prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Andrea inscribed a circle inside a regular pentagon and circumscribed a circle around it, calculating the area of the region between the two circles. Bethany performed the same operations with a regular heptagon (7 sides). The areas of the two regions were $A$ and $B$, respectively. Each polygon had a side length of $2... | 1 |
In the year $2001$, the United States will host the International Mathematical Olympiad. Let $I$, $M$, and $O$ be distinct positive integers such that the product $I \cdot M \cdot O = 2001$. Find the largest possible value of the sum $I + M + O$. | 671 |
Two integers have a sum of $26$. When two more integers are added to the first two integers, the sum is $41$. Finally, when two more integers are added to the sum of the previous four integers, the sum is $57$. Find the minimum number of even integers among the $6$ integers. | 1 |
In parallelogram \(ABCD\) of the accompanying diagram, line \(DP\) is drawn bisecting \(BC\) at \(N\) and meeting \(AB\) (extended) at \(P\). From vertex \(C\), line \(CQ\) is drawn bisecting side \(AD\) at \(M\) and meeting \(AB\) (extended) at \(Q\). Lines \(DP\) and \(CQ\) meet at \(O\). If the area of parallelogram... | 17 |
Let $h_1$ and $h_2$ be the altitudes of a triangle drawn to the sides with lengths $5$ and $2\sqrt{6}$, respectively. If $5 + h_1 \leq 2\sqrt{6} + h_2$, find the length of the third side of the triangle. | 7 |
A circular grass plot with a diameter of 12 feet is intersected by a straight gravel path that is 3 feet wide. One edge of the path passes through the center of the plot. Calculate the number of square feet in the remaining grass area. The original answer is in the form $k\pi - m\sqrt{n}$, please provide the value of k... | 42 |
If $a$ and $b$ are digits for which: \[ \begin{array}{ccc} & 2 & a \\ \times & b & 3 \\ \hline & 6 & 9 \\ 9 & 2 \\ \hline 9 & 8 & 9 \end{array} \] Find the value of $a + b$. | 7 |
$(a_n)_{n = 0}^\infty$ is a sequence of integers. For every $n \ge 0$, $a_{n + 1} = a_n^3 + a_n^2$. Find the maximum number of distinct residues of $a_i$ in $\pmod{11}$. | 3 |
The stronger Goldbach conjecture states that any even integer greater than 7 can be written as the sum of two different prime numbers. For such representations of the even number 126, find the largest possible difference between the two primes. | 100 |
Polynomial $f(x)$ satisfies $(x - 1)f(x + 1) - (x + 2)f(x) = 0$ for every $x \in \mathbb{R}$. If $f(2) = 6$, find the value of $f\left(\frac{3}{2}\right)$. The original answer is in \frac{k}{m} format, please give the value of k + m. | 23 |
A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last two marbles. Find the probability that Cheryl gets $2$ marbles of the same color. The original answer is in... | 6 |
At each step, we change the positions of exactly two numbers from the sequence $1, 2, 3, 4, 5, 6, 7$. How many different arrangements can be formed after two steps? | 176 |
Find the value of $\frac{\log_5 9 \cdot \log_7 5 \cdot \log_3 7}{\log_2 \sqrt{6}} + \frac{1}{\log_9 \sqrt{6}}$. | 4 |
Let $D$ be a point on side $[BC]$ of triangle $ABC$ such that $[AD]$ is an angle bisector, $|BD|=4$, and $|DC|=3$. Let $E$ be a point on side $[AB]$ and different than $A$ such that $m(\widehat{BED})=m(\widehat{DEC})$. If the perpendicular bisector of segment $[AE]$ meets the line $BC$ at $M$, what is $|CM|$? | 9 |
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n$. Determine how many values of $n$ satisfy the equation \( n + S(n) + S(S(n)) = 2007 \). | 4 |
A circle with radius 2 is centered at $(2,0)$. Another circle with radius 1 is centered at $(5,0)$. A line is tangent to both circles at points located in the first quadrant. The original answer is in the form k\sqrt{m}. Please find the value of k + m, where k and m are integers. | 4 |
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. Instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he wil... | 13 |
How many ten-digit positive integers with distinct digits are multiples of $11111$? Provide your answer as an integer. | 3456 |
A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle $ABC$ of side length $3$. Find the perimeter of the remaining quadrilateral. | 8 |
What is the maximum area of an isosceles trapezoid that has legs of length \(1\) and one base twice as long as the other? The maximum area can be expressed in the form \(\frac{k}{m}\), where \(k\) and \(m\) are coprime integers. Please find the value of \(k + m\). | 5 |
$ABC$ and $A'B'C'$ are equilateral triangles with parallel sides and the same center, as shown in the figure. The distance between side $BC$ and side $B'C'$ is $\frac{1}{6}$ the altitude of $\triangle ABC$. Find the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$. The answer should be in the form... | 5 |
Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a - (b - (c - (d + e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a$, $b$, $c$, and... | 3 |
The triangle \(ABC\) has vertices such that \(AB = 3\), \(BC = 4\), and \(AC = 5\). The inscribed circle is tangent to \(AB\) at \(C'\), \(BC\) at \(A'\), and \(AC\) at \(B'\). Find the ratio between the area of the triangles \(A'B'C'\) and \(ABC\). The answer should be in the form \(\frac{k}{m}\). Please calculate the... | 6 |
In $\triangle ABC$, $AB = 8$, $BC = 7$, $CA = 6$, and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Find the length of $PC$. | 9 |
Let $S$ be the set of all those 2007-place decimal integers $\overline{2s_1a_2a_3 \ldots a_{2006}}$ which contain an odd number of digit $9$ in each sequence $a_1, a_2, a_3, \ldots, a_{2006}$. The cardinal number of $S$ is given in the form \frac{1}{x}(y^{2006} - z^{2006}). Please calculate the value of x + y + z. | 20 |
In the equation \(\frac{x(x - 1) - (m + 1)}{(x - 1)(m - 1)} = \frac{x}{m}\), find the value of \(m\) such that the roots are equal. The original answer is in the form \(-\frac{a}{b}\). Please provide the value of \(a + b\). | 3 |
A particle of mass $m$ moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$. The fractional energy transfer is the absolute value of the final kinetic energy ... | 2 |
A sphere with center \(O\) has a radius of \(6\). A triangle with sides of lengths \(15\), \(15\), and \(24\) is situated in space such that each of its sides is tangent to the sphere. The distance between \(O\) and the plane determined by the triangle is in the form \(m\sqrt{n}\). Find the value of \(m + n\). | 7 |
A vertical mass-spring oscillator is displaced $2.0\, \text{cm}$ from equilibrium. The $100\, \text{g}$ mass passes through the equilibrium point with a speed of $0.75\, \text{m/s}$. What is the spring constant of the spring? If the spring constant is $k\, \text{N/m}$, what is the value of $k$? | 140 |
If one can find a student with at least $k$ friends in any class which has 21 students such that at least two of any three of these students are friends, what is the largest possible value of $k$? | 10 |
If $ABCD$ is a $2 \times 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, find the area of quadrilateral $BEIH$. The original answer is in \(\frac{k}{m}\) format, ple... | 22 |
A frog is at the center of a circular shaped island with radius $r$. The frog jumps $\frac{1}{2}$ meters at first. After the first jump, it turns right or left at exactly $90^\circ$, and it always jumps one half of its previous jump. After a finite number of jumps, what is the least $r$ that ensures the frog can never ... | 8 |
Find the value of $2^{-(2k+1)} - 2^{-(2k-1)} + 2^{-2k}$. The original answer is in the form $-x*2^{-(y*z+k)}$, where x, y, and z are integers. Please provide the sum of x, y, and z. | 4 |
Find the value of $\left(1+\cos \frac{\pi}{5}\right)\left(1+\cos \frac{3 \pi}{5}\right)$. The original answer is in the form of \frac{k}{m}, please provide the value of k + m. | 9 |
$a_{n}=\left(n-\frac{1}{2}\right)\left(\frac{9}{10}\right)^{n}, b_{n}=a_{n+1}-a_{n}$. Find the minimum value of $\left\{\frac{b_{n+1}}{b_{n}}\right\}$ and multiply it by 10. What is the result? | -9 |
Pascal's triangle is an array of positive integers, as illustrated in the figure. The first row consists of $1$, the second row contains two $1$'s, and each row starts and ends with $1$. For any given row, the $k^\text{th}$ number, when it is not $1$, is the sum of the $k^\text{th}$ and $(k-1)^\text{th}$ numbers from t... | 12 |
A truck is initially moving at velocity $v$. The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force $F$ to the truck. The truck rolls a distance $x$ before coming to a stop, and the time it takes to stop is $t$. Find the expression that represents the initial kinetic energ... | 2 |
Let \(f(x)\) be a real-valued function satisfying \(af(x) + bf(-x) = px^2 + qx + r\). Here, \(a\) and \(b\) are distinct real numbers, and \(p\), \(q\), \(r\) are non-zero real numbers. Determine the condition under which \(f(x) = 0\) will have real solutions. The original answer is in the form \(\left(\frac{k(a+b)}{m(... | 2 |
How many complex numbers satisfy the equation $z^{5}=\overline{z}$, where $\overline{z}$ is the conjugate of the complex number $z$? | 7 |
For how many real values of $x$ is $\sqrt{120 - \sqrt{x}}$ an integer? | 11 |
If \( f(x) = \frac{x(x-1)}{2} \), then what is the expression for \( f(x+2) \)? The original answer is in the form \(\frac{(x+k)f(x+m)}{n*x}\). Please find the value of k + m + n. | 4 |
What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation? | 755 |
A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM, the machine has completed one-third of the day's job. At what time will the doughnut machine complete the job? If the answer is in the format "x:xx PM", please provide the numerical value of x:xx without the colon. | 430 |
Suppose $a$ and $b$ are digits which are not both nine and not both zero. If the repeating decimal $0.\overline{AB}$ is expressed as a fraction in lowest terms, find how many possible denominators there are. | 5 |
On an examination of $n$ questions, a student answers correctly $15$ of the first $20$. Of the remaining questions, he answers one third correctly. All the questions have the same credit. If the student's mark is $50\%$, how many different values of $n$ can there be? | 1 |
Given $0 \le x_0 <1$, let \[ x_n= \begin{cases} 2x_{n-1} & \text{if}\ 2x_{n-1} <1 \\ 2x_{n-1}-1 & \text{if}\ 2x_{n-1} \ge 1 \end{cases} \] for all integers $n>0$. Find the number of values of $x_0$ for which $x_0=x_5$. | 31 |
In the adjoining figure, $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, find the area of $CMN$ in square inches. The original answer is in the form \(k\sqrt{n} - m\), please give the value of k + n + m. | 8 |
A list of five positive integers has a mean of $12$ and a range of $18$. The mode and median are both $8$. Find how many different values are possible for the second largest element of the list. | 6 |
A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to \((1,0)\). Then it makes a \(90^\circ\) counterclockwise turn and travels \(\frac{1}{2}\) a unit to \(\left(1, \frac{1}{2} \right)\). If it continues in this fashion, each time making a \(90^\circ\) degree turn co... | 16 |
A function \(f\left(a \frac{b}{c}\right)\) for a simplified mixed fraction \(a \frac{b}{c}\) returns \(\frac{a + b}{c}\). For instance, \(f\left(2 \frac{5}{7}\right) = 1\) and \(f\left(\frac{4}{5}\right) = \frac{4}{5}\). Find the sum of the three smallest positive rational \(x\) where \(f(x) = \frac{2}{9}\). The origin... | 95 |
Let $f(x) = x^2 + 6x + 1$, and let $R$ denote the set of points $(x,y)$ in the coordinate plane such that \[ f(x) + f(y) \leq 0 \text{ and } f(x) - f(y) \leq 0 \] Find the integer that is closest to the area of $R$. | 25 |
Two circles of radius 1 are to be constructed as follows. The center of circle \(A\) is chosen uniformly and at random from the line segment joining \((0,0)\) and \((2,0)\). The center of circle \(B\) is chosen uniformly and at random, and independently of the first choice, from the line segment joining \((0,1)\) to \(... | 14 |
A convex polyhedron $Q$ has vertices $V_1, V_2, \ldots, V_n$, and 100 edges. The polyhedron is cut by planes $P_1, P_2, \ldots, P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $... | 300 |
Evaluate the following sum: \[ \binom{n}{1} \sin(a) + \binom{n}{2} \sin(2a) + \ldots + \binom{n}{n} \sin(na) \] Find the value of the expression in terms of \( n \) and \( a \). The answer is in the form \( k \cdot m^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{n \cdot a}{2}\right) \). Please provide the sum of k ... | 3 |
Let $K$ be the number of sequences $A_1$, $A_2$, \dots, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of \{1, 2, 3, \dots, 10\}, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, \{\}, \{5, 7\}, \{2, 5, 7\}, \{2, 5, 7\}, \{2, 5, 6... | 5 |
On a certain math exam, $10\%$ of the students got 70 points, $25\%$ got 80 points, $20\%$ got 85 points, $15\%$ got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam? | 1 |
A softball team played ten games, scoring $1, 2, 3, 4, 5, 6, 7, 8, 9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? | 45 |
How many pairs of positive integers $(a,b)$ are there such that the roots of the polynomial $x^2-ax-b$ are not greater than $5$? Provide the number of such pairs. | 50 |
Find the sum of the digits of the largest even three-digit number (in base ten representation) which is not changed when its units and hundreds digits are interchanged. | 25 |
Regular hexagon \(ABCDEF\) has an area of \(1\). Starting with edge \(AB\) and moving clockwise, a new point is drawn exactly one half of the way along each side of the hexagon. For example, on side \(AB\), the new point, \(G\), is drawn so that \(AG = \frac{1}{2} AB\). This forms hexagon \(GHIJKL\), as shown. What is ... | 7 |
In triangle $ABC$, lines $CE$ and $AD$ are drawn such that \( \frac{CD}{DB} = \frac{3}{1} \) and \( \frac{AE}{EB} = \frac{3}{2} \). Let \( r = \frac{CP}{PE} \), where $P$ is the intersection point of $CE$ and $AD$. Find the value of \( r \). | 5 |
If $a, b,$ and $c$ are digits for which \[ \begin{array}{cccc} & 7 & a & 2 \\ - & 4 & 8 & b \\ \hline & c & 7 & 3 \end{array} \] then find the value of $a + b + c$. | 17 |
How many ordered pairs of integers $(x,y)$ are there such that $2011y^2 = 2010x + 3$? | 0 |
In base $R_1$, the expanded fraction $F_1$ becomes $0.373737\ldots$, and the expanded fraction $F_2$ becomes $0.737373\ldots$. In base $R_2$, fraction $F_1$, when expanded, becomes $0.252525\ldots$, while fraction $F_2$ becomes $0.525252\ldots$. Find the sum of $R_1$ and $R_2$, each written in base ten. | 19 |
Fermat determines that since his final exam counts as two tests, he only needs to score a $28$ on it for his test average to be $70$. If he gets a perfect $100$ on the final exam, his average will be $88$. What is the lowest score Fermat can receive on his final and still have an average of $80$? | 68 |
Let's write \(p\), \(q\), and \(r\) as three distinct prime numbers, where 1 is not a prime. Find the smallest positive perfect cube that leaves \(n = pq^2r^4\) as a divisor. The original answer is in the form \((p^a q^b r^c)^3\), please give the value of a + b + c. | 4 |
How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order? Provide your answer as an integer. | 34 |
In the adjoining figure, $AB$ and $BC$ are adjacent sides of square $ABCD$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $BC$. The lines $AN$ and $CM$ intersect at $O$. Determine the ratio of the area of quadrilateral $AOCD$ to the area of square $ABCD$. The original answer is in \(\frac{k}{m}\) format, p... | 5 |
What is the sum of real roots of the equation $x^4-8x^3+13x^2 -24x + 9 = 0$? | 7 |
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. What is the area of the hexagon? If the area is expressed as $x\sqrt{3}$, what is the value of $x$? | 25 |
In triangle $ABC$, $AB = AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP = PC = CB$, then if the answer is $x^\circ$, what is the value of $x$? | 36 |
If the (convex) area bounded by the x-axis and the lines $y=mx+4$, $x=1$, and $x=4$ is $7$, find the value of $m$. The original answer is in \frac{k}{m} format, please give the value of k + m. | 1 |
What is the perimeter of the smallest rectangle with integer side lengths that fits three non-overlapping squares with areas $4$, $9$, and $16$? | 24 |
Al and Barb start their new jobs on the same day. Al's schedule is $3$ work-days followed by $1$ rest-day. Barb's schedule is $7$ work-days followed by $3$ rest-days. On how many of their first $1000$ days do both have rest-days on the same day? | 100 |
Find the last 5 digits of the number $M = 5^{2010}$. | 65625 |
A mass is attached to an ideal spring. At time \(t = 0\) the spring is at its natural length and the mass is given an initial velocity; the period of the ensuing (one-dimensional) simple harmonic motion is \(T\). At what time \(t\) is the power delivered to the mass by the spring first a maximum, in terms of \(T\)? The... | 9 |
Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? The original ... | 13 |
First, $a$ is chosen at random from the set $\{1, 2, 3, \ldots, 99, 100\}$, and then $b$ is chosen at random from the same set. Find the probability that the integer $3^a + 7^b$ has a units digit of $8$. The original answer is in the form of a common fraction \(\frac{k}{m}\). Please provide the value of k + m. | 19 |
Dave's Amazing Hotel has $3$ floors. If you press the up button on the elevator from the $3$rd floor, you are immediately transported to the $1$st floor. Similarly, if you press the down button from the $1$st floor, you are immediately transported to the $3$rd floor. Dave gets in the elevator at the $1$st floor and ran... | 803 |
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? Write your answer as an integer. | 121 |
A square $ABCD$ with sides of length 1 is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points $E, F, G$ where $E$ is the midpoint of $BC$, $F$ and $G$ are on $AB$ and $CD$, respectively. They are positioned such that $AF < FB$, $DG < GC$, and $F$... | 11 |
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a negligible height above the equator, find the best estimate of the number of hours of flight. | 50 |
Find the number of distinct integral solutions of $x^{4} + 2x^{3} + 3x^{2} - x + 1 \equiv 0 \pmod{30}$ where $0 \leq x < 30$. | 1 |
What is the minimum value of \((x^2+2x+8-4\sqrt{3})\cdot(x^2-6x+16-4\sqrt{3})\) where \(x\) is a real number? The answer should be in the form \(a - b\sqrt{3}\). Please find the value of \(a + b\). | 176 |
A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and \[ a_n = \frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}} \] for all $n \geq 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the value of $p+q$. | 8078 |
Circles with centers \(O\) and \(P\) have radii 2 and 4, respectively, and are externally tangent. Points \(A\) and \(B\) are on the circle centered at \(O\), and points \(C\) and \(D\) are on the circle centered at \(P\), such that \(\overline{AD}\) and \(\overline{BC}\) are common external tangents to the circles. Fi... | 26 |
$(a_n)_{n = 1}^\infty$ is defined on real numbers with $a_n \neq 0$, $a_n a_{n + 3} = a_{n + 2} a_{n + 5}$ and $a_1 a_2 + a_3 a_4 + a_5 a_6 = 6$. Find the value of $a_1 a_2 + a_3 a_4 + \cdots + a_{41} a_{42}$. | 42 |
Let $d$ be one of the common tangent lines of externally tangent circles $k_1$ and $k_2$. $d$ touches $k_1$ at $A$. Let $[AB]$ be a diameter of $k_1$. The tangent from $B$ to $k_2$ touches $k_2$ at $C$. If $|AB|=8$ and the diameter of $k_2$ is $7$, then what is the length of $|BC|$? | 8 |
The Minnesota Twins face the New York Mets in the 2006 World Series. Assuming the two teams are evenly matched (each has a \(0.5\) probability of winning any game), find the probability that the World Series (a best of 7 series of games which lasts until one team wins four games) will require the full seven games to de... | 21 |
Let \(f(x) = \frac{x+1}{x-1}\). Then for \(x^2 \neq 1\), find the expression for \(f(-x)\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 2 |
A circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. If the measure of the angle subtended at the center of the circle by the shortest side is $x^\circ$, what is the value of $x$? | 60 |
A point \(P\) lies in the same plane as a given square with side length \(1\). The vertices of the square, taken counterclockwise, are labeled as \(A\), \(B\), \(C\), and \(D\). The distances from point \(P\) to the vertices \(A\), \(B\), and \(C\) are \(u\), \(v\), and \(w\), respectively. Determine the greatest dista... | 4 |
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? Provide your answer in the form of \(\frac{m}{n}\) and find the value of m + n. | 20922789888256 |
Suppose that \(\cos(3x) + 3\cos(x) = -2\). Find the value of \(\cos(2x)\). The original answer is in the form \(\sqrt[3]{a} - b\). Please provide the value of a + b. | 3 |
At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, find the number of campers who are in exactly two classes. | 12 |
If Scott rolls four fair six-sided dice, find the probability that he rolls more 2's than 1's. The probability is given in the form \(\frac{m}{n}\). Please calculate the value of m + n. | 1717 |
If \(\frac{x}{x-1} = \frac{y^2+2y-1}{y^2-2y-2}\), find the value of \(x\). The original answer can be expressed in the form \(ay^2 + by + c\). Please find the value of \(a + b + c\). | 2 |
How many real roots of the equation \[x^2 - 18\lfloor x \rfloor + 77 = 0\] are not integer, where \(\lfloor x \rfloor\) denotes the greatest integer not exceeding the real number \(x\)? | 3 |
If \( |x - \log y| = x + \log y \) where \( x \) and \( \log y \) are real, find the expression that represents the relationship between \( x \) and \( y \). The original answer is in the form \( mx(y-n)=p \). Please provide the value of \( m+n+p \). | 2 |
The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. What is the sum of $d$ and $e$? \[ \begin{array}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0 \\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \\ \end{array} \] | 8 |
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