problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given a right circular cone with a base radius of \(1 \, \text{cm}\) and a slant height of \(3 \, \text{cm}\), point \(P\) is on the circumference of the base. Determine the shortest distance from the vertex \(V\) of the cone to the shortest path from \(P\) back to \(P\). | 1.5 |
[asy] draw((0,0)--(0,3)--(4,0)--cycle,dot); draw((4,0)--(7,0)--(7,10)--cycle,dot); draw((0,3)--(7,10),dot); MP("C",(0,0),SW);MP("A",(0,3),NW);MP("B",(4,0),S);MP("E",(7,0),SE);MP("D",(7,10),NE); [/asy]
Triangle $ABC$ has a right angle at $C, AC=3$ and $BC=4$. Triangle $ABD$ has a right angle at $A$ and $AD=12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If\[\frac{DE}{DB}=\frac{m}{n},\]where $m$ and $n$ are relatively prime positive integers, then $m+n=$
$\text{(A) } 25\quad \text{(B) } 128\quad \text{(C) } 153\quad \text{(D) } 243\quad \text{(E) } 256$
| 128 |
A trader made a profit of $960 after a week of sales. To raise $610 to pay for her next shipment of wares, she splits the profit in half and solicits donations from her family and friends. If she received a total donation of $310 and adds it to one-half of her profit, how much money did she make above her goal? | She splits her $960 profit in half to get $960 / 2 = $<<960/2=480>>480.
She receives a total donation of $310, so she now has $310 + $480 = $<<310+480=790>>790.
She needed only $610 so she has $790 - $610 = $180 more than she needs.
#### 180 |
A geometric sequence of positive integers is formed for which the first term is 3 and the fifth term is 375. What is the sixth term of the sequence? | 9375 |
Out of the 400 emails that Antonia received in her mail, 1/4 were spam emails, while 2/5 of the remaining emails were promotional messages. If the rest of the emails were important emails, calculate the total number of important emails in her inbox. | Out of the 400 emails, 1/4*400 = <<400*1/4=100>>100 emails were spam emails.
The number of emails that were not spam emails is 400-100 = <<400-100=300>>300
2/5 of the emails which were not spam were promotional emails, a total of 2/5*300=<<2/5*300=120>>120 emails.
If the rest of the emails were important emails, there were 300-120 = <<300-120=180>>180 important emails.
#### 180 |
Square $ABCD$ is constructed along diameter $AB$ of a semicircle, as shown. The semicircle and square $ABCD$ are coplanar. Line segment $AB$ has a length of 6 centimeters. If point $M$ is the midpoint of arc $AB$, what is the length of segment $MC$? Express your answer in simplest radical form. [asy]
size(4cm);
dotfactor = 4;
defaultpen(linewidth(1)+fontsize(10pt));
pair A,B,C,D,M;
A = (0,1);
B = (1,1);
C = (1,0);
D = (0,0);
M = (.5,1.5);
draw(A..M..B--C--D--cycle);
draw(A--B);
dot("A",A,W);
dot("M",M,N);
dot("B",B,E);
dot("C",C,E);
dot("D",D,W);
draw(M--C,linetype("0 4"));
[/asy] | 3\sqrt{10} |
Amy writes down four integers \(a > b > c > d\) whose sum is 50. The pairwise positive differences of these numbers are \(2, 3, 4, 5, 7,\) and \(10\). What is the sum of the possible values for \(a\)? | 35 |
A regular $n$-gon $P_{1} P_{2} \ldots P_{n}$ satisfies $\angle P_{1} P_{7} P_{8}=178^{\circ}$. Compute $n$. | 630 |
Paul made two bank transfers of $90 and $60 respectively. A service charge of 2% was added to each transaction. If the second transaction was reversed (without the service charge), what is his account balance now if it was $400 before he made any of the transfers? | 2% of $90 is (2/100)*$90 = $<<(2/100)*90=1.8>>1.8
2% of $60 is (2/100)*$60 = $<<(2/100)*60=1.2>>1.2
The second transaction was reversed without the service charge so only a total of $90+$1.8+$1.2 = $<<90+1.8+1.2=93>>93 was deducted from his account
He will have a balance of $400-$93 = $<<400-93=307>>307
#### 307 |
A heptagonal prism has ____ faces and ____ vertices. | 14 |
Andrea needs 45 rhinestones to finish an art project. She bought a third of what she needed and found a fifth of what she needed in her supplies. How many rhinestones does she still need? | Andrea bought 45 / 3 = <<45/3=15>>15 rhinestones.
She found 45 / 5 = <<45/5=9>>9 rhinestones in her supplies.
Thus, Andrea still needs 45 - 15 - 9 = <<45-15-9=21>>21 rhinestones.
#### 21 |
Carla's order at Mcdonald's costs $7.50, but she has a coupon for $2.50. If she gets an additional 20% discount for being a senior citizen, how much does she pay for her order in dollars total? | First subtract the coupon amount from the price of the food: $7.50 - $2.50 = $<<7.5-2.5=5.00>>5.00
Then subtract 20% from 100% to find out what percentage of the reduced price Carla pays: 100% - 20% = 80%
Now multiply the price after the coupon by the percentage Carla pays to find the final price: $5.00 * .8 = $<<5*.8=4.00>>4.00
#### 4 |
Let $x_1,$ $x_2,$ $x_3$ be positive real numbers such that $x_1 + 2x_2 + 3x_3 = 60.$ Find the smallest possible value of
\[x_1^2 + x_2^2 + x_3^2.\] | \frac{1800}{7} |
Given point $P(2,2)$, and circle $C$: $x^{2}+y^{2}-8y=0$. A moving line $l$ passing through point $P$ intersects circle $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $M$, and $O$ being the origin.
$(1)$ Find the equation of the trajectory of point $M$;
$(2)$ When $|OP|=|OM|$, find the equation of line $l$ and the area of $\Delta POM$. | \frac{16}{5} |
The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$. | 262 |
A burger at Ricky C's now weighs 180 grams, of which 45 grams are filler. What percent of the burger is not filler? Additionally, what percent of the burger is filler? | 25\% |
How many continuous paths from $A$ to $B$, along segments of the figure, do not revisit any of the six labeled points?
[asy]
draw((0,0)--(3,0)--(3,2)--(0,2)--(0,0)--cycle,linewidth(2));
draw((0,2)--(1,0)--(3,2)--(0,2)--cycle,linewidth(2));
draw((0,2)--(1.5,3.5)--(3,2),linewidth(2));
label("$A$",(1.5,3.5),N);
label("$B$",(0,0),SW);
label("$C$",(0,2),W);
label("$D$",(3,2),E);
label("$E$",(3,0),SE);
label("$F$",(1,0),S);
[/asy] | 10 |
Find the unit vector $\mathbf{v},$ lying in the $xz$-plane, which makes an angle of $45^\circ$ with $\begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix},$ and an angle of $60^\circ$ with $\begin{pmatrix} 0 \\ 1 \\ - 1 \end{pmatrix}.$ | \begin{pmatrix} \sqrt{2}/2 \\ 0 \\ -\sqrt{2}/2 \end{pmatrix} |
Arrange 3 boys and 4 girls in a row. Calculate the number of different arrangements under the following conditions:
(1) Person A and Person B must stand at the two ends;
(2) All boys must stand together;
(3) No two boys stand next to each other;
(4) Exactly one person stands between Person A and Person B. | 1200 |
Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? | 968 |
There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? | 80 |
A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit will be less than 5? Express your answer as a common fraction. | \frac{1}{2} |
A circle $\omega_{1}$ of radius 15 intersects a circle $\omega_{2}$ of radius 13 at points $P$ and $Q$. Point $A$ is on line $P Q$ such that $P$ is between $A$ and $Q$. $R$ and $S$ are the points of tangency from $A$ to $\omega_{1}$ and $\omega_{2}$, respectively, such that the line $A S$ does not intersect $\omega_{1}$ and the line $A R$ does not intersect $\omega_{2}$. If $P Q=24$ and $\angle R A S$ has a measure of $90^{\circ}$, compute the length of $A R$. | 14+\sqrt{97} |
Find the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length $11$ . | 20 |
There are 21 different pairs of digits (a, b) such that $\overline{5a68} \times \overline{865b}$ is divisible by 824. | 19 |
If $\theta$ is an acute angle, and $\sin 2\theta=a$, then $\sin\theta+\cos\theta$ equals | $\sqrt{a+1}$ |
Compute $\begin{pmatrix} -4 \\ -1 \end{pmatrix} \cdot \begin{pmatrix} 6 \\ 8 \end{pmatrix}$. | -32 |
Below is a portion of the graph of a quadratic function, $y=q(x)=ax^2+bx+c$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=10.5,ymin=-5.5,ymax=5.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);
Label laxis; laxis.p=fontsize(10);
xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f1(real x){return 4-(x-8)*(x-7)/8;}
draw(graph(f1,-0.99,10.5),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
label("$y=q(x)$",(10.75,2.5),E);
[/asy] The value of $q(15)$ is an integer. What is that integer? | -3 |
On the shore of a circular island (viewed from above) are the cities $A$, $B$, $C$, and $D$. The straight asphalt road $AC$ divides the island into two equal halves. The straight asphalt road $BD$ is shorter than the road $AC$ and intersects it. The cyclist's speed on any asphalt road is 15 km/h. The island also has straight dirt roads $AB$, $BC$, $CD$, and $AD$, where the cyclist's speed on any dirt road is the same. The cyclist travels from point $B$ to each of the points $A$, $C$, and $D$ along a straight road in 2 hours. Find the area enclosed by the quadrilateral $ABCD$. | 450 |
A bag contains 15 red marbles, 10 blue marbles, and 5 green marbles. Four marbles are selected at random and without replacement. What is the probability that two marbles are red, one is blue, and one is green? Express your answer as a common fraction. | \frac{350}{1827} |
Find all integers $n \ge 3$ such that among any $n$ positive real numbers $a_1$ , $a_2$ , $\dots$ , $a_n$ with \[\max(a_1, a_2, \dots, a_n) \le n \cdot \min(a_1, a_2, \dots, a_n),\] there exist three that are the side lengths of an acute triangle. | \(\{n \ge 13\}\) |
Stephanie is decorating 24 cupcakes for a birthday party, but she needs more candles. She currently has a total of 30 candles. She wants to decorate half of the cupcakes with 1 candle each and the other half of the cupcakes with 2 candles each. How many additional candles does Stephanie need to complete the cupcakes? | For half of the cupcakes, Stephanie wants to use 1 candle each. Since half of the cupcakes is 24/2 and she plans to use 1 candle each for this half of the cupcakes, Stephanie needs (24/2)*1 = <<24/2*1=12>>12 candles for this half of the cupcakes.
For the other half of the cupcakes, Stephanie wants to use 2 candles. Therefore, she will need (24/2)*2 =<<24/2*2=24>>24 candles for this half of the cupcakes.
Because Stephanie needs 12 candles for half of the cupcakes and 24 candles for the other half, she needs a total of 12+24=<<12+24=36>>36 candles.
Since Stephanie needs 36 candles to decorate all the cupcakes and she currently has 30 candles, Stephanie needs 36-30= <<36-30=6>>6 additional candles.
#### 6 |
How many positive integers, not exceeding 200, are multiples of 5 or 7 but not 10? | 43 |
When three friends sold their video games, Ryan, the first of the three friends, received $50 more than Jason from his sales. Jason received 30% more money than Zachary from selling his video games. If Zachary sold 40 games at $5 each, calculate the total amount of money the three friends received together from the sale of the video games. | The total amount of money that Zachary received from selling his video games is $40*5 = $<<40*5=200>>200
Jason received 30/100*$200 = $<<30/100*200=60>>60 more than the amount Zachary received after selling his games.
The total amount of money that Jason received is $60+$200 = $<<60+200=260>>260
Together, Zachary and Jason received $260+$200 = $<<260+200=460>>460 from the sale of their video games.
Ryan received $260+$50=$<<260+50=310>>310 from the sale of his video games.
The three friends made a total of $310+$460 = $<<310+460=770>>770 from the sale of the video games.
#### 770 |
How many ways are there to arrange the letters of the word $\text{BA}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3$, in which the three A's and the two N's are considered different? | 720 |
Given that $x^2 + y^2 = 14x + 6y + 6,$ find the largest possible value of $3x + 4y.$ | 73 |
Find the number of ordered integer tuples \((k_1, k_2, k_3, k_4)\) that satisfy \(0 \leq k_i \leq 20\) for \(i = 1, 2, 3, 4\), and \(k_1 + k_3 = k_2 + k_4\). | 6181 |
Consider the ellipse \[9(x-1)^2 + y^2 = 36.\]Let $A$ be one of the endpoints of its major axis, and let $B$ be one of the endpoints of its minor axis. Find the distance $AB.$ | 2\sqrt{10} |
Daphney buys 5 kg of potatoes at the supermarket. If 2 kg of potatoes costs $6, how much will she pay? | The price per kilogram of potatoes is $6 / 2 = $<<6/2=3>>3.
To buy 5 kg of potatoes, she will have to pay 5 * $3 = $<<5*3=15>>15.
#### 15 |
Find all \( t \) such that \( x-t \) is a factor of \( 10x^2 + 21x - 10 \). | -\frac{5}{2} |
Our school's girls volleyball team has 14 players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters with no restrictions? (The triplets are treated as distinguishable.) | 3003 |
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$. | \pm 1/n! |
What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A( -2, 0)$ and $B(-1,4)$? | \frac{13}{8} |
Given: The lengths of the three sides of a triangle, $a$, $b$, and $c$, are integers, and $a \leq b < c$, where $b = 5$. Calculate the number of such triangles. | 10 |
The inclination angle of the line given by the parametric equations \[
\begin{cases}
x=1+t \\
y=1-t
\end{cases}
\]
calculate the inclination angle. | \frac {3\pi }{4} |
A group of children held a grape-eating contest. When the contest was over, the winner had eaten $n$ grapes, and the child in $k$-th place had eaten $n+2-2k$ grapes. The total number of grapes eaten in the contest was $2009$. Find the smallest possible value of $n$. | 89 |
Let $P=\{1,2,\ldots,6\}$, and let $A$ and $B$ be two non-empty subsets of $P$. Find the number of pairs of sets $(A,B)$ such that the maximum number in $A$ is less than the minimum number in $B$. | 129 |
Find the value of $k$ so that
\[3 + \frac{3 + k}{4} + \frac{3 + 2k}{4^2} + \frac{3 + 3k}{4^3} + \dotsb = 8.\] | 9 |
Find four-thirds of $\frac{9}{2}$. | 6 |
Anna can read 1 page in 1 minute. Carole can read as fast as Anna but at half the speed of Brianna. How long does it take Brianna to read a 100-page book? | If Anna can read 1 page in 1 minute, then her speed is 1/1 = <<1/1=1>>1 page per min
If Carole can read at Anna's same speed then her speed is also 1 page per min
If Carole is only half the speed of Brianna, then that means Brianna is twice as fast as Carole, so 2 * 1 page per min = <<1*2=2>>2 pages per min
If a book is 100 pages long and Brianna can read 2 pages per minute, then it will take her 100/2 = <<100/2=50>>50 minutes to read the book
#### 50 |
Manya has a stack of $85=1+4+16+64$ blocks comprised of 4 layers (the $k$ th layer from the top has $4^{k-1}$ blocks). Each block rests on 4 smaller blocks, each with dimensions half those of the larger block. Laura removes blocks one at a time from this stack, removing only blocks that currently have no blocks on top of them. Find the number of ways Laura can remove precisely 5 blocks from Manya's stack (the order in which they are removed matters). | 3384 |
Let $f(x)=x^3+3$ and $g(x) = 2x^2 + 2x +1$. What is $g(f(-2))$? | 41 |
The PTA had saved $400 set aside after a fundraising event. They spent a fourth of the money on school supplies. Then they spent half of what was left on food for the faculty. How much money did they have left? | They spent $400/4=$<<400/4=100>>100 on school supplies.
They spent $300/2=$<<300/2=150>>150 on food.
They had $300-150=$<<300-150=150>>150 left.
#### 150 |
Let \[f(x) = \left\{
\begin{array}{cl}
\sqrt{x} &\text{ if }x>4, \\
x^2 &\text{ if }x \le 4.
\end{array}
\right.\]Find $f(f(f(2)))$. | 4 |
Inside the cube \( ABCD A_1B_1C_1D_1 \), there is a center \( O \) of a sphere with a radius of 10. The sphere intersects the face \( AA_1D_1D \) along a circle of radius 1, the face \( A_1B_1C_1D_1 \) along a circle of radius 1, and the face \( CDD_1C_1 \) along a circle of radius 3. Find the length of the segment \( OD_1 \). | 17 |
Expand the product ${4(x-5)(x+8)}$. | 4x^2 + 12x - 160 |
The profit from a business transaction is shared among 2 business partners, Mike and Johnson in the ratio 2:5 respectively. If Johnson got $2500, how much will Mike have after spending some of his share on a shirt that costs $200? | According to the ratio, for every 5 parts that Johnson gets, Mike gets 2 parts
Since Johnson got $2500, each part is therefore $2500/5 = $<<2500/5=500>>500
Mike will get 2*$500 = $<<2*500=1000>>1000
After buying the shirt he will have $1000-$200 = $<<1000-200=800>>800 left
#### 800 |
An apartment complex has 4 identical buildings. Each building has 10 studio apartments, 20 2 person apartments, and 5 4 person apartments. How many people live in the apartment complex if it has 75% of its maximum occupancy? | First, we find out how many people live in each building. In addition to 10 1 person apartments, as a studio apartment is for a single person, there are 20 2 person apartments meaning there are 20*2=<<20*2=40>>40 people capable of living there.
We then find out how many people can live in the 4 person apartments by multiplying the number of available 4 person apartments by 4, performing 5*4=20 people capable of living there.
So, in 1 building there is room for 10+40+20=<<10+40+20=70>>70 people.
We multiply this by the 4 buildings in total, finding a maximum occupancy of 4*70=<<4*70=280>>280 people
We then multiply this by .75 to find out how many people would live there at 75% of maximum occupancy, performing 280*.75= <<210=210>>210 people.
#### 210 |
Tony works $2$ hours a day and is paid $\$0.50$ per hour for each full year of his age. During a six month period Tony worked $50$ days and earned $\$630$. How old was Tony at the end of the six month period? | 13 |
Lucy started with a bag of 180 oranges. She sold $30\%$ of them to Max. From the remaining, she then sold $20\%$ to Maya. Of the oranges left, she donated 10 to a local charity. Find the number of oranges Lucy had left. | 91 |
Let $Q$ be the product of the first $50$ positive odd integers. Find the largest integer $m$ such that $Q$ is divisible by $5^m.$ | 12 |
Let \(z=\frac{1+i}{\sqrt{2}}.\)What is \(\left(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}\right) \cdot \left(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}}\right)?\) | 36 |
When the square root of $x$ is cubed, the answer is 64. What is the value of $x$? | 16 |
A four-digit integer \( p \) and the four-digit integer obtained by reversing the order of the digits of \( p \) are both divisible by \( 63 \) (product of primes \( 7 \) and \( 9 \)). Additionally, \( p \) must be divisible by \( 19 \). Determine the greatest possible value of \( p \). | 5985 |
Rachel is twice as old as Rona and Collete's age is half the age of Rona's. If Rona is 8 years old, what is the difference between the age of Collete and Rachel? | Rachel's age is 8 x 2 = <<8*2=16>>16.
Collete's age is 8 / 2 = <<8/2=4>>4.
Therefore the difference between their age is 16 - 4 = <<16-4=12>>12.
#### 12 |
How many four-digit numbers, formed using the digits 0, 1, 2, 3, 4, 5 without repetition, are greater than 3410? | 132 |
What is the number of positive integers $p$ for which $-1<\sqrt{p}-\sqrt{100}<1$? | 39 |
Roberto recently received a 20% raise from his previous salary, which was already 40% higher than his starting salary. If Roberto's starting salary was $80,000, what is his current salary? | Roberto has received 2 raises. On his first raise he received a 40% bump, therefore his new salary was $80,000 * 140% = $<<80000*140*.01=112000>>112,000
On his second raise, Roberto received a 20% bump, therefore his current salary is $112,000 * 120% = $<<112000*120*.01=134400>>134,400
#### 134400 |
John needs to replace his shoes so he decides to buy a $150 pair of Nikes and a $120 pair of work boots. Tax is 10%. How much did he pay for everything? | The shoes cost $150 + $120 = $<<150+120=270>>270
The tax was $270 * .1 = $<<270*.1=27>>27
So the total cost was $270 + $27 = $<<270+27=297>>297
#### 297 |
From the set \( M = \{1, 2, \cdots, 2008\} \) of the first 2008 positive integers, a \( k \)-element subset \( A \) is chosen such that the sum of any two numbers in \( A \) cannot be divisible by the difference of those two numbers. What is the maximum value of \( k \)? | 670 |
Compute $3 \begin{pmatrix} 2 \\ -8 \end{pmatrix} - 2 \begin{pmatrix} 1 \\ -7 \end{pmatrix}$. | \begin{pmatrix} 4 \\ -10 \end{pmatrix} |
The function $g$, defined on the set of ordered pairs of positive integers, satisfies the following properties:
\[
g(x,x) = x, \quad
g(x,y) = g(y,x), \quad
(x + y) g(x,y) = yg(x, x + y).
\]
Calculate $g(18,63)$. | 126 |
$k, a_2, a_3$ and $k, b_2, b_3$ are both nonconstant geometric sequences with different common ratios. We have $$a_3-b_3=3(a_2-b_2).$$Find the sum of the common ratios of the two sequences. | 3 |
The three-digit number $2a3$ is added to the number $326$ to give the three-digit number $5b9$. If $5b9$ is divisible by 9, then $a+b$ equals | 6 |
$(Ⅰ){0.064^{-\frac{1}{3}}}+\sqrt{{{(-2)}^4}}-{(π+e)^0}-{9^{\frac{3}{2}}}×{({\frac{{\sqrt{3}}}{3}})^4}$；<br/>$(Ⅱ)\frac{{lg18+lg5-lg60}}{{{{log}_2}27×lg2-lg8}}$. | \frac{1}{3} |
Dr. Jones earns $6,000 a month. His house rental is $640 each month; his monthly food expense is $380; his electric and water bill costs 1/4 of what he makes, and his insurances cost 1/5 of what he makes. How much money does he have left after paying those four bills? | Dr. Jones pays $6000 / 4 =$ <<6000/4=1500>>1500 for his electric and water bill.
He pays $6000 / 5 = $<<6000/5=1200>>1200 for his insurances.
The total amount he pays for the bills and insurances is $1500 + $1200 = $<<1500+1200=2700>>2700.
The remaining money from his earning after paying off the bills and insurance is $6000 - $2700 = $<<6000-2700=3300>>3300.
The total amount he needs to pay for the rent and food expenses is $640 + $380 = $<<640+380=1020>>1020.
So, Dr. Jones has $3300 - $1020 = $<<3300-1020=2280>>2280 left
#### 2280 |
A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have? | 9 |
Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden? | 336 |
Given a general triangle \(ABC\) with points \(K, L, M, N, U\) on its sides:
- Point \(K\) is the midpoint of side \(AC\).
- Point \(U\) is the midpoint of side \(BC\).
- Points \(L\) and \(M\) lie on segments \(CK\) and \(CU\) respectively, such that \(LM \parallel KU\).
- Point \(N\) lies on segment \(AB\) such that \(|AN| : |AB| = 3 : 7\).
- The ratio of the areas of polygons \(UMLK\) and \(MLKNU\) is 3 : 7.
Determine the length ratio of segments \(LM\) and \(KU\). | \frac{1}{2} |
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8: 7$. Find the minimum possible value of their common perimeter.
| 676 |
Bob's password consists of a positive single-digit number followed by a letter and another positive single-digit number. What is the probability that Bob's password consists of an even single-digit number followed by a vowel (from A, E, I, O, U) and a number greater than 5? | \frac{40}{1053} |
In the right square prism $M-ABCD$, the base $ABCD$ is a rectangle, $MD \perp$ face $ABCD$ with $MD$ being an integer, and the lengths of $MA$, $MC$, and $MB$ are three consecutive even numbers. What is the volume of the right square prism $M-ABCD$? | $24 \sqrt{5}$ |
In the diagram, points $U$, $V$, $W$, $X$, $Y$, and $Z$ lie on a straight line with $UV=VW=WX=XY=YZ=5$. Semicircles with diameters $UZ$, $UV$, $VW$, $WX$, $XY$, and $YZ$ create the shape shown. What is the area of the shaded region?
[asy]
size(5cm); defaultpen(fontsize(9));
pair one = (1, 0);
pair u = (0, 0); pair v = u + one; pair w = v + one; pair x = w + one; pair y = x + one; pair z = y + one;
path region = u{up}..{down}z..{up}y..{down}x..{up}w..{down}v..{up}u--cycle;
filldraw(region, gray(0.75), linewidth(0.75));
draw(u--z, dashed + linewidth(0.75));
// labels
label("$U$", u, W); label("$Z$", z, E);
label("$V$", v, 0.8 * SE); label("$X$", x, 0.8 * SE);
label("$W$", w, 0.8 * SW); label("$Y$", y, 0.8 * SW);
[/asy] | \frac{325}{4}\pi |
What is the probability of rolling eight standard, six-sided dice and getting exactly three pairs of identical numbers, while the other two numbers are distinct from each other and from those in the pairs? Express your answer as a common fraction. | \frac{525}{972} |
Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have? | 61 |
Given a set of points in space, a *jump* consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$ . Find the smallest number $n$ such that for any set of $n$ lattice points in $10$ -dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
*Author: Anderson Wang* | 1025 |
A large shopping mall designed a lottery activity to reward its customers. In the lottery box, there are $8$ small balls of the same size, with $4$ red and $4$ black. The lottery method is as follows: each customer draws twice, picking two balls at a time from the lottery box each time. Winning is defined as drawing two balls of the same color, while losing is defined as drawing two balls of different colors.
$(1)$ If it is specified that after the first draw, the balls are put back into the lottery box for the second draw, find the distribution and mathematical expectation of the number of wins $X$.
$(2)$ If it is specified that after the first draw, the balls are not put back into the lottery box for the second draw, find the distribution and mathematical expectation of the number of wins $Y$. | \frac{6}{7} |
Let $a_1,$ $a_2,$ $\dots,$ $a_{12}$ be positive real numbers such that $a_1 + a_2 + \dots + a_{12} = 1.$ Find the minimum value of
\[\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_{12}}.\] | 144 |
Given an isosceles triangle with side lengths of $4x-2$, $x+1$, and $15-6x$, its perimeter is ____. | 12.3 |
A $3 \times 3$ table is initially filled with zeros. In one move, any $2 \times 2$ square in the table is chosen, and all zeros in it are replaced with crosses, and all crosses with zeros. Let's call a "pattern" any arrangement of crosses and zeros in the table. How many different patterns can be obtained as a result of such moves? Patterns that can be transformed into each other by a $90^\circ$ or $180^\circ$ rotation are considered different. | 16 |
A spider is on the edge of a ceiling of a circular room with a radius of 65 feet. The spider moves straight across the ceiling to the opposite edge, passing through the circle's center. It then moves directly to another point on the edge of the circle, not passing through the center. The final segment of the journey is straight back to the starting point and is 90 feet long. How many total feet did the spider travel during the entire journey? | 220 + 20\sqrt{22} |
A container holds one liter of wine, and another holds one liter of water. From the first container, we pour one deciliter into the second container and mix thoroughly. Then, we pour one deciliter of the mixture back into the first container. Calculate the limit of the amount of wine in the first container if this process is repeated infinitely many times, assuming perfect mixing with each pour and no loss of liquid. | \frac{1}{2} |
Consider the system of equations:
\begin{align*}
8x - 5y &= a, \\
10y - 15x &= b.
\end{align*}
If this system has a solution \((x, y)\) where both \(x\) and \(y\) are nonzero, calculate \(\frac{a}{b}\), assuming \(b\) is nonzero. | \frac{8}{15} |
Two trucks are transporting identical sacks of flour from France to Spain. The first truck carries 118 sacks, and the second one carries only 40. Since the drivers of these trucks lack the pesetas to pay the customs duty, the first driver leaves 10 sacks with the customs officers, after which they only need to pay 800 pesetas. The second driver does similarly, but he leaves only 4 sacks and the customs officer pays him an additional 800 pesetas.
How much does each sack of flour cost, given that the customs officers take exactly the amount of flour needed to pay the customs duty in full? | 1600 |
Let the odd function $f(x)$ defined on $\mathbb{R}$ satisfy $f(2-x) = f(x)$, and it is monotonically decreasing on the interval $[0, 1)$. If the equation $f(x) = -1$ has real roots in the interval $[0, 1)$, then the sum of all real roots of the equation $f(x) = 1$ in the interval $[-1, 7]$ is $\_\_\_\_\_\_$. | 12 |
Xiaoming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, due to some reasons, Xiaoming first took the subway and then transferred to the bus, taking 40 minutes to reach the school. The transfer process took 6 minutes. Calculate the time Xiaoming spent on the bus that day. | 10 |
The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have. | 294 |
In a kindergarten class, there are two (small) Christmas trees and five children. The teachers want to divide the children into two groups to form a ring around each tree, with at least one child in each group. The teachers distinguish the children but do not distinguish the trees: two configurations are considered identical if one can be converted into the other by swapping the trees (along with the corresponding groups) or by rotating each group around its tree. In how many ways can the children be divided into groups? | 50 |
A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,2), (2,2), and (2,0). What is the probability that $x+y < 3$? | \dfrac{7}{8} |
One hundred points labeled 1 to 100 are arranged in a $10 \times 10$ grid such that adjacent points are one unit apart. The labels are increasing left to right, top to bottom (so the first row has labels 1 to 10 , the second row has labels 11 to 20, and so on). Convex polygon $\mathcal{P}$ has the property that every point with a label divisible by 7 is either on the boundary or in the interior of $\mathcal{P}$. Compute the smallest possible area of $\mathcal{P}$. | 63 |
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