq 0$$ | nvidia |
Find the smallest positive integer that has exactly seven distinct positive factors. | nvidia |
Find the number of divisors of 500 that are also powers of prime numbers. | nvidia |
Consider the set $M$ of all nonnegative integers. Let $g : M \to M$ be a function such that
\[g(x) + g(y) = g(x^2+y^2+g(x-y))\]for all $x,$ $y \in M$ such that $x \geq y$.
Let $m$ be the number of possible values of $g(5),$ and let $t$ be the sum of all possible values of $g(5).$ Find $m \times t.$ | nvidia |
A positive five-digit integer is divisible by 7 and has exactly one digit 3. Find the number of possible positions of the digit 3 in the integer. | nvidia |
Compute the product of all the roots of $(x^2-2x-3)(x^2+4x+5)=0$, given that one of the roots of the quadratic $x^2-2x-3$ is also a root of the quadratic $x^2+4x+5$. | nvidia |
Let $P$ be the product of the first $100$ positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k .$
| nvidia |
The sum of the squares of two consecutive positive integers is 50, and their product is 6. What are the integers? | nvidia |
How many positive three-digit integers have exactly two distinct positive factors? | nvidia |
Let $f$ be a non-constant polynomial such that
\[f(x - 1) + f(x) + f(x + 1) = \frac{[f(x)]^2}{2013x}\]for all nonzero real numbers $x.$ Find the sum of all possible values of $f(1).$ | nvidia |
Find the number of integers between $(10.2)^3$ and $(10.3)^3$ that are divisible by 5. | nvidia |
Let $A$ and $B$ be real numbers such that $\frac{A}{x-5}+B(x+1)=\frac{-3x^2+12x+22}{x-5}$. What is $A+B$?
Write another problem inspired by this one:
Find the sum of the coefficients $a$, $b$, and $c$ in the equation $ax^2 + bx + c = 0$, given that the equation has two distinct real roots $r_1$ and $r_2$, and that th... | nvidia |
Find a nonzero $q$ such that $qx^2+8x+1=0$ has only one solution. | nvidia |
The functions $f(x)$ and $g(x)$ are defined for all real numbers $x$ and satisfy
\[f(x + y) + f(x - y) = g(x) + g(y)\]for all $x$ and $y.$ Given that $f(0) = 0,$ find $f(x).$ | nvidia |
Find the sum of the reciprocals of the divisors of 180. | nvidia |
Find all integer values of $a$ so that the polynomial
\[x^3 + 3x^2 + ax + 7 = 0\]has at least one integer root. Enter all possible values of $a,$ separated by commas. | nvidia |
Determine the largest positive integer less than $100$ that has no divisors other than $1$ and itself in common with $72$ and is such that the sum of its digits is a perfect square. | nvidia |
Let $a$ be a real number such that $\lceil{\arctan{a}}\rceil=\lceil{a^3}\rceil$, find the largest possible value of $a$. | nvidia |
Find the number of positive integers that are not perfect squares between 1 and 500. | nvidia |
Find the positive integer $M$ for which $$7^3 \times 14^3 = 2^3 \times M^3$$ | nvidia |
If $a$, $b$, and $c$ are digits and $0.ab0c$ can be expressed as $\frac{1}{y}$ where $y$ is an integer such that $0<y\le9$, then what is the largest possible value of $a+b+c$? | nvidia |
The roots of the equation
$$x^4-10ax^3+21a^2x^2-12a^3x+9a^4=0$$
are all real and positive.
Find the value of $a$ for which the sum of the squares of the roots is equal to 5. | nvidia |
Find the smallest positive integer $n$ such that the sequence $\{n^2+1, n^2+2, n^2+3, \ldots\}$ contains three consecutive terms that are each a sum of two perfect squares. | nvidia |
What is the smallest positive integer $n$ such that the sums of the digits of $n^2$ and those of $n$ differ by $6$? | nvidia |
The square root of a number is equal to the sum of its digits. Find all such numbers. | nvidia |
Find the smallest positive integer $n$ such that $n(2n+1)(5n-2)$ is always divisible by a perfect square greater than 1, regardless of the value of $n$. | nvidia |
Find the value of $a$ that satisfies the equation $293_{a}+468_{a}=73B_{a}$, where $B_{a}=11_{10}$. | nvidia |
Find all real numbers $x$ such that \[|x^2-3x+2|<|x-1|.\] (Give your answer in interval notation.) | nvidia |
Determine the number of positive real numbers $a$ such that the line $y=x^2+a^2$ is tangent to the parabola $y=e^x+a$. | nvidia |
When three positive integers are divided by $24$, the remainders are $10,$ $4,$ and $12,$ respectively.
When the sum of the three integers is divided by $24$, what is the remainder? | nvidia |
Determine the number of positive integers less than or equal to 1000 that are not divisible by any of the prime numbers 2, 3, or 5. | nvidia |
A cubic equation $x^3+ax^2+bx+c=0$ has two equal roots. If the sum of the roots is 12, find the product of the roots. | nvidia |
A circle with center $C$ and radius $2$ intersects the circle with center $D$ and radius $3$ at two points $E$ and $F$. Find the smallest possible value of $CE+DF$ if $C$ and $D$ lie on the parabola $y^2=x$. | nvidia |
Find the smallest two-digit prime number such that reversing the digits of the number produces a composite number. | nvidia |
The decimal number 1234 can be written in base 7 as $a_7b_7c_7d_7$, where $a$, $b$, $c$, and $d$ are the digits of the base 7 representation. Find the sum of all possible values of $a+b+c+d$ that satisfy this condition. | nvidia |
Find the function $f(x)$, with real coefficients, such that $f(1) = 2$ and
\[f(x) + f(y) = f(xy) + f\left(\frac{x}{y}\right)\]for all positive real numbers $x$ and $y$. | nvidia |
Find the sum of all real values of $k$ for which the equation $\frac{x^2+1}{x^2-4}=\frac{k}{x+2}$ has exactly two distinct real solutions. | nvidia |
Ava wrote 4 different numbers, one on each side of 2 cards, and laid the cards on a table. The products of the two numbers on each of the two cards are equal. The two numbers on the hidden sides are perfect squares. What is the sum of the hidden perfect squares?
[asy]
path box=(0,0)--(1,0)--(1,1.5)--(0,1.5)--cycle;
dr... | nvidia |
Given a set of six points on a circle, no three of which are collinear, what is the minimum number of lines that can be drawn through pairs of points in the set such that no two lines intersect inside the circle? | nvidia |
Let $n$ be the integer such that there exists a real square matrix of order $n$ with exactly 2000 entries that are equal to 1 and the rest of the entries are zero. Determine the largest possible value of $n$. | nvidia |
For the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, find the distance between the directrices, given that the asymptotes of the hyperbola are perpendicular to each other and the area of the rectangle formed by the vertices of the hyperbola and the intersection points of the asymptotes is $4\sqrt{2}$ square units... | nvidia |
If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? | nvidia |
In the year 2025, the International Mathematical Olympiad will take place in France. Let $F$, $R$, and $A$ be distinct positive integers such that the product $F\cdot R\cdot A=2025$. What is the largest possible value of the sum $F+R+A$? | nvidia |
The sequence $(b_n)$ is defined by $b_{n+2} = b_{n+1} + \frac{b_n}{1+b_n}$ for $n \ge 1$, where $b_1$ and $b_2$ are positive integers. Find the maximum possible value of $b_1$ such that $b_3$ is also an integer. | nvidia |
Find the smallest positive integer $n$ such that $n^2$ is divisible by each of the first ten positive perfect squares. | nvidia |
The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$ and $y$ are integers, is:
\[\textbf{(A)} \ 0 \qquad\textbf{(B)} \ 1 \qquad \textbf{(C)} \ 2 \qquad\textbf{(D)} \ 3\qquad \textbf{(E)} \ \text{More than three, but finite}\]
| nvidia |
Let $\sigma (n)$ denote the sum of the divisors of $n$ (including $1$ and $n$). Find the smallest positive integer $k$ such that there exist two consecutive positive integers $n$ and $n+1$ satisfying $\sigma (n) + \sigma (n+1) = k^2$. | nvidia |
A sequence of positive real numbers $a_1,a_2,\ldots,a_n$ is called a "harmonic sequence" if the reciprocals of the terms form an arithmetic sequence. For example, the sequence $1,2,\frac{2}{3},\frac{1}{2}$ is harmonic because $\frac{1}{1},\frac{1}{2},\frac{3}{2},2$ is arithmetic. Find the minimum value of the sum of a ... | nvidia |
Given that there are an infinite number of triangles whose perimeter is a whole number and whose sides are in arithmetic progression, find the smallest perimeter. | nvidia |
The sum of the squares of two positive integers is 100, and the sum of their reciprocals is 1/4. What is the value of the smaller of the two integers? | nvidia |
Let $S$ be a subset of $\{1,2,3,...,50\}$ such that no pair of distinct elements in $S$ has a sum divisible by $7$. What is the maximum number of elements in $S$?
$\text{(A) } 6\quad \text{(B) } 7\quad \text{(C) } 14\quad \text{(D) } 22\quad \text{(E) } 23$
| nvidia |
The three roots of the cubic $ 30 x^3 - 50x^2 + 22x - 1$ are distinct real numbers strictly between $ 0$ and $ 1$. If the roots are $p$, $q$, and $r$, what is the sum
\[ \frac{1}{1-p} + \frac{1}{1-q} +\frac{1}{1-r} ?\] | nvidia |
There are three real numbers $x$ that are not in the domain of $$f(x) = \frac{1}{1+\frac{1}{1+\frac 1x}}.$$ What is the sum of those three numbers? | nvidia |
Given that the points (3,7) and (-1,9) are the endpoints of a diameter of a circle, find the radius of the circle. | nvidia |
Find the largest number of non-overlapping unit circles that can be drawn inside a circle of radius 3. | nvidia |
Find a positive integer $n$ such that the sequence $1, n, n-1, \ldots$ where each term after the first two is obtained by subtracting the preceding term from the term before that, has a maximum length before reaching a term that is not a positive integer. | nvidia |
Determine the smallest base $b$ such that the number $2468_b$ is divisible by 4. | nvidia |
Find the number of ordered pairs $(a,b)$ of complex numbers such that
\[a^3 b^5 = a^7 b^2 = 1.\] | nvidia |
Let $x_1,$ $x_2,$ $\dots$ be a sequence of positive real numbers such that
\[x_n = \sqrt{x_{n - 1}} + \sqrt{x_{n - 2}}\]for all $n > 2.$ Find the largest possible value of $x_1 + x_2.$ | nvidia |
Determine the sum of all single-digit replacements for $z$ such that the number ${24{,}z38}$ is divisible by 6. | nvidia |
Problem:
Given that $\triangle ABC \sim \triangle BCD$, and that the area of $\triangle ABC$ is $12$ times the area of $\triangle BCD$, determine the ratio of the lengths of the corresponding sides. | nvidia |
Find the number of positive integers $n$ such that $2^n$ is a factor of at least one of the numbers from 1 to 1000, but not all of them. | nvidia |
Determine the value of $a$ such that the equation
\[2x^3 - 5x^2 + ax + 2 = 0\]
has exactly two real solutions that are additive inverses of each other. | nvidia |
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