RockToken/qwen3_30b_a3b_to_4b_onpolicy_5k_src30k-35k_cont
Text Generation • 4B • Updated • 433
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[
{
"from": "human",
"value": "A parabola is tangent to the lines $y = x + 2$ and $y = 2x - 1$. The focus of the parabola lies on the line $y = -x + 1$. Find the equation of the parabola."
}
] |
[
{
"from": "human",
"value": "In a bag, there are 5 cards, each representing a different flavor of ice cream, with prices ranging from $1 to $5. Each time Emily draws a card, she has the option to buy the corresponding ice cream if she has enough money. Emily has a $10 bill and 5 one-dollar coins. If the ice... |
[
{
"from": "human",
"value": "Consider the expansion of $(x+2)^n$, which contains 4 consecutive terms with coefficients in the ratio $2:3:5:7$, expressed as ${n\\choose k} : {n\\choose k+1} : {n \\choose k+2} : {n \\choose k+3}$.\nFind the sum of all possible values of $n+k$."
}
] |
[
{
"from": "human",
"value": "Let $a$, $b$ be two distinct positive integers. It is given that their modular inverses sum to $47$, modulo $51$. Find the value of $ab$, modulo $51$."
}
] |
[
{
"from": "human",
"value": "What is the largest positive integer $n$ such that $\\frac{25n+3}{27n+7}$ can be expressed as a terminating decimal?"
}
] |
[
{
"from": "human",
"value": "In $\\triangle ABC$, $\\measuredangle BAC=60^{\\circ}$ and $\\measuredangle ACB=45^{\\circ}$. $BC=1$. Let $D$ be the midpoint of segment $\\overline{AC}$. Point $E$ lies on side $\\overline{AB}$ such that $\\overline{CE}\\perp\\overline{BD}$. Extend segment $\\overline{AB}$ thro... |
[
{
"from": "human",
"value": "In the following cryptogram, each of the digits $1$, $2$, $3$, and $4$ is replaced by a distinct letter of the alphabet:\n$\\ds\\begin{array}{cccc} A & B & C & D\\\\ + & B & C & D\\\\ \\hline C & D & D & A\\\\ \\end{array}$\n\nWhat is the value of $A+B+C+D$?"
}
] |
[
{
"from": "human",
"value": "Let $\\mathbf{p},$ $\\mathbf{q},$ $\\mathbf{r},$ $\\mathbf{s},$ $\\mathbf{t}$ be five distinct unit vectors in space such that\n\\[\\mathbf{p} \\cdot \\mathbf{q} = \\mathbf{p} \\cdot \\mathbf{r} = \\mathbf{q} \\cdot \\mathbf{r} = \\mathbf{p} \\cdot \\mathbf{s} = \\mathbf{q} \\cd... |
[
{
"from": "human",
"value": "A set of three positive real numbers $a, b, c$ satisfies the equation\n\n\\[\\frac{(a^2 + 1)}{b + c - 1} + \\frac{(b^2 + 4)}{c + a - 4} + \\frac{(c^2 + 9)}{a + b - 9} = 30.\\]\n\nDetermine the ordered triple $(a, b, c)$."
}
] |
[
{
"from": "human",
"value": "Given that the polynomial $2x^3 - 5x^2 + 6x - 7$ has a root of multiplicity greater than 1, find the value of the coefficient of the quadratic term if the polynomial can be factored as $(x - a)^2(x + b)$, where $a$ and $b$ are integers."
}
] |
[
{
"from": "human",
"value": "Let $a$ and $b$ be two real numbers such that\n\\[a^2 + 2b^2 = 7\\]\nand $a + bi$ is a root of the equation\n\\[z^3 - 2z^2 + 5z - 1 = 0.\\]\nFind $b.$"
}
] |
[
{
"from": "human",
"value": "A sequence of real numbers $\\{b_n\\}$ has the property that for any positive integer $n$, the sum of the squares of the terms in the $n$th window of length 3 is constant. In other words,\n\n$$b_n^2 + b_{n+1}^2 + b_{n+2}^2 = C$$for all $n \\ge 1$, where $C$ is a constant. If the... |
[
{
"from": "human",
"value": "Let $x, y, z$ be complex numbers, with $y$ imaginary, such that\n$x + y + z = xy + xz + yz = xyz = 5$.\nFind $y$."
}
] |
[
{
"from": "human",
"value": "At a party, 24 guests are seated in a circle and participate in a game with 3 dice. Initially, one die shows a 2, another a 5, and the last one a 3. The first guest rolls the die showing 2, adds 1 to the die showing 5, and subtracts 1 from the die showing 3. Each subsequent gues... |
[
{
"from": "human",
"value": "The median of the set $\\{-2, 4, x\\}$ is equal to the mode of the set $\\{-2, x, 4, 7, y\\}$. If $y$ is a whole number, what is the largest possible value of $y$?"
}
] |
[
{
"from": "human",
"value": "Let $f(x)$ be the function defined by\n\\[f(x)=\\frac{x^2-2x+6}{x^3+3x^2+bx+c},\\]where $b$ and $c$ are constants. Suppose that $f(x)$ has two horizontal asymptotes. Find the sum of all possible values of $c$."
}
] |
[
{
"from": "human",
"value": "The graph of the function $f(x) = ax^3 + bx^2 + cx + d$ has a point of inflection at $(2, 6)$ and passes through the point $(0, 0)$. If the function is symmetric with respect to the line $y = x$, find the value of $a + b$."
}
] |
[
{
"from": "human",
"value": "What is the smallest positive integer $a$ such that $a^{-1}$ is undefined $\\pmod{105}$ and $a^2 \\equiv 1 \\pmod{105}$?"
}
] |
[
{
"from": "human",
"value": "In $\\triangle{ABC}$, $AB=10$ and $AC=15$. Let $D$ be the foot of the perpendicular from $B$ to $\\overline{AC}$, $O$ the center of the circle circumscribed about $\\triangle{ADC}$, and $E$ the other end of the diameter which goes through $B$. Find the radius of the circle."
}... |
[
{
"from": "human",
"value": "Consider sequences consisting of only 0's and 1's, where every run of consecutive 0's has a length that is a multiple of 3, and every run of consecutive 1's has a length that is a multiple of 2. Examples of such sequences are 000, 11, and 000110000, while 1010 is not such a sequ... |
[
{
"from": "human",
"value": "Let $a$, $b$, $c$, $d$, $e$, $f$, and $g$ be positive integers such that $\\gcd(a,b,c,d,e,f,g) = 1$. The system of equations \\begin{align*}\n12x^4 + ax^3 + bx^2 + cx + d &= 0,\\\\\n21x^3 + ex^2 + fx + g &= 0\n\\end{align*}has a common irrational root $r$. If the system of equat... |
[
{
"from": "human",
"value": "Consider the sequence of real numbers defined by $x_1 = 1$ and\n$x_{n+1} = x_n^3 - \\frac{3}{2}x_n^2 + \\frac{1}{2}$\nfor all $n \\ge 1.$ In the interval $[0, 1]$, what is the limit of $x_n$ as $n$ tends to infinity?"
}
] |
[
{
"from": "human",
"value": "A function $f(x)$ is defined by $f(0) = 1$ and $f(x) = \\frac{1}{2} + f(x - 1)^2$ for $x > 0.$ Find the sum of the infinite series $\\sum_{n=0}^\\infty \\frac{1}{f(n)}$."
}
] |
[
{
"from": "human",
"value": "In triangle $ABC$, $AB = 10$ cm, $BC = 12$ cm, and $CD = CE$. The area of triangle $CDE$ is one-third the area of triangle $ABC$. What is the length in centimeters of segment $DE$? Express your answer in simplest radical form.\n\n[asy]\ndraw((0,0)--(0,36)--(48,0)--cycle);\ndraw(... |
[
{
"from": "human",
"value": "A group of friends plan to travel from City A to City B by car, a distance of 240 miles. On the first day, they drive for 4 hours at an average speed of 40 miles per hour. On the second day, they drive for 3 hours at an average speed of 60 miles per hour. How many miles do they ... |
[
{
"from": "human",
"value": "Consider two functions $h(x) = x^3 - 3x^2 + 2x + 1$ and $j(x) = 2x^2 - 4x - 5$. Determine the value of $x$ for which $h(j(x)) - j(h(x)) = 15$."
}
] |
[
{
"from": "human",
"value": "What is the largest number $m$ such that for every positive integer $n$, the number of integers between $n^2$ and $(n+1)^2$ that are divisible by $m$ is at most $2$?"
}
] |
[
{
"from": "human",
"value": "Let $S$ be the sum of all positive integer divisors of $2004!$. Determine the remainder when $S$ is divided by $2004$."
}
] |
[
{
"from": "human",
"value": "Let $C$ be the circle with center $(0,0)$ and radius $16$.\nLet $R = (6,12)$. There are real numbers $r$ and $s$ such that the line through $R$\nwith slope $m$ is tangent to $C$ if and only if $r < m < s$.\nWhat is $s-r$?"
}
] |
[
{
"from": "human",
"value": "A sphere with center $O$ has a radius of 3 units. Let $A=(1,2,3)$, $B=(3,2,1)$, and $C=(1,3,2)$ be three points on the surface of the sphere. Find the smallest possible area of triangle $ABC$ if point $A$ is fixed and points $B$ and $C$ can be rotated along the surface of the sp... |
[
{
"from": "human",
"value": "In triangle $ABC$, let $A'$ and $B'$ denote the feet of the perpendiculars dropped from points $A$ and $B$, respectively, to side $BC$. If $A'B'=6$ and the measure of angle $A$ is twice that of angle $B$, what is the area of triangle $A'B'C$?"
}
] |
[
{
"from": "human",
"value": "A parametric graph is given by\n\\begin{align*}\nx &= \\sin t - \\frac{t}{3}, \\\\\ny &= \\cos t.\n\\end{align*}How many times does the graph intersect itself between $x = 2$ and $x = 60$?"
}
] |
[
{
"from": "human",
"value": "The diagonal of a rectangular prism is 25 units long, and the length of the longest edge is 15 units. If the surface area of the prism is 600 square units, find the volume of the prism."
}
] |
[
{
"from": "human",
"value": "Let $g$ be a function from the positive integers to the positive integers such that\n\n\\[g(l \\times m) + g(l + m) = g(l) \\times g(m) - 1\\]\n\nfor all positive integers $l$ and $m$. \n\nFind the number of possible values of $g(3)$ and the product of all possible values of $g(... |
[
{
"from": "human",
"value": "If $\\prod_{n=1}^{\\infty} (1 + \\sin^n \\theta) = 2$, what is the value of $\\cos \\theta$?"
}
] |
[
{
"from": "human",
"value": "In the Cartesian plane, two curves are given by the equations $y = \\sin x$ and $y = \\cos x$. For every point $(x, y)$ on the sine curve, a line segment can be drawn to the nearest point on the cosine curve. Find the maximum possible length of one of these line segments."
}
] |
[
{
"from": "human",
"value": "A rectangle consists of two congruent squares and two non-square rectangles whose dimensions are in the ratio 4:5.\nIf the diagonal of one of the squares is equal to 10 cm,\nwhat is the area of the rectangle?"
}
] |
[
{
"from": "human",
"value": "A circle is divided into nine congruent regions, as shown below. Each region is bounded by two radii and an arc of the circle.\n\n[asy]\npair center = (0,0);\nreal radius = 1;\ndraw(Circle(center, radius));\nfor(int i = 0; i < 9; i++){\npair point1 = rotate(i * 360/9, center) * ... |
[
{
"from": "human",
"value": "Let $P$ denote the product of all four-digit positive integers with digits that form an arithmetic progression. Compute the remainder when $P$ is divided by $10000$."
}
] |
[
{
"from": "human",
"value": "Consider the equation\n\n\\[2^{1/z} = 2\\cos \\frac{\\pi}{4z} + i \\sin \\frac{\\pi}{4z},\\]\n\nwhere $z$ is a complex number. For what values of $z$ is this equation true?\n\n(Your solution should include the correct values of $z$, as well as a clear explanation of how you arr... |
[
{
"from": "human",
"value": "Let $g$ be a non-constant polynomial such that\n\\[g(x^2) + g(x) + g(x^{-2}) = [g(x)]^2\\]for all nonzero real numbers $x.$ Find the sum of all possible values of $g(1)$."
}
] |
[
{
"from": "human",
"value": "Let\n\\[T=\\sum_{n=1}^{2007}\\sqrt{1+\\frac{1}{n^2}+\\frac{1}{(n+1)^2}+\\frac{1}{(n+2)^2}}.\\]\nCompute $\\lfloor T^3\\rfloor$."
}
] |
[
{
"from": "human",
"value": "Let $g$ be a function mapping the real numbers to the real numbers such that\n\n\\[g(xy) = g(x)g(y) + x + y\\]\n\nfor all real numbers $x$ and $y$. If $a$ and $b$ are the only possible values of $g(1)$, compute the product $ab$."
}
] |
[
{
"from": "human",
"value": "A circle is inscribed in a regular hexagon. A segment is drawn from a point on the circle, tangent to the circle, and ending on the circle, such that it divides the hexagon into two regions of equal area. What is the ratio of the length of the segment to the side length of the... |
[
{
"from": "human",
"value": "In the adjoining figure, $AC$ and $BD$ are two chords of the circle that intersect at $E$, with $\\angle AEB = \\alpha$. The ratio of the area of $\\triangle CDE$ to that of $\\triangle ABE$ is $\\cos^2\\alpha$. Find the ratio of the lengths of the segments $CE$ and $AE$."
}
] |
[
{
"from": "human",
"value": "Problem:\nA billiard ball is rolling on a rectangular billiard table $ABCD$, where $A(0,0)$, $B(a,0)$, $C(a,b)$, and $D(0,b)$. The ball reflects off the edges of the table according to the law of reflection, where the angle of incidence is equal to the angle of reflection. Suppo... |
[
{
"from": "human",
"value": "Three friends - Alex, Ben, and Chris - are playing a game where they take turns rolling a fair six-sided die. The game starts with Alex, followed by Ben, and then Chris. The first player to roll a 6 wins the game. What is the probability that Chris wins the game?"
}
] |
[
{
"from": "human",
"value": "Let\n\\[x^6 + 59x^3 + 1 = r(x) s(x),\\]\nwhere $r(x)$ and $s(x)$ are monic, non-constant polynomials with integer coefficients. Find $r(-1) + s(-1)$."
}
] |
[
{
"from": "human",
"value": "Let $a_n$ and $b_n$ be two nonconstant arithmetic sequences with different common differences. If\n$$\\frac{a_{n+1}}{a_n}=\\frac{b_{n+1}}{b_n}$$\nfor all positive integers $n$, then what is the sum of the common differences of the two sequences?"
}
] |
[
{
"from": "human",
"value": "Consider a set of 512 switches, each with four positions: $P, Q, R$, and $S$. The switches are initially set to position $P$ and are labeled with the integers $(2^{x})(3^{y})$, where $x$ and $y$ take on the values $0, 1, \\ldots, 8$. During a 512-step process, at step $i$, the $... |
[
{
"from": "human",
"value": "Let $a_1 < a_2 < \\cdots < a_{100}$ be an arithmetic sequence of positive integers such that $a_1 + a_2 + \\cdots + a_{100} = 128.$ How many different values of $k$ are there such that $(x - a_1)(x - a_2) \\cdots (x - a_{100}) = 0$ has exactly $k$ distinct positive integer solut... |
[
{
"from": "human",
"value": "A positive integer $m$ has the property that the sum of its factors is equal to the sum of the factors of its square. If $m$ has $5$ distinct prime divisors, what is the smallest possible value of $m$?"
}
] |
[
{
"from": "human",
"value": "Let $A$ be a point on the circle $x^2 + y^2 - 12x + 31 = 0,$ and let $C$ be a point on the hyperbola $x^2 - y^2 = 4.$ Find the largest possible distance $AC$ such that the segment $AC$ is perpendicular to the line $y = x.$"
}
] |
[
{
"from": "human",
"value": "In the diagram below, $ABCD$ is a trapezoid such that $\\overline{AB}\\parallel \\overline{CD}$ and $\\overline{AD}\\perp\\overline{CD}$. If $CD = 15$, $\\tan D = 2.5$, and $\\tan A = 3$, then what is $AD$?\n\n[asy]\n\npair A,B,C,D;\n\nC = (0,0);\n\nD = (15,0);\n\nA = (15,37.5)... |
[
{
"from": "human",
"value": "In quadrilateral $ABCD$, $AB$ is parallel to $DC$, and $AC$ is parallel to $BD$. Let $O$ be the point where the diagonals $AC$ and $BD$ intersect. If $\\angle AOB = 110^\\circ$ and $\\angle ACD = 30^\\circ$, find $\\angle BCD$, in degrees."
}
] |
[
{
"from": "human",
"value": "A positive integer has the same remainder when divided by 7 and 11. If the number is divided by 77, the remainder is 32. What is the least possible value of this integer?"
}
] |
[
{
"from": "human",
"value": "Let $a$, $b$ and $c$ be integers and consider a congruence $ax+by\\equiv c \\pmod {n}$. Find the smallest possible value of $n$ for which this congruence has no solutions, given that $n$ is prime, $\\gcd(a,b)=1$, $ab\\equiv b \\pmod n$ and $ac\\equiv 0 \\pmod n$."
}
] |
[
{
"from": "human",
"value": "The square pyramid $ABCDO$ is constructed from equilateral triangles $ABO$, $ACO$, $BCO$, $ADO$, $BDO$, and $CDO$, each with side length $\\sqrt{111}$. If $M$ is the midpoint of $\\overline{AD}$, and $P$ is the midpoint of $\\overline{CO}$, find the length of the altitude of $\\... |
[
{
"from": "human",
"value": "The polynomial $Q(x) = x^3 + px^2 + qx + r$ has the property that the median of its zeros, the sum of the squares of its zeros, and the product of its coefficients are all equal. If the $y$-intercept of the graph of $y = Q(x)$ is 3, what is $q$?"
}
] |
[
{
"from": "human",
"value": "Solve the following system of equations:\n$$\\left\\{ \\begin{array}{l}\nx^3+y^3+z^3=3xyz\\\\\nx+y+z=1\\\\\nx^2+y^2+z^2=3\\\\\n\\end{array} \\right.$$"
}
] |
[
{
"from": "human",
"value": "A polynomial is expressed as the product of two quadratic factors, with the coefficient of $x^2$ being equal to 12.\nOne factor is $ax^2+bx+c$, with $a$ and $b$ being equal in absolute value but having opposite signs.\nThe other factor has the same absolute values of the coeffic... |
[
{
"from": "human",
"value": "Find the maximum possible area of the intersection of a cone with its axis parallel to the z-axis and a plane passing through the origin, if the vertex of the cone is located at the point (2,3,5) and the angle between its axis and the z-axis is 60 degrees."
}
] |
[
{
"from": "human",
"value": "Find the quadratic function $f(x) = x^2 + cx + d$ such that\n\\[\\frac{f(f(x) - 2x)}{f(x)} = x^2 + 1234x + 1987.\\]"
}
] |
[
{
"from": "human",
"value": "Let $S$ be a set containing $n$ elements. Suppose that two elements $x$ and $y$ are removed from $S$. Two elements $a$ and $b$ are then randomly selected from the remaining elements. Let $P$ be the probability that $a$ and $b$ are both greater than $x$ and $y$. If $P=\\frac{1}{1... |
[
{
"from": "human",
"value": "A bookshelf has 6 shelves, and the number of books on each shelf is an integer. The total number of books is 72, and if we take all the books off the shelves and arrange them in a row in increasing order of their weights, we will see that every three consecutive books in the row... |
[
{
"from": "human",
"value": "Find the maximum possible distance between the centers of two circles, each with a diameter of 8 cm, that can be inscribed in a right-angled triangle with legs of 15 cm and 20 cm."
}
] |
[
{
"from": "human",
"value": "Given a positive integer n>3 find the number of values k can have so that the angles of a triangle with sides n-1, n and k, in some order, are in an arithmetic progression."
}
] |
[
{
"from": "human",
"value": "Let $f(x) = \\left\\lceil\\dfrac{1}{x+2}\\right\\rceil$ for $x > -2$, and $f(x) = \\left\\lfloor\\dfrac{1}{x+2}\\right\\rfloor$ for $x < -2$. ($f(x)$ is not defined at $x = -2$.) Which integer is not in the range of $f(x)$?\n\nNew problem:\n\nDefine $g(x) = \\left\\lfloor x^2 + ... |
[
{
"from": "human",
"value": "In $\\triangle ABC$, we have $AC=BC=7$ and $AB=2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$. What is $BD$?\n\nAnother problem inspired by this one:\nA right-angled triangle $ABC$ has side lengths $AB = 15$, $BC = 8$, and $AC =... |
[
{
"from": "human",
"value": "A company has five employees, and each employee is assigned to work on a project for a different number of hours. The total number of hours worked by all employees is 120 hours. The number of hours worked by each employee is a different positive integer, and no employee worked m... |
[
{
"from": "human",
"value": "The sides of triangle $ABC$ have lengths $8, 15,$ and $17.$ An altitude of the triangle is extended to meet the extension of the side of length 8. If the triangle formed by this intersection, the vertex of the altitude, and the opposite vertex of the triangle is similar to the o... |
[
{
"from": "human",
"value": "Find the largest integer $n$ such that for any non-negative integer $m$, the number $2^n + 3^m$ is never a perfect square."
}
] |
[
{
"from": "human",
"value": "Counting numbers a and b leave a remainder of 2 when divided by some positive integer c, where a and b are coprime.\nWhat is the greatest possible value of c?"
}
] |
[
{
"from": "human",
"value": "Given a triangle $ABC$ with vertices $A(0, 0)$, $B(1, 1)$, and $C(1, -1)$, a line with equation $y = mx$ intersects segments $AB$ and $AC$ at points $D$ and $E$, respectively, forming a triangle $\\triangle ADE$. If the area of $\\triangle ADE$ is $\\frac{1}{3}$ the area of $\\t... |
[
{
"from": "human",
"value": "Let $x$ be a positive number such that\n$$\\sqrt{2x+1}\\cdot\\sqrt{5x+2}\\cdot\\sqrt{x+1}\\cdot\\sqrt{3x+1}=90.$$\nFind all possible values for $x$."
}
] |
[
{
"from": "human",
"value": "Consider a pyramid of spheres, where each sphere in a layer rests on exactly three spheres in the layer below it. The bottom layer has 15 spheres, with each sphere colored either red or blue. The color of each sphere in the layers above is determined by the majority color of the... |
[
{
"from": "human",
"value": "Let $g(x)$ be a function such that $g(0) = 2$ and\n\\[g(x+y) = g \\left( \\frac{x^2 - y^2}{x+y} \\right) + \\frac{2xy}{x+y}\\]for all real numbers $x$ and $y$ where $x+y\\neq 0.$ Find $g(x).$"
}
] |
[
{
"from": "human",
"value": "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 $y^2=x$."
}
] |
[
{
"from": "human",
"value": "Let $T$ be the set of all positive integers. The function $g : T \\to T$ satisfies the following two properties:\n\n(i) First,\n\\[g \\left( x + 1 \\right) = g(x) + x + 1\\]for all $x \\in T.$\n\n(ii) Second,\n\\[g \\left( xy \\right) = g(x) \\times g(y)\\]for all $x \\in T$ an... |
[
{
"from": "human",
"value": "Compute the unique positive integer $m$ such that the series\n\\[1 \\cdot m^2 + 2 \\cdot m^3 + 3 \\cdot m^4 + \\dots + n \\cdot m^{n+1} = \\frac{m^{n+2} - m}{m - 1}.\\]\nholds true for all positive integers $n$."
}
] |
[
{
"from": "human",
"value": "Solve for $x$ in the equation $\\frac{2^x}{3^x}=\\frac{2x+1}{3x-1}$.\n\n(Note: This problem requires a different approach, using logarithms and algebraic manipulations, to solve for x)"
}
] |
[
{
"from": "human",
"value": "Last year, Olivia took 8 science quizzes and received 8 different scores, each an integer between 75 and 90, inclusive. After each quiz, she noticed that the average of her quiz scores was an integer. Her score on the eighth quiz was 80. What was her score on the fifth quiz?\n\n... |
[
{
"from": "human",
"value": "In quadrilateral $ABCD$, $\\angle A$ is a right angle, $BC = 5$, $CD = 4$, and $AD = 10$. Given that $AB = \\sqrt{p} + q$, where $p$ and $q$ are positive integers, find $p + q$."
}
] |
[
{
"from": "human",
"value": "In trapezoid $ABCD$, the diagonals $AC$ and $BD$ intersect at point $E$, and $AB = CD = 20$ units. If the area of triangle $ABE$ is twice the area of triangle $CDE$, find the length of base $AD$."
}
] |
[
{
"from": "human",
"value": "A certain number is written in base 7 and consists of only two digits. When this number is multiplied by a certain power of 7, the base 7 representation of the result has exactly three digits. If the first and last digits of the resulting three-digit number are 4 and 1, respecti... |
[
{
"from": "human",
"value": "A cylindrical log has a height of 8 meters. If a wedge-shaped piece with an angle of $\\theta$ degrees is cut from the log and then placed back into the log but rotated 180 degrees, the volume of the resulting solid is $\\frac{1}{16}$ of the original volume of the log. Find the ... |
[
{
"from": "human",
"value": "Find the greatest value of $b$ such that the quadratic function $f(x) = x^2 + bx + 16$ has no real roots in the interval $(-2, 4)$."
}
] |
[
{
"from": "human",
"value": "In an equilateral triangle, the sides are $\\sin 2x,$ $\\sin 2x,$ and $\\sin 5x,$ and one of the angles is $3x.$ (All angle measurements are in degrees.) Enter all possible values of $x,$ separated by commas."
}
] |
[
{
"from": "human",
"value": "In a trapezoid $ABCD$, where $AB$ is parallel to $CD$, the diagonals $AC$ and $BD$ intersect at a point $E$. The circle with center $E$ and radius $r$ is tangent to the sides $AD$ and $BC$ of the trapezoid. If the area of the trapezoid is $24$ and the height of the trapezoid is ... |
[
{
"from": "human",
"value": "Find all integers n such that the number 2^n is a palindrome number in base 3."
}
] |
[
{
"from": "human",
"value": "A bookshelf has 5 shelves, and the number of books on each shelf forms a geometric sequence. If the total number of books on the bookshelf is 341 and the number of books on the second shelf is 9, what is the number of books on the fifth shelf?"
}
] |
[
{
"from": "human",
"value": "In a game of chance, two players simultaneously roll two dice. The first player wins if the product of the two numbers on their dice is a perfect square. The second player wins if the product of the two numbers on their dice is not a perfect square. If the game is played many ti... |
[
{
"from": "human",
"value": "Determine all positive integer pairs $(a,b)$ such that the greatest common divisor of $a^2 + 3ab + 2b^2$ and $2a^2 + 3ab + b^2$ is $1$."
}
] |
[
{
"from": "human",
"value": "If $z$ and $w$ are complex numbers such that\n\\begin{align*}\n2|z|^2 + 3|w|^2 &= 1, \\\\\n2z + 3w &= 1.\n\\end{align*}\nFind $|z+2w|$.\n\n(Note: Assume that $z$ and $w$ satisfy the condition $|z|, |w| \\leq 1.$)"
}
] |
[
{
"from": "human",
"value": "A robot is programmed to move in a two-dimensional space according to the following parametric equations:\n\\begin{align*}\nx &= 2 \\sin t + \\cos 2t, \\\\\ny &= \\cos t + 2 \\sin 2t.\n\\end{align*}\nDetermine the number of times the robot's path intersects the circle $x^2 + y^2... |
[
{
"from": "human",
"value": "Josh and Adam are running at a constant speed along a circular track. After 20 minutes, they are at the same spot on the track but on opposite sides of the starting point. After 40 minutes, they meet at a spot 150 meters from the starting point. What is the length of the circula... |
[
{
"from": "human",
"value": "Consider a regular $n$-gon, which is a polygon with $n$ sides of equal length and $n$ internal angles of equal size. For a given number $n$, determine the number of non-similar regular $n$-gons that can be constructed using a specific set of rules: each internal angle must be le... |
[
{
"from": "human",
"value": "Triangle $ABC$ is inscribed in a circle with center $O$. We have $\\angle BOC = 60^\\circ$, $\\angle AOC = 120^\\circ$, and $\\tan A = \\frac{\\tan B}{2}$. What is the radius of the circle?"
}
] |
[
{
"from": "human",
"value": "In a music competition, 7 judges scored a contestant's performance with the following scores: 78, 82, 95, 88, 76, 90, and 92. However, it was later discovered that one of the judges mistakenly wrote down the score as 5 points lower than intended. If the corrected median score is... |
[
{
"from": "human",
"value": "Points $A$, $B$, and $C$ are consecutive vertices of a regular polygon with $n$ sides connected by line segments, and line segment $\\overline{AB}$ is extended past $B$ to point $P$ such that $AP:PB = k:\\ell,$ where $k$ and $\\ell$ are positive integers. Determine the set of po... |
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