problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
Calculate \(3 \cdot 15 + 20 \div 4 + 1\).
Then add parentheses to the expression so that the result is:
1. The largest possible integer,
2. The smallest possible integer. | 13 | 0.1875 | 7,912.5 | 6,908.333333 | 8,144.230769 | |
Fill the digits 1 to 9 into the boxes in the following equation such that each digit is used exactly once and the equality holds true. It is known that the two-digit number $\overrightarrow{\mathrm{DE}}$ is not a multiple of 3. Determine the five-digit number $\overrightarrow{\mathrm{ABCDE}}$. | 85132 | 0 | 8,192 | -1 | 8,192 | |
Let $\overline{MN}$ be a diameter of a circle with diameter 1. Let $A$ and $B$ be points on one of the semicircular arcs determined by $\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\frac{3}5$. Point $C$ lies on the other semicircular arc. Let $d$ be the length of the line segment whose endpoin... | 14 | 0 | 8,192 | -1 | 8,192 | |
Given the sequence $\{a_n\}$ with the sum of the first $n$ terms $S_n=1-5+9-13+17-21+......+{(-1)}^{n-1}(4n-3)$, calculate the value of $S_{15}$. | 29 | 0.625 | 5,920.875 | 4,558.2 | 8,192 | |
How many nonzero terms are in the expansion of $$(x+4)(2x^2+3x+9)-3(x^3-2x^2+7x)?$$ | 3 | 0.5625 | 3,316.5625 | 2,911.444444 | 3,837.428571 | |
A circle is inscribed in a square, then a square is inscribed in this circle. Following this, a regular hexagon is inscribed in the smaller circle and finally, a circle is inscribed in this hexagon. What is the ratio of the area of the smallest circle to the area of the original largest square? | \frac{3\pi}{32} | 0.1875 | 5,018.0625 | 6,768.333333 | 4,614.153846 | |
Mac is trying to fill 2012 barrels with apple cider. He starts with 0 energy. Every minute, he may rest, gaining 1 energy, or if he has \( n \) energy, he may expend \( k \) energy \((0 \leq k \leq n)\) to fill up to \( n(k+1) \) barrels with cider. What is the minimal number of minutes he needs to fill all the barrels... | 46 | 0 | 8,192 | -1 | 8,192 | |
Find the number of rationals $\frac{m}{n}$ such that
(i) $0 < \frac{m}{n} < 1$ ;
(ii) $m$ and $n$ are relatively prime;
(iii) $mn = 25!$ . | 256 | 0.4375 | 6,950.25 | 5,353.714286 | 8,192 | |
Define a $\textit{tasty residue}$ of $n$ to be an integer $1<a<n$ such that there exists an integer $m>1$ satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of $2016$ . | 831 | 0 | 8,143.6875 | -1 | 8,143.6875 | |
Let the real numbers \( x_{1}, x_{2}, \cdots, x_{2008} \) satisfy the condition \( \left|x_{1} - x_{2}\right| + \left|x_{2} - x_{3}\right| + \cdots + \left|x_{2007} - x_{2008}\right| = 2008 \). Define \( y_{k} = \frac{1}{k} (x_{1} + x_{2} + \cdots + x_{k}) \) for \( k = 1, 2, \cdots, 2008 \). Find the maximum value of ... | 2007 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Find the product of all the divisors of $36$. | 10077696 | 1 | 3,394.1875 | 3,394.1875 | -1 | |
There are 3 boys and 4 girls, all lined up in a row. How many ways are there for the following situations?
- $(1)$ Person A is neither at the middle nor at the ends;
- $(2)$ Persons A and B must be at the two ends;
- $(3)$ Boys and girls alternate. | 144 | 0.875 | 3,626.4375 | 3,772.928571 | 2,601 | |
The number $18!=6,402,373,705,728,000$ has many positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
A) $\frac{1}{16}$
B) $\frac{1}{18}$
C) $\frac{1}{15}$
D) $\frac{1}{20}$
E) $\frac{1}{21}$ | \frac{1}{16} | 0 | 8,192 | -1 | 8,192 | |
Regular decagon `ABCDEFGHIJ` has its center at `K`. Each of the vertices and the center are to be associated with one of the digits `1` through `10`, with each digit used once, in such a way that the sums of the numbers on the lines `AKE`, `BKF`, `CKG`, `DLH` and `EJI` are all equal. In how many ways can this be done? | 3840 | 0 | 8,178.5 | -1 | 8,178.5 | |
Round to the nearest hundredth: 18.4851 | 18.49 | 0.9375 | 355.3125 | 354.533333 | 367 | |
A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
\[a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\]for all $n \geq 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$? | 8078 | 1. **Base Case Verification**:
- We are given $a_1 = 1$ and $a_2 = \frac{3}{7}$.
- We check if these values satisfy the proposed formula $a_n = \frac{3}{4n-1}$:
- For $n=1$, $\frac{3}{4(1)-1} = \frac{3}{3} = 1$, which matches $a_1$.
- For $n=2$, $\frac{3}{4(2)-1} = \frac{3}{7}$, which matches $a_2$.
... | 0.75 | 6,154.875 | 5,475.833333 | 8,192 |
The nine points of this grid are equally spaced horizontally and vertically. The distance between two neighboring points is 1 unit. What is the area, in square units, of the region where the two triangles overlap?
[asy]
size(80);
dot((0,0)); dot((0,1));dot((0,2));dot((1,0));dot((1,1));dot((1,2));dot((2,0));dot((2,1)... | 1 | 0 | 8,128.3125 | -1 | 8,128.3125 | |
Given $f\left( \alpha \right)=\frac{\cos \left( \frac{\pi }{2}+\alpha \right)\cdot \cos \left( 2\pi -\alpha \right)\cdot \sin \left( -\alpha +\frac{3}{2}\pi \right)}{\sin \left( -\pi -\alpha \right)\sin \left( \frac{3}{2}\pi +\alpha \right)}$.
$(1)$ Simplify $f\left( \alpha \right)$; $(2)$ If $\alpha$ is an angle in t... | \frac{2 \sqrt{6}}{5} | 0 | 4,585.75 | -1 | 4,585.75 | |
A person's age at the time of their death was one 31st of their birth year. How old was this person in 1930? | 39 | 0 | 1,215.75 | -1 | 1,215.75 | |
Two circles have centers at $(3,5)$ and $(20,15)$. Both circles are tangent to the x-axis. Determine the distance between the closest points of the two circles. | \sqrt{389} - 20 | 0 | 8,145.25 | -1 | 8,145.25 | |
From the set $\{1,2,3, \cdots, 10\}$, six distinct integers are chosen at random. What is the probability that the second smallest number among the chosen integers is 3? | $\frac{1}{3}$ | 0 | 6,027.875 | -1 | 6,027.875 | |
Express $7.\overline{123}$ as a common fraction in lowest terms. | \frac{2372}{333} | 1 | 4,191.9375 | 4,191.9375 | -1 | |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace and the second card is a King? | \dfrac{4}{663} | 0.6875 | 5,402.25 | 4,582.363636 | 7,206 | |
Given that $x$ is a root of the equation $x^{2}+x-6=0$, simplify $\frac{x-1}{\frac{2}{{x-1}}-1}$ and find its value. | \frac{8}{3} | 0.0625 | 8,171.9375 | 7,871 | 8,192 | |
Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2 + y^2 = 2^k$ for some integer $k \leq n$. Determine, as a function of $n$, ... | 5n+1 | The answer is $5n+1$.
We first determine the set $P_n$. Let $Q_n$ be the set of points in $\mathbb{Z}^2$ of the form $(0, \pm 2^k)$ or $(\pm 2^k, 0)$ for some $k \leq n$. Let $R_n$ be the set of points in $\mathbb{Z}^2$ of the form $(\pm 2^k, \pm 2^k)$ for some $k \leq n$ (the two signs being chosen independently). We... | 0 | 8,192 | -1 | 8,192 |
In triangle $\triangle ABC$, $a$, $b$, $c$ are the opposite sides of the internal angles $A$, $B$, $C$, respectively, and $\sin ^{2}A+\sin A\sin C+\sin ^{2}C+\cos ^{2}B=1$.
$(1)$ Find the measure of angle $B$;
$(2)$ If $a=5$, $b=7$, find $\sin C$. | \frac{3\sqrt{3}}{14} | 0 | 5,503.625 | -1 | 5,503.625 | |
Consider the largest solution to the equation \[\log_{5x^3} 5 + \log_{25x^4} 5 = -1.\] Find the value of \( x^{10} \). | 0.0000001024 | 0 | 7,081.6875 | -1 | 7,081.6875 | |
Given the geometric sequence $\{a_n\}$, $a_3a_4a_5 = 3$ and $a_6a_7a_8 = 24$, calculate the value of $a_9a_{10}a_{11}$. | 192 | 1 | 3,422.375 | 3,422.375 | -1 | |
A wholesaler gives different discounts based on the size of the customer's order. The steps to calculate the amount payable by the customer are as follows:
S1 Input the order amount x (unit: pieces); input the unit price A (unit: yuan);
S2 If x < 250, then the discount rate d=0;
If 250 ≤ x < 500, then the discoun... | 980 | 0.5625 | 6,684.5 | 5,640.666667 | 8,026.571429 | |
For any positive integer $x$ , let $f(x)=x^x$ . Suppose that $n$ is a positive integer such that there exists a positive integer $m$ with $m \neq 1$ such that $f(f(f(m)))=m^{m^{n+2020}}$ . Compute the smallest possible value of $n$ .
*Proposed by Luke Robitaille* | 13611 | 0.25 | 8,134.6875 | 7,962.75 | 8,192 | |
Two solutions of
\[x^4 - 3x^3 + 5x^2 - 27x - 36 = 0\]are pure imaginary. Enter these solutions, separated by commas. | 3i,-3i | 0 | 3,464.8125 | -1 | 3,464.8125 | |
Let $n$ be the number of ways $10$ dollars can be changed into dimes and quarters, with at least one of each coin being used. Then $n$ equals: | 19 | To solve this problem, we need to find the number of ways to express $10$ dollars using dimes ($0.10$ dollars) and quarters ($0.25$ dollars), with the condition that at least one of each type of coin is used.
1. **Convert the total amount into cents for simplicity**: $10$ dollars = $1000$ cents.
2. **Set up the equat... | 1 | 4,910.6875 | 4,910.6875 | -1 |
Bob is coloring lattice points in the coordinate plane. Find the number of ways Bob can color five points in $\{(x, y) \mid 1 \leq x, y \leq 5\}$ blue such that the distance between any two blue points is not an integer. | 80 | We can see that no two blue points can have the same $x$ or $y$ coordinate. The blue points then must make a permutation of $1,2,3,4,5$ that avoid the pattern of $3-4-5$ triangles. It is not hard to use complementary counting to get the answer from here. There are 8 possible pairs of points that are a distance of 5 apa... | 0 | 8,192 | -1 | 8,192 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $4b\sin A= \sqrt {7}a$.
(I) Find the value of $\sin B$;
(II) If $a$, $b$, and $c$ form an arithmetic sequence with a common difference greater than $0$, find the value of $\cos A-\cos C$. | \frac { \sqrt {7}}{2} | 0 | 7,996.625 | -1 | 7,996.625 | |
On side \(BC\) of square \(ABCD\), point \(E\) is chosen such that it divides the segment into \(BE = 2\) and \(EC = 3\). The circumscribed circle of triangle \(ABE\) intersects the diagonal \(BD\) a second time at point \(G\). Find the area of triangle \(AGE\). | 43.25 | 0 | 5,711.5625 | -1 | 5,711.5625 | |
Inside the ellipse $4x^2+9y^2=144$, there exists a point $P(3, 2)$. Find the slope of the line that contains the chord with point P as its midpoint. | -\frac{2}{3} | 0.875 | 5,324.375 | 4,914.714286 | 8,192 | |
In preparation for a game of Fish, Carl must deal 48 cards to 6 players. For each card that he deals, he runs through the entirety of the following process: 1. He gives a card to a random player. 2. A player Z is randomly chosen from the set of players who have at least as many cards as every other player (i.e. Z has t... | \frac{5}{6} | After any number of cards are dealt, we see that the difference between the number of cards that any two players hold is at most one. Thus, after the first 47 cards have been dealt, there is only one possible distribution: there must be 5 players with 8 cards and 1 player with 7 cards. We have two cases: - Carl gives t... | 0 | 7,915.25 | -1 | 7,915.25 |
Divide all coins into two parts of 20 coins each and weigh them. Since the number of fake coins is odd, one of the piles will be heavier. Thus, there is at most one fake coin in that pile. Divide it into two piles of 10 coins and weigh them. If the balance is even, then all 20 coins weighed are genuine. If one of the p... | 16 | 0.0625 | 7,498.5625 | 4,974 | 7,666.866667 | |
The diagram shows two 10 by 14 rectangles which are edge-to-edge and share a common vertex. It also shows the center \( O \) of one rectangle and the midpoint \( M \) of one edge of the other. What is the distance \( OM \)?
A) 12
B) 15
C) 18
D) 21
E) 24 | 15 | 0 | 7,949.75 | -1 | 7,949.75 | |
In a cone with a height of 4 and a slant height of 8, three spheres of radius \( r \) are placed. They touch each other (externally), the lateral surface of the cone, and the first two spheres touch the base of the cone. Find the maximum value of \( r \). | \frac{12}{5+2\sqrt{3}} | 0 | 8,192 | -1 | 8,192 | |
Square $ABCD$ has side length $1$ ; circle $\Gamma$ is centered at $A$ with radius $1$ . Let $M$ be the midpoint of $BC$ , and let $N$ be the point on segment $CD$ such that $MN$ is tangent to $\Gamma$ . Compute $MN$ .
*2018 CCA Math Bonanza Individual Round #11* | \frac{5}{6} | 0.625 | 6,328.8125 | 5,210.9 | 8,192 | |
A sphere is centered at a point with integer coordinates and passes through the three points $(2,0,0)$, $(0,4,0),(0,0,6)$, but not the origin $(0,0,0)$. If $r$ is the smallest possible radius of the sphere, compute $r^{2}$. | 51 | Let $(x, y, z)$ be the center of the sphere. By the given condition, we have $$(x-2)^{2}+y^{2}+z^{2}=x^{2}+(y-4)^{2}+z^{2}=x^{2}+y^{2}+(z-6)^{2}$$ Subtracting $x^{2}+y^{2}+z^{2}$ yields $$x^{2}-(x-2)^{2}=y^{2}-(y-4)^{2}=z^{2}-(z-6)^{2}$$ or $$4(x-1)=8(y-2)=12(z-3)$$ Therefore $(x-1, y-2, z-3)$ must be $(6 t, 3 t, 2 t)$... | 0.375 | 7,051.0625 | 5,159 | 8,186.3 |
In a box, there are 8 identical balls, including 3 red balls, 4 white balls, and 1 black ball.
$(1)$ If two balls are drawn consecutively without replacement from the box, one at a time, find the probability of drawing a red ball on the first draw and then drawing a red ball on the second draw.
$(2)$ If three balls... | \frac{3}{2} | 0.9375 | 2,375.375 | 2,394.066667 | 2,095 | |
An infinite geometric series has a first term of $15$ and a second term of $5$. A second infinite geometric series has the same first term of $15$, a second term of $5+n$, and a sum of three times that of the first series. Find the value of $n$. | \frac{20}{3} | 0.9375 | 2,355.625 | 2,338.866667 | 2,607 | |
Points on a square with side length $ c$ are either painted blue or red. Find the smallest possible value of $ c$ such that how the points are painted, there exist two points with same color having a distance not less than $ \sqrt {5}$ . | $ \frac {\sqrt {10} }{2} $ | 0 | 8,083.5625 | -1 | 8,083.5625 | |
How many non- empty subsets $S$ of $\{1,2,3,\ldots ,15\}$ have the following two properties?
$(1)$ No two consecutive integers belong to $S$.
$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$.
$\mathrm{(A) \ } 277\qquad \mathrm{(B) \ } 311\qquad \mathrm{(C) \ } 376\qquad \mathrm{(D) \ } 377\... | 405 | 0 | 7,814.5 | -1 | 7,814.5 | |
The five-digit number $246\underline{\hspace{5pt}}8$ is divisible by 9. What is the missing digit? | 7 | 1 | 465 | 465 | -1 | |
The value of $\log_{10}{17}$ is between the consecutive integers $a$ and $b$. Find $a+b$. | 3 | 0.9375 | 2,914.5 | 2,562.666667 | 8,192 | |
Let $S$ be the sum of all integers $b$ for which the polynomial $x^2+bx+2008b$ can be factored over the integers. Compute $|S|$.
| 88352 | 0 | 8,192 | -1 | 8,192 | |
Linda is tasked with writing a report on extracurricular activities at her school. The school offers two clubs: Robotics and Science. Linda has a list of 30 students who are members of at least one club. She knows that 22 students are in the Robotics club and 24 students are in the Science club. If Linda picks two stud... | \frac{392}{435} | 0 | 1,722.5625 | -1 | 1,722.5625 | |
Given the equation of the parabola $y^{2}=4x$, and a line $l$ passing through its focus $F$ intersecting the parabola at points $A$ and $B$. If $S_{\triangle AOF}=3S_{\triangle BOF}$ (where $O$ is the origin), calculate the length of $|AB|$. | \dfrac {16}{3} | 0.5625 | 6,825.4375 | 5,843.222222 | 8,088.285714 | |
Allison, Brian, and Noah each have a die. All of the faces on Allison's die have a 5. The faces on Brian's die are numbered 1, 2, 3, 4, 4, 5, 5, and 6. Noah's die has an 8-sided die with the numbers 2, 2, 6, 6, 3, 3, 7, and 7. All three dice are rolled. What is the probability that Allison's roll is greater than each o... | \frac{5}{16} | 0.75 | 3,259.125 | 2,926.333333 | 4,257.5 | |
Let $n$ be a 5-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by 50. Determine the number of values of $n$ for which $q+r$ is divisible by 7. | 12600 | 0.125 | 6,470.5625 | 4,395 | 6,767.071429 | |
Evaluate the following expression as a simplified fraction. $$1+\cfrac{3}{4+\cfrac{5}{6}}$$ | \frac{47}{29} | 1 | 1,706.1875 | 1,706.1875 | -1 | |
Find the matrix that corresponds to rotating about the origin by an angle of $120^\circ$ counter-clockwise. | \begin{pmatrix} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2 & -1/2 \end{pmatrix} | 0 | 5,373.625 | -1 | 5,373.625 | |
Given the function $y = x^2 - 8x + 12$, find the vertex of the parabola and calculate the value of $y$ when $x = 3$. | -3 | 1 | 828.4375 | 828.4375 | -1 | |
30 beads (blue and green) were arranged in a circle. 26 beads had a neighboring blue bead, and 20 beads had a neighboring green bead. How many blue beads were there? | 18 | 0.125 | 8,014.5625 | 6,772.5 | 8,192 | |
The lengths of the sides of a triangle are 6 cm, 7 cm and 9 cm. In a similar triangle whose perimeter is 110 cm, what is the length of the longest side, in centimeters? | 45 | 1 | 1,465.4375 | 1,465.4375 | -1 | |
Let $p,$ $q,$ $r$ be the roots of the cubic polynomial $x^3 - 3x - 2 = 0.$ Find
\[p(q - r)^2 + q(r - p)^2 + r(p - q)^2.\] | 12 | 0 | 4,767.1875 | -1 | 4,767.1875 | |
The first three stages of a pattern are shown below, in which each line segment represents a toothpick. If the pattern continues such that at each successive stage, three toothpicks are added to the previous arrangement, how many toothpicks are necessary to create the arrangement for the 250th stage? [asy]
size(150);
d... | 751 | 0.875 | 4,268.3125 | 4,046.285714 | 5,822.5 | |
Given the numbers \( x, y, z \in \left[0, \frac{\pi}{2}\right] \), find the maximum value of the expression
\[ A = \sin(x-y) + \sin(y-z) + \sin(z-x). \] | \sqrt{2} - 1 | 0 | 8,192 | -1 | 8,192 | |
On a certain island, only knights, who always tell the truth, and liars, who always lie, live. One time, 1001 inhabitants of this island stood in a circle, and each one said: "All ten people following me in a clockwise direction are liars." How many knights could there be among those standing in the circle? | 91 | 0.375 | 7,605.125 | 6,673.333333 | 8,164.2 | |
Let \( F \) be the left focus of the ellipse \( E: \frac{x^{2}}{3}+y^{2}=1 \). A line \( l \) with a positive slope passes through point \( F \) and intersects \( E \) at points \( A \) and \( B \). Through points \( A \) and \( B \), lines \( AM \) and \( BN \) are drawn such that \( AM \perp l \) and \( BN \perp l \)... | \sqrt{6} | 0.8125 | 6,418.3125 | 6,009 | 8,192 | |
Albert now decides to extend his list to the 2000th digit. He writes down positive integers in increasing order with a first digit of 1, such as $1, 10, 11, 12, \ldots$. Determine the three-digit number formed by the 1998th, 1999th, and 2000th digits. | 141 | 0 | 7,060.0625 | -1 | 7,060.0625 | |
Calculate the area of the crescent moon enclosed by the portion of the circle of radius 4 centered at (0,0) that lies in the first quadrant, the portion of the circle with radius 2 centered at (0,1) that lies in the first quadrant, and the line segment from (0,0) to (4,0). | 2\pi | 0 | 7,850.3125 | -1 | 7,850.3125 | |
Given that $x_{1}$ and $x_{2}$ are two real roots of the one-variable quadratic equation $x^{2}-6x+k=0$, and (choose one of the conditions $A$ or $B$ to answer the following questions).<br/>$A$: $x_1^2x_2^2-x_1-x_2=115$;<br/>$B$: $x_1^2+x_2^2-6x_1-6x_2+k^2+2k-121=0$.<br/>$(1)$ Find the value of $k$;<br/>$(2)$ Solve thi... | -11 | 0.125 | 2,971.9375 | 3,145 | 2,947.214286 | |
Determine how much money the Romanov family will save by using a multi-tariff meter over three years.
The cost of the meter is 3500 rubles. The installation cost is 1100 rubles. On average, the family's electricity consumption is 300 kWh per month, with 230 kWh used from 23:00 to 07:00.
Electricity rates with a mult... | 3824 | 0 | 2,797.125 | -1 | 2,797.125 | |
Find the distance between the foci of the hyperbola given by the equation \(x^2 - 4x - 12y^2 + 24y = -36.\) | \frac{2\sqrt{273}}{3} | 0 | 6,390.1875 | -1 | 6,390.1875 | |
Find the only real number that can be expressed in the form \[(a + bi)^3 - 107i,\]where $i^2 = -1,$ and $a$ and $b$ are positive integers. | 198 | 0.9375 | 2,722.25 | 2,654.6 | 3,737 | |
Homer started peeling a pile of 60 potatoes at a rate of 4 potatoes per minute. Five minutes later, Christen joined him peeling at a rate of 6 potatoes per minute. After working together for 3 minutes, Christen took a 2-minute break, then resumed peeling at a rate of 4 potatoes per minute. Calculate the total number of... | 23 | 0.0625 | 7,857.375 | 8,192 | 7,835.066667 | |
Find the value of \( \cos \frac{\pi}{11} + \cos \frac{3\pi}{11} + \cos \frac{5\pi}{11} + \cos \frac{7\pi}{11} + \cos \frac{9\pi}{11} \). | \frac{1}{2} | 0.0625 | 8,192 | 8,192 | 8,192 | |
Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of $n$? | 21 | 1. **Identify the total amount of money Casper has:**
Casper can buy $12$ pieces of red candy, $14$ pieces of green candy, $15$ pieces of blue candy, or $n$ pieces of purple candy. Let the cost of each type of candy be $r$, $g$, $b$, and $p$ cents respectively. Since a piece of purple candy costs $20$ cents, we ha... | 1 | 2,707.375 | 2,707.375 | -1 |
A cube \(ABCDA_1B_1C_1D_1\) has edge length 1. Point \(M\) is taken on the side diagonal \(A_1D\), and point \(N\) is taken on \(CD_1\), such that the line segment \(MN\) is parallel to the diagonal plane \(A_1ACC_1\). Find the minimum length of \(MN\). | \frac{\sqrt{3}}{3} | 0 | 5,634.25 | -1 | 5,634.25 | |
Find the number of positive divisors $d$ of $15!=15 \cdot 14 \cdots 2 \cdot 1$ such that $\operatorname{gcd}(d, 60)=5$. | 36 | Since $\operatorname{gcd}(d, 60)=5$, we know that $d=5^{i} d^{\prime}$ for some integer $i>0$ and some integer $d^{\prime}$ which is relatively prime to 60. Consequently, $d^{\prime}$ is a divisor of $(15!) / 5$; eliminating common factors with 60 gives that $d^{\prime}$ is a factor of $\left(7^{2}\right)(11)(13)$, whi... | 0.1875 | 7,354.8125 | 5,073 | 7,881.384615 |
Let \( l \) and \( m \) be two skew lines. On line \( l \), there are three points \( A, B, C \) such that \( AB = BC \). Perpendiculars \( AD, BE, CF \) are drawn from points \( A, B, \) and \( C \) to line \( m \) respectively, with feet of perpendiculars at \( D, E, \) and \( F \). Given that \( AD = \sqrt{15} \), \... | \sqrt{6} | 0 | 8,192 | -1 | 8,192 | |
Select two distinct numbers a, b from the set {0,1,2,3,4,5,6} to form a complex number a+bi, and determine the total number of such complex numbers with imaginary parts. | 36 | 0.75 | 4,311.8125 | 3,718.75 | 6,091 | |
Define a function $f$ from nonnegative integers to real numbers, with $f(1) = 1$ and the functional equation:
\[ f(m+n) + f(m-n) = 3(f(m) + f(n)) \]
for all nonnegative integers $m \ge n$. Determine $f(10)$. | 100 | 0 | 7,973 | -1 | 7,973 | |
From a set consisting of three blue cards labeled $X$, $Y$, $Z$ and three orange cards labeled $X$, $Y$, $Z$, two cards are randomly drawn. A winning pair is defined as either two cards with the same label or two cards of the same color. What is the probability of drawing a winning pair?
- **A)** $\frac{1}{3}$
- **B)**... | \frac{3}{5} | 0 | 7,220.5625 | -1 | 7,220.5625 | |
What is the remainder when $1492\cdot 1999$ is divided by $500$? | 8 | 1 | 3,934.25 | 3,934.25 | -1 | |
Find the product of the least common multiple (LCM) of $8$ and $6$ and the greatest common divisor (GCD) of $8$ and $6$. | 48 | 1 | 1,304.375 | 1,304.375 | -1 | |
Given that the focus of the parabola $y^{2}=2px\left(p \gt 0\right)$ is $F\left(4,0\right)$, and $O$ is the origin.
$(1)$ Find the equation of the parabola.
$(2)$ A line with a slope of $1$ passes through point $F$ and intersects the parabola at points $A$ and $B$. Find the area of $\triangle AOB$. | 32\sqrt{2} | 0.9375 | 4,504.375 | 4,441.8 | 5,443 | |
Two concentric squares share the same center $O$, each with sides of length 2. The length of the line segment $\overline{AB}$ is $\frac{1}{3}$, and the area of the octagon $ABCDEFGH$ formed by the intersection of lines drawn from the vertices of the inner square to the midpoints of sides of the outer square is sought. ... | 11 | 0 | 8,192 | -1 | 8,192 | |
In a modified design of Wei's logo, nine identical circles are placed within a square of side length 24 inches. Each circle is tangent to two sides of the square and to its adjacent circles wherever possible. How many square inches will be shaded in this new design? | 576 - 144\pi | 0.375 | 6,475.25 | 6,271.5 | 6,597.5 | |
Two different numbers are randomly selected from the set $\{-3, -1, 0, 2, 4\}$ and then multiplied together. What is the probability that the product of the two numbers chosen is 0? | \frac{2}{5} | There are ten possible pairs of numbers that can be chosen: -3 and $-1$; -3 and $0$; -3 and 2; -3 and $4$; -1 and $0$; -1 and $2$; -1 and $4$; 0 and $2$; 0 and $4$; 2 and 4. Each pair is equally likely to be chosen. Pairs that include 0 (4 pairs) have a product of 0; pairs that do not include 0 (6 of them) do not have ... | 0.9375 | 3,470.25 | 3,155.466667 | 8,192 |
Let \( N \) be the positive integers. The function \( f : N \to N \) satisfies \( f(1) = 5 \), \( f(f(n)) = 4n + 9 \), and \( f(2n) = 2n+1 + 3 \) for all \( n \). Find \( f(1789) \). | 3581 | 0 | 8,192 | -1 | 8,192 | |
In a vertical vessel with straight walls closed by a piston, there is water. Its height is $h=2$ mm. There is no air in the vessel. To what height must the piston be raised for all the water to evaporate? The density of water is $\rho=1000$ kg / $\mathrm{m}^{3}$, the molar mass of water vapor is $M=0.018$ kg/mol, the p... | 24.258 | 0 | 7,401.1875 | -1 | 7,401.1875 | |
A special six-sided die has its faces numbered 1 through 6 and has the property that rolling each number \( x \) is \( x \) times as likely as rolling a 1. For example, the probability of rolling a 5 is 5 times the probability of rolling a 1, while the probability of rolling a 2 is 2 times the probability of rolling a ... | 96 | 0.3125 | 7,044.75 | 4,575.8 | 8,167 | |
$A B C$ is an acute triangle with incircle $\omega$. $\omega$ is tangent to sides $\overline{B C}, \overline{C A}$, and $\overline{A B}$ at $D, E$, and $F$ respectively. $P$ is a point on the altitude from $A$ such that $\Gamma$, the circle with diameter $\overline{A P}$, is tangent to $\omega$. $\Gamma$ intersects $\o... | \frac{675}{4} | By the Law of Sines we have $\sin \angle A=\frac{X Y}{A P}=\frac{4}{5}$. Let $I, T$, and $Q$ denote the center of $\omega$, the point of tangency between $\omega$ and $\Gamma$, and the center of $\Gamma$ respectively. Since we are told $A B C$ is acute, we can compute $\tan \angle \frac{A}{2}=\frac{1}{2}$. Since $\angl... | 0 | 8,192 | -1 | 8,192 |
Find a unit vector that is orthogonal to both $\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}.$ | \begin{pmatrix} -2/3 \\ 2/3 \\ 1/3 \end{pmatrix} | 0 | 2,790.1875 | -1 | 2,790.1875 | |
In rhombus $ABCD$, $\angle BAD=60^{\circ}$, $\overrightarrow{DE}=\overrightarrow{EC}$, $AB=2$, calculate $\overrightarrow{AE}\cdot\overrightarrow{DB}$. | -1 | 1 | 4,068.3125 | 4,068.3125 | -1 | |
A hexagon inscribed in a circle has three consecutive sides, each of length 4, and three consecutive sides, each of length 7. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length 4, and the other with three sides each of length 7, has length equal to $p/q$, where $p$... | 1017 | 0 | 6,995.9375 | -1 | 6,995.9375 | |
Find the smallest value of $x$ that satisfies the equation $|3x+7|=26$. | -11 | 1 | 1,315.0625 | 1,315.0625 | -1 | |
Each of the eight letters in "GEOMETRY" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "ANGLE"? Express your answer as a common fraction. | \frac{1}{4} | 0.4375 | 2,816.5625 | 1,567.142857 | 3,788.333333 | |
My date tonight made reservations for us at his favorite Italian restaurant. He didn't know that I am a vegetarian with a gluten allergy. There are four dishes here without meat in them. That's just one fifth of the menu. And three of those meatless dishes are made with pasta, which contains gluten. What fraction ... | \frac{1}{20} | 0.9375 | 1,775.3125 | 1,830.666667 | 945 | |
Given \( x_{0} > 0 \), \( x_{0} \neq \sqrt{3} \), a point \( Q\left( x_{0}, 0 \right) \), and a point \( P(0, 4) \), the line \( PQ \) intersects the hyperbola \( x^{2} - \frac{y^{2}}{3} = 1 \) at points \( A \) and \( B \). If \( \overrightarrow{PQ} = t \overrightarrow{QA} = (2-t) \overrightarrow{QB} \), then \( x_{0}... | \frac{\sqrt{2}}{2} | 0 | 8,125.375 | -1 | 8,125.375 | |
\( AB \) and \( AC \) are tangents to a circle with center \( O \), \( M \) is the point of intersection of the line \( AO \) with the circle; \( DE \) is the segment of the tangent drawn through the point \( M \) between \( AB \) and \( AC \). Find \( DE \), if the radius of the circle is 15 and \( AO = 39 \). | 20 | 0.375 | 7,275.1875 | 6,442.5 | 7,774.8 | |
Given the set $A=\{x|x=a_0+a_1\times3+a_2\times3^2+a_3\times3^3\}$, where $a_k\in\{0,1,2\}$ ($k=0,1,2,3$), and $a_3\neq0$, calculate the sum of all elements in set $A$. | 2889 | 0.125 | 7,867.25 | 5,782.5 | 8,165.071429 | |
The first operation divides the bottom-left square of diagram $\mathrm{a}$ into four smaller squares, as shown in diagram b. The second operation further divides the bottom-left smaller square of diagram b into four even smaller squares, as shown in diagram c; continuing this process, after the sixth operation, the res... | 29 | 0 | 2,638.5 | -1 | 2,638.5 | |
The cost of two pencils and three pens is $4.10, and the cost of three pencils and one pen is $2.95. What is the cost of one pencil and four pens? | 4.34 | 0.75 | 6,968.5 | 6,560.666667 | 8,192 | |
There is one odd integer \( N \) between 400 and 600 that is divisible by both 5 and 11. What is the sum of the digits of \( N \)? | 18 | If \( N \) is divisible by both 5 and 11, then \( N \) is divisible by \( 5 \times 11=55 \). This is because 5 and 11 have no common divisor larger than 1. Therefore, we are looking for a multiple of 55 between 400 and 600 that is odd. One way to find such a multiple is to start with a known multiple of 55, such as 550... | 1 | 1,944.5 | 1,944.5 | -1 |
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