prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Let $f(x) = \sqrt{\sin^4 x + 4\cos^2 x} - \sqrt{\cos^4 x + 4\sin^2 x}$. Find an equivalent form of $f(x)$ in the format $k\cos mx$. Please provide the value of $k + m$. | 3 |
How many integer pairs \((x, y)\) are there such that: \[0 \leq x < 165, \quad 0 \leq y < 165 \text{ and } y^2 \equiv x^3 + x \pmod{165}?\] Find the number of such integer pairs. | 99 |
In a class, some pupils learn German, while others learn French. The number of girls learning French and the number of boys learning German total to 16. There are 11 pupils learning French, and there are 10 girls in the class. Besides the girls learning French, there are 16 pupils. How many pupils are there in the clas... | 23 |
Let $A$, $B$ be the number of digits of $2^{1998}$ and $5^{1998}$ in the decimal system. Find the value of $A + B$. | 1999 |
A digital watch displays hours and minutes with $\text{AM}$ and $\text{PM}$. What is the largest possible sum of the digits in the display? | 23 |
What is the probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ white, and $2$ blue balls? Provide your answer in the form \(\frac{k}{m}\), where the fraction is in simplest form. Calculate the value of \(k + m\). | 31 |
Find the number of solution-pairs in the positive integers of the equation $3x + 5y = 501$. | 33 |
In a ten-mile race, First beats Second by $2$ miles and First beats Third by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third? Provide your answer in the form of a fraction \frac{k}{m}, and then calculate the value of k + m. | 7 |
In $\triangle ABC$, $AB = 10$, $AC = 8$, and $BC = 6$. Circle $P$ is the circle with the smallest radius that passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$, of circle $P$ with sides $AC$ and $BC$, respectively. If the length of segment $QR$ is $x$, what is t... | 48 |
Albert, Bob, and Charlie are each thinking of a number. Albert's number is one more than twice Bob's. Bob's number is one more than twice Charlie's, and Charlie's number is two more than twice Albert's. What number is Albert thinking of? The original answer is in \frac{k}{m} format, please give the value of k + m. | -4 |
Pikachu, Charmander, and Vulpix are three of the four equally-skilled players in a Pokemon bracket tournament. Because they are equally skilled, whenever any two of the players battle, they are equally likely to win. In the bracket tournament, the four players are randomly paired into two rounds, each round consisting ... | 9 |
Vessel $A$ contains liquids $X$ and $Y$ in the ratio $X:Y = 8:7$. Vessel $B$ holds a mixture of $X$ and $Y$ in the ratio $X:Y = 5:9$. In what ratio should you mix the liquids from both vessels to achieve a mixture with the ratio $X:Y = 1:1$? Express your answer in the form of $a:b$, and provide the value of $a + b$. | 37 |
Let \( ABC \) be an acute triangle and \( O \) be its circumcenter. Let \( D \) be the midpoint of \( [AB] \). The circumcircle of \( \triangle ADO \) meets \( [AC] \) at \( A \) and \( E \). If \( |AE|=7 \), \( |DE|=8 \), and \( m(\widehat{AOD}) = 45^\circ \), what is the area of \( \triangle ABC \) in the form \( x \... | 56 |
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of $50$ miles per hour, and Joe can run at a speed of $10$ miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe... | 144 |
Let $(1+\sqrt{2})^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers. Find the greatest common divisor of $b$ and $81$. | 3 |
Two fair dice, each with at least 6 faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of 7 is $\frac{3}{4}$ of the probability of rolling a sum of 10, and the probability of rolling a sum of 12 is $\frac{1}{12... | 17 |
Find the sum of all positive integers $n$ such that \(\frac{n^3+8n^2+8n+80}{n+7}\) is an integer. | 66 |
The addition below is incorrect. What is the largest digit that can be changed to make the addition correct? \[ \begin{array}{cccc} & 6 & 4 & 1 \\ & 8 & 5 & 2 \\ + & 9 & 7 & 3 \\ \hline 2 & 4 & 5 & 6 \end{array} \] Provide the largest digit that can be changed. | 7 |
What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{123,456,789}{2^{26}\cdot 5^4}$ as a decimal? | 26 |
Let $E$ be a point outside of square $ABCD$. If the distance of $E$ to $AC$ is $6$, to $BD$ is $17$, and to the nearest vertex of the square is $10$, what is the area of the square? | 162 |
Let $f(x)=x^3+7x^2+9x+10$. Find the value of $p$ such that for every integer $a,b$, the statement \[ f(a) \equiv f(b) \ (\text{mod } p) \Rightarrow a \equiv b \ (\text{mod } p) \] is satisfied. | 11 |
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, how many campers are in exactly two classes? | 12 |
The volume of a pyramid whose base is an equilateral triangle with a side length of 6, and whose other edges are each of length $\sqrt{15}$, is _______. | 9 |
Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Find the value of $n$. | 1 |
If $\frac{m}{n} = \frac{4}{3}$ and $\frac{r}{t} = \frac{9}{14}$, find the value of $\frac{3mr - nt}{4nt - 7mr}$. The original answer is in \frac{a}{b} format, please give the value of a + b. | 3 |
Let \(M\) be the intersection of the diagonals \(AC\) and \(BD\) of cyclic quadrilateral \(ABCD\). If \(|AB|=5\), \(|CD|=3\), and \(m(\widehat{AMB}) = 60^\circ\), find the circumradius of the quadrilateral. If the circumradius is expressed in the form \(\frac{a\sqrt{b}}{c}\), where \(a\), \(b\), and \(c\) are integers,... | 13 |
If $(x^2+1)(y^2+1)+9=6(x+y)$ where $x,y$ are real numbers, find the value of $x^2+y^2$. | 7 |
In the adjoining figure, triangle $ABC$ is such that $AB = 4$ and $AC = 8$. If $M$ is the midpoint of $BC$ and $AM = 3$, what is the length of $BC$? Provide your answer in the form $a\sqrt{b}$, where $a$ and $b$ are integers, and find the value of $a + b$. | 33 |
A polynomial $P$ is called *level* if it has integer coefficients and satisfies the conditions $P(0) = P(2) = P(5) = P(6) = 30$. Find the largest positive integer $d$ such that for any level polynomial $P$, $d$ is a divisor of $P(n)$ for all integers $n$. | 2 |
Evaluate the following expression: \[ 1+\frac{1+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}{2+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2... | 8 |
We have a half-circle with endpoints $A$ and $B$ and center $S$. The points $C$ and $D$ lie on the half-circle such that $\angle BAC = 20^\circ$ and the lines $AC$ and $SD$ are perpendicular to each other. If the angle between the lines $AC$ and $BD$ is $x^\circ$, what is the value of $x$? | 55 |
When $(a-b)^n$, $n \geq 2$, $ab \neq 0$, is expanded by the binomial theorem, it is found that, when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. The original answer is in the form of $mk+n$. Please find the value of $m+n$. | 3 |
A dealer bought $n$ radios for $d$ dollars, where $d$ is a positive integer. He contributed two radios to a community bazaar at half their cost. The rest he sold at a profit of $8$ dollars on each radio sold. If the overall profit was $72$ dollars, find the least possible value of $n$ for the given information. | 12 |
The diameter \(\overline{AB}\) of a circle with radius \(2\) is extended to a point \(D\) outside the circle such that \(BD=3\). Point \(E\) is chosen so that \(ED=5\) and the line \(ED\) is perpendicular to the line \(AD\). Segment \(\overline{AE}\) intersects the circle at point \(C\) between \(A\) and \(E\). Find th... | 177 |
A watch loses $2\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. Let $n$ be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows $9$ A.M. on March 21, what is the value of $n$ in the form of an improper fraction \frac{k}{m}? Please p... | 359 |
If $3a = 1 + \sqrt{2}$, find the largest integer not exceeding $9a^4 - 6a^3 + 8a^2 - 6a + 9$. | 10 |
Inside square \(ABCD\) with side \(s\), quarter-circle arcs with radii \(s\) and centers at \(A\) and \(B\) are drawn. These arcs intersect at point \(X\) inside the square. The distance from \(X\) to side \(CD\) is in the form \(\frac{k}{m}s(n-\sqrt{p})\). Please calculate the value of \(k + m + n + p\). | 8 |
Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(5681) = 6 + 8 = 14$. Find the value of $E(1) + E(2) + E(3) + \cdots + E(100)$. | 400 |
If a planet of radius $R$ spins with an angular velocity $\omega$ about an axis through the North Pole, find the ratio of the normal force experienced by a person at the equator to that experienced by a person at the North Pole. Assume a constant gravitational field $g$ and that both people are stationary relative to t... | 3 |
Two obvious approximations to the length of the perimeter of the ellipse with semi-axes $a$ and $b$ are $\pi(a+b)$ and $2 \pi(a b)^{1/2}$. Which one comes nearer the truth when the ratio $b / a$ is very close to 1? Write your answer in the form of $\pi(x+y)$, where $x$ and $y$ are integers. Find the value of $x + y$. | 2 |
If the ratio of the legs of a right triangle is $1:2$, then what is the ratio of the corresponding segments of the hypotenuse made by a perpendicular from the vertex? Express your answer in the form $x:y$, and calculate the value of x + y. | 5 |
If $b$ men take $c$ days to lay $f$ bricks, find the number of days it will take $c$ men working at the same rate to lay $b$ bricks. The original answer is in the format \(\frac{x*b^2}{y*f}\). Please provide the value of x + y. | 2 |
On the same side of a straight line, three circles are drawn as follows: a circle with a radius of $4$ inches is tangent to the line. The other two circles are equal, and each is tangent to the line and to the other two circles. What is the radius of the equal circles in inches? | 16 |
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest... | 7 |
From a point $P$ outside of a circle with center $O$, tangent segments $\overline{PA}$ and $\overline{PB}$ are drawn. If $\frac{1}{|\overline{OA}|^2} + \frac{1}{|\overline{PA}|^2} = \frac{1}{16}$, find the length of $|\overline{AB}|$. | 8 |
There are lily pads in a row numbered 0 to 11, in that order. There are predators on lily pads 3 and 6, and a morsel of food on lily pad 10. Fiona the frog starts on pad 0, and from any given lily pad, has a \(\frac{1}{2}\) chance to hop to the next pad, and an equal chance to jump 2 pads. Find the probability that Fio... | 271 |
The solution of the equation \(7^{x+7} = 8^x\) can be expressed in the form \(x = \log_b 7^7\). The original answer is in the format \(\frac{k}{m}\). Please find the value of \(k + m\). | 15 |
Let \(M\) be the intersection of diagonals of the convex quadrilateral \(ABCD\), where \(m(\widehat{AMB})=60^\circ\). Let the points \(O_1\), \(O_2\), \(O_3\), \(O_4\) be the circumcenters of the triangles \(ABM\), \(BCM\), \(CDM\), \(DAM\), respectively. The original answer is in the form \(\frac{k}{m}\), where k and ... | 5 |
A person standing on the edge of a fire escape simultaneously launches two apples, one straight up with a speed of $7\,\text{m/s}$ and the other straight down at the same speed. How far apart are the two apples $2$ seconds after they were thrown, assuming that neither has hit the ground? If the answer is $x\,\text{m}$,... | 28 |
Equilateral triangle \(ABC\) has been creased and folded so that vertex \(A\) now rests at \(A'\) on \(\overline{BC}\) as shown. If \(BA' = 1\) and \(A'C = 2\), then find the length of the crease \(\overline{PQ}\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\), please give the value of k + m + n. | 48 |
Given a right triangle $ABC$ with legs $BC=3$ and $AC=4$, find the length of the shorter angle trisector from $C$ to the hypotenuse. Provide your answer in the form \(\frac{k\sqrt{m}-n}{p}\), and then find the value of k + m + n + p. | 72 |
A rise of $600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. Find the additional length of track required to reduce the grade from $3\%$ to $2\%$. If the answer is $x$ ft., what is the value of $x$? | 10000 |
What is the maximum value of the function \(\frac{1}{|x+1| + |x+2| + |x-3|}?\) The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 6 |
Find the coefficient of $x^7$ in the polynomial expansion of \[ (1+2x-x^2)^4 \]. | -8 |
How many integer quadruples \(a, b, c, d\) are there such that \(7\) divides \(ab - cd\) where \(0 \leq a, b, c, d < 7\)? Find the total number of such quadruples. | 385 |
A farmer bought $749$ sheep. He sold $700$ of them for the price paid for the $749$ sheep. The remaining $49$ sheep were sold at the same price per head as the other $700$. Based on the cost, what is the percent gain on the entire transaction? Provide your answer as a number. | 7 |
A palindrome, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x + 32$ are three-digit and four-digit palindromes, respectively. Find the sum of the digits of $x$. | 24 |
A barn with a flat roof is rectangular in shape, $10$ yd. wide, $13$ yd. long, and $5$ yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. What is the total number of square yards to be painted? | 590 |
Chord $[AD]$ is perpendicular to the diameter $[BC]$ of a circle. Let $E$ and $F$ be the midpoints of the arcs $AC$ and $CD$, respectively. If $AD \cap BE = \{G\}$, $AF \cap BC = \{H\}$ and $m(AC) = \alpha$, find the measure of angle $BHC$ in terms of $\alpha$. The answer is in the form $x^\circ$, where $x = k - \frac{... | 91 |
Which value of $x$ satisfies the equation $\sqrt[3]{6+\sqrt{x}} + \sqrt[3]{6-\sqrt{x}} = \sqrt[3]{3}$? | 45 |
With a rational denominator, express \(\frac{\sqrt{2}}{\sqrt{2} + \sqrt{3} - \sqrt{5}}\) in the form of a simplified fraction. The original answer is in the form \(\frac{k + m\sqrt{6} + n\sqrt{15}}{p}\). Please find the value of k + m + n + p. | 11 |
Find the number of digits in the number \( N=2^{12}\times 5^8 \). | 10 |
An \(8 \times 2\sqrt{2}\) rectangle has the same center as a circle of radius \(2\). Find the area of the region common to both the rectangle and the circle. The original answer is in the form \(k\pi + m\). Please provide the value of \(k + m\). | 6 |
Consider two concentric circles with radii \(17\) and \(19\). The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle? The original answer is in the form \(k\sqrt{m}\). Please find the value of \(k + m\). | 14 |
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. Find the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,... | 359 |
Three digits are selected at random and without replacement from the set of digits $0$ through $9$. Find the probability that the three digits can be arranged to form a multiple of $5$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 23 |
If \(b > 1\), \(x > 0\) and \((2x)^{\log_b 2} - (3x)^{\log_b 3} = 0\), find the value of \(x\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 7 |
Determine the number of digits in $4^{16} \times 5^{25}$ when written in the usual base 10 form. | 28 |
$A$ gives $B$ as many cents as $B$ has and $C$ as many cents as $C$ has. Similarly, $B$ then gives $A$ and $C$ as many cents as each then has. $C$, similarly, then gives $A$ and $B$ as many cents as each then has. If each finally has $16$ cents, with how many cents does $A$ start? | 26 |
How many real pairs $(x, y)$ are there such that \[ x^2 + 2y = 2xy \\ x^3 + x^2y = y^2 \] Find the number of real pairs $(x, y)$ that satisfy the equations. | 3 |
For every real number $x$, let $\lfloor x \rfloor$ denote the greatest integer not exceeding $x$, and let: \[f(x) = \lfloor x \rfloor (2014^{x-\lfloor x \rfloor} - 1).\] The set of all numbers $x$ such that $1 \leq x < 2014$ and $f(x) \leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those int... | 1 |
How many triples of positive integers $(a,b,c)$ are there such that $a!+b^3 = 18+c^3$? | 1 |
A \(4 \times 4 \times h\) rectangular box contains a sphere of radius \(2\) and eight smaller spheres of radius \(1\). The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. The original answer is in the form \(k + m\sqrt{n}\). Please find the va... | 11 |
For how many primes $p$, does there exist an integer $m$ such that $m^3+3m-2 \equiv 0 \pmod{p}$ and $m^2+4m+5 \equiv 0 \pmod{p}$? Write your answer as an integer. | 2 |
Let $a$ and $b$ satisfy the conditions \( \begin{cases} a^3 - 6a^2 + 15a = 9 \\ b^3 - 3b^2 + 6b = -1 \end{cases} \). Find the value of \((a - b)^{2014}\). | 1 |
Two different positive numbers \(a\) and \(b\) each differ from their reciprocals by 1. The original answer is in the form \(\sqrt{k}\). Please find the value of \(k\). | 5 |
If \(a^2 + b^2 + c^2 + d^2 - ab - bc - cd - d + \frac{2}{5} = 0\) where \(a, b, c, d\) are real numbers, find the value of \(a\). The answer should be in the form \(\frac{k}{m}\), please provide the value of \(k + m\). | 6 |
Nine congruent spheres are packed inside a unit cube such that one sphere is centered at the cube's center, and each of the remaining spheres is tangent to the central sphere and three faces of the cube. What is the radius of each sphere? Provide your answer in the form of \(\frac{k\sqrt{n}-m}{p}\), where \(k\), \(m\),... | 10 |
The expression $1 - \frac{1}{1 + \sqrt{3}} + \frac{1}{1 - \sqrt{3}}$ equals what value? The answer should be in the form k - \sqrt{m}. Please find the value of k + m. | 4 |
Let $D$ be a point on side $[BC]$ of triangle $ABC$ where $|BC|=11$ and $|BD|=8$. The circle passing through the points $C$ and $D$ touches $AB$ at $E$. Let $P$ be a point on the line which is passing through $B$ and is perpendicular to $DE$. If $|PE|=7$, then what is $|DP|$? | 5 |
A thin, uniform rod has mass \(m\) and length \(L\). Let the acceleration due to gravity be \(g\). The rotational inertia of the rod about its center is \(md^2\). The rod is suspended from a distance \(kd\) from the center and undergoes small oscillations with an angular frequency \(\beta \sqrt{\frac{g}{d}}\). Find the... | 3 |
Find the total number of solutions of the equation $xyz=2520$. | 540 |
If \( a \pm bi \) (\( b \neq 0 \)) are imaginary roots of the equation \( x^3 + qx + r = 0 \), where \( a \), \( b \), \( q \), and \( r \) are real numbers, find \( q \) in terms of \( a \) and \( b \). The original answer is in the form \( kb^2 - ma^2 \). Please calculate the value of k + m. | 4 |
If $ \log 2 = 0.3010$ and $ \log 3 = 0.4771$, find the value of $ x$ when $ 3^{x+3} = 135$. If the answer is $y$, what is the value of $100y$? | 147 |
A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars would increase sales. If the diameter of the jars is increased by $25\%$ without altering the volume, by what percent must the height be decreased? If the answer is $y\%$, what is the value of $y$? | 36 |
Let \(P\) be an arbitrary point inside \(\triangle ABC\) with sides \(3, 7, 8\). Find the probability that the distance of \(P\) to at least one vertex of the triangle is less than \(1\). The original answer is in the form \(\frac{\pi}{k}\sqrt{m}\). Please find the value of \(k + m\). | 39 |
Three balls marked 1, 2, and 3, are placed in an urn. One ball is drawn, its number is recorded, then the ball is returned to the urn. This process is repeated twice more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is 6, find the probability that the ball numbered 2... | 8 |
If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A, B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\o... | 29 |
Let $M$ be the least common multiple of all the integers $10$ through $30$, inclusive. Let $N$ be the least common multiple of $M$, $32$, $33$, $34$, $35$, $36$, $37$, $38$, $39$, and $40$. What is the value of $\frac{N}{M}$? | 74 |
Line segment $\overline{AB}$ is a diameter of a circle with $AB=24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle{ABC}$ traces out a closed curve missing two points. To the nearest positive integer, find the area of the region ... | 50 |
What is the sum \( \frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot 3 \cdot 4} + \cdots + \frac{1}{1996 \cdot 1997 \cdot 1998} \)? Provide your answer in the form of \(\frac{k}{m} - \frac{n}{p \cdot q \cdot r}\). Please find the value of \(k + m + n + p + q + r\). | 4003 |
Let $C_1$ and $C_2$ be circles defined by: \[ (x - 10)^2 + y^2 = 36 \] and \[ (x + 15)^2 + y^2 = 81 \] respectively. Find the length of the shortest line segment $\overline{PQ}$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$. | 20 |
For how many integers $n$, does the equation system \[\begin{array}{rcl} 2x+3y &=& 7 \\ 5x + ny &=& n^2 \end{array}\] have a solution over integers? Write your answer as an integer. | 8 |
A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a > b > 0$. The area of $PQRS$ is $16$. What is the value of $a+b$? | 4 |
A rhombus is formed by two radii and two chords of a circle whose radius is $16$ feet. What is the area of the rhombus in square feet? The original answer is in the form k\sqrt{m}, please give the value of k + m. | 131 |
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1 \le x \le 4$ and $1 \le y \le 4$? Provide your answer as an integer. | 516 |
Let $a$, $b$, $c$, $d$, and $e$ be distinct integers such that \[ (6 - a)(6 - b)(6 - c)(6 - d)(6 - e) = 45. \] Find the value of $a + b + c + d + e$. | 25 |
If \( t = \frac{1}{1 - \sqrt[4]{2}} \), find the expression that \( t \) equals to. The original answer is in the form of \(-(k + \sqrt[m]{n})(p + \sqrt{q})\). Please give the value of k + m + n + p + q. | 10 |
Let $\lfloor x \rfloor$ be the greatest integer less than or equal to $x$. Determine the number of real solutions to the equation: $$4x^2 - 40 \lfloor x \rfloor + 51 = 0$$ Fill in the blank with the number of real solutions. | 4 |
Carl is exploring a cave and finds three types of rocks: - 5-pound rocks worth $\$14 each, - 4-pound rocks worth $\$11 each, - 1-pound rocks worth $\$2 each. There are at least 20 of each size available. Carl can carry at most 18 pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave? | 50 |
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