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In a triangle $ABC$, the angle $BAC$ has a measure of $35^\circ$. Given that $AB = 13$ cm and $BC = 14$ cm, determine the number of possible points on side $AC$ such that the line passing through that point and perpendicular to side $AB$ bisects angle $BAC$. | nvidia |
The volume of a rectangular box is equal to the surface area of the box. If the length, width, and height of the box are all different positive integers, find the smallest possible length of the longest edge of the box. | nvidia |
Let's assume that $a$, $b$, $c$, $d$ are four positive integer numbers satisfying $d=a+b+c$, and $(a,b)=1$ , $(b,c)=1$, $(c,a)=1$. Let's also assume that none of these four numbers divide the other three.
Prove that there exists a pair $(x,y)$ of positive integers such that $a \cdot x+b \cdot y=d$ and $a \cdot y+c \c... | nvidia |
Find the sum of the proper divisors of all numbers from 1 to 10. | nvidia |
Point $P$ lies on the line $x= -3$ and is 10 units from the point $(5,2)$. Find the product of all possible $y$-coordinates that satisfy the given conditions. | nvidia |
What is the maximum number of rectangles that can be inscribed in a 1000 x 1000 square grid, such that each rectangle's vertices are lattice points, and no two rectangles share a side or a vertex? | nvidia |
Find the largest possible value of $b$ such that the quadratic equation $x^2 + bx + 1 = 0$ has only real solutions. | nvidia |
A cylinder is circumscribed about a sphere so that the height of the cylinder is equal to its diameter. Given that the radius of the sphere is 3 inches, find the volume of the space between the sphere and the cylinder. Express your answer in terms of $\pi$. | nvidia |
Find the positive integer $m$ such that the sum of the coefficients in the expansion of $(x+y)^m$ is equal to the sum of the coefficients in the expansion of $(x-y)^m$. | nvidia |
Find the smallest positive integer $n$ such that the equation $x^2 + 6x \equiv n \pmod{11}$ has a solution for $x$. | nvidia |
Find the number of positive integers less than 1000 that are divisible by 3, 5, and 7. | nvidia |
Find the minimum value of the sum of the products of pairs of consecutive elements in a sequence of five real numbers, where each number is restricted to the interval [0, 1]. | nvidia |
In the triangle $ABC$ with acute angles, the tangent of the sum of the angles $A$ and $B$ equals $\frac{6}{7}$. Determine the value of the tangent of the angle $C$. | nvidia |
Find the product of the values of $x$ which satisfy $x^2 + 1992x + 1993 = 0$, given that the roots are in the form of $x = a \pm \sqrt{b}$, where $a$ and $b$ are integers. | nvidia |
Find the product of all positive integer values of $c$ such that $8x^2+15x+c=0$ has two real roots. | nvidia |
Find the product of the roots of the equation $\sin^2x-3\sin x+2=0$ that lie in the interval $[0, 4\pi]$ radians. | nvidia |
Find the largest number that is certain to divide the sum of the five numbers that are visible when a standard six-sided die is rolled. | nvidia |
Let $x$ and $y$ be natural numbers such that $x^2+y^2=100$, and $xy$ has the greatest possible number of distinct natural-number factors. Determine $x+y$ and $xy$. | nvidia |
Let $x$ be a number such that
\[
\begin{array}{ccc}
\sqrt{x}+\sqrt{x+2}+\sqrt{x+4}+\cdots+\sqrt{x+2004} & = & 2004\sqrt{2} \medskip \\
\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}+\cdots+\sqrt{x+2004} & = & 2005\sqrt{2}
\end{array}
\]
Determine $\lfloor x\rfloor.$ | nvidia |
Find the product of three consecutive positive integers if their sum is equal to their product minus 16. | nvidia |
Find the maximum value of
\[\tan^2 x + \frac{4}{3} \cot^2 x,\]as $x$ varies over all real numbers. | nvidia |
Find the sum of all positive divisors of $72$ that are also divisors of $24$. | nvidia |
Given a positive integer n, find the smallest possible value of the sum of two positive integers x and y such that their product is equal to n and their greatest common divisor is greater than 1. | 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 $A+B$? | nvidia |
Find the sum of all odd positive integers that can be represented in the form $4k^2+3$, where $k$ is a positive integer, and $4k^2+3<500$. | nvidia |
What is the largest positive integer $n$ such that $n!$ can be expressed as a product of $n$ consecutive integers? | nvidia |
Find the number of positive integers whose prime factorization has exactly three distinct prime factors and whose square is divisible by exactly four distinct prime factors. | nvidia |
Given that $A$ is a point on a circle and $B$ is a point outside the circle, what is the maximum number of points of intersection of the circle and the locus of all points equidistant from $A$ and $B$? | nvidia |
Let $\mathbf{A}$ be a matrix with infinitely many rows and columns, where each entry $a_{ij}$ is the product of the reciprocals of the two numbers $i$ and $j$. For example, $\mathbf{A}$ begins with the following entries:
\begin{align*}
\mathbf{A}=\begin{pmatrix}
1 & \frac 12 & \frac 13 & \cdots\\
\frac 12 & \frac 14 & ... | nvidia |
Let $\mathcal{S}$ be the set of all nonzero real numbers. Let $f : \mathcal{S} \to \mathcal{S}$ be a function such that
\[f(x) + f(y) = f(xyf(x + y))\]for all $x,$ $y \in \mathcal{S}$ such that $x + y \neq 0.$
Let $n$ be the number of possible values of $f(4),$ and let $s$ be the sum of all possible values of $f(4).... | nvidia |
A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\frac{1}{3}$, independently of its previous moves. The prob... | nvidia |
Find the largest integer that can be subtracted from 945 to result in a number that is divisible by 7. | nvidia |
Find the smallest positive integer n such that n^3 has exactly 3 digits. | nvidia |
The sequences of positive integers $1,a_2, a_3, \dots$ and $1,b_2, b_3, \dots$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $$abc=a+b+c.$$
What is the minimum value of $|a-b|+|b-c|$? | nvidia |
How many positive integers $n$ satisfy $200 < n^2 < 900$? | nvidia |
On the coordinate plane, there are three points $A(1,0)$, $B(a,b)$, and $C(c,d)$ such that $\triangle ABC$ is obtuse at vertex $A$ and acute at vertex $C$. Also, no triangle with positive area can be formed with side lengths $OA$, $OB$, and $OC$, where $O$ is the origin. Find the smallest possible value of $d$.
(Here,... | nvidia |
Let $f(x)$ be a function from the set of positive real numbers to the set of real numbers, such that
\[f(xy)=f(x)+f(y)+(x-1)(y-1)\]
for all positive $x,y.$ Find $f(1)+f(2)+f(3).$ | nvidia |
Let
\[\begin{array}{ll}
P(x) & =x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0, \\
Q(x) & =x^m+b_{m-1}x^{m-1}+\cdots+b_1x+b_0,
\end{array}\]be polynomials with integer coefficients, and let $\omega$ be an integer such that
\[\frac{P(\omega)}{Q(\omega)}=\frac{7}{5}.\]Show that if $7 | b_0$ and $5 \nmid b_0$, then $7 \nmid a... | nvidia |
Find the number of four-digit numbers that are palindromes and consist only of even digits. | nvidia |
Find the value of $a$ such that the sum of the cubes of the roots of the equation $x^3 + ax^2 + x + a = 0$ is equal to $-1$. | nvidia |
Consider the sequence of positive integers that are formed by concatenating all the integers from 1 to 9999 in increasing order. Find the smallest prime number that appears in this sequence after the 1000th digit. | nvidia |
Determine the value of $x$ such that the sum of the digits of $x$ is equal to the sum of the digits of $x$ when written in base $2$. | nvidia |
Given the right triangles $T_1, T_2, \dots , T_n$ such that the acute angles of $T_{k+1}$ are half the acute angles of $T_k$ for $k=1,2,\dots , n-1$, what is the greatest integer $n$ for which all these triangles can be constructed using straightedge and compass? | nvidia |
Find all values of $x$ that satisfy the equation:
$$\sqrt{x} + \sqrt{x + \sqrt{x}} = 4$$ | nvidia |
The equations $x+2y=3$ and $x^2+4xy+3y^2=4$ are equivalent in the sense that any ordered pair $(x,y)$ that satisfies one also satisfies the other. How many ordered pairs of real numbers $(x,y)$ satisfy these equations? | nvidia |
Let $g(x)$ be a function that satisfies
\[g(x+y) = g(x)+g(y)\]
for all real numbers $x$ and $y$. Find all possible values of $g(0)$.
Enter all the possible values, separated by commas. | nvidia |
Let $a_1,a_2,a_3,...$ be a sequence of real numbers such that for any positive integer $n$, $\displaystyle\sum_{k=1}^{n} a_k=2^n-1$.
Find $\displaystyle\sum_{n=1}^{\infty} \frac{a_n}{2^n}$. | 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 |
What is the largest integer whose decimal representation has all digits even, and the sum of any two adjacent digits is at most 10? | nvidia |
If the positive numbers $x$, $y$, $z$ and $w$ satisfy
$x+y+z+w=4$,
then compute the maximum value of
$\ds\f{x}{y} \cdot \ds\f{y}{z} \cdot \ds\f{z}{w} \cdot \ds\f{w}{x}$. | nvidia |
Find the smallest positive integer $k$ such that for any two relatively prime positive integers $a$ and $b$ with $a < b$, the number $a^{bk} + b^{ak}$ is divisible by $a + b$. | nvidia |
Find the largest integer n that satisfies the inequality \(\frac{4n+7}{n+4} > 3.\) | nvidia |
Consider a cube with dimensions 3 x 3 x 3, where each vertex is assigned a unique integer value from 1 to 27, inclusive, such that the sum of the values of each pair of adjacent vertices (connected by an edge) is always even. What is the smallest possible sum of the values of all the vertices that have odd coordinates ... | nvidia |
A list of five positive integers has all of the following properties:
$\bullet$ The only integer in the list that occurs more than once is $8,$
$\bullet$ its median is $9,$ and
$\bullet$ its average (mean) is $10.$
What is the largest possible integer that could appear in the list? | nvidia |
A line passes through the points $(k, -2)$ and $(4, 7)$. Find the value of $k$ such that the line makes an angle of 30 degrees with the x-axis. | nvidia |
A chef is preparing a special dish that requires one of four different types of meat and one of six different spices. However, two of the spices are incompatible with one of the meats. Otherwise, there are no restrictions on the combinations of meats and spices. In how many ways can the chef prepare the dish? | nvidia |
A $\textit{composite number}$ is a number that has two or more prime factors. The number 87 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers? | nvidia |
Let $a,$ $b,$ $c$ be non-zero real numbers such that $a + b + c = 0.$ Find all possible values of
\[\frac{a^3 + b^3 + c^3}{abc}.\]Enter all the possible values, separated by commas. | nvidia |
The number $8$ is written on a blackboard. On each step, any one of the following operations is performed:
(i) multiply the number by $5$;
(ii) multiply the number by $-4$.
How many different integers can be written on the blackboard after ten steps? | nvidia |
A triangle is inscribed in a circle with a radius of $5\sqrt{2}$ cm. The triangle has two right angles and its area is three times the area of the circle. Find the length of the hypotenuse of the triangle. Express your answer in terms of $\pi$. | nvidia |
Determine the complex number $w$ satisfying the equation $4w-2\bar{w}=15+36i$. Note that $\bar{w}$ denotes the conjugate of $w$. | nvidia |
A rectangle has integer dimensions such that its area is four times greater than its perimeter. Find the dimensions of this rectangle and determine the smallest possible area among all rectangles satisfying these conditions. | nvidia |
Find the smallest positive integer $x$ such that $x^2 \equiv 1 \pmod{37}$. | nvidia |
Find the number of ways to arrange the letters in the word "mathematics" such that the first letter is "m" and the last letter is "s". | nvidia |
Given a $3\times 3$ matrix whose entries are all integers, find the smallest possible non-zero determinant of the matrix such that the sum of the absolute values of all its entries is 12. | nvidia |
The difference between the squares of the solutions of $y^2+|y|=3$ is equal to 5. Find the sum of the solutions. | 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 |
Find the interval of all $x$ such that both $x^2$ and $x^3$ are in the interval $(1,4)$. | nvidia |
How many different 4-digit positive integers exist, where the integer cannot start with 0, and the last digit must be even? | nvidia |
Find the sum of the positive even divisors of 60 that are not divisible by 4. | nvidia |
Find $x$ such that $\log_{12}3x=2$. | nvidia |
A certain positive integer has the property that its prime factorization contains exactly three distinct prime factors, and when divided by any of its prime factors, the resulting quotient has the same number of divisors. Find the smallest such integer. | nvidia |
Find the largest integer $n$ such that the increasing sequence $\left\lfloor \frac{2^{k_1}+2^{k_2}+\ldots+2^{k_n}}{n}\right\rfloor$, where $k_i\geq 1$ are distinct integers for all $i$, contains all positive integers. | nvidia |
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that
\[\begin{aligned} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7 &= 1 \\
4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7 &= 12 \\
9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7 &= 123. \end{aligned}\]Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$. | nvidia |
Find the number of ordered triples $(x,y,z)$ of real numbers such that $x + y = 2$ and $xy - z^2 = 1.$ | nvidia |
Find the sum of all integer values of $x$ such that $x^2-6x+9$ is a perfect square and $4<x<20$. | nvidia |
The polynomial $x^3 - 2ax^2 + bx + c$ has three distinct positive integer roots, and the sum of its roots is equal to twice the product of its roots. Determine the value of $a$. | nvidia |
Let $g(x) = \frac{x}{x+1}$ be a function. Determine all real numbers $a$ such that the graphs of $y=g(x)$ and $y=g^{-1}(x)$ intersect at the point with x-coordinate $a$. | nvidia |
Let $c$ and $d$ be nonzero real numbers such that $(3 + 5i)(c + di)$ is a real number. Find $\frac{c}{d}$. | nvidia |
On a $4\times 4$ chessboard, four distinct pawns are placed such that each column and row contains exactly one pawn. How many ways are there to move one of the pawns to an adjacent square (horizontally or vertically) so that the resulting configuration still satisfies the condition that each column and row contains exa... | nvidia |
Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, and let $P$ be the midpoint of $\overline{CD}$. Let $M$ be a point on $\overline{AP}$ such that the volume of the tetrahedron $ABMP$ is $\frac{1}{3}$ of the volume of the tetrahedron $ABCP$. Find the ratio $\frac{AM}{AP}$. | nvidia |
Let $g(x)=\dfrac{x^2+1}{x+1}$. Find the value of $x$ such that $g(x)=2$. | nvidia |
Compute the sum of the roots of the equation \[x\sqrt{x} - 6x + 7\sqrt{x} - 1 = 0,\]given that all of the roots are real and nonnegative. | nvidia |
Let $m$ and $n$ be positive integers. What is the largest possible value of $m$ such that the product of all the positive integers from $m$ to $n$, inclusive, has only one prime factor? | nvidia |
Let $S_n$ denote the sum of the first $n$ terms of a geometric sequence with common ratio $r$. If $\frac{S_{2n}}{S_n}$ is a constant that does not depend on $n$, for all positive integers $n$, then find the value of $r$. | nvidia |
Let $F$ be a function that maps positive integers to real numbers. If $F(m) + F(n) = F(mn)$ for all positive integers $m$ and $n$, and $F(4) = 9$, find $F(16)$. | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x + y) f(x - y) = (f(x) + f(y))^2 + 2x^2 f(y)\]for all real numbers $x$ and $y.$ | nvidia |
Find the number of positive integers $n,$ $1 \le n \le 1000,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
The sum of the squares of two positive integers is 193. The product of the two integers is 84. What is the sum of the two integers? | nvidia |
The corner points of a parallelogram lie on the axes. One side of the parallelogram is fixed by the points (0;2) and (3;2). Among all possible positions of the parallelogram determine the position for which the area is maximum. | nvidia |
Find a positive integer that is divisible by 21 and has a cube root between 27 and 27.5. | nvidia |
Find the smallest positive integer that has exactly four different prime factors, and whose total number of divisors is equal to the square of the number of its prime factors.
(Note: This problem requires a different approach, as it involves the relationship between the number of divisors and the number of prime facto... | nvidia |
Determine all real numbers $a$ such that there exists a complex number $z$ satisfying $|z| = 1$ and $\left| z + az^2 \right| = 1.$ | nvidia |
Find the largest integer $m$ such that the product of the first $m$ prime numbers is divisible by $2^m$. | nvidia |
In a triangle ABC, the sum of the lengths of the sides AB and BC is 25 cm, the sum of the lengths of the sides BC and CA is 32 cm, and the sum of the lengths of the sides CA and AB is 28 cm. What is the perimeter of the triangle ABC? | nvidia |
Two integers $a$ and $b$ are such that $a+b=10$ and $a^2+b^2=52$. Compute $a^3+b^3$. | 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 |
Simplify the expression $\sqrt{2a^3} \cdot \sqrt{8a^4}$ and find the value of $a$ such that the resulting expression is equal to $12a^2$. | nvidia |
Find the value of $y$ such that $\sqrt{y-3} = 5$. | nvidia |
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