problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Evaluate $(a^b)^a - (b^a)^b$ for $a = 2$ and $b = 3$. | -665 |
Calculate the sum of the series $1-2-3+4+5-6-7+8+9-10-11+\cdots+1998+1999-2000-2001$. | 2001 |
How many different positive integers can be represented as a difference of two distinct members of the set $\{1, 2, 3, \ldots, 14, 15, 16 \}?$ | 15 |
The expression $\circ \ 1\ \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
*Proposed by Aaron Lin* | 1006 |
Given the ellipse $\Gamma$: $\dfrac {x^{2}}{a^{2}}+y^{2}=1(a > 1)$, its left focus is $F_{1}$, the right vertex is $A_{1}$, and the top vertex is $B_{1}$. The circle $P$ that passes through points $F_{1}$, $A_{1}$, and $B_{1}$ has its center coordinates at $\left( \dfrac { \sqrt {3}- \sqrt {2}}{2}, \dfrac {1- \sqrt {6}}{2}\right)$.
(Ⅰ) Find the equation of the ellipse;
(Ⅱ) If the line $l$: $y=kx+m$ ($k,m$ are constants, $k\neq 0$) intersects the ellipse $\Gamma$ at two distinct points $M$ and $N$.
(i) When the line $l$ passes through $E(1,0)$, and $\overrightarrow{EM}+2 \overrightarrow{EN}= \overrightarrow{0}$, find the equation of the line $l$;
(ii) When the distance from the origin $O$ to the line $l$ is $\dfrac { \sqrt {3}}{2}$, find the maximum area of $\triangle MON$. | \dfrac { \sqrt {3}}{2} |
Let $A_{11}$ denote the answer to problem 11. Determine the smallest prime $p$ such that the arithmetic sequence $p, p+A_{11}, p+2 A_{11}, \ldots$ begins with the largest possible number of primes. | 7 |
Simplify $(2 \times 10^9) - (6 \times 10^7) \div (2 \times 10^2)$. | 1999700000 |
The Eat "N Go Mobile Sausage Sandwich Shop specializes in the sale of spicy sausage sandwiches served on a toasted bun. Each sausage sandwich comes with four strips of jalapeno pepper, one sausage patty, and a tablespoon of Stephen's famous special sauce. If a single jalapeno pepper makes 8 slices, and the Sandwich Shop serves a sandwich every 5 minutes, how many jalapeno peppers are required by the Sandwich Shop to serve all customers during an 8-hour day? | If each sandwich comes with 4 strips of pepper, and each pepper makes 8 slices, then each sandwich uses 4/8=<<4/8=0.5>>0.5 pepper.
If the Sandwich Shop serves a sandwich every 5 minutes, then per hour the Shop serves 60/5=<<60/5=12>>12 sandwiches.
Thus, per hour, the Shop uses 12*0.5=<<12*0.5=6>>6 jalapeno peppers.
And in an 8-hour day, the Sandwich Shop uses 8*6=<<8*6=48>>48 jalapeno peppers.
#### 48 |
Point $P$ is inside equilateral $\triangle ABC$. Points $Q$, $R$, and $S$ are the feet of the perpendiculars from $P$ to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. Given that $PQ=1$, $PR=2$, and $PS=3$, what is $AB$ in terms of radicals? | 4\sqrt{3} |
Simplify $((5p+1)-2p\cdot4)(3)+(4-1\div3)(6p-9)$ to a much simpler expression of the form $ap-b$ , where $a$ and $b$ are positive integers. | 13p-30 |
If $f(x) = x^2$ and $g(x) = 3x + 4$, what is $f(g(-3))$? | 25 |
What is the product of all real numbers that are tripled when added to their reciprocals? | -\frac{1}{2} |
A family has three adults and children, both girls and boys. They went out for a family trip and prepared 3 dozen boiled eggs for the trip. Every adult got 3 eggs, and the boys each received 1 more egg than each girl since the girls had some snacks. How many boys went on the trip if the number of girls was 7 and each girl received an egg? | The total number of eggs the family had is 3 * 12 = <<3*12=36>>36 eggs
If each adult received 3 eggs, the total number of eggs they got is 3 * 3 = <<3*3=9>>9 eggs.
The children shared 36 - 9 = <<36-9=27>>27 eggs
Since each girl received an egg, the boys shared 27 - 7 = <<27-7=20>>20 eggs
If each boy received 1 egg more than each girl, each received 1+1 = <<1+1=2>>2 eggs
The boys received 20 eggs, and if each got 2 eggs, then 20/2 = <<20/2=10>>10 boys went on the trip
#### 10 |
A sign painter paints numbers for each of 150 houses, numbered consecutively from 1 to 150. How many times does the digit 9 appear in total on all the house numbers? | 25 |
A two-digit positive integer $x$ has the property that when 109 is divided by $x$, the remainder is 4. What is the sum of all such two-digit positive integers $x$? | 71 |
The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 25$, $b_9 = 125$, and for $n \ge 3$, $b_n$ is the geometric mean of the first $n - 1$ terms. Find $b_2$. | 625 |
What is the perimeter of the shaded region in a \( 3 \times 3 \) grid where some \( 1 \times 1 \) squares are shaded? | 10 |
One caterer charges a basic fee of $\$100$ plus $\$15$ per person. A second caterer charges a basic fee of $\$200$ plus $\$12$ per person. What is the least number of people for which the second caterer is cheaper? | 34 |
Given an integer sequence \(\{a_i\}\) defined as follows:
\[ a_i = \begin{cases}
i, & \text{if } 1 \leq i \leq 5; \\
a_1 a_2 \cdots a_{i-1} - 1, & \text{if } i > 5.
\end{cases} \]
Find the value of \(\sum_{i=1}^{2019} a_i^2 - a_1 a_2 \cdots a_{2019}\). | 1949 |
A crew of workers was tasked with pouring ice rinks on a large and a small field, where the area of the large field is twice the area of the small field. The part of the crew working on the large field had 4 more workers than the part of the crew working on the small field. When the pouring on the large rink was completed, the group working on the small field was still working. What is the maximum number of workers that could have been in the crew? | 10 |
A circle passes through the three vertices of an isosceles triangle that has two sides of length 5 and a base of length 4. What is the area of this circle? Express your answer in terms of $\pi$. | \frac{13125}{1764}\pi |
The pattern of Pascal's triangle is illustrated in the diagram shown. What is the fourth element in Row 15 of Pascal's triangle? $$
\begin{array}{ccccccccccccc}\vspace{0.1in}
\textrm{Row 0}: & \qquad & & & & & 1 & & & & & & \\ \vspace{0.1in}
\textrm{Row 1}: & \qquad & & & & 1 & & 1 & & & & &\\ \vspace{0.1in}
\textrm{Row 2}: & \qquad & & & 1 & & 2 & & 1 & & & &\\ \vspace{0.1in}
\textrm{Row 3}: & \qquad & & 1 && 3 && 3 && 1&& \\ \vspace{0.1in}
\textrm{Row 4}: & \qquad & 1&& 4 && 6 && 4 && 1
\end{array}
$$ | 455 |
Tom weighs 150 kg. He manages to hold 1.5 times his weight in each hand while wearing a weight vest weighing half his weight. How much total weight was he moving with? | He was carrying 150*1.5=<<150*1.5=225>>225 kg in each hand
So he had 225*2=<<225*2=450>>450 kg in his hands
The weight vest weighed 150*.5=<<150*.5=75>>75 kg
So he had a total of 450+75=<<450+75=525>>525 kg he was moving
#### 525 |
Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitute for $e$? | 3 |
A group of friends are sharing a bag of candy. On the first day, they eat $ rac{1}{2}$ of the candies in the bag. On the second day, they eat $ rac{2}{3}$ of the remaining candies. On the third day, they eat $ rac{3}{4}$ of the remaining candies. On the fourth day, they eat $ rac{4}{5}$ of the remaining candies. On the fifth day, they eat $ rac{5}{6}$ of the remaining candies. At the end of the fifth day, there is 1 candy remaining in the bag. How many candies were in the bag before the first day? | 720 |
In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle? | 50 |
If $a = \log 8$ and $b = \log 25,$ compute
\[5^{a/b} + 2^{b/a}.\] | 2 \sqrt{2} + 5^{2/3} |
Let $n$ be the answer to this problem. The polynomial $x^{n}+ax^{2}+bx+c$ has real coefficients and exactly $k$ real roots. Find the sum of the possible values of $k$. | 10 |
Mrs. Riley revised her data after realizing that there was an additional score bracket and a special bonus score for one of the brackets. Recalculate the average percent score for the $100$ students given the updated table:
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
100&5\\\hline
95&12\\\hline
90&20\\\hline
80&30\\\hline
70&20\\\hline
60&8\\\hline
50&4\\\hline
40&1\\\hline
\end{tabular}
Furthermore, all students scoring 95% receive a 5% bonus, which effectively makes their score 100%. | 80.2 |
The scores on a $110$-point test were organized in the stem-and-leaf plot shown. $9 | 6$ represents $96$ points. What is the mode of the scores? \begin{tabular}{c|lllllll}
\multicolumn{8}{c}{\underline{Points on the Test}}\\
5 &0 & 0 & & & & &\\
6 &3 & & & & & &\\
7 &7 & 8 & & & & &\\
8 &2 & 6 & 7 & 9 & 9 & 9 & 9\\
9 &1 & 4 & 4 & 4 & 6 & &\\
10 &0 & 0 & 0 & & & &\\
\end{tabular} | 89 |
In the diagram, $ABCD$ is a trapezoid with bases $AB$ and $CD$ such that $AB$ is parallel to $CD$ and $CD$ is three times the length of $AB$. The area of $ABCD$ is $27$. Find the area of $\triangle ABC$.
[asy]
draw((0,0)--(3,6)--(9,6)--(12,0)--cycle);
draw((3,6)--(0,0));
label("$A$",(0,0),W);
label("$B$",(3,6),NW);
label("$C$",(9,6),NE);
label("$D$",(12,0),E);
[/asy] | 6.75 |
Evaluate the expression \[ \frac{a+2}{a+1} \cdot \frac{b-1}{b-2} \cdot \frac{c + 8}{c+6} , \] given that $c = b-10$, $b = a+2$, $a = 4$, and none of the denominators are zero. | 3 |
In the diagram below, we have $AB = 24$ and $\angle ADB =90^\circ$. If $\sin A = \frac23$ and $\sin C = \frac13$, then what is $DC$?
[asy]
pair A,B,C,D;
A = (0,0);
B = (8*sqrt(5),16);
D = (8*sqrt(5),0);
C = (8*sqrt(5) + 32*sqrt(2),0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,S);
draw(rightanglemark(B,D,A,63));
[/asy] | 32\sqrt{2} |
Let $a$ and $b$ be positive real numbers. Find the minimum value of
\[a^2 + b^2 + \frac{1}{(a + b)^2}.\] | \sqrt{2} |
A rectangular floor that is $12$ feet wide and $19$ feet long is tiled with rectangular tiles that are $1$ foot by $2$ feet. Find the number of tiles a bug visits when walking from one corner to the diagonal opposite corner. | 30 |
Given that all vertices of the tetrahedron S-ABC are on the surface of sphere O, SC is the diameter of sphere O, and if plane SCA is perpendicular to plane SCB, with SA = AC and SB = BC, and the volume of tetrahedron S-ABC is 9, find the surface area of sphere O. | 36\pi |
Given $0 < \beta < \frac{\pi}{2} < \alpha < \pi$ and $\cos \left(\alpha- \frac{\beta}{2}\right)=- \frac{1}{9}, \sin \left( \frac{\alpha}{2}-\beta\right)= \frac{2}{3}$, calculate the value of $\cos (\alpha+\beta)$. | -\frac{239}{729} |
A factory produces a certain product for the Brazilian Olympic Games with an annual fixed cost of 2.5 million yuan. For every $x$ thousand units produced, an additional cost of $C(x)$ (in ten thousand yuan) is incurred. When the annual production is less than 80 thousand units, $C(x)=\frac{1}{3}x^2+10x$; when the annual production is not less than 80 thousand units, $C(x)=51x+\frac{10000}{x}-1450$. The selling price per product is 0.05 ten thousand yuan. Through market analysis, it is determined that all the products produced by the factory can be sold.
(1) Write the analytical expression of the annual profit $L$ (in ten thousand yuan) as a function of the annual production $x$ (in thousand units);
(2) At what annual production volume (in thousand units) does the factory maximize its profit from this product? | 100 |
Point $C(0,p)$ lies on the $y$-axis between $Q(0,12)$ and $O(0,0)$ as shown. Determine an expression for the area of $\triangle COB$ in terms of $p$. Your answer should be simplified as much as possible. [asy]
size(5cm);defaultpen(fontsize(9));
pair o = (0, 0); pair q = (0, 12); pair b = (12, 0);
pair a = (2, 12); pair t = (2, 0); pair c = (0, 9);
draw((-2, 0)--(15, 0), Arrow);
draw((0, -2)--(0, 15), Arrow);
draw(q--a--b);
//draw(a--t);
draw(a--c--b);
label("$Q(0, 12)$", q, W);
label("$A(2, 12)$", a, NE);
label("$B(12, 0)$", b, S);
label("$O(0, 0)$", o, SW);
label("$x$", (15, 0), E);
label("$y$", (0, 15), N);
//label("$T(2, 0)$", t, S + 0.6 * E);
label("$C(0, p)$", c, W);
[/asy] | 6p |
Triangle $ABC$ has its vertices $A$, $B$, and $C$ on the sides of a rectangle 4 units by 5 units as shown. What is the area of triangle $ABC$ in square units?
[asy]
fill((0,1)--(4,0)--(2,5)--cycle,lightgray);
for(int i=1; i < 5; ++i){
for(int k=1; k < 4; ++k){
draw((0,i)--(4,i),dashed);
draw((k,0)--(k,5),dashed);
} }
draw((0,0)--(4,0)--(4,5)--(0,5)--(0,0));
draw((0,1)--(4,0)--(2,5)--(0,1));
label("$A$",(0,1),W);
label("$B$",(4,0),SE);
label("$C$",(2,5),N);
[/asy] | 9 |
Given that $2$ boys and $4$ girls are lined up, calculate the probability that the boys are neither adjacent nor at the ends. | \frac{1}{5} |
Find the projection of the vector $\begin{pmatrix} 3 \\ 0 \\ -2 \end{pmatrix}$ onto the line
\[\frac{x}{2} = y = \frac{z}{-1}.\] | \begin{pmatrix} 8/3 \\ 4/3 \\ -4/3 \end{pmatrix} |
Three real numbers $x, y, z$ are chosen randomly, and independently of each other, between 0 and 1, inclusive. What is the probability that each of $x-y$ and $x-z$ is greater than $-\frac{1}{2}$ and less than $\frac{1}{2}$? | \frac{7}{12} |
In coordinate space, $A = (1,2,3),$ $B = (5,3,1),$ and $C = (3,4,5).$ Find the orthocenter of triangle $ABC.$ | \left( \frac{5}{2}, 3, \frac{7}{2} \right) |
A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken than on the first round. How many people were sitting at the table? | 10 |
Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
$\circ$ Art's cookies are trapezoids:
$\circ$ Roger's cookies are rectangles:
$\circ$ Paul's cookies are parallelograms:
$\circ$ Trisha's cookies are triangles:
Each friend uses the same amount of dough, and Art makes exactly $12$ cookies. Art's cookies sell for $60$ cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents? | 40 |
Solve for $c$: \[\frac{c-23}{2} = \frac{2c +5}{7}.\] | 57 |
Given a hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. One of its asymptotes is $x+\sqrt{2}y=0$. Point $M$ lies on the hyperbola, and $MF\_1 \perp x$-axis. If $F\_2$ is also a focus of the parabola $y^{2}=12x$, find the distance from $F\_1$ to line $F\_2M$. | \frac{6}{5} |
Find a constant \( A > 0 \) such that for all real \( x \geqslant A \), we have
\[ x^{3} + x + 1 \leq \frac{x^{4}}{1000000} \] | 1000000 |
In a trapezoid, the lengths of the diagonals are known to be 6 and 8, and the length of the midsegment is 5. Find the height of the trapezoid. | 4.8 |
At Academic Academy, to pass an algebra test you must score at least $80\%$. If there are 35 problems on the test, what is the greatest number you can miss and still pass? | 7 |
A set \( \mathcal{T} \) of distinct positive integers has the property that for every integer \( y \) in \( \mathcal{T}, \) the arithmetic mean of the set of values obtained by deleting \( y \) from \( \mathcal{T} \) is an integer. Given that 2 belongs to \( \mathcal{T} \) and that 3003 is the largest element of \( \mathcal{T}, \) what is the greatest number of elements that \( \mathcal{T} \) can have? | 30 |
There are 300 children in the "Young Photographer" club. In a session, they divided into 100 groups of 3 people each, and in every group, each member took a photograph of the other two members in their group. No one took any additional photographs. In total, there were 100 photographs of "boy+boy" and 56 photographs of "girl+girl." How many "mixed" groups were there, that is, groups containing both boys and girls? | 72 |
Krishna and Callum are playing a game where they earn 10 points if they win any round. If they played eight matches and Krishna won 3/4 of the matches, what is the total number of points that Callum earned? | Krishna won 3/4 * 8 rounds = <<3/4*8=6>>6 rounds.
If they played eight rounds, Callum won 8 rounds - 6 rounds = <<8-6=2>>2 rounds
If a round earns one 10 points, Callum got 2 rounds * 10 points/round = <<2*10=20>>20 points.
#### 20 |
Simplify the following expression: $(x^5+x^4+x+10)-(x^5+2x^4-x^3+12).$ Express your answer as a polynomial with the degrees of the terms in decreasing order. | -x^4+x^3+x-2 |
Barbi lost 1.5 kilograms each month for a year. Luca lost 9 kilograms every year for 11 years. How many more kilograms did Luca lose than Barbi? | Barbi = 1.5 * 12 months = <<1.5*12=18>>18 kg
Luca = 9 * 11 = <<9*11=99>>99 kg
99 - 18 = <<99-18=81>>81 kg
Luca lost 81 kg more than Barbi.
#### 81 |
A game board is constructed by shading two of the regions formed by the altitudes of an equilateral triangle as shown. What is the probability that the tip of the spinner will come to rest in a shaded region? Express your answer as a common fraction. [asy]
import olympiad; size(100); defaultpen(linewidth(0.8));
pair A = (0,0), B = (1,0), C = (0.5,sqrt(3)/2);
pair D = (A + B)/2, E = (B + C)/2, F = (C + A)/2;
pair M = intersectionpoint(A--E,B--F);
draw(A--B--C--cycle);
draw(A--E^^B--F^^C--D);
filldraw(D--M--B--cycle,fillpen=gray(0.6));
filldraw(F--M--C--cycle,fillpen=gray(0.6));
draw(M--(0.4,0.5),EndArrow(size=10));
[/asy] | \frac{1}{3} |
Let \[f(x) =
\begin{cases}
x/2 &\quad \text{if } x \text{ is even}, \\
3x+1 &\quad \text{if } x \text{ is odd}.
\end{cases}
\]What is $f(f(f(f(1))))$? | 4 |
In the country of Taxland, everyone pays a percentage of their salary as tax that is equal to the number of thousands of Tuzrics their salary amounts to. What salary is the most advantageous to have?
(Salary is measured in positive, not necessarily whole number, Tuzrics) | 50000 |
Determine the required distance between the pins and the length of the string in order to draw an ellipse with a length of 12 cm and a width of 8 cm. Find a simple rule that allows for constructing an ellipse of predetermined dimensions. | 24 |
Calculate the lengths of arcs of curves given by equations in polar coordinates.
$$
\rho = 3(1 + \sin \varphi), -\frac{\pi}{6} \leq \varphi \leq 0
$$ | 6(\sqrt{3} - \sqrt{2}) |
The quadratic equation $ax^2+20x+c=0$ has exactly one solution. If $a+c=29$, and $a<c$ find the ordered pair $(a,c)$. | (4,25) |
If \( AC = 1.5 \, \text{cm} \) and \( AD = 4 \, \text{cm} \), what is the relationship between the areas of triangles \( \triangle ABC \) and \( \triangle DBC \)? | 3/5 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, where $|\overrightarrow{a}|= \sqrt {2}$, $|\overrightarrow{b}|=2$, and $(\overrightarrow{a}-\overrightarrow{b})\perp \overrightarrow{a}$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{4} |
Given a sufficiently large positive integer \( n \), which can be divided by all the integers from 1 to 250 except for two consecutive integers \( k \) and \( k+1 \), find \( k \). | 127 |
Two farmers agree that pigs are worth $\$300$ and that goats are worth $\$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $\$390$ debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way? | \$30 |
For the function $f(x)= \sqrt {2}(\sin x+\cos x)$, the following four propositions are given:
$(1)$ There exists $\alpha\in\left(- \frac {\pi}{2},0\right)$, such that $f(\alpha)= \sqrt {2}$;
$(2)$ The graph of the function $f(x)$ is symmetric about the line $x=- \frac {3\pi}{4}$;
$(3)$ There exists $\phi\in\mathbb{R}$, such that the graph of the function $f(x+\phi)$ is centrally symmetric about the origin;
$(4)$ The graph of the function $f(x)$ can be obtained by shifting the graph of $y=-2\cos x$ to the left by $ \frac {\pi}{4}$.
Among these, the correct propositions are \_\_\_\_\_\_. | (2)(3) |
Given the function $f\left( x \right)=\frac{{{e}^{x}}-a}{{{e}^{x}}+1}\left( a\in R \right)$ defined on $R$ as an odd function.
(1) Find the range of the function $y=f\left( x \right)$;
(2) When ${{x}_{1}},{{x}_{2}}\in \left[ \ln \frac{1}{2},\ln 2 \right]$, the inequality $\left| \frac{f\left( {{x}_{1}} \right)+f\left( {{x}_{2}} \right)}{{{x}_{1}}+{{x}_{2}}} \right| < \lambda \left( \lambda \in R \right)$ always holds. Find the minimum value of the real number $\lambda$. | \frac{1}{2} |
In quadrilateral $EFGH$, $EF = 6$, $FG = 18$, $GH = 6$, and $HE = x$ where $x$ is an integer. Calculate the value of $x$. | 12 |
When rolling a fair 6-sided die, what is the probability of a 2 or 4 being rolled? | \frac{1}{3} |
Calculate $(2.1)(50.5 + 0.15)$ after increasing $50.5$ by $5\%$. What is the product closest to? | 112 |
Jim buys a wedding ring for $10,000. He gets his wife a ring that is twice that much and sells the first one for half its value. How much is he out of pocket? | The second ring cost 2*10000=$<<2*10000=20000>>20,000
He sells the first ring for 10,000/2=$<<10000/2=5000>>5000
So he is out of pocket 10000-5000=$<<10000-5000=5000>>5000
So that means he is out of pocket 20,000+5000=$<<20000+5000=25000>>25,000
#### 25000 |
Define two binary operations on real numbers where $a \otimes b = \frac{a+b}{a-b}$ and $b \oplus a = \frac{b-a}{b+a}$. Compute the value of $(8\otimes 6) \oplus 2$.
A) $\frac{5}{9}$
B) $\frac{7}{9}$
C) $\frac{12}{9}$
D) $\frac{1}{9}$
E) $\frac{14}{9}$ | \frac{5}{9} |
Toby is in a juggling contest with a friend. The winner is whoever gets the most objects rotated around in 4 minutes. Toby has 5 baseballs and each one makes 80 rotations. His friend has 4 apples and each one makes 101 rotations. How many total rotations of objects are made by the winner? | Toby gets 400 full rotations because 5 x 80 = <<5*80=400>>400
His friend get 404 rotations because 4 x 101 = <<4*101=404>>404
The winner rotated 404 objects because 404 > 400
#### 404 |
Calvin buys a pack of chips, for $0.50, from the vending machine at lunch, 5 days a week. After 4 weeks, how much money has Calvin spent on chips? | The chips cost $0.50 a bag and he buys them 5 days a week so that's .50*5 = $<<0.50*5=2.50>>2.50 a week
If he spends $2.50 on chips a week then over 4 weeks he will spend 2.50*4 = $<<2.50*4=10.00>>10.00 on chips
#### 10 |
In an organization, there are five leaders and a number of regular members. Each year, the leaders are expelled, followed by each regular member recruiting three new members to become regular members. After this, five new leaders are elected from outside the organization. Initially, the organisation had twenty people total. How many total people will be in the organization six years from now? | 10895 |
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $W$?
[asy]
path a=(0,0)--(10,0)--(10,10)--(0,10)--cycle;
path b = (0,10)--(6,16)--(16,16)--(16,6)--(10,0);
path c= (10,10)--(16,16);
path d= (0,0)--(3,13)--(13,13)--(10,0);
path e= (13,13)--(16,6);
draw(a,linewidth(0.7));
draw(b,linewidth(0.7));
draw(c,linewidth(0.7));
draw(d,linewidth(0.7));
draw(e,linewidth(0.7));
draw(shift((20,0))*a,linewidth(0.7));
draw(shift((20,0))*b,linewidth(0.7));
draw(shift((20,0))*c,linewidth(0.7));
draw(shift((20,0))*d,linewidth(0.7));
draw(shift((20,0))*e,linewidth(0.7));
draw((20,0)--(25,10)--(30,0),dashed);
draw((25,10)--(31,16)--(36,6),dashed);
draw((15,0)--(10,10),Arrow);
draw((15.5,0)--(30,10),Arrow);
label("$W$",(15.2,0),S);
label("Figure 1",(5,0),S);
label("Figure 2",(25,0),S);
[/asy] | \frac{1}{12} |
A group of adventurers displays their loot. It is known that exactly 9 adventurers have rubies; exactly 8 have emeralds; exactly 2 have sapphires; exactly 11 have diamonds. Additionally, it is known that:
- If an adventurer has diamonds, they either have rubies or sapphires (but not both simultaneously);
- If an adventurer has rubies, they either have emeralds or diamonds (but not both simultaneously).
What is the minimum number of adventurers that could be in this group? | 17 |
The positive divisors of the integer 630 (including 1 and 630) total how many? | 24 |
A store is having a sale for a change of season, offering discounts on a certain type of clothing. If each item is sold at 40% of the marked price, there is a loss of 30 yuan per item, while selling it at 70% of the marked price yields a profit of 60 yuan per item.
Find:
(1) What is the marked price of each item of clothing?
(2) To ensure no loss is incurred, what is the maximum discount that can be offered on this clothing? | 50\% |
It takes 320 rose petals to make an ounce of perfume. If each rose produces 8 petals, and there are 12 roses per bush, how many bushes will Fern have to harvest to make 20 12-ounce bottles of perfume? | First find the number of roses needed for one ounce of perfume: 320 petals / 8 petals/rose = <<320/8=40>>40 roses
Then multiply the number of roses needed for one ounce by the number of ounces per bottle to find the number of roses per bottle: 40 roses/ounce * 12 ounces/bottle = <<40*12=480>>480 roses/bottle
Then multiply that number by the number of bottles to find the total number of roses needed: 480 roses/bottle * 20 bottles = 9600 roses
Then divide that number by the number of roses per bush to find the number of bushes Fern needs to harvest: 9600 roses / 12 roses/bush = <<9600/12=800>>800 bushes
#### 800 |
In the beginning, Justine had 10 more rubber bands than Bailey but 2 fewer bands than Ylona. Bailey decided to give two bands each to Justine and Ylona so that he is left with only 8 rubber bands. How many rubber bands did Ylona have in the beginning? | Bailey gave 2 + 2 = <<2+2=4>>4 bands.
So Bailey had 8 + 4 = <<8+4=12>>12 bands in the beginning.
Justine had 12 + 10 = <<12+10=22>>22 bands in the beginning.
Thus, Ylona had 22 + 2 = <<22+2=24>>24 bands in the beginning.
#### 24 |
Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 10$ and
\[ab + ac + ad + bc + bd + cd = 20.\]
Find the largest possible value of $d.$ | \frac{5 + \sqrt{105}}{2} |
What is the distance, in units, between the points $(-3, -4)$ and $(4, -5)$? Express your answer in simplest radical form. | 5\sqrt{2} |
In triangle $ABC,\,$ angle $C$ is a right angle and the altitude from $C\,$ meets $\overline{AB}\,$ at $D.\,$ The lengths of the sides of $\triangle ABC\,$ are integers, $BD=29^3,\,$ and $\cos B=m/n\,$, where $m\,$ and $n\,$ are relatively prime positive integers. Find $m+n.\,$
| 450 |
Reagan's school has a fish tank with a total of 280 fish of two types, koi fish and goldfish. Over the next 3 weeks, the school added 2 koi fish and 5 goldfish per day. If the tank had 200 goldfish at the end of the three weeks, what's the total number of koi fish in the tank after the three weeks? | The total number of days in 3 weeks is 3 * 7 = <<3*7=21>>21 days.
If 2 koi fish are added every day, the total after 21 days is 21 * 2 = <<21*2=42>>42 koi fish.
if 5 goldfish are added every day, the total after 21 days is 5 * 21 = <<5*21=105>>105 goldfish.
The total number of fish in the tank after 3 weeks is 280 + 42 + 105 = <<280+42+105=427>>427 fish.
If there were a total of 200 goldfish in the tank after 3 weeks, then the number of koi in the tank is 427 - 200 = 227 koi fish.
#### 227 |
A number is composed of 6 millions, 3 tens of thousands, and 4 thousands. This number is written as ____, and when rewritten in terms of "ten thousands" as the unit, it becomes ____ ten thousands. | 603.4 |
Factor $46x^3-115x^7.$ | -23x^3(5x^4-2) |
According to the notice from the Ministry of Industry and Information Technology on the comprehensive promotion of China's characteristic enterprise new apprenticeship system and the strengthening of skills training, our region clearly promotes the new apprenticeship system training for all types of enterprises, deepens the integration of production and education, school-enterprise cooperation, and the apprenticeship training goal is to cultivate intermediate and senior technical workers that meet the needs of business positions. In the year 2020, a certain enterprise needs to train 200 apprentices. After the training, an assessment is conducted, and the statistics of obtaining corresponding job certificates are as follows:
| Job Certificate | Junior Worker | Intermediate Worker | Senior Worker | Technician | Senior Technician |
|-----------------|---------------|---------------------|--------------|-----------|------------------|
| Number of People | 20 | 60 | 60 | 40 | 20 |
$(1)$ Now, using stratified sampling, 10 people are selected from these 200 people to form a group for exchanging skills and experiences. Find the number of people in the exchange group who have obtained job certificates in the technician category (including technicians and senior technicians).
$(2)$ From the 10 people selected in (1) for the exchange group, 3 people are randomly chosen as representatives to speak. Let the number of technicians among these 3 people be $X$. Find the probability distribution and the mathematical expectation of the random variable $X$. | \frac{9}{10} |
Suppose that $f$ is a function and $f^{-1}$ is the inverse of $f$. If $f(3)=4$, $f(5)=1$, and $f(2)=5$, evaluate $f^{-1}\left(f^{-1}(5)+f^{-1}(4)\right)$. | 2 |
Given real numbers $a$, $b$, $c$, and $d$ satisfy $(b + 2a^2 - 6\ln a)^2 + |2c - d + 6| = 0$, find the minimum value of $(a - c)^2 + (b - d)^2$. | 20 |
Given that $\alpha$ and $\beta$ are the roots of the equation $x^2 - 3x - 2 = 0,$ find the value of $5 \alpha^4 + 12 \beta^3.$ | 672.5 + 31.5\sqrt{17} |
Given that the probability of Team A winning a single game is $\frac{2}{3}$, calculate the probability that Team A will win in a "best of three" format, where the first team to win two games wins the match and ends the competition. | \frac{16}{27} |
Let's call a number \( \mathrm{X} \) "50-supportive" if for any 50 real numbers \( a_{1}, \ldots, a_{50} \) whose sum is an integer, there is at least one number for which \( \left|a_{i} - \frac{1}{2}\right| \geq X \).
Indicate the greatest 50-supportive \( X \), rounded to the nearest hundredth based on standard mathematical rules. | 0.01 |
Carrie is trying to sneak some mashed turnips into her kids' mashed potatoes. She knows they didn't notice when she mixed 2 cups of turnips with 5 cups of potatoes. If she has 20 cups of potatoes, how many cups of turnips can she add? | First find the number of cups of potatoes Carrie adds per one cup of turnips: 5 cups potatoes / 2 cups turnips = <<5/2=2.5>>2.5 cups potatoes/cup turnips
Then divide the number of cups of potatoes she has that that ratio to find the number of cups of turnips she adds: 20 cups potatoes / 2.5 cups potatoes/cup turnips = <<20/2.5=8>>8 cups turnips
#### 8 |
Calculate $(3^5 \cdot 6^5)^2$. | 3570467226624 |
Given a hexagon \( A B C D E F \) with an area of 60 that is inscribed in a circle \( \odot O \), where \( AB = BC, CD = DE, \) and \( EF = AF \). What is the area of \( \triangle B D F \)? | 30 |
The graph of the function in the form \( y=\frac{b}{|x|-a} \) (where \( a, b > 0 \)) resembles the Chinese character "唄". It is referred to as the "唄 function", and the point symmetric to its intersection with the y-axis about the origin is called the "目 point". A circle with its center at the 明 point that intersects the 唄 function is called the "唄 circle". For \( a=b=1 \), the minimum area of all 唄 circles is . | 3\pi |
The three-digit positive integer $N$ has a ones digit of 3. What is the probability that $N$ is divisible by 3? Express your answer as a common fraction. | \frac{1}{3} |
Knowing that the system
\[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\]
has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution. | 83 |
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