problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
A woman needs to buy 16 oz of butter for a dessert recipe. She can either buy a single 16 oz package of the store-brand butter for $7, or she can buy an 8oz package of store butter for $4 and use a coupon to get an additional two 4 oz packages that normally sell for $2.0 each at a 50% discount (which is applied to eac... | If she uses the coupon, she will pay 2*0.5 = $<<2*0.5=1>>1 for each 4oz package.
Thus, she will pay 4 + 2*1 = $<<4+2*1=6>>6 in total for the 8oz package and the two 4oz packages she needs to reach 16 oz of butter when using the coupon.
Therefore, the lowest amount she can pay is $6, as this is cheaper than the $7 price... |
Let \(a\) and \(b\) be constants. The parabola \(C: y = (t^2 + t + 1)x^2 - 2(a + t)^2 x + t^2 + 3at + b\) passes through a fixed point \(P(1,0)\) for any real number \(t\). Find the value of \(t\) such that the chord obtained by intersecting the parabola \(C\) with the x-axis is the longest. | -1 |
Tom's algebra notebook consists of 50 pages, 25 sheets of paper. Specifically, page 1 and page 2 are the front and back of the first sheet of paper, page 3 and page 4 are the front and back of the second sheet of paper, and so on. One day, Tom left the notebook on the table while he went out, and his roommate took away... | 13 |
Given the function $f(x)=ax^{3}-4x+4$, where $a\in\mathbb{R}$, $f′(x)$ is the derivative of $f(x)$, and $f′(1)=-3$.
(1) Find the value of $a$;
(2) Find the extreme values of the function $f(x)$. | -\frac{4}{3} |
In trapezoid $PQRS$, leg $\overline{QR}$ is perpendicular to bases $\overline{PQ}$ and $\overline{RS}$, and diagonals $\overline{PR}$ and $\overline{QS}$ are perpendicular. Given that $PQ=\sqrt{23}$ and $PS=\sqrt{2023}$, find $QR^2$. | 100\sqrt{46} |
Ryan has 30 stickers. Steven has thrice as many stickers as Ryan. Terry has 20 more stickers than Steven. How many stickers do they have altogether? | Steven has 3 x 30 = <<3*30=90>>90 stickers.
And, Terry has 90 + 20 = <<90+20=110>>110 stickers.
Therefore, they have 30 +90 + 110 = <<30+90+110=230>>230 stickers altogether.
#### 230 |
Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 12$ and $X$ is an integer, what is the smallest possible value of $X$? | 925 |
Call a positive integer $n$ quixotic if the value of $\operatorname{lcm}(1,2,3, \ldots, n) \cdot\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}\right)$ is divisible by 45 . Compute the tenth smallest quixotic integer. | 573 |
Find the smallest prime number that can be expressed as the sum of five different prime numbers. | 43 |
Each twin from the first 4 sets shakes hands with all twins except his/her sibling and with one-third of the triplets; the remaining 8 sets of twins shake hands with all twins except his/her sibling but does not shake hands with any triplet; and each triplet shakes hands with all triplets except his/her siblings and wi... | 394 |
Given the function $f(x)=\sin x\cos x- \sqrt {3}\cos ^{2}x.$
(I) Find the smallest positive period of $f(x)$;
(II) When $x\in[0, \frac {π}{2}]$, find the maximum and minimum values of $f(x)$. | - \sqrt {3} |
Shari walks at a constant rate of 3 miles per hour. After 1.5 hours, how many miles did she walk? Express your answer as a decimal to the nearest tenth. | 4.5 |
Six students participate in an apple eating contest. The graph shows the number of apples eaten by each participating student. Aaron ate the most apples and Zeb ate the fewest. How many more apples than Zeb did Aaron eat?
[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[8];
for( int i = 0 ; i <... | 5 |
Let $x$ be the largest root of $x^4 - 2009x + 1$ . Find the nearest integer to $\frac{1}{x^3-2009}$ . | -13 |
A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the fie... | 702 |
Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around ... | \frac{12}{13} |
For any real number $x$, the symbol $[x]$ represents the largest integer not greater than $x$. For example, $[2]=2$, $[2.1]=2$, $[-2.2]=-3$. The function $y=[x]$ is called the "floor function", which has wide applications in mathematics and practical production. Then, the value of $[\log _{3}1]+[\log _{3}2]+[\log _{3}3... | 12 |
When the length of a rectangle is increased by $20\%$ and the width increased by $10\%$, by what percent is the area increased? | 32 \% |
Find the sum of all possible values of $s$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \[(\cos 30^\circ, \sin 30^\circ), (\cos 45^\circ, \sin 45^\circ), \text{ and } (\cos s^\circ, \sin s^\circ)\] is isosceles and its area is greater than $0.1$.
A) 15
B) 30
C) 45
D) 60 | 60 |
Coordinate System and Parametric Equation
Given the ellipse $(C)$: $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$, which intersects with the positive semi-axis of $x$ and $y$ at points $A$ and $B$ respectively. Point $P$ is any point on the ellipse. Find the maximum area of $\triangle PAB$. | 6(\sqrt{2} + 1) |
Let \( M = \{1, 2, \ldots, 20\} \) and \( A_1, A_2, \ldots, A_n \) be distinct non-empty subsets of \( M \). When \( i \neq j \), the intersection of \( A_i \) and \( A_j \) has at most two elements. Find the maximum value of \( n \). | 1350 |
A right circular cone is placed on a table, pointing upwards. The vertical cross-section triangle, perpendicular to the base, has a vertex angle of 90 degrees. The diameter of the cone's base is 16 inches. A sphere is placed inside the cone so that it touches the sides of the cone and rests on the table. Find the volum... | \frac{256}{3}\pi |
When $x^9-x$ is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is: | 5 |
Given a triangle $ABC$ with the sides opposite to angles $A$, $B$, $C$ denoted by $a$, $b$, $c$ respectively, let vectors $\overrightarrow{m}=(1-\cos(A+B), \cos \frac{A-B}{2})$ and $\overrightarrow{n}=(\frac{5}{8}, \cos \frac{A-B}{2})$, and it's known that $\overrightarrow{m} \cdot \overrightarrow{n} = \frac{9}{8}$.
1.... | -\frac{3}{8} |
The coefficient of $x^{2}$ in the expansion of $\left( \frac {3}{x}+x\right)\left(2- \sqrt {x}\right)^{6}$ is ______. | 243 |
What is the sum of all integer values $n$ for which $\binom{26}{13}+\binom{26}{n}=\binom{27}{14}$? | 26 |
Given vectors $\mathbf{v}$ and $\mathbf{w}$ such that $\|\mathbf{v}\| = 3,$ $\|\mathbf{w}\| = 7,$ and $\mathbf{v} \cdot \mathbf{w} = 10,$ then find $\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|.$ | \frac{10}{7} |
Po is trying to solve the following equation by completing the square: $$49x^2+56x-64 = 0.$$He successfully rewrites the above equation in the following form: $$(ax + b)^2 = c,$$where $a$, $b$, and $c$ are integers and $a > 0$. What is the value of $a + b + c$? | 91 |
Given that point $O$ is the origin of coordinates, point $A$ in the first quadrant lies on the graph of the inverse proportional function $y=\frac{1}{x}$ for $x>0$, and point $B$ in the second quadrant lies on the graph of the inverse proportional function $y=-\frac{4}{x}$ for $x<0$, and $O A$ is perpendicular to $O B$... | $\frac{1}{2}$ |
The area of the lunar crescent shape bounded by the portion of the circle of radius 5 and center (0,0), the portion of the circle with radius 2 and center (0,2), and the line segment from (0,0) to (5,0). | \frac{21\pi}{4} |
Contessa is taking a random lattice walk in the plane, starting at $(1,1)$. (In a random lattice walk, one moves up, down, left, or right 1 unit with equal probability at each step.) If she lands on a point of the form $(6 m, 6 n)$ for $m, n \in \mathbb{Z}$, she ascends to heaven, but if she lands on a point of the for... | \frac{13}{22} |
Compute $\dbinom{5}{3}$. | 10 |
Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of $5$ heads before one encounters a run of $2$ tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
| 37 |
Given that $A$, $B$, $C$ are the three internal angles of $\triangle ABC$, and their respective opposite sides are $a$, $b$, $c$, and $2\cos ^{2} \frac {A}{2}+\cos A=0$.
(1) Find the value of angle $A$;
(2) If $a=2 \sqrt {3},b+c=4$, find the area of $\triangle ABC$. | \sqrt {3} |
You walk twice as fast as Mr. Harris, and Mr. Harris took 2 hours to walk to the store. If your destination is 3 times further away than the store Mr. Harris walked to, how many hours will it take you to get there? | First we find out how long it would take Mr. Harris to walk to your destination, which we find by multiplying his 2 hour walk by 3 since your destination is 3 times further away, meaning it would take him 2*3= <<2*3=6>>6 hours.
Since you walk twice as fast, it would take you half the time. Half of 6, 6/2=<<6/2=3>>3 hou... |
Ten positive integers include the numbers 3, 5, 8, 9, and 11. What is the largest possible value of the median of this list of ten positive integers? | 11 |
In right triangle $XYZ$ with $\angle Z = 90^\circ$, the length $XY = 15$ and the length $XZ = 8$. Find $\sin Y$. | \frac{\sqrt{161}}{15} |
If $a$,$b$, and $c$ are positive real numbers such that $a(b+c) = 152$, $b(c+a) = 162$, and $c(a+b) = 170$, then find $abc.$ | 720 |
Let $T_{L}=\sum_{n=1}^{L}\left\lfloor n^{3} / 9\right\rfloor$ for positive integers $L$. Determine all $L$ for which $T_{L}$ is a square number. | L=1 \text{ or } L=2 |
Compute the remainder when $$\sum_{k=1}^{30303} k^{k}$$ is divided by 101. | 29 |
If $x$ and $y$ are non-zero numbers such that $x=1+\frac{1}{y}$ and $y=1+\frac{1}{x}$, then $y$ equals | x |
There are $64$ booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered $1$ to $64$ in this order. At the center of the table there are $1996$ light bulbs which are all turned off. Every minute the chips move simultaneously in a circular way (following ... | 64 |
An equilateral triangle has two vertices at $(0,5)$ and $(8,5)$. If the third vertex is in the first quadrant, what is the y-coordinate? Express your answer in simplest radical form. [asy]
draw((-1,0)--(11,0),Arrows);
draw((0,-1)--(0,12),Arrows);
for(int i=0;i<11;++i)
{draw((i,-0.1)--(i,0.1));}
for(int j=0;j<11;++j)
{d... | 5+4\sqrt{3} |
Suppose that for some $a,b,c$ we have $a+b+c = 6$, $ab+ac+bc = 5$ and $abc = -12$. What is $a^3+b^3+c^3$? | 90 |
Hannah collects mugs. She already has 40 different mugs in 4 different colors. She has three times more blue mugs than red mugs and 12 yellow mugs. Considering that she has only half as many red mugs as yellow mugs, how many mugs of another color than mentioned does she have? | Hannah has half as many red mugs as yellow mugs, which means she has 12 * 0.5 = <<12*0.5=6>>6 red mugs.
Hannas has three times more blue mugs than red ones, which means 3 * 6 = <<3*6=18>>18 blue mugs.
So there are 40 - 18 - 6 - 12 = <<40-18-6-12=4>>4 mugs in a different color than mentioned.
#### 4 |
Given $f(x) = \begin{cases} x^{2}+1 & (x>0) \\ 2f(x+1) & (x\leq 0) \end{cases}$, find $f(2)$ and $f(-2)$. | 16 |
In how many ways can 7 people sit around a round table, considering that two seatings are the same if one is a rotation of the other, and additionally, one specific person must sit between two particular individuals? | 240 |
Percy wants to save up for a new PlayStation, which costs $500. He gets $200 on his birthday and $150 at Christmas. To make the rest of the money, he's going to sell his old PlayStation games for $7.5 each. How many games does he need to sell to reach his goal? | He needs to earn $150 more dollars because 500 - 200 - 150 = <<500-200-150=150>>150
He needs to sell 20 games because 150 / 7.5 = <<150/7.5=20>>20
#### 20 |
Billy Bones has two coins - one gold and one silver. One of them is symmetrical, and the other is not. It is unknown which coin is asymmetrical, but it is known that the asymmetrical coin lands heads with a probability of $p=0.6$.
Billy Bones tossed the gold coin, and it landed heads immediately. Then Billy Bones star... | 5/9 |
Let $A$ , $M$ , and $C$ be nonnegative integers such that $A+M+C=10$ . Find the maximum value of $A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A$. | 69 |
Given the function $f\left( x \right)=\sqrt{3}\sin\left( \omega x-\frac{\pi }{6} \right)(\omega > 0)$, the distance between two adjacent highest points on the graph is $\pi$.
(1) Find the value of $\omega$ and the equation of the axis of symmetry for the function $f\left( x \right)$;
(2) If $f\left( \frac{\alpha }{2}... | \frac{3\sqrt{5}-1}{8} |
Mitzel spent 35% of her allowance. If she spent $14, how much money is left in her allowance? | If 35% represents 14, then 100-35 = 65 represents the amount left of her allowance.
The amount she didn't use is 65*14/35 = <<65*14/35=26>>26
#### 26 |
The numbers $5,6,10,17$, and 21 are rearranged so that the sum of the first three numbers is equal to the sum of the last three numbers. Which number is in the middle of this rearrangement? | 5 |
Find the coefficient of \(x^9\) in the polynomial expansion of \((1+3x-2x^2)^5\). | 240 |
Rectangle $ABCD$ has area $2006.$ An ellipse with area $2006\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$. What is the perimeter of the rectangle? | 8\sqrt{1003} |
Point $C(0,p)$ lies on the $y$-axis between $Q(0,15)$ and $O(0,0)$. Point $B$ has coordinates $(15,0)$. Determine an expression for the area of $\triangle COB$ in terms of $p$, and compute the length of segment $QB$. Your answer should be simplified as much as possible. | 15\sqrt{2} |
Which of the following integers cannot be written as a product of two integers, each greater than 1: 6, 27, 53, 39, 77? | 53 |
In a community of 50 families, 15 families own 2 dogs, 20 families own 1 dog, while the remaining families own 2 cats each. How many dogs and cats are there in all? | The 15 families own 15x2=<<15*2=30>>30 dogs.
The 20 families own 20x1=<<20*1=20>>20 dogs.
There are 50-15-20=<<50-15-20=15>>15 cat owners.
The remaining families own 15x2=<<15*2=30>>30 cats.
There are 30+20+30=<<30+20+30=80>>80 dogs and cats in the village.
#### 80 |
Calculate the sum $\frac{3}{50} + \frac{5}{500} + \frac{7}{5000}$.
A) $0.0714$
B) $0.00714$
C) $0.714$
D) $0.0357$
E) $0.00143$ | 0.0714 |
Angles $A$ and $B$ are supplementary. If the measure of angle $A$ is $8$ times angle $B$, what is the measure of angle A? | 160 |
An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A *tour route* is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of ... | 15 |
Three balls are drawn simultaneously from the urn (as described in Problem 4). Find the probability that all the drawn balls are blue (event $B$). | 1/12 |
Robots Robert and Hubert assemble and disassemble coffee grinders. Each of them assembles a grinder four times faster than they disassemble one. When they arrived at the workshop in the morning, several grinders were already assembled.
At 7:00 AM, Hubert started assembling and Robert started disassembling. Exactly at ... | 15 |
Find the maximum value of the expression for \( a, b > 0 \):
$$
\frac{|4a - 10b| + |2(a - b\sqrt{3}) - 5(a\sqrt{3} + b)|}{\sqrt{a^2 + b^2}}
$$ | 2 \sqrt{87} |
Find $\left(\sqrt{(\sqrt3)^3}\right)^4$. | 27 |
Let the three-digit number \( n = abc \). If \( a, b, \) and \( c \) as the lengths of the sides can form an isosceles (including equilateral) triangle, then how many such three-digit numbers \( n \) are there? | 165 |
Determine the number of ways to arrange the letters of the word "SUCCESS". | 420 |
Given Karl's rectangular garden measures \(30\) feet by \(50\) feet with a \(2\)-feet wide uniformly distributed pathway and Makenna's garden measures \(35\) feet by \(55\) feet with a \(3\)-feet wide pathway, compare the areas of their gardens, assuming the pathways take up gardening space. | 225 |
What is the sum of all positive integers less than 500 that are fourth powers of even perfect squares? | 272 |
An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of length 1 unit either up or to the right. How many up-right paths from $(0,0)$ to $(7,7)$, when drawn in the plane with the line $y=x-2.021$, enclose exactly one bounded region below that line? | 637 |
If Sarah is leading a class of 35 students, and each time Sarah waves her hands a prime number of students sit down, determine the greatest possible number of students that could have been standing before her third wave. | 31 |
Add $-45.367$, $108.2$, and $23.7654$, then round your answer to the nearest tenth. | 86.6 |
An o-Pod MP3 player stores and plays entire songs. Celeste has 10 songs stored on her o-Pod. The time length of each song is different. When the songs are ordered by length, the shortest song is only 30 seconds long and each subsequent song is 30 seconds longer than the previous song. Her favorite song is 3 minutes, 30... | \dfrac{79}{90} |
Five blue beads, three green beads, and one red bead are placed in line in random order. Calculate the probability that no two blue beads are immediately adjacent to each other. | \frac{1}{126} |
Nathan is buying decorations for his wedding reception. The reception hall will have 20 tables. Each table needs a linen tablecloth ($25 to rent), 4 place settings ($10 each to rent), and a centerpiece. Each centerpiece will have 10 roses ($5 each) and 15 lilies ($4 each). How much will the decorations cost? | First find how much the roses for one centerpiece cost: $5/rose * 10 roses = $<<5*10=50>>50
Then find how much the lilies for one centerpiece cost: $4/rose * 15 roses = $<<4*15=60>>60
Then find how much the table settings for one table cost: 4 settings * $10/setting = $<<4*10=40>>40
Then add the cost of both types of f... |
Jessica is making an apple pie. She knows that each serving requires 1.5 apples and she has 12 quests. She plans to make 3 pies, which each contain 8 servings. If her guests finish all the pie, on average , how many apples does each guest eat? | She is making 24 servings because 3 x 8 = <<3*8=24>>24
Each guest has 2 servings because 24 / 12 = <<24/12=2>>2
Each guest has 3 apples because 2 x 1.5 = <<2*1.5=3>>3
#### 3 |
Given $3^{7} + 1$ and $3^{15} + 1$ inclusive, how many perfect cubes lie between these two values? | 231 |
A sphere is inscribed in a right circular cylinder. The height of the cylinder is 12 inches, and the diameter of its base is 10 inches. Find the volume of the inscribed sphere. Express your answer in terms of $\pi$. | \frac{500}{3} \pi |
Tyler has entered a buffet line in which he chooses one kind of meat, two different vegetables and one dessert. If the order of food items is not important, how many different meals might he choose?
$\bullet$ Meat: beef, chicken, pork
$\bullet$ Vegetables: baked beans, corn, potatoes, tomatoes
$\bullet$ Dessert:... | 72 |
Given the ellipse C₁: $$\frac {x^{2}}{a^{2}}$$+ $$\frac {y^{2}}{b^{2}}$$\=1 (a＞b＞0) with one focus coinciding with the focus of the parabola C₂: y<sup>2</sup>\=4 $$\sqrt {2}$$x, and the eccentricity of the ellipse is e= $$\frac { \sqrt {6}}{3}$$.
(I) Find the equation of C₁.
(II) A moving line l passes through the poin... | \frac{\sqrt{3}}{2} |
A motorcycle travels due west at $\frac{5}{8}$ mile per minute on a long straight road. At the same time, a circular storm, whose radius is $60$ miles, moves southwest at $\frac{1}{2}$ mile per minute. At time $t=0$, the center of the storm is $100$ miles due north of the motorcycle. At time $t=t_1$ minutes, the motorc... | 160 |
For all composite integers $n$, what is the largest integer that always divides into the difference between $n$ and the cube of $n$? | 6 |
Let $y=(x-a)^2+(x-b)^2, a, b$ constants. For what value of $x$ is $y$ a minimum? | \frac{a+b}{2} |
Let $ABC$ be a triangle with circumcenter $O$ such that $AC=7$. Suppose that the circumcircle of $AOC$ is tangent to $BC$ at $C$ and intersects the line $AB$ at $A$ and $F$. Let $FO$ intersect $BC$ at $E$. Compute $BE$. | \frac{7}{2} |
A triangle has a base of 20 inches. Two lines are drawn parallel to the base, intersecting the other two sides and dividing the triangle into four regions of equal area. Determine the length of the parallel line closer to the base. | 10 |
At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values... | 60 |
When $\sqrt[4]{2^7\cdot3^3}$ is fully simplified, the result is $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$? | 218 |
Given the equation $a + b = 30$, where $a$ and $b$ are positive integers, how many distinct ordered-pair solutions $(a, b)$ exist? | 29 |
In trapezoid \(A B C D\), the bases \(A D\) and \(B C\) are 8 and 18, respectively. It is known that the circumscribed circle of triangle \(A B D\) is tangent to lines \(B C\) and \(C D\). Find the perimeter of the trapezoid. | 56 |
Compute $\sin(-60^\circ)$. | -\frac{\sqrt{3}}{2} |
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ that has center $I$ . If $IA = 5, IB = 7, IC = 4, ID = 9$ , find the value of $\frac{AB}{CD}$ . | 35/36 |
Solve for $y$: $3y+7y = 282-8(y-3)$. | 17 |
In a town, there is a multi-story parking lot, which has room for 425 cars. The parking lot has 5 levels, each of the same size. How many more cars can one level fit if there are already 23 parked cars on that level? | When all levels are empty, each level can fit 425 / 5 = <<425/5=85>>85 cars.
When there are already 23 parked cars on a level, there is still a place for 85 - 23 = 62 cars.
#### 62 |
The solution to the equation \(\arcsin x + \arcsin 2x = \arccos x + \arccos 2x\) is | \frac{\sqrt{5}}{5} |
Given Josie makes lemonade by using 150 grams of lemon juice, 200 grams of sugar, and 300 grams of honey, and there are 30 calories in 100 grams of lemon juice, 386 calories in 100 grams of sugar, and 304 calories in 100 grams of honey, determine the total number of calories in 250 grams of her lemonade. | 665 |
Simplify $\dfrac{30}{45} \cdot \dfrac{75}{128} \cdot \dfrac{256}{150}$. | \frac{1}{6} |
There are constants $\alpha$ and $\beta$ such that $\frac{x-\alpha}{x+\beta} = \frac{x^2-80x+1551}{x^2+57x-2970}$. What is $\alpha+\beta$? | 137 |
If $M = 2007 \div 3$, $N = M \div 3$, and $X = M - N$, then what is the value of $X$? | 446 |
What is the maximum number of bishops that can be placed on an $8 \times 8$ chessboard such that at most three bishops lie on any diagonal? | 38 |
Let the function $f(x)= \sqrt{3}\cos^2\omega x+\sin \omega x\cos \omega x+a$ where $\omega > 0$, $a\in\mathbb{R}$, and the graph of $f(x)$ has its first highest point on the right side of the y-axis at the x-coordinate $\dfrac{\pi}{6}$.
(Ⅰ) Find the smallest positive period of $f(x)$; (Ⅱ) If the minimum value of $f(x)... | \dfrac{ \sqrt{3}+1}{2} |
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