problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given a finite arithmetic sequence \(\left\{a_{n}\right\}\) with the first term equal to 1 and the last term \(a_{n} = 1997\) (where \(n > 3\)), and the common difference being a natural number, find the sum of all possible values of \(n\). | 3501 |
In the tetrahedron $ABCD$, $\triangle ABC$ is an equilateral triangle, $AD = BD = 2$, $AD \perp BD$, and $AD \perp CD$. Find the distance from point $D$ to the plane $ABC$. | \frac{2\sqrt{3}}{3} |
A 12-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Only six slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms? | 4 |
Aliens and Martians have different numbers of limbs. Aliens have three arms and eight legs, while Martians have half as many legs and twice as many arms. How many more limbs will five aliens have than five Martians? | Martians have 2 * 3 = <<2*3=6>>6 arms each.
They have 8 / 2 = <<8/2=4>>4 legs each.
Five martians will have 5 * 6 + 5 * 4 = 30 + 20 = <<5*6+5*4=50>>50 limbs.
Five aliens will have 5 * 3 + 5 * 8 = 15 + 40 = <<5*3+5*8=55>>55 limbs.
Five aliens will have 55 - 50 = <<55-50=5>>5 more limbs than five Martians.
#### 5 |
Let the ordered triples $(x,y,z)$ of complex numbers that satisfy
\begin{align*}
x + yz &= 7, \\
y + xz &= 10, \\
z + xy &= 10.
\end{align*}be $(x_1,y_1,z_1),$ $(x_2,y_2,z_2),$ $\dots,$ $(x_n,y_n,z_n).$ Find $x_1 + x_2 + \dots + x_n.$ | 7 |
Find the largest natural number consisting of distinct digits such that the product of its digits equals 2016. | 876321 |
In triangle \( \triangle ABC \), \( BD \) is a median, \( CF \) intersects \( BD \) at \( E \), and \( BE = ED \). Point \( F \) is on \( AB \), and if \( BF = 5 \), then the length of \( BA \) is: | 15 |
Three male students and two female students stand in a row. The total number of arrangements where the female students do not stand at the ends is given by what total count. | 36 |
A number is called a visible factor number if it is divisible by each of its non-zero digits. For example, 102 is divisible by 1 and 2, so it is a visible factor number. How many visible factor numbers are there from 100 through 150, inclusive? | 19 |
Given that $min\{ a,b\}$ represents the smaller value between the real numbers $a$ and $b$, and the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ satisfy $(\vert\overrightarrow{a}\vert=1,\vert\overrightarrow{b}\vert=2,\overrightarrow{a}\cdot\overrightarrow{b}=0,\overrightarrow{c}=\lambda\overrigh... | \frac{2\sqrt{5}}{5} |
The hypotenuse of a right triangle measures $9$ inches, and one angle is $30^{\circ}$. What is the number of square inches in the area of the triangle? | 10.125\sqrt{3} |
Given that A and B are any two points on the line l, and O is a point outside of l. If there is a point C on l that satisfies the equation $\overrightarrow {OC}= \overrightarrow {OA}cosθ+ \overrightarrow {OB}cos^{2}θ$, find the value of $sin^{2}θ+sin^{4}θ+sin^{6}θ$. | \sqrt {5}-1 |
In \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively. Given that \\(a=2\\), \\(c=3\\), and \\(\cos B= \dfrac {1}{4}\\),
\\((1)\\) find the value of \\(b\\);
\\((2)\\) find the value of \\(\sin C\\). | \dfrac {3 \sqrt {6}}{8} |
In the diagram, $AOB$ is a sector of a circle with $\angle AOB=60^\circ.$ $OY$ is drawn perpendicular to $AB$ and intersects $AB$ at $X.$ What is the length of $XY
?$ [asy]
draw((0,0)--(12,0),black+linewidth(1));
draw((0,0)--(10.3923,-6)..(12,0)..(10.3923,6)--(0,0),black+linewidth(1));
draw((10.3923,-6)--(10.3923,6),b... | 12 - 6\sqrt{3} |
Jon drinks a bottle of water that is 16 ounces every 4 hours for the 16 hours he is awake. Twice a day he also drinks a bottle that is 25% larger than those bottles. How much fluid does he drink a week? | He drinks 16/4=<<16/4=4>>4 of the small bottles each day
That is 4*16=<<4*16=64>>64 ounces
The larger bottles are 16*.25=<<16*.25=4>>4 ounces larger
So the large bottles are 16+4=<<16+4=20>>20 ounces
So he drinks 20*2=<<20*2=40>>40 ounces of those a day
So he drinks 64+40=<<64+40=104>>104 ounces a day
That means he dri... |
Find the value of $\cos(-\pi - \alpha)$ given a point $P(-3, 4)$ on the terminal side of angle $\alpha$. | -\dfrac{3}{5} |
In a football championship with 16 teams, each team played with every other team exactly once. A win was awarded 3 points, a draw 1 point, and a loss 0 points. A team is considered successful if it scored at least half of the maximum possible points. What is the maximum number of successful teams that could have partic... | 15 |
In a right trapezoid \(ABCD\), the sum of the lengths of the bases \(AD\) and \(BC\) is equal to its height \(AB\). In what ratio does the angle bisector of \(\angle ABC\) divide the lateral side \(CD\)? | 1:1 |
Matt's four cousins are coming to visit. There are four identical rooms that they can stay in. If any number of the cousins can stay in one room, how many different ways are there to put the cousins in the rooms? | 15 |
Each pair of vertices of a regular $67$ -gon is joined by a line segment. Suppose $n$ of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of $n$ for which, regardless of which $n$ segments were selected and how they were painted, there will alw... | 2011 |
A flagpole is 12 feet tall. It breaks, folding over in half, such that what was the tip of the flagpole is now dangling two feet above the ground. How far from the base, in feet, did the flagpole break? | The break occurred (12-2)/2=5 feet from the top.
Then it was 12-5=<<12-5=7>>7 feet from the base.
#### 7 |
A child builds towers using identically shaped cubes of different colors. How many different towers with a height 8 cubes can the child build with 2 red cubes, 3 blue cubes, and 4 green cubes? (One cube will be left out.) | 1260 |
Equilateral triangle $T$ is inscribed in circle $A$, which has radius $10$. Circle $B$ with radius $3$ is internally tangent to circle $A$ at one vertex of $T$. Circles $C$ and $D$, both with radius $2$, are internally tangent to circle $A$ at the other two vertices of $T$. Circles $B$, $C$, and $D$ are all externally ... | 32 |
In $\triangle ABC, AB = 8, BC = 7, CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\triangle PAB$ is similar to $\triangle PCA$. The length of $PC$ is
[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, P=(1.5,5), B=(8,0), C=P+2.5*dir(P--B); draw(A--P--C--A--B--C); label("A"... | 9 |
If
\[\frac{\sin x}{\cos y} + \frac{\sin y}{\cos x} = 1 \quad \text{and} \quad \frac{\cos x}{\sin y} + \frac{\cos y}{\sin x} = 6,\]then find $\frac{\tan x}{\tan y} + \frac{\tan y}{\tan x}.$ | \frac{124}{13} |
What is the greatest integer less than 150 for which the greatest common divisor of that integer and 18 is 6? | 138 |
Two dice are rolled consecutively, and the numbers obtained are denoted as $a$ and $b$.
(Ⅰ) Find the probability that the point $(a, b)$ lies on the graph of the function $y=2^x$.
(Ⅱ) Using the values of $a$, $b$, and $4$ as the lengths of three line segments, find the probability that these three segments can form... | \frac{7}{18} |
Mark plants some strawberries in his backyard. Every month, the number of strawberry plants doubles. After 3 months, Mark digs up 4 strawberry plants and gives them to his friend. If he still has 20 strawberry plants, how many did he initially plant? | First, add the 4 plants Mark gave away to the 20 he has left: 4 + 20 = <<4+20=24>>24
Then divide this number by 2 to find how many plants Mark had after two months: 24 / 2 = <<24/2=12>>12
Then divide that number by 2 to find how many plants Mark had after one month: 12 / 2 = <<6=6>>6
Finally, divide that number by 2 to... |
For breakfast, Anna bought a bagel for $0.95 and a glass of orange juice for $0.85. At lunch, Anna spent $4.65 on a sandwich and $1.15 on a carton of milk. How much more money did Anna spend on lunch than on breakfast? | The total cost of breakfast is $0.95 + $0.85 = $<<0.95+0.85=1.80>>1.80.
The total cost of lunch is $4.65 + $1.15 = $<<4.65+1.15=5.80>>5.80.
Anna spent $5.80 − $1.80 = $4 more on lunch than breakfast.
#### 4 |
Let's define the distance between two numbers as the absolute value of their difference. It is known that the sum of the distances from twelve consecutive natural numbers to a certain number \(a\) is 358, and the sum of the distances from these same twelve numbers to another number \(b\) is 212. Find all possible value... | \frac{190}{3} |
$A B C$ is a right triangle with $\angle A=30^{\circ}$ and circumcircle $O$. Circles $\omega_{1}, \omega_{2}$, and $\omega_{3}$ lie outside $A B C$ and are tangent to $O$ at $T_{1}, T_{2}$, and $T_{3}$ respectively and to $A B, B C$, and $C A$ at $S_{1}, S_{2}$, and $S_{3}$, respectively. Lines $T_{1} S_{1}, T_{2} S_{2... | \frac{\sqrt{3}+1}{2} |
Calculate the value of
$$
\sqrt{2018 \times 2021 \times 2022 \times 2023 + 2024^{2}} - 2024^{2}
$$ | -12138 |
A painting $18$" X $24$" is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is: | 2:3 |
Given \\(a, b, c > 0\\), the minimum value of \\(\frac{a^{2} + b^{2} + c^{2}}{ab + 2bc}\\) is \_\_\_\_\_\_. | \frac{2 \sqrt{5}}{5} |
Find the last 3 digits of \(1 \times 3 \times 5 \times 7 \times \cdots \times 2005\). | 375 |
Find the smallest $n$ such that $n$! ends in 290 zeroes. | 1170 |
An engraver makes plates with letters. He engraves the same letters in the same amount of time, but different letters may take different times. On two plates, "ДОМ МОДЫ" (DOM MODY) and "ВХОД" (VKHOD), together he spent 50 minutes, and one plate "В ДЫМОХОД" (V DYMOHOD) took him 35 minutes. How much time will it take him... | 20 |
How many of the divisors of $8!$ are larger than $7!$? | 7 |
What is the result if you add the largest odd two-digit number to the smallest even three-digit number? | 199 |
The diagram shows two 10 by 14 rectangles which are edge-to-edge and share a common vertex. It also shows the center \( O \) of one rectangle and the midpoint \( M \) of one edge of the other. What is the distance \( OM \)?
A) 12
B) 15
C) 18
D) 21
E) 24 | 15 |
The graph of the equation $10x + 270y = 2700$ is drawn on graph paper where each square represents one unit in each direction. A second line defined by $x + y = 10$ also passes through the graph. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below both graphs and entirely in the first qua... | 50 |
Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? | 156 |
Lee mows one lawn and charges $33. Last week he mowed 16 lawns and three customers each gave him a $10 tip. How many dollars did Lee earn mowing lawns last week? | 33 * 16 = $<<33*16=528>>528
3 * 10 = $<<3*10=30>>30
528 + 30 = $<<528+30=558>>558
Lee earned $558 mowing lawns last week.
#### 558 |
Given the sequence defined by $O = \begin{cases} 3N + 2, & \text{if } N \text{ is odd} \\ \frac{N}{2}, & \text{if } N \text{ is even} \end{cases}$, for a given integer $N$, find the sum of all integers that, after being inputted repeatedly for 7 more times, ultimately result in 4. | 1016 |
Coins are arranged in a row from left to right. It is known that two of them are fake, they lie next to each other, the left one weighs 9 grams, the right one weighs 11 grams, and all the remaining are genuine and each weighs 10 grams. The coins are weighed on a balance scale, which either shows which of the two sides ... | 28 |
For what value of $a$ does the equation $3(2x-a) = 2(3x+12)$ have infinitely many solutions $x$? | -8 |
If a vehicle is driven 12 miles on Monday, 18 miles on Tuesday, and 21 miles on Wednesday. What is the average distance traveled per day? | The total distance covered from Monday to Wednesday is 12 + 18 + 21 = <<12+18+21=51>>51 miles.
So the average distance traveled per day is 51/3 = <<51/3=17>>17 miles.
#### 17 |
A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $9$, $16$, and $36$, respectively. Find the area of $\triangle DEF$. | 169 |
Compute
\[\begin{pmatrix} 1 & 1 & -2 \\ 0 & 4 & -3 \\ -1 & 4 & 3 \end{pmatrix} \begin{pmatrix} 2 & -2 & 0 \\ 1 & 0 & -3 \\ 4 & 0 & 0 \end{pmatrix}.\] | \begin{pmatrix} -5 & -2 & -3 \\ -8 & 0 & -12 \\ 14 & 2 & -12 \end{pmatrix} |
Suppose that $x^{10} + x + 1 = 0$ and $x^100 = a_0 + a_1x +... + a_9x^9$ . Find $a_5$ . | -252 |
Given a geometric series \(\left\{a_{n}\right\}\) with the sum of its first \(n\) terms denoted by \(S_{n}\), and satisfying the equation \(S_{n}=\frac{\left(a_{n}+1\right)^{2}}{4}\), find the value of \(S_{20}\). | 400 |
Simon and Peter have a big stamp collection. Simon collects red stamps and Peter collects white stamps. Simon has 30 red stamps and Peter has 80 white stamps. If the red stamps are then sold for 50 cents each and the white stamps are sold for 20 cents each, what is the difference in the amount of money they make in dol... | Red stamps are 50 cents each and for 30 stamps, Simon would get 50 * 30 = <<50*30=1500>>1500 cents
White stamps are 20 cents each and for 80 stamps, Peter would get 20 * 80 = <<20*80=1600>>1600 cents
The difference in cents is 1600 - 1500 = <<1600-1500=100>>100 cents
One dollar equals 100 cents, so the difference in do... |
Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of... | 126 |
5 years ago, a mother was twice as old as her daughter. If the mother is 41 years old now, how old will the daughter be in 3 years? | The mother is 41 years old now, so 5 years ago she was 41-5=<<41-5=36>>36 years old
At that time the daughter's age was half hers so the daughter was 36/2=<<36/2=18>>18 years old
Now (5 years after that time) the daughter is 18+5=23 years old
In 3 years, the daughter will be 23+3=<<23+3=26>>26 years old
#### 26 |
Given 1985 sets, each consisting of 45 elements, and the union of any two sets contains exactly 89 elements. How many elements are in the union of all these 1985 sets? | 87341 |
Points $A$ and $C$ lie on a circle centered at $P$, which is inside $\triangle ABC$ such that $\overline{AP}$ is perpendicular to $\overline{BC}$ and $\triangle ABC$ is equilateral. The circle intersects $\overline{BP}$ at $D$. If $\angle BAP = 45^\circ$, what is $\frac{BD}{BP}$?
A) $\frac{2}{3}$
B) $\frac{2 - \sqrt{2}... | \frac{2 - \sqrt{2}}{2} |
Compute the product of $0.\overline{123}$ and $9$, and write your result as a fraction in simplified form. | \frac{41}{37} |
Jesse received $50 as a gift to buy what she wants. She goes to the mall and falls in love with a novel that costs her $7. Then she went to lunch because she was very hungry and spent twice as much as the novel cost her. How much money did Jesse have left after going to the mall? | Jesse bought a novel for $<<7=7>>7.
She then bought lunch for twice the value of the novel, so lunch cost her $7 x 2 = $<<7*2=14>>14.
To find out how much money she has left from her gift she must know the total amount she spent $7 + $14 = $<<7+14=21>>21.
Her gift was $50, so she was left with $50 - $21 = $<<50-21=29>>... |
For what value of $x$ does $10^{x} \cdot 100^{2x}=1000^{5}$? | 3 |
Given that the coefficients of the first three terms of the expansion of $(x+ \frac {1}{2})^{n}$ form an arithmetic sequence. Let $(x+ \frac {1}{2})^{n} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots + a_{n}x^{n}$. Find:
(1) The value of $n$;
(2) The value of $a_{5}$;
(3) The value of $a_{0} - a_{1} + a_{2} - a_{3} + \ld... | \frac {1}{256} |
Given $\sin\left( \frac{\pi}{3} + a \right) = \frac{5}{13}$, and $a \in \left( \frac{\pi}{6}, \frac{2\pi}{3} \right)$, find the value of $\sin\left( \frac{\pi}{12} + a \right)$. | \frac{17\sqrt{2}}{26} |
Winston has 14 quarters. He then spends half a dollar on candy. How many cents does he have left? | Winston has 350 cents because 14 x 25 = <<14*25=350>>350
He spends 50 cents because $1 / 2 = 50 cents
He has 300 cents left at the end.
#### 300 |
A baseball league has nine teams. During the season, each of the nine teams plays exactly three games with each of the other teams. What is the total number of games played? | 108 |
Given the parametric equation of line $l$ as $$\begin{cases} x= \sqrt {3}+t \\ y=7+ \sqrt {3}t\end{cases}$$ ($t$ is the parameter), a coordinate system is established with the origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of curve $C$ is $\rho \sqrt {a^{2}\sin^{2}\theta+... | \frac{2\sqrt{21}}{3} |
How many solutions does the equation \[\frac{(x-1)(x-2)(x-3)\dotsm(x-100)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-100^2)} = 0\]have for $x$? | 90 |
The sum
$$
\frac{1}{1 \times 2 \times 3}+\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\cdots+\frac{1}{100 \times 101 \times 102}
$$
can be expressed as $\frac{a}{b}$, a fraction in its simplest form. Find $a+b$. | 12877 |
In a Go championship participated by three players: A, B, and C, the matches are conducted according to the following rules: the first match is between A and B; the second match is between the winner of the first match and C; the third match is between the winner of the second match and the loser of the first match; th... | 0.162 |
Find the value of the first term in the geometric sequence $a,b,c,32,64$. | 4 |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{\begin{array}{l}{x=2+\sqrt{5}\cos\theta,}\\{y=\sqrt{5}\sin\theta}\end{array}\right.$ ($\theta$ is the parameter). Line $l$ passes through point $P(1,-1)$ with a slope of $60^{\circ}$ and intersects curve $C$ at points $A$ and $B... | \frac{\sqrt{16+2\sqrt{3}}}{3} |
Determine all real numbers $ a$ such that the inequality $ |x^2 + 2ax + 3a|\le2$ has exactly one solution in $ x$. | 1, 2 |
Solve for the largest value of $x$ such that $5(9x^2+9x+10) = x(9x-40).$ Express your answer as a simplified common fraction. | -\dfrac{10}{9} |
When simplified, what is the value of $$(10^{0.5})(10^{0.3})(10^{0.2})(10^{0.1})(10^{0.9})?$$ | 100 |
Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians. | 3 \pi |
For real numbers $t,$ the point
\[(x,y) = (\cos^2 t, \sin^2 t)\]is plotted. All the plotted points lie on what kind of curve?
(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. | \text{(A)} |
On a clock, there are two hands: the hour hand and the minute hand. At a random moment in time, the clock stops. Find the probability that the angle between the hands on the stopped clock is acute. | 1/2 |
Compute $\tan (-3645^\circ)$. | -1 |
What is the greatest prime factor of $3^7+6^6$? | 67 |
Given that a match between two people is played with a best-of-five-games format, where the winner is the first to win three games, and that the probability of person A winning a game is $\dfrac{2}{3}$, calculate the probability that person A wins with a score of $3:1$. | \dfrac{8}{27} |
In the set \(\{1, 2, 3, \cdots, 99, 100\}\), how many numbers \(n\) satisfy the condition that the tens digit of \(n^2\) is odd?
(45th American High School Mathematics Examination, 1994) | 20 |
In acute $\triangle A B C$ with centroid $G, A B=22$ and $A C=19$. Let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ to $A C$ and $A B$ respectively. Let $G^{\prime}$ be the reflection of $G$ over $B C$. If $E, F, G$, and $G^{\prime}$ lie on a circle, compute $B C$. | 13 |
The Diving Club offers 2 beginning diving classes on weekdays and 4 beginning classes on each day of the weekend. Each class has room for 5 people. How many people can take classes in 3 weeks? | There are 2 classes x 5 days = <<2*5=10>>10 classes on weekdays.
There are 4 classes x 2 days = <<4*2=8>>8 classes on weekends.
There are a total of 10 + 8 = <<10+8=18>>18 classes per week.
So each week, 18 x 5 = <<18*5=90>>90 people can take classes.
Thus, 90 x 3 = <<90*3=270>>270 people can take classes in 3 weeks.
#... |
If $a \div b = 2$ and $b \div c = \frac{3}{4}$, what is the value of $c \div a$? Express your answer as a common fraction. | \frac{2}{3} |
Find all pairs of distinct rational numbers $(a,b)$ such that $a^a=b^b$. | \left(\left(\frac{u}{v}\right)^{\frac{u}{v-u}}, \left(\frac{u}{v}\right)^{\frac{v}{v-u}}\right) |
A city has a population of 300,000. 50,000 people immigrate to the country and then 30,000 people leave. After that 1/8 of the population gets pregnant and 1/4 of those people have twins. How many people are there after the births? | There was a net immigration to the country of 50,000-30,000=<<50000-30000=20000>>20,000
So the total number of people before birth is 300,000+20,000=<<300000+20000=320000>>320,000
That means 320,000/8=<<320000/8=40000>>40,000 get pregnant
So 40,000/4=<<40000/4=10000>>10,000 have twins
So that means they have 10,000*2=<... |
Which one of the following points is not on the graph of $y=\dfrac{x}{x+1}$? | (-1,1) |
A department store offers two promotions. Promotion A says, "Buy one pair of shoes, get the second pair for half the price." Promotion B says, "Buy one pair of shoes, get $\$10$ off the second pair." Jane wants to buy two pairs of shoes that cost $\$30$ each. She can only use one of the promotions, A or B. Jane decides... | 5 |
Rosie can make two pies out of nine apples. How many pies can she make out of twenty-seven apples? | 6 |
If $a\ast b = 3a+4b-ab$, what is the value of $5\ast2$? | 13 |
An ellipse and a hyperbola have the same foci $F\_1(-c,0)$, $F\_2(c,0)$. One endpoint of the ellipse's minor axis is $B$, and the line $F\_1B$ is parallel to one of the hyperbola's asymptotes. If the eccentricities of the ellipse and hyperbola are $e\_1$ and $e\_2$, respectively, find the minimum value of $3e\_1^2+e\_2... | 2\sqrt{3} |
All positive integers whose digits add up to 14 are listed in increasing order: $59, 68, 77, ...$. What is the fifteenth number in that list? | 266 |
Anton colors a cell in a \(4 \times 50\) rectangle. He then repeatedly chooses an uncolored cell that is adjacent to at most one already colored cell. What is the maximum number of cells that can be colored? | 150 |
What is the value of the least positive base ten number which requires six digits for its binary representation? | 32 |
Compute: $\displaystyle \frac{66,\!666^4}{22,\!222^4}$. | 81 |
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2023 \left( \frac{2}{x} + \frac{2}{y} + 2022 \right)\] for all $x, y > 0.$
Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s.... | 2023 |
What is the tens digit of $2015^{2016}-2017?$ | 0 |
The base of a prism is an equilateral triangle $ABC$. The lateral edges of the prism $AA_1$, $BB_1$, and $CC_1$ are perpendicular to the base. A sphere, whose radius is equal to the edge of the base of the prism, touches the plane $A_1B_1C_1$ and the extensions of the segments $AB_1$, $BC_1$, and $CA_1$ beyond the poin... | \sqrt{44} - 6 |
What is the coefficient of $x^5$ when $$2x^5 - 4x^4 + 3x^3 - x^2 + 2x - 1$$ is multiplied by $$x^3 + 3x^2 - 2x + 4$$ and the like terms are combined? | 24 |
A survey of $150$ teachers determined the following:
- $90$ had high blood pressure
- $60$ had heart trouble
- $50$ had diabetes
- $30$ had both high blood pressure and heart trouble
- $20$ had both high blood pressure and diabetes
- $10$ had both heart trouble and diabetes
- $5$ had all three conditions
What percent ... | 3.33\% |
On a "prime date," both the month and the day are prime numbers. For example, Feb. 7 or 2/7 is a prime date. How many prime dates occurred in 2007? | 52 |
Given $x= \frac {\pi}{12}$ is a symmetry axis of the function $f(x)= \sqrt {3}\sin(2x+\varphi)+\cos(2x+\varphi)$ $(0<\varphi<\pi)$, after shifting the graph of function $f(x)$ to the right by $\frac {3\pi}{4}$ units, find the minimum value of the resulting function $g(x)$ on the interval $\left[-\frac {\pi}{4}, \frac {... | -1 |
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