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Antonio is preparing a meal of spaghetti and meatballs for his family. His recipe for meatballs calls for 1/8 of a pound of hamburger per meatball. Antonio has 8 family members, including himself. If he uses 4 pounds of hamburger to make meatballs, and each member of the family eats an equal number of meatballs, how many meatballs will Antonio eat?
|
If one meatball is made from 1/8 pound of hamburger meat, then 4 pounds of hamburger meat will make 4/(1/8)=4*8=<<4/(1/8)=32>>32 meatballs.
32 meatballs divided amongst 8 family members is 32/8=<<32/8=4>>4 meatballs per family member.
#### 4
|
Children get in for half the price of adults. The price for $8$ adult tickets and $7$ child tickets is $42$. If a group buys more than $10$ tickets, they get an additional $10\%$ discount on the total price. Calculate the cost of $10$ adult tickets and $8$ child tickets.
|
46
|
Mrs. Toad has a class of 2017 students, with unhappiness levels $1,2, \ldots, 2017$ respectively. Today in class, there is a group project and Mrs. Toad wants to split the class in exactly 15 groups. The unhappiness level of a group is the average unhappiness of its members, and the unhappiness of the class is the sum of the unhappiness of all 15 groups. What's the minimum unhappiness of the class Mrs. Toad can achieve by splitting the class into 15 groups?
|
1121
|
Jerry owes a loan shark $200. The loan shark charges Jerry 10% interest compounded monthly. (This means that every month the loan shark adds Jerry's interest amount to his loan, and the next month Jerry has to pay 10% on the new total). If Jerry doesn't make any payments, how much (in dollars) does the loan shark charge in interest the second month?
|
First calculate how much the loan shark charges in interest the first month by multiplying the loan amount by the interest rate: $200 * 10% = $<<200*10*.01=20>>20
Now add the interest amount to the original amount of the loan to find Jerry's new total: $200 + $20 = $<<200+20=220>>220
Now multiply the new total by 10% to find out how much Jerry owes the loan shark in interest the second month: $220 * 10% = $<<220*10*.01=22>>22
#### 22
|
In an addition problem where the digits were written on cards, two cards were swapped, resulting in an incorrect expression: $37541 + 43839 = 80280$. Find the error and write the correct value of the sum.
|
81380
|
A carton contains 12 boxes. If each box has 10 packs of cheese cookies, what's the price of a pack of cheese cookies if a dozen cartons cost $1440?
|
If 1 carton contains 12 boxes, then a dozen cartons contain 12*12 = 144 boxes
If 1 box has 10 packs of cookies, then 144 boxes have 144*10 = <<144*10=1440>>1440 packs
If a dozen cartons cost $1440 and contain 1440 packs of cookies, then each pack costs 1440/1440 = $<<1440/1440=1>>1
#### 1
|
Three positive integers have a sum of 72 and are in the ratio 1:3:4. What is the least of these three integers?
|
9
|
Simplify the fraction $\dfrac{88}{7744}.$
|
\dfrac{1}{88}
|
The numbers $1, 2, \dots, 16$ are randomly placed into the squares of a $4 \times 4$ grid. Each square gets one number, and each of the numbers is used once. Find the probability that the sum of the numbers in each row and each column is even.
|
\frac{36}{20922789888000}
|
In ancient China, soldiers positioned in beacon towers along the Great Wall would send smoke signals to warn of impending attacks. Since the towers were located at 5 kilometer intervals, they could send a signal the length of the Great Wall. If the Great wall was 7300 kilometers long, and every tower had two soldiers, what was the combined number of soldiers in beacon towers on the Great Wall?
|
If there were beacon towers every 5 kilometers along the 7300 kilometer length of the Great Wall, then there were 7300/5=<<7300/5=1460>>1460 beacon towers.
If every tower had two soldiers, then there were a total of 1460*2=<<1460*2=2920>>2920 soldiers in beacon towers along the Great Wall.
#### 2920
|
In $\triangle PQR$, point $T$ is on side $QR$ such that $QT=6$ and $TR=10$. What is the ratio of the area of $\triangle PQT$ to the area of $\triangle PTR$? [asy]
size(6cm);
pair q = (0, 0); pair t = (6, 0); pair r = (16, 0);
pair p = (4, 8);
draw(p--q--r--cycle--t);
label("$P$", p, N);
label("$Q$", q, SW);
label("$T$", t, S);
label("$R$", r, SE);
label("$6$", midpoint(q--t), S, fontsize(10));
label("$10$", midpoint(t--r), S, fontsize(10));
[/asy] Write your answer in the form $x:y$, where $x$ and $y$ are relatively prime positive integers.
|
3:5
|
Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first $40$ miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of $0.02$ gallons per mile. On the whole trip he averaged $55$ miles per gallon. How long was the trip in miles?
|
440
|
Solve for $r$: $r = \displaystyle\frac{\sqrt{5^2+12^2}}{\sqrt{16+9}}$. Express as a common fraction.
|
\frac{13}{5}
|
For the "Skillful Hands" club, Pavel needs to cut several identical pieces of wire (the length of each piece is an integer number of centimeters). Initially, Pavel took a wire piece of 10 meters and was able to cut only 15 pieces of the required length. Then, Pavel took a piece that was 40 centimeters longer, but it was also enough for only 15 pieces. What length should the pieces be? Express the answer in centimeters.
|
66
|
Given the inequality $x\ln x - kx > 3$, which holds true for any $x > 1$, determine the maximum value of the integer $k$.
|
-3
|
Find the smallest square in which 5 circles, each with a radius of 1, can be arranged so that no two circles share any interior points.
|
2\sqrt{2} + 2
|
If $2^{10} \cdot 2^{15}$ is expressed as some integer to the fifth power, what is that integer?
|
32
|
A bulk warehouse is offering 48 cans of sparkling water for $12.00 a case. The local grocery store is offering the same sparkling water for $6.00 and it only has 12 cans. How much more expensive, per can, in cents, is this deal at the grocery store?
|
The bulk warehouse has 48 cans for $12.00 so that's 12/48 = $0.25 a can
The local grocery store has 12 cans for $6.00 so that's 6/12 = $0.50 a can
The grocery store offer is $0.50 a can and the warehouse is $0.25 a can so the grocery store is .50-.25 = $<<0.50-.25=0.25>>0.25 more expensive per can
#### 25
|
There are five students: A, B, C, D, and E.
1. In how many different ways can they line up in a row such that A and B must be adjacent, and C and D cannot be adjacent?
2. In how many different ways can these five students be distributed into three classes, with each class having at least one student?
|
150
|
A grocery store has 4 kinds of jelly. They sell grape jelly twice as much as strawberry jelly, and raspberry jelly twice as much as plum jelly. The raspberry jelly sells a third as much as the grape jelly. If they sold 6 jars of plum jelly today, how many jars of strawberry jelly did they sell?
|
They sell twice as much raspberry jelly as plum jelly, so they sold 2 * 6 = <<2*6=12>>12 jars of raspberry jelly today.
The raspberry jelly sells a third as much as the grape jelly, so they sold 12 * 3 = <<12*3=36>>36 jars of grape jelly today.
The grape jelly sells twice as much as the strawberry jelly, so they sold 36 / 2 = <<36/2=18>>18 jars of strawberry jelly today.
#### 18
|
Given the function $f(x)= \begin{cases} \sin \frac {π}{2}x,-4\leqslant x\leqslant 0 \\ 2^{x}+1,x > 0\end{cases}$, find the zero point of $y=f[f(x)]-3$.
|
x=-3
|
In triangle $ABC$, $AB = 3$, $AC = 5$, and $BC = 4$. The medians $AD$, $BE$, and $CF$ of triangle $ABC$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$.
[asy]
import geometry;
unitsize(1 cm);
pair A, B, C, D, E, F, G, P, Q, R;
A = (0,3);
B = (0,0);
C = (4,0);
D = (B + C)/2;
E = (C + A)/2;
F = (A + B)/2;
G = (A + B + C)/3;
P = (G + reflect(B,C)*(G))/2;
Q = (G + reflect(C,A)*(G))/2;
R = (G + reflect(A,B)*(G))/2;
draw(A--B--C--cycle);
draw(A--G);
draw(B--G);
draw(C--G);
draw(G--P);
draw(G--Q);
draw(G--R);
label("$A$", A, dir(90));
label("$B$", B, SW);
label("$C$", C, SE);
label("$G$", G, SE);
label("$P$", P, S);
label("$Q$", Q, NE);
label("$R$", R, W);
[/asy]
|
\frac{47}{15}
|
A circle with diameter $\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The length of $\overline{AB}$ can be written in the form $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.
Note: The diagram was not given during the actual contest.
|
312
|
Call a positive integer 'mild' if its base-3 representation never contains the digit 2. How many values of $n(1 \leq n \leq 1000)$ have the property that $n$ and $n^{2}$ are both mild?
|
7
|
There are 5 houses on a street, and each of the first four houses has 3 gnomes in the garden. If there are a total of 20 gnomes on the street, how many gnomes does the fifth house have?
|
In the first four houses, there are a total of 4 houses * 3 gnomes = <<4*3=12>>12 gnomes.
Therefore, the fifth house had 20 total gnomes – 12 gnomes = <<20-12=8>>8 gnomes.
#### 8
|
An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 2 \angle D$ and $\angle BAC = k \pi$ in radians, then find $k$.
[asy]
import graph;
unitsize(2 cm);
pair O, A, B, C, D;
O = (0,0);
A = dir(90);
B = dir(-30);
C = dir(210);
D = extension(B, B + rotate(90)*(B), C, C + rotate(90)*(C));
draw(Circle(O,1));
draw(A--B--C--cycle);
draw(B--D--C);
label("$A$", A, N);
label("$B$", B, SE);
label("$C$", C, SW);
label("$D$", D, S);
[/asy]
|
3/7
|
Let point $O$ be the origin of a three-dimensional coordinate system, and let points $A,$ $B,$ and $C$ be located on the positive $x,$ $y,$ and $z$ axes, respectively. If $OA = \sqrt[4]{75}$ and $\angle BAC = 30^\circ,$ then compute the area of triangle $ABC.$
|
\frac{5}{2}
|
James takes up dancing for fitness. He loses twice as many calories per hour as he did when he was walking. He dances twice a day for .5 hours each time and he does this 4 times a week. He burned 300 calories an hour walking. How many calories does he lose a week from dancing?
|
He burns 2*300=<<2*300=600>>600 calories an hour
He dances for .5*2=<<.5*2=1>>1 hour per day
So he burns 600*1=<<600=600>>600 calories per day
So he burns 600*4=<<600*4=2400>>2400 calories per week
#### 2400
|
Evaluate $\left\lceil\sqrt{140}\right\rceil$.
|
12
|
How many three-digit numbers are there in which each digit is greater than the digit to its right?
|
84
|
How many ways are there to choose 3 cards from a standard deck of 52 cards, if all three cards must be of different suits? (Assume that the order of the cards does not matter.)
|
8788
|
Suppose that the angles of triangle $PQR$ satisfy
\[\cos 3P + \cos 3Q + \cos 3R = 1.\]Two sides of the triangle have lengths 12 and 15. Find the maximum length of the third side.
|
27
|
Lily bought a Russian nesting doll as a souvenir. The biggest doll is 243 cm tall, and each doll is 2/3rd the size of the doll that contains it. How big is the 6th biggest doll?
|
First find the size of the second-biggest doll: 243 cm * 2/3 = <<243*2/3=162>>162 cm
Then find the size of the third-biggest doll: 162 cm * 2/3 = <<162*2/3=108>>108 cm
Then find the size of the fourth-biggest doll: 108 cm * 2/3 = <<108*2/3=72>>72 cm
Then find the size of the fifth-biggest doll: 72 cm * 2/3 = <<72*2/3=48>>48 cm
Then find the size of the sixth-biggest doll: 48 cm * 2/3 = <<48*2/3=32>>32 cm
#### 32
|
Donna can watch 8 episodes each day during weekdays. On each weekend day, she can watch three times the number of episodes she can watch on the weekdays. How many episodes can she watch in a week?
|
Donna can finish a total of 8 x 5 = <<8*5=40>>40 episodes during the weekdays.
Meanwhile, she can finish 8 x 3 = <<8*3=24>>24 episodes each day during the weekend.
The total number of episodes she can watch every weekend is 24 x 2 = <<24*2=48>>48
Therefore, Donna can watch 40 + 48 = <<40+48=88>>88 episodes every week.
#### 88
|
Transport Teams A and B need to deliver a batch of relief supplies to an earthquake-stricken area. Team A can transport 64.4 tons per day, which is 75% more than Team B can transport per day. If both teams transport the supplies simultaneously, when Team A has transported half of the total supplies, it has transported 138 tons more than Team B. How many tons of relief supplies are there in total?
|
644
|
A triangle $\bigtriangleup ABC$ has vertices lying on the parabola defined by $y = x^2 + 4$. Vertices $B$ and $C$ are symmetric about the $y$-axis and the line $\overline{BC}$ is parallel to the $x$-axis. The area of $\bigtriangleup ABC$ is $100$. $A$ is the point $(2,8)$. Determine the length of $\overline{BC}$.
|
10
|
The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$?
|
799
|
Find constants $b_1, b_2, b_3, b_4, b_5, b_6, b_7$ such that
\[
\cos^7 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta + b_7 \cos 7 \theta
\]
for all angles $\theta$, and compute $b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2 + b_7^2$.
|
\frac{1716}{4096}
|
If $\frac{1}{8}$ of $2^{32}$ equals $8^y$, what is the value of $y$?
|
9.67
|
Find the remainder when $2^{2^{2^2}}$ is divided by $500$.
|
536
|
Sanya can wash 7 bath towels in one wash, which will take 1 hour. She only has 2 hours in a day to do this task. If she has 98 bath towels, how many days will she need to wash all of them?
|
Every day, Sanya can wash 7 x 2 = <<7*2=14>>14 bath towels.
To wash all the towels, she will need 98 / 14 = <<98/14=7>>7 days.
#### 7
|
The germination rate of cotton seeds is $0.9$, and the probability of developing into strong seedlings is $0.6$,
$(1)$ If two seeds are sown per hole, the probability of missing seedlings in this hole is _______; the probability of having no strong seedlings in this hole is _______.
$(2)$ If three seeds are sown per hole, the probability of having seedlings in this hole is _______; the probability of having strong seedlings in this hole is _______.
|
0.936
|
The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 120^{\circ}$ and $\angle AOB =
140^{\circ}$, as shown. What is the degree measure of $\angle
ABC$?
[asy]
pair A,B,C;
draw(Circle((0,0),20),linewidth(0.7));
label("$O$",(0,0),S);
A=(-16,-12);
C=(16,-12);
B=(3,19.7);
draw(A--B--C--cycle,linewidth(0.7));
label("$140^{\circ}$",(0,0),W);
label("$120^{\circ}$",(0,0.3),E);
draw(C--(0,0)--B);
draw(A--(0,0));
label("$A$",A,SW);
label("$B$",B,NE);
label("$C$",C,SE);
[/asy]
|
50^{\circ}
|
If $3^{x}=5$, what is the value of $3^{x+2}$?
|
45
|
How many numbers can you get by multiplying two or more distinct members of the set $\{1,2,3,5,11\}$ together?
|
15
|
When Cheenu was a young man, he could run 20 miles in 4 hours. In his middle age, he could jog 15 miles in 3 hours and 45 minutes. Now, as an older man, he walks 12 miles in 5 hours. What is the time difference, in minutes, between his current walking speed and his running speed as a young man?
|
13
|
Calculate:<br/>$(Ⅰ)(1\frac{9}{16})^{0.5}+0.01^{-1}+(2\frac{10}{27})^{-\frac{2}{3}}-2π^0+\frac{3}{16}$;<br/>$(Ⅱ)\log _{2}(4\times 8^{2})+\log _{3}18-\log _{3}2+\log _{4}3\times \log _{3}16$.
|
12
|
Doris works at the Widget Factory in the packing department. She puts 3 widgets in each carton, which are 4 inches wide, 4 inches long, and 5 inches tall. She then packs those cartons into a shipping box before sending it to the loading bay. The shipping boxes are 20 inches wide, 20 inches long, and 20 inches high. How many widgets get shipped in each shipping box?
|
Each carton has an area of 4*4*5 = <<4*4*5=80>>80 square inches.
Each shipping box has an area of 20*20*20 = <<20*20*20=8000>>8000 square inches
The total number of cartons that will fit into each box is 8000/80 = <<8000/80=100>>100
Since there are 3 widgets in each carton, the total number of cartons in each box will be 3*100 = <<3*100=300>>300
#### 300
|
Three people are sitting in a row of eight seats. If there must be empty seats on both sides of each person, then the number of different seating arrangements is.
|
24
|
Given the function $f(x)=x^{2}\cos \frac {πx}{2}$, the sequence {a<sub>n</sub>} is defined as a<sub>n</sub> = f(n) + f(n+1) (n ∈ N*), find the sum of the first 40 terms of the sequence {a<sub>n</sub>}, denoted as S<sub>40</sub>.
|
1680
|
When a polynomial $p(x)$ is divided by $x + 1,$ the remainder is 5. When $p(x)$ is divided by $x + 5,$ the remainder is $-7.$ Find the remainder when $p(x)$ is divided by $(x + 1)(x + 5).$
|
3x + 8
|
Given set \( S \) such that \( |S| = 10 \). Let \( A_1, A_2, \cdots, A_k \) be non-empty subsets of \( S \), and the intersection of any two subsets contains at most two elements. Find the maximum value of \( k \).
|
175
|
Elena intends to buy 7 binders priced at $\textdollar 3$ each. Coincidentally, a store offers a 25% discount the next day and an additional $\textdollar 5$ rebate for purchases over $\textdollar 20$. Calculate the amount Elena could save by making her purchase on the day of the discount.
|
10.25
|
Let $p>q$ be primes, such that $240 \nmid p^4-q^4$ . Find the maximal value of $\frac{q} {p}$ .
|
2/3
|
One of the following 8 figures is randomly chosen. What is the probability that the chosen figure is a triangle? [asy]
size(8cm);
path tri = (0, 0)--(1, 0)--(0.5, Sin(60))--cycle;
path circ = shift((0.5, 0.5)) * (scale(0.5) * unitcircle);
path sq = unitsquare;
pair sf = (1.9, 0); // Shift factor
draw(sq); draw(shift(sf) * tri); draw(shift(2 * sf) * circ); draw(shift(3 * sf) * tri);
draw(shift(4 * sf) * sq); draw(shift(5 * sf) * circ); draw(shift(6 * sf) * tri); draw(shift(7 * sf) * sq);
[/asy]
|
\frac38
|
The set of points with spherical coordinates of the form
\[(\rho, \theta, \phi) = \left( 1, \theta, \frac{\pi}{6} \right)\]forms a circle. Find the radius of this circle.
|
\frac{1}{2}
|
What is the nearest integer to $(2+\sqrt3)^4$?
|
194
|
a) Vanya flips a coin 3 times, and Tanya flips a coin 2 times. What is the probability that Vanya gets more heads than Tanya?
b) Vanya flips a coin $n+1$ times, and Tanya flips a coin $n$ times. What is the probability that Vanya gets more heads than Tanya?
|
\frac{1}{2}
|
Given that point $P$ is a moving point on circle $C$: $x^{2}+y^{2}-2x-4y+1=0$, the maximum distance from point $P$ to a certain line $l$ is $6$. If a point $A$ is taken arbitrarily on line $l$ to form a tangent line $AB$ to circle $C$, with $B$ being the point of tangency, then the minimum value of $AB$ is _______.
|
2\sqrt{3}
|
Select 5 different numbers from $0,1,2,3,4,5,6,7,8,9$ to form a five-digit number such that this five-digit number is divisible by $3$, $5$, $7$, and $13$. What is the largest such five-digit number?
|
94185
|
Determine the greatest number m such that the
system $x^2$ + $y^2$ = 1; | $x^3$ - $y^3$ |+|x-y|= $m^3$ has a solution.
|
\sqrt[3]{2}
|
Mrs. Smith wanted to buy wears worth $500. She went to a boutique with the $500 but by the time she had picked out everything she liked, she realized that she would need two-fifths more money than she had. If the shop owner gave her a discount of 15%, how much more money will she still need?
|
Two fifths of $500 is (2/5)*$500 = $<<(2/5)*500=200>>200
She needed $200 more than $500 which is $200+ $500 = $<<200+500=700>>700
15% of $700 is (15/100)*$700 = $<<(15/100)*700=105>>105
She was given a $105 discount so she has to pay $700-$105 = $595
She would still need $595-$500= $<<595-500=95>>95
#### 95
|
In the process of making a steel cable, it was determined that the length of the cable is the same as the curve given by the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=10 \\
x y+y z+x z=18
\end{array}\right.
$$
Find the length of the cable.
|
4 \pi \sqrt{\frac{23}{3}}
|
There are 60 ridges on a vinyl record. Jerry has 4 cases, each with 3 shelves that can hold 20 records each. If his shelves are 60% full, how many ridges are there on all his records?
|
First find the total number of records that can fit on all the shelves: 4 cases * 3 shelves/case * 20 records/shelf = <<4*3*20=240>>240 records
Then multiply that number by 60% to find the number of records actually on the shelves: 240 records * 60% = <<240*60*.01=144>>144 records
Then multiply that number by 60 to find the number of ridges on all the records: 144 records * 60 ridges/record = <<144*60=8640>>8640 ridges
#### 8640
|
\(5, 6, 7\)
|
21
|
A ball is dropped from a height of 150 feet. Each time it hits the ground, it rebounds to 40% of the height it fell. How many feet will the ball have traveled when it hits the ground the fifth time?
|
344.88
|
Solomon collected three times as many cans as Juwan. Levi collected half of what Juwan collected. Solomon collected 66 cans. How many cans did the boys collect in all?
|
Solomon is 3*Juwan so Juwan is 1/3 of Solomon = 66/3 = 22
Levi collected half of 22 = <<22-11=11>>11
Combined the boys collected 66 + 22 + 11 = <<66+22+11=99>>99 cans in all.
#### 99
|
The mayor of a city wishes to establish a transport system with at least one bus line, in which:
- each line passes exactly three stops,
- every two different lines have exactly one stop in common,
- for each two different bus stops there is exactly one line that passes through both.
Determine the number of bus stops in the city.
|
$3,7$
|
In how many ways can 8 identical rooks be placed on an $8 \times 8$ chessboard symmetrically with respect to the diagonal that passes through the lower-left corner square?
|
139448
|
Let $0 \le a,$ $b,$ $c,$ $d \le 1.$ Find the possible values of the expression
\[\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - a)^2}.\]
|
[2 \sqrt{2},4]
|
Let $f$ be a function satisfying $f(xy) = f(x)/y$ for all positive real numbers $x$ and $y$. If $f(500) = 3$, what is the value of $f(600)$?
|
\frac{5}{2}
|
There are 14 kindergartners and 24 first graders and some second graders in Ms. Watson's class. Overall Ms. Watson has 42 students. How many second graders are in Ms. Watson’s class?
|
There are 14+24 = <<14+24=38>>38 kindergartners and first graders in Ms. Watson’s class.
There are 42-38 = <<42-38=4>>4 second graders in Ms. Watson’s class.
#### 4
|
Given circle $O$, points $E$ and $F$ are located such that $E$ and $F$ are on opposite sides of the diameter $\overline{AB}$. If $\angle AOE = 60^\circ$ and $\angle BOF = 30^\circ$, find the ratio of the area of sector $EOF$ to the area of the circle.
|
\frac{3}{4}
|
In each cell of a $5 \times 5$ table, a natural number is written with invisible ink. It is known that the sum of all the numbers is 200, and the sum of the three numbers inside any $1 \times 3$ rectangle is 23. What is the value of the central number in the table?
|
16
|
In how many ways can the digits of $45,\!502,\!2$ be arranged to form a 6-digit number? (Remember, numbers cannot begin with 0.)
|
150
|
What is $\log_{7}{2400}$ rounded to the nearest integer?
|
4
|
Evaluate $i^{11} + i^{111}$.
|
-2i
|
The sequences where each term is a real number are denoted as $\left\{a_{n}\right\}$, with the sum of the first $n$ terms recorded as $S_{n}$. Given that $S_{10} = 10$ and $S_{30} = 70$, what is the value of $S_{40}$?
|
150
|
A box contains $6$ balls, of which $3$ are yellow, $2$ are blue, and $1$ is red. Three balls are drawn from the box.
$(1)$ If the $3$ yellow balls are numbered as $A$, $B$, $C$, the $2$ blue balls are numbered as $d$, $e$, and the $1$ red ball is numbered as $x$, use $(a,b,c)$ to represent the basic event. List all the basic events of this experiment.
$(2)$ Calculate the probability of having exactly two yellow balls.
$(3)$ Calculate the probability of having at least $1$ blue ball.
|
\frac{4}{5}
|
Two distinct positive integers from 1 to 60 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 7?
|
\frac{148}{590}
|
Seven couples are at a social gathering. If each person shakes hands exactly once with everyone else except their spouse and one other person they choose not to shake hands with, how many handshakes were exchanged?
|
77
|
How many natural numbers between 150 and 300 are divisible by 9?
|
17
|
Given that the first tank is $\tfrac{3}{4}$ full of oil and the second tank is empty, while the second tank becomes $\tfrac{5}{8}$ full after oil transfer, determine the ratio of the volume of the first tank to that of the second tank.
|
\frac{6}{5}
|
Jared counted 15% fewer cars than his sister Ann while they were watching the road from the school, and Ann counted 7 more cars than their friend Alfred. If Jared counted 300 cars, how many cars did all of them count?
|
If Jared counted 300 cars, and Ann counted 15% more, then Ann counted 1.15 * 300 = 345 cars.
If Ann counted 7 cars more than their friend Alfred, then Alfred counted 345 - 7 = 338 cars.
The total number of cars counted by all of them will be 338 + 345 + 300 = <<338+345+300=983>>983 cars
#### 983
|
For any real number $x$, the symbol $[x]$ represents the integer part of $x$, i.e., $[x]$ is the largest integer not exceeding $x$. For example, $[2]=2$, $[2.1]=2$, $[-2.2]=-3$. This function $[x]$ is called the "floor function", which has wide applications in mathematics and practical production. Given that the function $f(x) (x \in \mathbb{R})$ satisfies $f(x)=f(2-x)$, and when $x \geqslant 1$, $f(x)=\log _{2}x$, find the value of $[f(-16)]+[f(-15)]+\ldots+[f(15)]+[f(16)]$.
|
84
|
Two mathematicians were both born in the last 500 years. Each lives (or will live) to be 100 years old, then dies. Each mathematician is equally likely to be born at any point during those 500 years. What is the probability that they were contemporaries for any length of time?
|
\frac{9}{25}
|
In triangle \(ABC\), points \(M\) and \(N\) are the midpoints of \(AB\) and \(AC\), respectively, and points \(P\) and \(Q\) trisect \(BC\). Given that \(A\), \(M\), \(N\), \(P\), and \(Q\) lie on a circle and \(BC = 1\), compute the area of triangle \(ABC\).
|
\frac{\sqrt{7}}{12}
|
Given $3\vec{a} + 4\vec{b} + 5\vec{c} = 0$ and $|\vec{a}| = |\vec{b}| = |\vec{c}| = 1$, calculate $\vec{b} \cdot (\vec{a} + \vec{c})$.
|
-\dfrac{4}{5}
|
Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a,b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, then $x=$
|
27a^2
|
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction.
|
\frac{335}{2011}
|
Given the linear function \( y = ax + b \) and the hyperbolic function \( y = \frac{k}{x} \) (where \( k > 0 \)) intersect at points \( A \) and \( B \), with \( O \) being the origin. If the triangle \( \triangle OAB \) is an equilateral triangle with an area of \( \frac{2\sqrt{3}}{3} \), find the value of \( k \).
|
\frac{2}{3}
|
Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, suppose $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
|
682
|
Simplify $\frac{84}{144}.$
|
\frac{7}{12}
|
John builds a box. The box is 26 inches by 26 inches by 14 inches. The walls are 1 inch thick on each side. How much is the internal volume in cubic feet?
|
The walls take away 2*1=<<2*1=2>>2 inches from each dimension
So the longer sides are 26-2=<<26-2=24>>24 inches
That is 24/12=<<24/12=2>>2 feet
The smaller dimension is 14-2=<<14-2=12>>12 inches
That is 12/12=<<12/12=1>>1 foot
So the internal volume is 2*2*1=<<2*2*1=4>>4 cubic feet
#### 4
|
Given that there are 6 male doctors and 3 female nurses who need to be divided into three medical teams, where each team consists of two male doctors and 1 female nurse, find the number of different arrangements.
|
540
|
Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
|
12
|
Given that $\alpha$ is an angle in the second quadrant, simplify the function $f(\alpha) = \frac{\tan(\alpha - \pi) \cos(2\pi - \alpha) \sin(-\alpha + \frac{3\pi}{2})}{\cos(-\alpha - \pi) \tan(\pi + \alpha)}$. Then, if $\cos(\alpha + \frac{\pi}{2}) = -\frac{1}{5}$, find $f(\alpha)$.
|
-\frac{2\sqrt{6}}{5}
|
When Diane turns 30, she will be half the age of Alex and twice as old as Allison. Diane is 16 years old now. What is the sum of the ages of Alex and Allison now?
|
When Dianne turns 30, Alex will be 30 x 2 = <<30*2=60>>60 years old.
Allison will be 30/2 = <<30/2=15>>15 years old then.
Dianne will turn 30 in 30 - 16 = <<30-16=14>>14 years.
Currently, the age of Alex is 60 - 14 = <<60-14=46>>46.
Currently, Allison is 15 - 14 = <<15-14=1>>1 year old.
So, the sum of Alex and Allison's ages now is 46 + 1 = <<46+1=47>>47 years old.
#### 47
|
There are 12 bananas and 4 apples in the blue basket. The red basket holds half as many fruits as the blue basket. How many fruits are in the red basket?
|
In the blue basket, there are 12 bananas + 4 apples = <<12+4=16>>16 fruits.
The red basket holds half of this so there are 16 fruits in the blue basket / 2 = <<16/2=8>>8 fruits in the red basket.
#### 8
|
Stuart is going on a cross-country trip and wants to find the fastest route. On one route the total distance is 1500 miles and the average speed is 75 MPH. On the second trip, the total distance is 750 but the average speed is 25 MPH. How long does his trip take if he goes on the fastest route?
|
The first route will take 20 hours because 1,500 / 75 = <<1500/75=20>>20
The second route will take 30 hours because 750 / 25 = <<750/25=30>>30
The fastest route takes 20 hours because 20 < 30
#### 20
|
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