problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
What is the area enclosed by the graph of $|3x|+|4y|=12$? | 24 |
In a bag, there are $5$ balls of the same size, including $3$ red balls and $2$ white balls.<br/>$(1)$ If one ball is drawn with replacement each time, and this process is repeated $3$ times, with the number of times a red ball is drawn denoted as $X$, find the probability distribution and expectation of the random var... | \frac{108}{625} |
For any positive integer \( n \), the value of \( n! \) is the product of the first \( n \) positive integers. Calculate the greatest common divisor of \( 8! \) and \( 10! \). | 40320 |
Given the function $y=\cos(2x+\frac{\pi}{4})$, determine the $x$-coordinate of one of the symmetric centers of the translated graph after translating it to the left by $\frac{\pi}{6}$ units. | \frac{11\pi}{24} |
There are 66 goats at a farm and twice as many chickens. If the number of ducks is half of the total of the goats and chickens, and the number of pigs is a third of the number of ducks, how many more goats are there than pigs? | There are 66 x 2 = <<66*2=132>>132 chickens
There are 66 + 132 = <<66+132=198>>198 goats and chickens
There are are 198/2 = <<198/2=99>>99 ducks
There are 99/3 = <<99/3=33>>33 pigs
There are 66 - 33 = <<66-33=33>>33 more goats than pigs
#### 33 |
What is the largest number, all of whose digits are 3 or 2, and whose digits add up to $11$? | 32222 |
In the expression $c \cdot a^b - d$, the values of $a$, $b$, $c$, and $d$ are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result? | 9 |
Find the sum of all positive integers $n$ such that $1.2n-4.4<5.2$. | 28 |
The points $P(3,-2), Q(3,1), R(7,1)$, and $S$ form a rectangle. What are the coordinates of $S$? | (7,-2) |
Triangle $ABC$ is isosceles with $AB = AC = 2$ and $BC = 1.5$. Points $E$ and $G$ are on segment $\overline{AC}$, and points $D$ and $F$ are on segment $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$, trapezoid $DFGE$, and trapezoid $FBCG$ ... | \frac{19}{6} |
If farmer Steven can use his tractor to plow up to 10 acres of farmland per day, or use the same tractor to mow up to 12 acres of grassland per day, how long would it take him to plow his 55 acres of farmland and mow his 30 acres of grassland? | At a rate of 10 acres per day, he can plow his 55 acres of farmland in 55/10=<<55/10=5.5>>5.5 days.
At a rate of 12 acres per day, he can mow his 30 acres of grassland in 30/12=2.5 days.
Altogether, it would take him 5.5+2.5=<<5.5+2.5=8>>8 days to completely mow and plow his land.
#### 8 |
What is the largest possible value of \(| |a_1 - a_2| - a_3| - \ldots - a_{1990}|\), where \(a_1, a_2, \ldots, a_{1990}\) is a permutation of \(1, 2, 3, \ldots, 1990\)? | 1989 |
In a chorus performance, there are 6 female singers (including 1 lead singer) and 2 male singers arranged in two rows.
(1) If there are 4 people per row, how many different arrangements are possible?
(2) If the lead singer stands in the front row and the male singers stand in the back row, with again 4 people per row, ... | 5760 |
There are six unicorns in the Enchanted Forest. Everywhere a unicorn steps, four flowers spring into bloom. The six unicorns are going to walk all the way across the forest side-by-side, a journey of 9 kilometers. If each unicorn moves 3 meters forward with each step, how many flowers bloom because of this trip? | First convert the total length of the journey from kilometers to meters: 9 kilometers * 1000 meters/kilometer = <<9*1000=9000>>9000 meters
Then divide this by the length of one step to find the number of steps in the journey: 9000 meters / 3 meters/step = <<9000/3=3000>>3000 steps
Then multiply the number of steps by t... |
A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\text{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate t... | 1925 |
Evaluate the infinite geometric series: $$\frac{1}{3}+\frac{1}{6}+\frac{1}{12}+\frac{1}{24}+\dots$$ | \frac{2}{3} |
Let $p$, $q$, $r$, $s$, $t$, and $u$ be positive integers with $p+q+r+s+t+u = 2023$. Let $N$ be the largest of the sum $p+q$, $q+r$, $r+s$, $s+t$ and $t+u$. What is the smallest possible value of $N$? | 810 |
The legs \( AC \) and \( CB \) of the right triangle \( ABC \) are 15 and 8, respectively. A circular arc with radius \( CB \) is drawn from center \( C \), cutting off a part \( BD \) from the hypotenuse. Find \( BD \). | \frac{128}{17} |
Kimothy starts in the bottom-left square of a 4 by 4 chessboard. In one step, he can move up, down, left, or right to an adjacent square. Kimothy takes 16 steps and ends up where he started, visiting each square exactly once (except for his starting/ending square). How many paths could he have taken? | 12 |
Burt spent $2.00 on a packet of basil seeds and $8.00 on potting soil. The packet of seeds yielded 20 basil plants. He sells each basil plant for $5.00 at the local farmer's market. What is the net profit from his basil plants? | He spent $2.00 on seeds and $8.00 on soil for a total of 2+8 = $<<2+8=10.00>>10.00
He sells each of the 20 basil plants for $5.00 so he makes 20*5 = $<<20*5=100.00>>100.00
He made $100.00 from selling basil plants and he spent $10.00 to buy and grow the seeds. His net profit is 100-10 = $<<100-10=90.00>>90.00
#### 90 |
The maximum value of the function $f(x) = \frac{\frac{1}{6} \cdot (-1)^{1+ C_{2x}^{x}} \cdot A_{x+2}^{5}}{1+ C_{3}^{2} + C_{4}^{2} + \ldots + C_{x-1}^{2}}$ ($x \in \mathbb{N}$) is ______. | -20 |
If income of $5$ yuan is denoted as $+5$ yuan, then expenses of $5$ yuan are denoted as what? | -5 |
Given integers $a$ and $b$ whose sum is $20$, determine the number of distinct integers that can be written as the sum of two, not necessarily different, fractions $\frac{a}{b}$. | 14 |
Joan collects rocks. In her collection of rock samples, she had half as many gemstones as minerals yesterday. Today, she went on a rock collecting trip and found 6 more mineral samples. If she has 48 minerals now, how many gemstone samples does she have? | Yesterday before her rock collecting trip, she had 48 - 6 = <<48-6=42>>42 mineral samples.
She has 1/2 the number of gemstones, so she has 42 / 2 = <<42/2=21>>21 gemstone samples.
#### 21 |
Given the line $l$: $x=my+1$ passes through the right focus $F$ of the ellipse $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, the focus of the parabola $x^{2}=4\sqrt{3}y$ is the upper vertex of the ellipse $C$, and the line $l$ intersects the ellipse $C$ at points $A$ and $B$.
1. Find the equation of th... | -\frac{8}{3} |
Given the sets
$$
\begin{array}{l}
A=\left\{(x, y) \mid x=m, y=-3m+2, m \in \mathbf{Z}_{+}\right\}, \\
B=\left\{(x, y) \mid x=n, y=a\left(a^{2}-n+1\right), n \in \mathbf{Z}_{+}\right\},
\end{array}
$$
find the total number of integers $a$ such that $A \cap B \neq \varnothing$. | 10 |
Let $\mathbf{a}$ and $\mathbf{b}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 3 \\ -3 \end{pmatrix} = \begin{pmatrix} -\frac{3}{5} \\ -\frac{6}{5} \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 3 \\ -3 \end{pmatrix}.$ | \begin{pmatrix} \frac{18}{5} \\ -\frac{9}{5} \end{pmatrix} |
Lola and Tara decide to race to the top of a 20 story building. Tara takes the elevator and Lola runs up the stairs. Lola can run up 1 story in 10 seconds. The elevator goes up a story in 8 seconds but stops for 3 seconds on every single floor. How long would it take for the slower one of Lola and Tara to reach the top... | Lola climbs 20 stories in 20 stories x 10 seconds/story = <<20*10=200>>200 seconds
Tara can go up the elevator in 8 seconds/story x 20 stories = <<8*20=160>>160 seconds without the stops at each floor.
The elevator stops for 3 seconds on each floor for a total of 20 stories x 3 seconds/story = <<20*3=60>>60 seconds.
In... |
Two numbers \( x \) and \( y \) satisfy the equation \( 280x^{2} - 61xy + 3y^{2} - 13 = 0 \) and are respectively the fourth and ninth terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression. | -5 |
Seven students are standing in a row for a graduation photo. Among them, student A must stand in the middle, and students B and C must stand together. How many different arrangements are there? | 192 |
In an $h$-meter race, Sunny is exactly $d$ meters ahead of Windy when Sunny finishes the race. The next time they race, Sunny sportingly starts $d$ meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finis... | \frac {d^2}{h} |
A pie shop charges $5 for a slice of pie. They cut each whole pie into 4 slices. How much money will the pie shop make if they sell 9 pies? | A pie contains 4 slices so the pie shop makes 5*4=$<<4*5=20>>20 for a pie.
If they sell 9 pies, they make 9*20=$<<9*20=180>>180
#### 180 |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{\sqrt{2}}{2}t,\\ y=\frac{\sqrt{2}}{2}t\end{array}\right.$ (where $t$ is the parameter). Using $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinat... | \frac{5}{2} |
If $e^{i \alpha} = \frac{3}{5} +\frac{4}{5} i$ and $e^{i \beta} = -\frac{12}{13} + \frac{5}{13} i,$ then find $\sin (\alpha + \beta).$ | -\frac{33}{65} |
In Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points-per-game average for the $10$ ga... | 40 |
For a positive integer \( n \), define \( s(n) \) as the smallest positive integer \( t \) such that \( n \) is a factor of \( t! \). Compute the number of positive integers \( n \) for which \( s(n) = 13 \). | 792 |
In $\triangle ABC$, where $C=60 ^{\circ}$, $AB= \sqrt {3}$, and the height from $AB$ is $\frac {4}{3}$, find the value of $AC+BC$. | \sqrt {11} |
The lengths of the diagonals of a rhombus and the length of its side form a geometric progression. Find the sine of the angle between the side of the rhombus and its longer diagonal, given that it is greater than \( \frac{1}{2} \). | \sqrt{\frac{\sqrt{17}-1}{8}} |
Marcel bought a pen for $4, and a briefcase for five times the price. How much did Marcel pay for both items? | The cost of the briefcase was 5 * 4 = $<<5*4=20>>20.
So for both items Marcel paid 4 + 20 = $<<4+20=24>>24.
#### 24 |
400 adults and 200 children go to a Broadway show. The price of an adult ticket is twice that of a child's ticket. What is the price of an adult ticket if the total amount collected is $16,000? | Let X be the price of a child ticket.
So the price of an adult ticket is X*2.
We know from the problem that 400*X*2 + 200*X = 16,000.
After multiplying, we have 800*X+ 200*X = 16,000.
Combining like terms, we have 1000* X = 16,000.
Dividing both sides by 1000, we have X = $16.
The price of an adult ticket is $16 * 2 = ... |
Given a geometric sequence $\{a_n\}$ with a common ratio $q > 1$, $a_1 = 2$, and $a_1, a_2, a_3 - 8$ form an arithmetic sequence. The sum of the first n terms of the sequence $\{b_n\}$ is denoted as $S_n$, and $S_n = n^2 - 8n$.
(I) Find the general term formulas for both sequences $\{a_n\}$ and $\{b_n\}$.
(II) Let $c_n... | \frac{1}{162} |
Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_3 x & \quad \text{ if } \log_3 x \text{ is an integer} \\ 1 + g(x + 2) & \quad \text{ otherwise}. \end{aligned} \right.\] Compute $g(50).$ | 20 |
Let \(ABCD\) be a convex quadrilateral, and let \(M_A,\) \(M_B,\) \(M_C,\) \(M_D\) denote the midpoints of sides \(BC,\) \(CA,\) \(AD,\) and \(DB,\) respectively. Find the ratio \(\frac{[M_A M_B M_C M_D]}{[ABCD]}.\) | \frac{1}{4} |
How many 7-digit positive integers are made up of the digits 0 and 1 only, and are divisible by 6? | 11 |
The Bank of Oslo issues two types of coin: aluminum (denoted A) and bronze (denoted B). Marianne has $n$ aluminum coins and $n$ bronze coins arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k \leq 2n$, Gilberty repeatedl... | {n \leq k \leq \lceil \tfrac32n \rceil} |
Given a $9 \times 9$ chess board, we consider all the rectangles whose edges lie along grid lines (the board consists of 81 unit squares, and the grid lines lie on the borders of the unit squares). For each such rectangle, we put a mark in every one of the unit squares inside it. When this process is completed, how man... | 56 |
Xiao Hua needs to attend an event at the Youth Palace at 2 PM, but his watch gains 4 minutes every hour. He reset his watch at 10 AM. When Xiao Hua arrives at the Youth Palace according to his watch at 2 PM, how many minutes early is he actually? | 16 |
Mr. Roberts can buy a television for $400 cash or $120 down payment and $30 a month for 12 months. How much can he save by paying cash? | Mr. Roberts will make a total payment of $30 x 12 = $<<30*12=360>>360 for 12 months.
Thus, the television costs $360 + $120 = $<<360+120=480>>480 when not paid in cash.
Therefore, Mr. Roberts can save $480 - $400 = $<<480-400=80>>80 by paying cash.
#### 80 |
A tourist is learning an incorrect way to sort a permutation $(p_{1}, \ldots, p_{n})$ of the integers $(1, \ldots, n)$. We define a fix on two adjacent elements $p_{i}$ and $p_{i+1}$, to be an operation which swaps the two elements if $p_{i}>p_{i+1}$, and does nothing otherwise. The tourist performs $n-1$ rounds of fix... | 1009! \cdot 1010! |
Let $A B C$ be a triangle and $D$ a point on $B C$ such that $A B=\sqrt{2}, A C=\sqrt{3}, \angle B A D=30^{\circ}$, and $\angle C A D=45^{\circ}$. Find $A D$. | \frac{\sqrt{6}}{2} |
A Ferris wheel rotates at a constant speed, completing one revolution every 12 minutes. The lowest point of the Ferris wheel is 2 meters above the ground, and the highest point is 18 meters above the ground. Let P be a fixed point on the circumference of the Ferris wheel. Starting the timing when P is at the lowest poi... | 14 |
There are 94 safes and 94 keys. Each key can open only one safe, and each safe can be opened by only one key. We place randomly one key into each safe. 92 safes are then randomly chosen, and then locked. What is the probability that we can open all the safes with the two keys in the two remaining safes?
(Once a safe i... | 1/47 |
Haley has 5 more necklaces than Jason. Josh has half the number of necklaces as Jason. If Haley has 25 necklaces, how many more necklaces does Haley have than Josh? | Jason has 25-5 = <<25-5=20>>20 necklaces.
Josh has 20/2 = <<20/2=10>>10 necklaces.
Haley has 25-10 = <<25-10=15>>15 more necklaces than Josh.
#### 15 |
Amanda is figuring out how many bags of grass seed she needs to buy to cover a new lot the city is turning into a park. The lot measures 120 feet by 60 feet. One section that measures 40 feet by 40 feet will be covered with concrete, and the rest needs to be covered in grass seeds. Each bag of grass seeds covers 56 squ... | First find the area of the lot by multiplying the width and the length: 120 ft * 60 ft = <<120*60=7200>>7200 sq ft.
Then find the area of the square covered in concrete: 40 ft * 40 ft = <<40*40=1600>>1600 sq ft.
Then subtract the concrete area from the total area to find the grassy area: 7200 sq ft - 1600 sq ft = <<720... |
A park is 1000 feet long and 2000 feet wide. If there is 1 tree per 20 square feet, how many trees are there in the park? | The park is 2,000,000 square feet because 1000*2000 =<<1000*2000=2000000>>2,000,000.
Thus, there are 100,000 trees in the park because 2,000,000/20 = <<2000000/20=100000>>100,000.
#### 100,000 |
In \(\triangle ABC\), the external angle bisector of \(\angle BAC\) intersects line \(BC\) at \(D\). \(E\) is a point on ray \(\overrightarrow{AC}\) such that \(\angle BDE=2 \angle ADB\). If \(AB=10, AC=12\), and \(CE=33\), compute \(\frac{DB}{DE}\). | \frac{2}{3} |
Let \( A = \{1, 2, 3, 4, 5, 6\} \). Find the number of distinct functions \( f: A \rightarrow A \) such that \( f(f(f(n))) = n \) for all \( n \in A \). | 81 |
Given a triangular pyramid \( S-ABC \) with vertex \( S \). The projection of \( S \) onto the base \( \triangle ABC \) is the orthocenter \( H \) of \( \triangle ABC \). Additionally, \( BC = 2 \), \( SB = SC \), and the dihedral angle between the face \( SBC \) and the base is \( 60^\circ \). Determine the volume of ... | \frac{\sqrt{3}}{3} |
Given the function $f(x)=e^{-x}+ \frac {nx}{mx+n}$.
$(1)$ If $m=0$, $n=1$, find the minimum value of the function $f(x)$.
$(2)$ If $m > 0$, $n > 0$, and the minimum value of $f(x)$ on $[0,+\infty)$ is $1$, find the maximum value of $\frac {m}{n}$. | \frac {1}{2} |
Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\left(\frac58, \frac38 \right)$ is greater than or equal to $\frac12$ can be written as $\frac{m}{n}$, where $... | 171 |
Nikita schematically drew the graph of the quadratic polynomial \( y = ax^{2} + bx + c \). It turned out that \( AB = CD = 1 \). Consider the four numbers \(-a, b, c\), and the discriminant of the polynomial. It is known that three of these numbers are equal to \( \frac{1}{4}, -1, -\frac{3}{2} \) in some order. Find th... | -1/2 |
George is planning a dinner party for three other couples, his wife, and himself. He plans to seat the four couples around a circular table for 8, and wants each husband to be seated opposite his wife. How many seating arrangements can he make, if rotations and reflections of each seating arrangement are not considered... | 24 |
Let $p$ and $q$ be the two distinct solutions to the equation $$(x-5)(2x+9) = x^2-13x+40.$$What is $(p + 3)(q + 3)$? | -112 |
Let \(ABCD\) be a square of side length 1. \(P\) and \(Q\) are two points on the plane such that \(Q\) is the circumcentre of \(\triangle BPC\) and \(D\) is the circumcentre of \(\triangle PQA\). Find the largest possible value of \(PQ^2\). Express the answer in the form \(a + \sqrt{b}\) or \(a - \sqrt{b}\), where \(a\... | 2 + \sqrt{3} |
A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?
[asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(1,1/2,1/4); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white,... | \frac{7}{30} |
A set \( A \) consists of 40 elements chosen from \(\{1, 2, \ldots, 50\}\), and \( S \) is the sum of all elements in the set \( A \). How many distinct values can \( S \) take? | 401 |
The Hangzhou Asian Games are underway, and table tennis, known as China's "national sport," is receiving a lot of attention. In table tennis matches, each game is played to 11 points, with one point awarded for each winning shot. In a game, one side serves two balls first, followed by the other side serving two balls, ... | \frac{3}{4} |
Four carpenters were hired by a guest to build a yard. The first carpenter said: "If only I alone were to build the yard, I would complete it in one year." The second carpenter said: "If only I alone were to build the yard, I would complete it in two years." The third carpenter said: "If only I alone were to build the ... | 175.2 |
In the figure, the area of square $WXYZ$ is $25 \text{ cm}^2$. The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In $\triangle
ABC$, $AB = AC$, and when $\triangle ABC$ is folded over side $\overline{BC}$, point $A$ coincides with $O$, the center of sq... | \frac{27}{4} |
If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1,$ $\|\mathbf{a} + \mathbf{b}\| = \sqrt{3},$ and
\[\mathbf{c} - \mathbf{a} - 2 \mathbf{b} = 3 (\mathbf{a} \times \mathbf{b}),\]then find $\mathbf{b} \cdot \mathbf{c}.$ | \frac{5}{2} |
Let $r$ be a real number, $|r| < 2,$ and let $z$ be a complex number such that
\[z + \frac{1}{z} = r.\]Find $|z|.$ | 1 |
Seven test scores have a mean of $85$, a median of $88$, and a mode of $90$. Calculate the sum of the three lowest test scores. | 237 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is given that $\frac{-b + \sqrt{2}c}{\cos B} = \frac{a}{\cos A}$,
$(I)$ Find the size of angle $A$;
$(II)$ If $a=2$, find the maximum value of the area $S$. | \sqrt{2} + 1 |
The lateral surface of a cylinder unfolds into a square. What is the ratio of its lateral surface area to the base area. | 4\pi |
In any permutation of the numbers \(1, 2, 3, \ldots, 18\), we can always find a set of 6 consecutive numbers whose sum is at least \(m\). Find the maximum value of the real number \(m\). | 57 |
In the triangular pyramid \(ABCD\) with base \(ABC\), the lateral edges are pairwise perpendicular, \(DA = DB = 5, DC = 1\). A ray of light is emitted from a point on the base. After reflecting exactly once from each lateral face (the ray does not reflect from the edges), the ray hits a point on the pyramid's base. Wha... | \frac{10 \sqrt{3}}{9} |
An equilateral triangle has an area of $64\sqrt{3}$ $\text{cm}^2$. If each side of the triangle is decreased by 4 cm, by how many square centimeters is the area decreased? | 28\sqrt{3} |
The Bank of Springfield's Super High Yield savings account compounds annually at a rate of one percent. If Lisa invests 1000 dollars in one of these accounts, then how much interest will she earn after five years? (Give your answer to the nearest dollar.) | 51 |
The numbers \(a_{1}, a_{2}, \ldots, a_{n}\) are such that the sum of any seven consecutive numbers is negative, and the sum of any eleven consecutive numbers is positive. What is the largest possible \(n\) for which this is true? | 16 |
A school bought 20 cartons of pencils at the start of school. Pencils come in cartons of 10 boxes and each box costs $2. The school also bought 10 cartons of markers. A carton has 5 boxes and costs $4. How much did the school spend in all? | The school bought 20 x 10 = <<20*10=200>>200 boxes of pencils.
The pencils cost 200 x $2 = $<<200*2=400>>400.
The school bought 10 x 5 = <<10*5=50>>50 boxes of markers.
The markers cost 50 x $4 = $<<50*4=200>>200.
Thus, the school spent a total of $400 + $200 = $<<400+200=600>>600.
#### 600 |
Let $b_n$ be the integer obtained by writing down the integers from 1 to $n$ in reverse, from right to left. Compute the remainder when $b_{39}$ is divided by 125. | 21 |
Given that ${(1-2x)^{2016}}=a_{0}+a_{1}(x-2)+a_{2}(x-2)^{2}+\cdots+a_{2015}(x-2)^{2015}+a_{2016}(x-2)^{2016}$ $(x\in\mathbb{R})$, find the value of $a_{1}-2a_{2}+3a_{3}-4a_{4}+\cdots+2015a_{2015}-2016a_{2016}$. | 4032 |
Which of the following numbers is closest to 1: $ rac{11}{10}$, $ rac{111}{100}$, 1.101, $ rac{1111}{1000}$, 1.011? | 1.011 |
Given that the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ is 120°, and the magnitude of $\overrightarrow {a}$ is 2. If $(\overrightarrow {a} + \overrightarrow {b}) \cdot (\overrightarrow {a} - 2\overrightarrow {b}) = 0$, find the projection of $\overrightarrow {b}$ on $\overrightarrow {a}$. | -\frac{\sqrt{33} + 1}{8} |
Given 5 different letters from the word "equation", find the total number of different arrangements that contain "qu" where "qu" are consecutive and in the same order. | 480 |
There is a committee composed of 10 members who meet around a table: 7 women, 2 men, and 1 child. The women sit in indistinguishable rocking chairs, the men on indistinguishable stools, and the child on a bench, also indistinguishable from any other benches. How many distinct ways are there to arrange the seven rocking... | 360 |
Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -.
Tom presses a sequence of $5$ random keystrokes; at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, yielding a result of $E$ . Find the expected valu... | 1866 |
In triangle $PQR$, angle $R$ is a right angle and the altitude from $R$ meets $\overline{PQ}$ at $S$. The lengths of the sides of $\triangle PQR$ are integers, $PS=17^3$, and $\cos Q = a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. | 18 |
On square $ABCD$, point $E$ lies on side $AD$ and point $F$ lies on side $BC$, so that $BE=EF=FD=30$. Find the area of the square $ABCD$. | 810 |
Given $S$ is the set of the 1000 smallest positive multiples of $5$, and $T$ is the set of the 1000 smallest positive multiples of $9$, determine the number of elements common to both sets $S$ and $T$. | 111 |
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is denoted as $S_n$. It is known that $a_1=9$, $a_2$ is an integer, and $S_n \leqslant S_5$. The sum of the first $9$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$ is ______. | - \frac{1}{9} |
Darius, Matt, and Marius are friends, who played table football. During all the games they played, Marius scored 3 points more than Darius, and Darius scored 5 points less than Matt. How many points did all three friends score together, if Darius scored 10 points? | If Darius scored 10 points, then Marius scored 10 + 3 = <<10+3=13>>13 points in total.
Darius scored 5 points less than Matt, so Matt scored 10 + 5 = <<10+5=15>>15 points in total.
So all three friends gathered 13 + 15 + 10 = <<13+15+10=38>>38 points.
#### 38 |
A cube-shaped container has vertices $A,$ $B,$ $C,$ and $D,$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of faces of the cube, as shown. Vertex $A$ of the cube is set on a horizontal plane $\mathcal{P}$ so that the plane of the rectangl... | 751 |
Expand the product ${(x+2)(x+5)}$. | x^2 + 7x + 10 |
Find all values of \( x \) for which the greater of the numbers \( \sqrt{\frac{x}{2}} \) and \( \operatorname{tg} x \) is not greater than 1. Provide the total length of the intervals on the number line that satisfy this condition, rounding the result to the nearest hundredth if necessary. | 1.21 |
Given that the function $f\left(x\right)$ is an even function on $R$, and $f\left(x+2\right)$ is an odd function. If $f\left(0\right)=1$, then $f\left(1\right)+f\left(2\right)+\ldots +f\left(2023\right)=\_\_\_\_\_\_$. | -1 |
The diagram above shows several numbers in the complex plane. The circle is the unit circle centered at the origin.
One of these numbers is the reciprocal of $F$. Which one? | C |
Given that $a > 0$, $b > 0$, and $2a+b=1$, find the maximum value of $2 \sqrt {ab}-4a^{2}-b^{2}$. | \dfrac { \sqrt {2}-1}{2} |
A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the wa... | 52 |
In the triangle below, find $XY$. Triangle $XYZ$ is a right triangle with $XZ = 18$ and $Z$ as the right angle. Angle $Y = 60^\circ$.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (1,0);
R = (0.5,sqrt(3)/2);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(R,P,Q,3));
label("$X$",P,S);
label("$Y$",Q,S);
label("... | 36 |
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