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The integers $a$, $b$, $c$ and $d$ are positive and not greater than $20$. Each of the four products $ab$, $bc$, $cd$ and $ad$ is equal to the sum of two squares. Determine the smallest possible value of the sum of these four products. | nvidia |
Let $a$ be a positive real number. Define the function $g(x)=\frac{2x+1}{x-a}$. If the equation $g(x)=g^{-1}(x)$ has exactly two solutions, find the minimum value of $a$. | nvidia |
Let $C$ be the number of four-digit numbers that have exactly two distinct digits, and let $D$ be the number of four-digit numbers that are divisible by 9. Find $C-D$. | nvidia |
Let $f(n)$ be the base-10 logarithm of the sum of the elements of the $n$th row in Pascal's triangle. Express $\frac{f(n)}{\log_{10} 2}$ in terms of $n$. Recall that Pascal's triangle begins
\begin{tabular}{rccccccccc}
$n=0$:& & & & & 1\\\noalign{\smallskip\smallskip}
$n=1$:& & & & 1 & & 1\\\noalign{\smallskip\smallsk... | nvidia |
What is the sum of all positive integer divisors of 91? | nvidia |
Let $a$ and $b$ be positive integers such that $ab = 48$ and $\text{lcm}(a,b) = 24$. Determine the largest possible value of $\gcd(a,b)$. | nvidia |
Find the value of $a$ such that the expression $\ds\sqrt[3]{1 + \ds\frac{1}{\sqrt[3]{1 + \ds\frac{1}{a}}}} = \ds\frac{3}{2}.$ | nvidia |
Consider the function $g(x)$ that satisfies the equation:
\[g(3^x) - xg(3^{-x}) = 2\]
for all real numbers $x.$ Find $g(3).$ | nvidia |
Let $g(x)$ be a function satisfying $g(x)=g(-x)$ for all $x$ and such that the graph of $y=g(x)$ passes through the point $(3,4)$.
Determine the value of $g(-3)$. | nvidia |
Consider the set of all complex numbers of the form $a+bi$, where $a$ and $b$ are integers, that can be expressed in base $2+i$ using the digits $0,1,2,\ldots,4$. That is, the equation
\[a+bi=d_n(2+i)^n+d_{n-1}(2+i)^{n-1}+\cdots+d_1(2+i)+d_0\]is true for a unique choice of non-negative integer $n$ and digits $d_0,d_1,\... | nvidia |
The polynomial
$$ax^3+bx^2+cx+d$$
has integer coefficients, and its three roots are integers.
If the sum of the absolute values of the roots is 12,
determine the maximum possible value of $|d|$. | nvidia |
What is the maximum number of non-overlapping unit circles that can be packed around a circle of radius 6 units such that each unit circle touches the larger circle and at least two other unit circles? | nvidia |
Let
$(x_1,y_1)$,
$(x_2,y_2)$,
...
,
$(x_n,y_n)$
be points on the plane such that
$x_i$ and $y_i$
are integers from 1 through 50,
and no two points coincide.
Find the minimal value of
$\sum\limits_{i=1}^{n-1}
|y_{i+1}-y_i|$ | nvidia |
Find the largest three-digit number that can be expressed as the sum of three consecutive integers, and whose digits' product is a multiple of 5. | nvidia |
Count the number of permutations of the numbers 1 through 5 such that the sum of any two consecutive numbers is odd. | nvidia |
Find the sum of all integer divisors of $350$ that are congruent to $1$ modulo $4$. (Recall that the divisors of an integer may be positive or negative.) | nvidia |
Let $p$ be a real number. Find the value of $p$ such that
$$\sin\left(\arctan p\right) = -\frac{1}{\sqrt{5}}$$ | nvidia |
Let a and b be real numbers such that
|a+3i|+|b+4i|=7. Find |a+b| | nvidia |
Evaluate the product \[ (n-1) \cdot n \cdot (n+1) \cdot (n+2) \cdot (n+3), \] where $n=2$. | nvidia |
What is the sum of all values of $z$ such that $z^2=12z-7$? | nvidia |
A sequence $a_1, a_2, a_3, \ldots$ is called recursively centered if $a_{n+1} = a_n^2 + 1$ for all positive integers $n$, and $a_1$ is a positive integer. If $a_n$ is a recursively centered sequence, what is the maximum number of terms $a_n$ that can be equal to a perfect square? | nvidia |
Let $P(x,y,z)$ be a polynomial in three variables. Consider the set of points $(x,y,z)$ in space where $x,y,z$ are integers such that $P(x,y,z)$ is divisible by 7. Let $C$ be the largest value of $z$ such that there exist integers $x$ and $y$ satisfying $P(x,y,z) \equiv 0 \pmod{7}$.
Given a list of coefficients of $P(... | nvidia |
How many four-digit positive integers have at least one digit that is a 2 or a 3? | nvidia |
Find the largest value of \(c\) such that the inequality
\[|x^2 - 2x + c| \leq 2\]
holds for all real values of \(x\). | nvidia |
In a certain triangle, the coordinates of the vertices are all integers, and the triangle has no horizontal or vertical sides. The slope of one side is -2/3, and the slopes of the other two sides are reciprocals of each other. If the perimeter of the triangle is 30 units, find the area of the triangle. | nvidia |
Find the absolute difference between the sum of the cubes of the first 10 positive integers and the sum of the squares of the first 10 positive integers. | nvidia |
Given a rectangle $ABCD$ with $AB=8$, $BC=6$ and an interior point $O$ such that the circle centered at $O$ passes through points $A$, $B$, $C$ and $D$. Determine the area of triangle $ABC$ assuming that point $O$ has been randomly selected inside the rectangle $ABCD$. | nvidia |
In triangle $ABC$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O$, and that $\frac{AO}{OA'}+\frac{BO}{OB'}+\frac{CO}{OC'}=92$, find $\frac{AO}{OA'}\cdot \frac{BO}{OB'}\cdot \frac{CO}{OC'}$.
| nvidia |
Let $f$ be a function defined on the positive integers, such that
\[f(xy) = f(x) + f(y)\]for all positive integers $x$ and $y.$ Given $f(10) = 14$ and $f(40) = 20,$ find $f(500).$ | nvidia |
Consider the triangular array of numbers with 0, 1, 2, 3, $\dots$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.
\[
\begin{array}{ccccccccccc}
& & & & & 0 & & & & & \\
& & & & 1 & & 1 & & & & \\
& & & 2 & & 2 & & 2 & & & \\
& & 3 & & 4... | nvidia |
Find the maximum value of
\[x^2 + xy + y^2\]
over all real numbers $x$ and $y$ satisfying the condition that $x + y = 1.$ | nvidia |
In a circle, a chord divides the circle into two segments such that the sum of the lengths of the chord and the arc subtended by it at the center is 120 cm. The lengths of the two segments are in the ratio 7:4. Find the radius of the circle. | nvidia |
In a regular hexagon, a triangle is formed by connecting three vertices, such that two of its sides are also sides of the hexagon. If the length of the side of the hexagon is 8 cm, what is the area of the triangle? | nvidia |
Let $A$ be a matrix with entries being complex numbers such that
\[\det(A+zI)=\det(A)+az\det(A+z^2I)\]holds for every $z\in\mathbb C$, where $a$ is a given positive number.
Can we conclude that $A=0$ or $A=-\frac{1}{a}I$? | nvidia |
Let $a$ and $b$ be acute angles such that
$$\sum_{k=1}^{n} \tan(a+bk) = \cos \frac{c}{m},$$
where angles are measured in degrees, and $c$ and $m$ are relatively prime positive integers that satisfy $\frac{c}{m}<90,$ find $c+m.$ | nvidia |
Given a quadratic equation $ax^2 + bx + c = 0$, determine the conditions on $a$, $b$, and $c$ such that the sum of the roots is greater than the product of the roots. | nvidia |
Find the number of integers $n$ that satisfy the inequality $|n-2|+|n+5| \le 6$. | nvidia |
How many different three-digit numbers $\overline{xyz}$ are there such that $x$, $y$, and $z$ are all prime digits and $yzx$ is divisible by 5? | nvidia |
A sequence of positive integers $a_n$ is defined by $a_1=1$ and $a_{n+1} = \sqrt{2a_n^2 + 1}$ for all integers $n \geq 1.$ Find the least value of $n$ such that $a_n$ is not an integer. | nvidia |
Let $ f(x) = x^4 + 2x^2 + 1$. Suppose $ g$ is a polynomial of degree 6 such that $ g(0) = 1$, and the roots of $ g$ are the cubes of the roots of $ f$. Find $ g(-8)$. | nvidia |
Let $a$ and $b$ be complex numbers such that $|a|=|b|=1$ and $|a+b|=|a^2+b^2|=2$. Find the largest possible value of $|a^3+b^3|$. | nvidia |
Let $C$ be a circle of radius $r$ centered at the origin, and let $P$ be a point chosen uniformly at random from the interior of $C$. The probability that the distance from $P$ to the point $(r/2,0)$ is greater than or equal to $r/2$ can be written as $m/n$, where $m$ and $n$ are relatively prime positive integers. Fin... | nvidia |
Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 120$. Express $x$ as a decimal. | nvidia |
Find the largest real number $\mu$ such that
\[p^2 + q^2 + r^2 + s^2 \ge pq + \mu qr + rs\]
for all nonnegative real numbers $p,$ $q,$ $r,$ $s.$ | nvidia |
Let $a$, $b$, $c$, and $d$ be positive real numbers. Solve the following system of equations:
$$a^2+b^2=c^2+d^2,$$
$$a^3+b^3=3c^3-d^3.$$
Find all possible values of $a/b+c/d$. | nvidia |
Let $M$ and $N$ be two points inside the rhombus $ABCD$ such that $\angle MAN=\angle NBC$. The diagonals of the rhombus intersect at the point $O$. If $\overline{AM}$ and $\overline{NO}$ intersect at the point $P$ and $\overline{BN}$ and $\overline{MO}$ intersect at the point $Q$, and $PQ=6$, find the area of the rhomb... | nvidia |
What is the largest number that can be formed using only the digits 1 and 4, with the restriction that the number of 1's used is exactly twice the number of 4's used? | nvidia |
Find the largest integer $n$ such that the product of any two consecutive positive integers less than $n$ is less than $700$. | nvidia |
Find all possible values of $|a^2b+a^2c+ab^2+b^2c+ac^2+bc^2|$, where $a$, $b$, and $c$ are complex numbers such that $|a|=|b|=|c|=1$ and $a^2+b^2+c^2=0$. | nvidia |
The value of the repeating decimal $0.\overline{11}$ can be expressed as a fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Determine the sum of the numerator and denominator of this fraction. | nvidia |
The arithmetic mean of five numbers is 21. Three of the numbers are 17, 24, and 20, and the remaining two numbers are equal. What is the sum of the two equal numbers? | nvidia |
Let AABC' be a tetrahedron with vertices
$$A(0,0,0)\qquad A'(10,0,0)\qquad B(0,12,0)\qquad C(0,0,21)$$
and M,N,P,Q be the midpoints of AB,AC,A'B,A'C respectively.
Find MP•NQ . | nvidia |
Find the distance between the center of the circle with equation $x^2+y^2=-6x+8y+5$ and the point $(-2,11)$. | nvidia |
Find the largest three-digit number that is a palindrome and can be expressed as the sum of three consecutive primes. | nvidia |
How many positive three-digit integers have an even number of positive factors? | nvidia |
Find the number of ways to arrange 10 books on a shelf such that no two books of the same color are adjacent, given that there are 5 red books and 5 blue books. | nvidia |
The positive integers $a$ and $b$ have the same number of positive integer divisors, and for every prime number $p$ that divides $a$, there is exactly one prime number $q$ that divides $b$ such that $p \equiv q \pmod{2}$. What is the minimum possible value of $|a - b|$? | nvidia |
Let $n = p_1^{a_1} \cdot p_2^{a_2} \cdot... \cdot p_k^{a_k}$ be the prime factorization of a positive integer $n$, where $p_1, p_2,..., p_k$ are distinct primes and $a_1, a_2,..., a_k$ are positive integers. Determine $n$ such that the sum of the exponents of the prime factors of $n$ is equal to 20 and the product of t... | nvidia |
If $x$ is a real number and $k$ is a nonnegative integer, recall that the binomial coefficient $\binom{x}{k}$ is defined by the formula
\[
\binom{x}{k} = \frac{x(x - 1)(x - 2) \dots (x - k + 1)}{k!} \, .
\]Compute the value of
\[
\frac{\binom{1/2}{2014} \cdot 4^{2014}}{\binom{4028}{2014}} \, .
\] | nvidia |
A bag contains 18 marbles, some of which are blue and the rest of which are green. When three marbles are drawn at random at the same time, the probability that all three marbles are blue is $\frac{1}{120}$. How many marbles in the bag are blue? | nvidia |
Find the number of 5-digit numbers that start with 2 and have exactly three identical digits. | nvidia |
If $a$, $b$ are nonzero real numbers such that $a^2+b^2=8ab$, find the value of $\left|\frac{a+b}{a-b}\right|$. | nvidia |
Milton spilled some ink on his homework paper. He can't read the coefficient of $x$, but he knows that the equation has two distinct negative, integer solutions. What is the sum of all of the distinct possible integers that could be under the ink stain?
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)-... | nvidia |
Find the largest positive integer whose prime factorization has at least 7 distinct prime factors, such that the sum of the exponents of these prime factors is 10. | nvidia |
How many positive three-digit integers have a digit sum that is divisible by 9 and are also palindromes? | nvidia |
Find the maximum possible value of $a - b,$ given that $a$ and $b$ are positive real numbers such that each of the equations $x^2 + ax + 2b = 0$ and $x^2 + 2bx + a = 0$ has real roots and the same discriminant. | nvidia |
What is the least positive integer $n$ such that $n^2 - n$ is divisible by some but not all integer values of $k$ when $1\leq k \leq n$? | nvidia |
Find the smallest positive integer $b$ for which $x^2+bx+2015$ factors into a product of two binomials, each having integer coefficients. | nvidia |
In the equation $$\begin{array}{c@{}c@{}c@{}c@{}c@{}c} &&a&b&c&d_h\\ &+&e&f&g&i_h\\ \cline{2-6} &1&h&1&j&k_h,\\ \end{array}$$ where $a, b, c, d, e, f, g, i, j, k$ are all digits in base $h$, determine $h$ such that the difference between the sum of the digits of the first number and the sum of the digits of the second ... | nvidia |
Find the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,12]=84$. | nvidia |
Find the sum of the cubes of the solutions to
\[|x^3 - x + \frac{1}{n}| = \frac{1}{n},\]
where n is a positive integer. | nvidia |
Compute the sum of all the positive integer divisors of 25. | nvidia |
A bag contains $n$ red and $n$ blue marbles. Two marbles are randomly drawn without replacement. The probability that the two marbles are of the same color is $\frac{5}{12}$. Determine the value of $n$. | nvidia |
Suppose that $a$ and $b$ are positive integers such that $(a-bi)^2 = 8-6i$. What is $a-bi$? | nvidia |
Find the positive value of $x$ that satisfies $|z^2|=x-3$ given $z=2-i$. | nvidia |
What is the sum of the digits of the smallest positive integer $n$ such that $2^n$ contains the digit 7? | nvidia |
Let $x$ be a number in base 8 such that $xx=25_n$, where $n$ is a positive integer greater than 2. What is the smallest possible value of $x+n$? | nvidia |
Let $p(x)$ be a polynomial with integer coefficients, and let $q(x)$ be a quadratic polynomial with integer coefficients. If $p(x)$ has exactly $3$ real roots, and $q(x)$ has exactly $2$ real roots, what is the maximum number of real roots that the polynomial $p(x)q(x)$ can have? | nvidia |
Find all possible values of $k$ such that the quadratic equation $x^2 + kx + k = 0$ has roots that differ by a factor of 3. | nvidia |
Determine the range of the function
$$f(x) = \sin \left(\frac{1}{x}\right)$$
over the domain of all real numbers $x$. | nvidia |
Find the smallest integer $n$ such that $2^n$ is a perfect square and is divisible by 8, and lies between 60 and 100. | nvidia |
Consider a string of $n$ digits, where each digit is either $0$ or $1$. A segmentation of this string is a way of splitting it into non-overlapping substrings. For example, the string $110101$ can be segmented as $1+10+1+0+1$. For how many values of $n$ is it possible to segment the string in such a way that the sum of... | nvidia |
Let $F$ be the set of all non-constant polynomials with real coefficients. Let $g : F \to F$ be a function such that
\[g(P(x)) + g(Q(x)) = g((P(x) + Q(x))g(P(x)Q(x)))\]for all $P(x), Q(x) \in F.$
Let $m$ be the number of possible values of $g(x^2 + 1),$ and let $t$ be the sum of all possible values of $g(x^2 + 1).$ ... | nvidia |
A certain function $f(x)$ has the properties that $f(3x)=3f(x)$ for all positive real values of $x$, and that $f(x)=1-\frac{1}{x}$ for $1 \leq x < 3$. Find the smallest $x$ for which $f(x)=f(2001)$. | nvidia |
In a convex polygon with $n$ sides, the sum of the interior angles is 1620 degrees. Find the sum of the measures of all the exterior angles of this polygon. | nvidia |
Let $a$ and $b$ be positive real numbers such that each of the equations $x^3 + ax + b = 0$ and $x^3 + bx + a = 0$ has a real root in the interval $[-1, 1]$. Find the smallest possible value of $a + b$. | nvidia |
Find the greatest natural number $n$ such that $n\leq 2010$ and $n(n+1)(n+2)(n+3)$ is a perfect square.
Note: This problem requires a different approach, focusing on the product of consecutive numbers being a perfect square, rather than the sum of squares. | nvidia |
A circle is inscribed in a square with a side length of 10 units. The center of the circle is labeled as O, and two tangent lines to the circle are drawn from the vertices of the square, meeting at a point P outside the square. If the distance from P to one of the vertices is 5 units, find the area of the triangle form... | nvidia |
Let $m$ be a positive integer. A list of $m$ integers $x_1, x_2, \dots, x_m$ is called a $\textit{family}$ if for every $i$, $x_i$ is relatively prime to $x_{i+1}$ (with $x_{m+1}=x_1$). Determine the largest positive integer $m$ such that there exists a family of $m$ integers, all of whose members are between 1000 and ... | nvidia |
A regular hexagon $\mathcal{H}$ is inscribed in a circle of radius $r$. Let $A$ and $B$ be two adjacent vertices of the hexagon, and let $C$ be a point on the circle such that $AB$ is parallel to $OC$, where $O$ is the center of the circle. Find the constant $t$ such that
\[\frac{1}{AC^2} + \frac{1}{BC^2} = t\]
for all... | nvidia |
Find the number of ways to arrange 5 different colored candles on a round table, considering that two candles of the same color are indistinguishable from each other. | nvidia |
Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,...,\dfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?
$\textbf{(A) } 324 \qquad \textbf{... | nvidia |
In $\triangle PQR$, point $S$ is the midpoint of side $QR$. Point $T$ is on $PR$ such that $PT:TR = 2:3$. Point $U$ is on $PS$ such that $PU:US = 3:2$. If the area of $\triangle STU$ is 24, determine the area of $\triangle PQR$. | nvidia |
Triangle $ABC$ is inscribed in circle $\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $\overline{DE}$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Prove that th... | nvidia |
Find all real numbers $a$ for which there exists a nonzero, 3-dimensional vector $\mathbf{v}$ such that
\[\begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{pmatrix} \mathbf{v} = a \mathbf{v}.\] | nvidia |
What is the smallest positive integer $n$ such that $7^n$ ends in the digits 01? | nvidia |
Find the smallest positive integer $x$ which is greater than $1$ and relatively prime to $120$ (recall that relatively prime means that the GCD of $x$ and $120$ is $1$) | nvidia |
Evaluate the product \[ (a-10) \cdot (a-9) \cdot \dotsm \cdot (a-1) \cdot a, \] where $a=2$. | nvidia |
Triangle $ABC$ has sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ of length 43, 13, and 48, respectively. Let $\omega$ be the circle circumscribed around $\triangle ABC$ and let $D$ be the intersection of $\omega$ and the perpendicular bisector of $\overline{AC}$ that is not on the same side of $\overline{... | nvidia |
Find all values of $a$ such that the series $1+ar+ar^2+\cdots$ and $1+ar^2+ar^4+\cdots$ have the same sum. | nvidia |
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