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:<br/>$(1)-1^{2023}+8×(-\frac{1}{2})^{3}+|-3|$;<br/>$(2)(-25)×\frac{3}{2}-(-25)×\frac{5}{8}+(-25)÷8($simplified calculation). | -25 | 0.5625 | 2,179.125 | 3,164.777778 | 911.857143 | |
The sum of the coefficients of all rational terms in the expansion of $$(2 \sqrt {x}- \frac {1}{x})^{6}$$ is \_\_\_\_\_\_ (answer with a number). | 365 | 0.75 | 5,361.8125 | 4,749.5 | 7,198.75 | |
Let $a$ and $b$ each be chosen at random from the set $\{1, 2, 3, \ldots, 40\}$. Additionally, let $c$ and $d$ also be chosen at random from the same set. Calculate the probability that the integer $2^c + 5^d + 3^a + 7^b$ has a units digit of $8$. | \frac{3}{16} | 0 | 7,884.25 | -1 | 7,884.25 | |
Triangle $ABC$ has sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ of length 43, 13, and 48, respectively. Let $\omega$ be the circle circumscribed around $\triangle ABC$ and let $D$ be the intersection of $\omega$ and the perpendicular bisector of $\overline{AC}$ that is not on the same side of $\overline{... | 12 | 0.4375 | 6,842.3125 | 5,556.714286 | 7,842.222222 | |
Given $a\in R$, $b \gt 0$, $a+b=2$, then the minimum value of $\frac{1}{2|a|}+\frac{|a|}{b}$ is ______. | \frac{3}{4} | 0.5 | 7,080.5 | 6,603.75 | 7,557.25 | |
Expand $(x+2)(3x-6)$. | 3x^2-12 | 1 | 2,106.5 | 2,106.5 | -1 | |
Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where the upper vertex of $C$ is $A$, and the two foci are $F_{1}$ and $F_{2}$, with an eccentricity of $\frac{1}{2}$. A line passing through $F_{1}$ and perpendicular to $AF_{2}$ intersects $C$ at points $D$ and $E$, where $|DE| = 6$. Find the ... | 13 | 0 | 8,192 | -1 | 8,192 | |
Calculate: \(\frac{2 \times 4.6 \times 9 + 4 \times 9.2 \times 18}{1 \times 2.3 \times 4.5 + 3 \times 6.9 \times 13.5} =\) | \frac{18}{7} | 0.6875 | 6,314.75 | 5,629.181818 | 7,823 | |
There are six clearly distinguishable frogs sitting in a row. Two are green, three are red, and one is blue. Green frogs refuse to sit next to the red frogs, for they are highly poisonous. In how many ways can the frogs be arranged? | 24 | 0 | 8,192 | -1 | 8,192 | |
Let $x, y$ be complex numbers such that \frac{x^{2}+y^{2}}{x+y}=4$ and \frac{x^{4}+y^{4}}{x^{3}+y^{3}}=2$. Find all possible values of \frac{x^{6}+y^{6}}{x^{5}+y^{5}}$. | 10 \pm 2 \sqrt{17} | Let $A=\frac{1}{x}+\frac{1}{y}$ and let $B=\frac{x}{y}+\frac{y}{x}$. Then $$ \frac{B}{A}=\frac{x^{2}+y^{2}}{x+y}=4 $$ so $B=4 A$. Next, note that $$ B^{2}-2=\frac{x^{4}+y^{4}}{x^{2} y^{2}} \text { and } A B-A=\frac{x^{3}+y^{3}}{x^{2} y^{2}} $$ so $$ \frac{B^{2}-2}{A B-A}=2 $$ Substituting $B=4 A$ and simplifying, we fi... | 0 | 8,119.5625 | -1 | 8,119.5625 |
Let \( p(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants. Given \( p(1) = 1993 \), \( p(2) = 3986 \), \( p(3) = 5979 \), find \( \frac{1}{4} [p(11) + p(-7)] \). | 5233 | 0.625 | 6,435.6875 | 5,604.6 | 7,820.833333 | |
What is the value of $(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)$? | 127 | We are given the expression $(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)$ and need to evaluate it step by step.
1. **Start from the innermost expression**:
\[
2 + 1 = 3
\]
2. **Move to the next layer**:
\[
2(3) + 1 = 6 + 1 = 7
\]
3. **Continue to the next layer**:
\[
2(7) + 1 = 14 + 1 = 15
\]
4. **Pro... | 0.6875 | 2,799.8125 | 2,424.545455 | 3,625.4 |
A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles? | 6:\pi | 1. **Set up the proportion for $AD$ and $AB$:**
Given the ratio of $AD$ to $AB$ is $3:2$, and $AB = 30$ inches, we can write:
\[
\frac{AD}{AB} = \frac{3}{2}
\]
Substituting $AB = 30$ inches into the proportion:
\[
\frac{AD}{30} = \frac{3}{2}
\]
2. **Solve for $AD$:**
To find $AD$, cross-mult... | 0 | 4,718.6875 | -1 | 4,718.6875 |
From the center \( O \) of the inscribed circle of a right triangle, the half of the hypotenuse that is closer to \( O \) appears at a right angle. What is the ratio of the sides of the triangle? | 3 : 4 : 5 | 0.125 | 7,866.5 | 6,686 | 8,035.142857 | |
Two identical resistors $R_{0}$ are connected in series and connected to a DC voltage source. An ideal voltmeter is connected in parallel with one of the resistors. Its reading is $U=2 \text{V}$. If the voltmeter is replaced with an ideal ammeter, its reading will be $I=4 \text{A}$. Determine the value of $R_{0}$. | 0.5 | 0.5 | 6,754.0625 | 5,769.375 | 7,738.75 | |
Find the sum of $245_8$, $174_8$, and $354_8$ in base 8. | 1015_8 | 0.625 | 5,808.75 | 4,378.8 | 8,192 | |
A line with slope $2$ passes through the focus $F$ of the parabola $y^2 = 2px$ $(p > 0)$ and intersects the parabola at points $A$ and $B$. The projections of $A$ and $B$ on the $y$-axis are $D$ and $C$ respectively. If the area of trapezoid $\triangle BCD$ is $6\sqrt{5}$, then calculate the value of $p$. | 2\sqrt{2} | 0.0625 | 7,637.6875 | 6,222 | 7,732.066667 | |
There are two positive integers, \(A\) and \(B\). The sum of the digits of \(A\) is \(19\), the sum of the digits of \(B\) is \(20\), and their addition results in carrying over twice. What is the sum of the digits of \((\mathbf{A} + B)\)? | 21 | 0.5 | 6,554.4375 | 4,916.875 | 8,192 | |
Let
\[f(x) = \frac{ax}{x + 1}.\]Find the constant $a$ so that $f(f(x)) = x$ for all $x \neq -1.$ | -1 | 1 | 2,672.4375 | 2,672.4375 | -1 | |
In the expression \((x+y+z)^{2034}+(x-y-z)^{2034}\), the brackets were expanded, and like terms were combined. How many monomials of the form \(x^{a} y^{b} z^{c}\) have a non-zero coefficient? | 1036324 | 0.125 | 7,939.9375 | 6,175.5 | 8,192 | |
How many ways can 1995 be factored as a product of two two-digit numbers? (Two factorizations of the form $a\cdot b$ and $b\cdot a$ are considered the same). | 2 | 0.6875 | 6,641.8125 | 5,937.181818 | 8,192 | |
Let \(A\) and \(G\) be two opposite vertices of a cube with unit edge length. What is the distance between the plane determined by the vertices adjacent to \(A\), denoted as \(S_{A}\), and the plane determined by the vertices adjacent to \(G\), denoted as \(S_{G}\)? | \frac{\sqrt{3}}{3} | 0 | 6,021.5625 | -1 | 6,021.5625 | |
János, a secretary of a rural cooperative, travels to Budapest weekly. His wife leaves home at 4 o'clock to meet him at the station, arriving at exactly the same time as the train. They are home by 5 o'clock. One day, the train arrived earlier, unbeknownst to his wife, so she encountered him on the way home. They arriv... | 3.5 | 0 | 7,878.875 | -1 | 7,878.875 | |
Given two points A and B on a number line, their distance is 2, and the distance between point A and the origin O is 3. Then, the sum of all possible distances between point B and the origin O equals to . | 12 | 0.875 | 3,520.375 | 3,787.071429 | 1,653.5 | |
Write $(-5)^5\div5^3+3^{4}-6^{1}$ as an integer. | 50 | 1 | 1,904 | 1,904 | -1 | |
A set of six edges of a regular octahedron is called Hamiltonian cycle if the edges in some order constitute a single continuous loop that visits each vertex exactly once. How many ways are there to partition the twelve edges into two Hamiltonian cycles? | 6 | Call the octahedron $A B C D E F$, where $A, B$, and $C$ are opposite $D, E$, and $F$, respectively. Note that each Hamiltonian cycle can be described in terms of the order it visits vertices in exactly 12 different ways. Conversely, listing the six vertices in some order determines a Hamiltonian cycle precisely when n... | 0 | 8,015.3125 | -1 | 8,015.3125 |
Points \( M, N, \) and \( K \) are located on the lateral edges \( A A_{1}, B B_{1}, \) and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( \frac{A M}{A A_{1}} = \frac{5}{6}, \frac{B N}{B B_{1}} = \frac{6}{7}, \) and \( \frac{C K}{C C_{1}} = \frac{2}{3} \). Point \( P \) belongs to the ... | 10 | 0.0625 | 8,053.375 | 8,192 | 8,044.133333 | |
If $x$ and $y$ are positive real numbers such that $5x^2 + 10xy = x^3 + 2x^2 y,$ what is the value of $x$? | 5 | 0.9375 | 4,100.375 | 3,827.6 | 8,192 | |
Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is $60$, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$.
| 400 | 0 | 8,192 | -1 | 8,192 | |
In the figure, circle $O$ has radius 6 units. Chord $CD$ has length 8 units and is parallel to segment $KB$. If $KA$ = 12 units and points $K$, $A$, $O$ and $B$ are collinear, what is the area of triangle $KDC$? Express your answer in simplest radical form. [asy]
draw(Circle((0,0),6));
dot((0,0));
label("$O$",(0,0),S);... | 8\sqrt{5} | 0.875 | 4,492 | 3,963.428571 | 8,192 | |
Sherry starts at the number 1. Whenever she's at 1, she moves one step up (to 2). Whenever she's at a number strictly between 1 and 10, she moves one step up or one step down, each with probability $\frac{1}{2}$ . When she reaches 10, she stops. What is the expected number (average number) of steps that Sherry wil... | 81 | 0.8125 | 5,694.125 | 5,117.692308 | 8,192 | |
Consider the arithmetic sequence defined by the set $\{2, 5, 8, 11, 14, 17, 20\}$. Determine the total number of different integers that can be expressed as the sum of three distinct members of this set. | 13 | 0.8125 | 6,661 | 6,314 | 8,164.666667 | |
Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges... | 103 | 1. **Define the variables and equations:**
Let $x$ be the number of times Alex visits the first booth, and $y$ be the number of times he visits the second booth. Each visit to the first booth changes the token counts as follows: Alex loses 2 red tokens and gains 1 blue token and 1 silver token. Each visit to the sec... | 0.0625 | 8,036 | 6,412 | 8,144.266667 |
$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying:
(1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$.
(2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' ... | 79 |
Let \( S \) be a non-empty subset of the set \( \{ 1, 2, \ldots, 108 \} \) satisfying the following conditions:
1. For any two numbers \( a, b \in S \) (not necessarily distinct), there exists \( c \in S \) such that \( \gcd(a, c) = \gcd(b, c) = 1 \).
2. For any two numbers \( a, b \in S \) (not necessarily distinct)... | 0 | 8,192 | -1 | 8,192 |
Let $ABCD$ be a unit square. $E$ and $F$ trisect $AB$ such that $AE<AF. G$ and $H$ trisect $BC$ such that $BG<BH. I$ and $J$ bisect $CD$ and $DA,$ respectively. Let $HJ$ and $EI$ meet at $K,$ and let $GJ$ and $FI$ meet at $L.$ Compute the length $KL.$ | \frac{6\sqrt{2}}{35} | 0 | 4,130.0625 | -1 | 4,130.0625 | |
Given that the sum of all odd terms in the first 10 terms of a geometric sequence is $85 \frac{1}{4}$, and the sum of all even terms is $170 \frac{1}{2}$, find the value of $S=a_{3}+a_{6}+a_{9}+a_{12}$. | 585 | 1 | 3,903.75 | 3,903.75 | -1 | |
How many $5$ -digit numbers $N$ (in base $10$ ) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$ ? (The leading digit of $N$ cannot be zero.)
*Based on a proposal by Yannick Yao* | 256 | 0 | 8,160.75 | -1 | 8,160.75 | |
Simplify $(2 \times 10^9) - (6 \times 10^7) \div (2 \times 10^2)$. | 1999700000 | 0.375 | 5,127.1875 | 3,574.5 | 6,058.8 | |
Compute the sum $i^{-103} + i^{-102} + \cdots + i^{-1} + i^0 + i^1 + \cdots + i^{102} + i^{103} + \sum_{n=1}^{103} n$. | 5355 | 0.3125 | 7,215.8125 | 5,721 | 7,895.272727 | |
Given the point \( P \) inside the triangle \( \triangle ABC \), satisfying \( \overrightarrow{AP} = \frac{1}{3} \overrightarrow{AB} + \frac{1}{4} \overrightarrow{AC} \), let the areas of triangles \( \triangle PBC \), \( \triangle PCA \), and \( \triangle PAB \) be \( S_1 \), \( S_2 \), and \( S_3 \) respectively. Det... | 5:4:3 | 0.75 | 5,553.6875 | 4,910.166667 | 7,484.25 | |
For each positive digit $D$ and positive integer $k$, we use the symbol $D_{(k)}$ to represent the positive integer having exactly $k$ digits, each of which is equal to $D$. For example, $2_{(1)}=2$ and $3_{(4)}=3333$. There are $N$ quadruples $(P, Q, R, k)$ with $P, Q$ and $R$ positive digits, $k$ a positive integer w... | 11 | Suppose that $D$ is a digit and $k$ is a positive integer. Then $D_{(k)}=\underbrace{D D \cdots D D}_{k \text { times }}=D \cdot \underbrace{11 \cdots 11}_{k \text { times }}=D \cdot \frac{1}{9} \cdot \underbrace{99 \cdots 99}_{k \text { times }}=D \cdot \frac{1}{9} \cdot(\underbrace{00 \cdots 00}_{k \text { times }}-1... | 0 | 7,989.4375 | -1 | 7,989.4375 |
Sets $A$ and $B$, shown in the Venn diagram, are such that the total number of elements in set $A$ is twice the total number of elements in set $B$. Altogether, there are 3011 elements in the union of $A$ and $B$, and their intersection has 1000 elements. What is the total number of elements in set $A$?
[asy]
label("$... | 2674 | 1 | 1,628.1875 | 1,628.1875 | -1 | |
In a corridor 100 meters long, 20 carpet strips with a total length of 1000 meters are laid down. What could be the maximum number of uncovered sections (the width of the carpet strip is equal to the width of the corridor)? | 10 | 0 | 8,046.5 | -1 | 8,046.5 | |
A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square ... | 68 | To solve this problem, we need to calculate the probability that a randomly placed coin will cover part of the black region on the square. We start by determining the total possible region where the center of the coin can land and then calculate the area of the regions where the coin would overlap with the black region... | 0 | 7,868.75 | -1 | 7,868.75 |
Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$? | 30 | 1. **Define the sets and probabilities**: Let $A$ be the set of all configurations where the balls are distributed as $3{-}5{-}4{-}4{-}4$ among the bins, and let $B$ be the set of all configurations where the balls are distributed as $4{-}4{-}4{-}4{-}4$. Define $p = \frac{|A|}{N}$ and $q = \frac{|B|}{N}$, where $N$ is ... | 0 | 5,084.0625 | -1 | 5,084.0625 |
Suppose $P(x)$ is a polynomial such that $P(1)=1$ and $$\frac{P(2 x)}{P(x+1)}=8-\frac{56}{x+7}$$ for all real $x$ for which both sides are defined. Find $P(-1)$. | -5/21 | Cross-multiplying gives $(x+7) P(2 x)=8 x P(x+1)$. If $P$ has degree $n$ and leading coefficient $c$, then the leading coefficients of the two sides are $2^{n} c$ and $8 c$, so $n=3$. Now $x=0$ is a root of the right-hand side, so it's a root of the left-hand side, so that $P(x)=x Q(x)$ for some polynomial $Q \Rightarr... | 0 | 7,306.375 | -1 | 7,306.375 |
(1) Solve the inequality $$\frac {2x+1}{3-x}≥1$$
(2) Given $x>0$, $y>0$, and $x+y=1$, find the minimum value of $$\frac {4}{x} + \frac {9}{y}$$. | 25 | 1 | 3,854.9375 | 3,854.9375 | -1 | |
Circle $T$ has its center at point $T(-2,6)$. Circle $T$ is reflected across the $y$-axis and then translated 8 units down. What are the coordinates of the image of the center of circle $T$? | (2, -2) | 1 | 1,576.6875 | 1,576.6875 | -1 | |
Given a function $f(x)$ defined on $R$ such that $f(x) + x^{2}$ is an odd function and $f(x) + x^{3}$ is an even function, then $f(2)$ is ______. | -12 | 0.9375 | 2,954.75 | 2,605.6 | 8,192 | |
Count the number of sequences $1 \leq a_{1} \leq a_{2} \leq \cdots \leq a_{5}$ of integers with $a_{i} \leq i$ for all $i$. | 42 | $C$ (number of terms) $=C(5)=42$. | 0 | 8,064.8125 | -1 | 8,064.8125 |
In the Cartesian coordinate system, with the origin as the pole and the x-axis as the positive semi-axis, a polar coordinate system is established. The polar equation of circle C is $\rho=6\cos\theta$, and the parametric equation of line $l$ is
$$
\begin{cases}
x=3+ \frac {1}{2}t \\
y=-3+ \frac { \sqrt {3}}{2}t
\end... | 1:2 | 0.625 | 4,493.3125 | 4,169 | 5,033.833333 | |
What is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime? | 8 | 0.5625 | 6,951.9375 | 5,987.444444 | 8,192 | |
find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have $$ \{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\} $$ . | {0} | 0 | 7,752 | -1 | 7,752 | |
Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ and $d,$
\[\sqrt{\frac{a}{b + c + d}} + \sqrt{\frac{b}{a + c + d}} + \sqrt{\frac{c}{a + b + d}} + \sqrt{\frac{d}{a + b + c}} > m.\] | 2 | 0.125 | 7,964.1875 | 6,369.5 | 8,192 | |
In a regular 2019-gon, numbers are placed at the vertices such that the sum of the numbers in any nine consecutive vertices is 300. It is known that the number at the 19th vertex is 19, and the number at the 20th vertex is 20. What number is at the 2019th vertex? | 61 | 0.375 | 6,841.375 | 4,590.333333 | 8,192 | |
In a certain high school, there are 300 freshmen students, including 180 boys and 120 girls. In order to understand the height information of the freshmen students, a stratified random sampling method is used to select samples according to the proportion of the sample size. It is found that the average height of the bo... | 42 | 0 | 8,015.625 | -1 | 8,015.625 | |
If the seven digits 1, 1, 3, 5, 5, 5, and 9 are arranged to form a seven-digit positive integer, what is the probability that the integer is divisible by 25? | \frac{1}{14} | 0.0625 | 3,493.0625 | 1,708 | 3,612.066667 | |
Let \(ABCD\) be a quadrilateral circumscribed about a circle with center \(O\). Let \(O_1, O_2, O_3,\) and \(O_4\) denote the circumcenters of \(\triangle AOB, \triangle BOC, \triangle COD,\) and \(\triangle DOA\). If \(\angle A = 120^\circ\), \(\angle B = 80^\circ\), and \(\angle C = 45^\circ\), what is the acute angl... | 45 | 0 | 8,192 | -1 | 8,192 | |
For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 50$.
(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? | 6 | 1 | 4,710.0625 | 4,710.0625 | -1 | |
A rectangle has a length to width ratio of 5:2. Within this rectangle, a right triangle is formed by drawing a line from one corner to the midpoint of the opposite side. If the length of this line (hypotenuse of the triangle) is measured as $d$, find the constant $k$ such that the area of the rectangle can be expressed... | \frac{5}{13} | 0.3125 | 5,628.4375 | 4,464.4 | 6,157.545455 | |
Let the triangle $ABC$ have area $1$ . The interior bisectors of the angles $\angle BAC,\angle ABC, \angle BCA$ intersect the sides $(BC), (AC), (AB) $ and the circumscribed circle of the respective triangle $ABC$ at the points $L$ and $G, N$ and $F, Q$ and $E$ . The lines $EF, FG,GE$ intersect the bi... | 1/2 | 0.0625 | 7,934.75 | 6,232 | 8,048.266667 | |
Teams A, B, and C need to complete two projects, $A$ and $B$. The workload of project $B$ is $\frac{1}{4}$ more than the workload of project $A$. If teams A, B, and C work alone, they can finish project $A$ in 20 days, 24 days, and 30 days respectively. To complete these two projects simultaneously, team A is assigned ... | 15 | 0 | 7,612.4375 | -1 | 7,612.4375 | |
If $64^5 = 32^x$, what is the value of $2^{-x}$? Express your answer as a common fraction. | \frac{1}{64} | 1 | 1,669.0625 | 1,669.0625 | -1 | |
Suppose \( x_{1}, x_{2}, \ldots, x_{2011} \) are positive integers satisfying
\[ x_{1} + x_{2} + \cdots + x_{2011} = x_{1} x_{2} \cdots x_{2011} \]
Find the maximum value of \( x_{1} + x_{2} + \cdots + x_{2011} \). | 4022 | 0 | 8,192 | -1 | 8,192 | |
Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$. | 40\pi | 0.5625 | 6,291.9375 | 4,814.111111 | 8,192 | |
The perimeter of the quadrilateral formed by the four vertices of the ellipse $C: \frac {x^{2}}{4}+ \frac {y^{2}}{16}=1$ is equal to _____. | 8 \sqrt {5} | 0 | 4,617.9375 | -1 | 4,617.9375 | |
Given the operation defined as \(a \odot b \odot c = a \times b \times c + (a \times b + b \times c + c \times a) - (a + b + c)\), calculate \(1 \odot 43 \odot 47\). | 4041 | 0.5625 | 3,050.1875 | 3,737.555556 | 2,166.428571 | |
In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Find $x$. | 200 | Use the table from above. Obviously $c = 114$. Hence $a+e = 115$. Similarly, $1+a = 96 + e \Rightarrow a = 95+e$.
Substitute that into the first to get $2e = 20 \Rightarrow e=10$, so $a=105$, and so the value of $x$ is just $115+x = 210 + 105 \Rightarrow x = \boxed{200}$ | 0 | 7,952.25 | -1 | 7,952.25 |
For each positive integer $n$, the mean of the first $n$ terms of a sequence is $n$. What is the 2008th term of the sequence? | 4015 | 1 | 2,346.9375 | 2,346.9375 | -1 | |
A regular hexagon has a side length of 8 cm. Calculate the area of the shaded region formed by connecting two non-adjacent vertices to the center of the hexagon, creating a kite-shaped region.
[asy]
size(100);
pair A,B,C,D,E,F,O;
A = dir(0); B = dir(60); C = dir(120); D = dir(180); E = dir(240); F = dir(300); O = (0,... | 16\sqrt{3} | 0.25 | 7,360.5 | 5,571 | 7,957 | |
Maria invested $10,000 for 3 years at an annual interest rate of 5 percent compounded annually. Liam invested $10,000 for the same period of time, at the same interest rate, but the interest was compounded semi-annually. Calculate the difference in the amount earned by Liam's investment compared to Maria's, to the near... | 16 | 0 | 6,270.375 | -1 | 6,270.375 | |
Two riders simultaneously departed from points \( A \) and \( C \) towards point \( B \). Despite the fact that \( C \) was 20 km farther from \( B \) than \( A \) was from \( B \), both riders arrived at \( B \) at the same time. Find the distance from \( C \) to \( B \), given that the rider from \( C \) traveled eac... | 80 | 0.25 | 7,335.5 | 4,766 | 8,192 | |
Real numbers $a$ , $b$ , $c$ which are differ from $1$ satisfies the following conditions;
(1) $abc =1$ (2) $a^2+b^2+c^2 - \left( \dfrac{1}{a^2} + \dfrac{1}{b^2} + \dfrac{1}{c^2} \right) = 8(a+b+c) - 8 (ab+bc+ca)$ Find all possible values of expression $\dfrac{1}{a-1} + \dfrac{1}{b-1} + \dfrac{1}{c-1}$ . | -\frac{3}{2} | 0.4375 | 7,450.3125 | 6,496.714286 | 8,192 | |
On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$ . Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$ . $(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to... | \pi | 0.125 | 7,849.875 | 5,946 | 8,121.857143 | |
Solve for $x$ if $\frac{2}{x+3} + \frac{3x}{x+3} - \frac{4}{x+3} = 4$. | -14 | 0.9375 | 1,997 | 1,584 | 8,192 | |
Calculate \(3^{18} \div 27^2\) and multiply the result by 7. Write your answer as an integer. | 3720087 | 0.375 | 2,641 | 3,617.833333 | 2,054.9 | |
Find the probability that, when five different numbers are randomly chosen from the set $\{1, 2, \ldots, 20\}$, at least two of them are consecutive. | \frac{232}{323} | 0.5625 | 6,148.3125 | 4,558.777778 | 8,192 | |
An eight-sided die numbered from 1 to 8 is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$? | 48 | 0.625 | 6,744.625 | 6,189.4 | 7,670 | |
For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 16 | 0.25 | 7,821.4375 | 6,709.75 | 8,192 | |
In isosceles triangle $\triangle ABC$ we have $AB=AC=4$. The altitude from $B$ meets $\overline{AC}$ at $H$. If $AH=3(HC)$ then determine $BC$. | 2\sqrt{2} | 0.875 | 4,108.125 | 3,586.285714 | 7,761 | |
Given that Alice's car averages 30 miles per gallon of gasoline, and Bob's car averages 20 miles per gallon of gasoline, and Alice drives 120 miles and Bob drives 180 miles, calculate the combined rate of miles per gallon of gasoline for both cars. | \frac{300}{13} | 0.0625 | 609.3125 | 641 | 607.2 | |
Given that \(x\) is a real number, find the least possible value of \((x+2)(x+3)(x+4)(x+5)+3033\). | 3032 | 0.8125 | 5,165.0625 | 4,768.769231 | 6,882.333333 | |
Round $54.\overline{54}$ to the nearest hundredth. | 54.55 | 0.9375 | 3,944.5 | 3,661.333333 | 8,192 | |
Determine the number of scalene triangles where all sides are integers and have a perimeter less than 20. | 12 | 0 | 8,192 | -1 | 8,192 | |
Given vectors $\overrightarrow{m}=(1,\sqrt{3})$, $\overrightarrow{n}=(\sin x,\cos x)$, let function $f(x)=\overrightarrow{m}\cdot \overrightarrow{n}$
(I) Find the smallest positive period and maximum value of function $f(x)$;
(II) In acute triangle $\Delta ABC$, let the sides opposite angles $A$, $B$, $C$ be $a$, $b$... | \frac{8}{3} | 1 | 3,639.0625 | 3,639.0625 | -1 | |
Let S<sub>n</sub> represent the sum of the first n terms of the arithmetic sequence {a<sub>n</sub>}. If S<sub>5</sub> = 2S<sub>4</sub> and a<sub>2</sub> + a<sub>4</sub> = 8, find the value of a<sub>5</sub>. | 10 | 1 | 2,302.125 | 2,302.125 | -1 | |
Let $f(x) = x^2 - x + 2010$. What is the greatest common divisor of $f(100)$ and $f(101)$? | 10 | 0.9375 | 4,727.125 | 4,496.133333 | 8,192 | |
Let $a$, $b$, and $c$ be the 3 roots of the polynomial $x^3 - 2x + 4 = 0$. Find $\frac{1}{a-2} + \frac{1}{b-2} + \frac{1}{c-2}$. | -\frac{5}{4} | 0.75 | 6,035.75 | 5,317 | 8,192 | |
When the number "POTOP" was added together 99,999 times, the resulting number had the last three digits of 285. What number is represented by the word "POTOP"? (Identical letters represent identical digits.) | 51715 | 0.4375 | 6,499.6875 | 4,953.285714 | 7,702.444444 | |
Suppose $p$ and $q$ are inversely proportional. If $p=25$ when $q=6$, find the value of $p$ when $q=15$. | 10 | 1 | 1,435.1875 | 1,435.1875 | -1 | |
Let $x, y, z$ be real numbers such that
\[
x + y + z = 5,
\]
\[
x^2 + y^2 + z^2 = 11.
\]
Find the smallest and largest possible values of $x$, and compute their sum. | \frac{10}{3} | 1 | 3,768.4375 | 3,768.4375 | -1 | |
Given the function $f(x)=\sin (2x+ \frac {π}{3})- \sqrt {3}\sin (2x- \frac {π}{6})$
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) When $x\in\[- \frac {π}{6}, \frac {π}{3}\]$, find the maximum and minimum values of $f(x)$, and write out the values of the inde... | -\sqrt {3} | 0 | 6,781.1875 | -1 | 6,781.1875 | |
A uniform cubic die with faces numbered $1, 2, 3, 4, 5, 6$ is rolled three times independently, resulting in outcomes $a_1, a_2, a_3$. Find the probability of the event "$|a_1 - a_2| + |a_2 - a_3| + |a_3 - a_1| = 6$". | 1/4 | 0.25 | 7,491.6875 | 6,236.75 | 7,910 | |
Suppose in a right triangle $PQR$, $\cos Q = 0.5$. Point $Q$ is at the origin, and $PQ = 15$ units along the positive x-axis. What is the length of $QR$? | 30 | 0.3125 | 4,281.8125 | 4,564.4 | 4,153.363636 | |
In the Cartesian coordinate system, $O$ is the origin, and points $A(-1,0)$, $B(0, \sqrt{3})$, $C(3,0)$. A moving point $D$ satisfies $|\overrightarrow{CD}|=1$, then the maximum value of $|\overrightarrow{OA}+ \overrightarrow{OB}+ \overrightarrow{OD}|$ is ______. | \sqrt{7}+1 | 0 | 6,916.5625 | -1 | 6,916.5625 | |
Given that a full circle is 800 clerts on Venus and is 360 degrees, calculate the number of clerts in an angle of 60 degrees. | 133.\overline{3} | 0.25 | 1,666.25 | 862 | 1,934.333333 | |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. Given the equation $$b^{2}- \frac {2 \sqrt {3}}{3}bcsinA+c^{2}=a^{2}$$.
(I) Find the measure of angle $A$;
(II) If $b=2$, $c=3$, find the values of $a$ and $sin(2B-A)$. | \frac{3\sqrt{3}}{14} | 0 | 5,034.3125 | -1 | 5,034.3125 | |
Consider the statements:
$\textbf{(1)}\ p\wedge \sim q\wedge r \qquad\textbf{(2)}\ \sim p\wedge \sim q\wedge r \qquad\textbf{(3)}\ p\wedge \sim q\wedge \sim r \qquad\textbf{(4)}\ \sim p\wedge q\wedge r$
where $p,q$, and $r$ are propositions. How many of these imply the truth of $(p\rightarrow q)\rightarrow r$? | 4 | To solve this problem, we need to evaluate each statement to see if it implies the truth of $(p \rightarrow q) \rightarrow r$. We start by understanding the implication $p \rightarrow q$, which is logically equivalent to $\sim p \vee q$. Then, we evaluate $(p \rightarrow q) \rightarrow r$ for each statement.
#### Stat... | 0.875 | 5,572.6875 | 5,198.5 | 8,192 |
In the pattern of numbers shown, every row begins with a 1 and ends with a 2. Each of the numbers, not on the end of a row, is the sum of the two numbers located immediately above and to the right, and immediately above and to the left. For example, in the fourth row the 9 is the sum of the 4 and the 5 in the third row... | 12288 | 0.1875 | 7,545.6875 | 6,615 | 7,760.461538 | |
In how many ways can the numbers $1,2, \ldots, 2002$ be placed at the vertices of a regular 2002-gon so that no two adjacent numbers differ by more than 2? (Rotations and reflections are considered distinct.) | 4004 | 4004. There are 2002 possible positions for the 1. The two numbers adjacent to the 1 must be 2 and 3; there are two possible ways of placing these. The positions of these numbers uniquely determine the rest: for example, if 3 lies clockwise from 1, then the number lying counterclockwise from 2 must be 4; the number lyi... | 0 | 8,166.4375 | -1 | 8,166.4375 |
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