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 |
|---|---|---|---|---|---|---|
There are 10 people who want to choose a committee of 5 people among them. They do this by first electing a set of $1,2,3$, or 4 committee leaders, who then choose among the remaining people to complete the 5-person committee. In how many ways can the committee be formed, assuming that people are distinguishable? (Two ... | 7560 | There are $\binom{10}{5}$ ways to choose the 5-person committee. After choosing the committee, there are $2^{5}-2=30$ ways to choose the leaders. So the answer is $30 \cdot\binom{10}{5}=7560$. | 0.8125 | 5,318 | 4,777.076923 | 7,662 |
Given $α∈({\frac{π}{2},π})$, $sinα=\frac{3}{5}$, find $\tan 2\alpha$. | -\frac{24}{7} | 0.9375 | 3,136.6875 | 2,799.666667 | 8,192 | |
Let $P, Q$ be functions defined as $P(x) = 3\sqrt{x}$ and $Q(x) = x^2$. Calculate the value of $P(Q(P(Q(P(Q(5))))))$. | 135 | 0.75 | 4,065.875 | 3,645.083333 | 5,328.25 | |
Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have?
[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1)); draw((2,3)--(4,4)--(3,2)--cycle,linewi... | 36 | 1. **Identify the original structure**: The original figure is a rectangular prism, which has 8 vertices and 12 edges.
2. **Understand the modification**: Each corner (vertex) of the prism is cut off. The cut at each vertex is such that it does not intersect any other cuts inside the prism. This means each vertex is r... | 0.6875 | 6,763.375 | 6,191 | 8,022.6 |
The center of a circle with a radius of 5, circumscribed around an isosceles trapezoid, lies on the longer base, and the shorter base is equal to 6. Find the area of the trapezoid. | 32 | 0.1875 | 7,009.5 | 4,224.333333 | 7,652.230769 | |
The sum of five consecutive odd integers is 125. What is the smallest of these integers? | 21 | Suppose that the smallest of the five odd integers is $x$. Since consecutive odd integers differ by 2, the other four odd integers are $x+2, x+4, x+6$, and $x+8$. Therefore, $x + (x+2) + (x+4) + (x+6) + (x+8) = 125$. From this, we obtain $5x + 20 = 125$ and so $5x = 105$, which gives $x = 21$. Thus, the smallest of the... | 1 | 2,695 | 2,695 | -1 |
In a certain exam, 6 questions are randomly selected from 20 questions. If a student can correctly answer at least 4 of these questions, they pass the exam. If they can correctly answer at least 5 of these questions, they achieve an excellent grade. It is known that a certain student can correctly answer 10 of these qu... | \frac{13}{58} | 0.6875 | 5,730.25 | 5,218.090909 | 6,857 | |
If we express $2x^2 + 6x + 11$ in the form $a(x - h)^2 + k$, then what is $h$? | -\frac{3}{2} | 1 | 2,133.375 | 2,133.375 | -1 | |
Reduce the number $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$. | 1 | Observe that $(\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}})^{3}=(2+\sqrt{5})-3(\sqrt[3]{2+\sqrt{5}})-3(\sqrt[3]{2-\sqrt{5}})+(2-\sqrt{5})=4-3(\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}})$ Hence $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$ is a root of the cubic $x^{3}+3 x-4=(x-1)(x^{2}+x+4)$. The roots of $x^{2}+x+4$ are im... | 1 | 1,878.75 | 1,878.75 | -1 |
Find the number of partitions of the set $\{1,2,3,\cdots ,11,12\}$ into three nonempty subsets such that no subset has two elements which differ by $1$ .
[i]Proposed by Nathan Ramesh | 1023 | 0 | 7,904.4375 | -1 | 7,904.4375 | |
Given that $0 < α < \dfrac {π}{2}$, and $\cos (2π-α)-\sin (π-α)=- \dfrac { \sqrt {5}}{5}$,
(1) Find the value of $\sin α+\cos α$;
(2) Find the value of $\sin (2α- \dfrac {π}{4})$. | \dfrac {7 \sqrt {2}}{10} | 0 | 4,457.25 | -1 | 4,457.25 | |
Each number in the list $1,2,3,\ldots,10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positi... | 455 | 0 | 8,136.8125 | -1 | 8,136.8125 | |
What is the maximum value of $\frac{(3^t - 2t)t}{9^t}$ for integer values of $t$?
**A)** $\frac{1}{8}$
**B)** $\frac{1}{10}$
**C)** $\frac{1}{12}$
**D)** $\frac{1}{16}$
**E)** $\frac{1}{18}$ | \frac{1}{8} | 0 | 8,192 | -1 | 8,192 | |
For positive real numbers \(a\), \(b\), and \(c\), compute the maximum value of
\[
\frac{abc(a + b + c + ab)}{(a + b)^3 (b + c)^3}.
\] | \frac{1}{16} | 0 | 8,192 | -1 | 8,192 | |
Reading material: After studying square roots, Xiaoming found that some expressions containing square roots can be written as the square of another expression, such as: $3+2\sqrt{2}=(1+\sqrt{2})^{2}$. With his good thinking skills, Xiaoming conducted the following exploration:<br/>Let: $a+b\sqrt{2}=(m+n\sqrt{2})^2$ (wh... | 13 | 0.875 | 6,282.8125 | 6,422.642857 | 5,304 | |
How many integers $N$ less than $1000$ can be written as the sum of $j$ consecutive positive odd integers from exactly 5 values of $j\ge 1$? | 15 | Let the largest odd number below the sequence be the $q$th positive odd number, and the largest odd number in the sequence be the $p$th positive odd number. Therefore, the sum is $p^2-q^2=(p+q)(p-q)$ by sum of consecutive odd numbers. Note that $p+q$ and $p-q$ have the same parity, and $q$ can equal $0$. We then perfor... | 0 | 8,192 | -1 | 8,192 |
Natural numbers are arranged according to the following pattern:
\begin{tabular}{cccc}
$1-2$ & 5 & 10 & 17 \\
$\mid$ & $\mid$ & $\mid$ & $\mid$ \\
$4-3$ & 6 & 11 & 18 \\
& $\mid$ & $\mid$ & $\mid$ \\
$9-8-7$ & 12 & 19 \\
& $\mid$ & $\mid$ \\
$16-15-14-13$ & 20 \\
$25-24-23-22-21$
\end{tabular}
Then the number at the 2... | 2002 \times 2003 | 0 | 8,192 | -1 | 8,192 | |
Following the directions of the arrows, how many different paths are there from $A$ to $C$?
[asy]
pair A,B,C;
A=(0,0);
B=(5,0);
C=(10,0);
dot(A);
dot(B);
dot(C);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,S);
draw((0,0)--(2.5,1)--(5,0),Arrow);
draw((0,0)--(2.5,-1)--(5,0),Arrow);
draw(B--(7.5,1)--C,Arrow);
draw(... | 5 | 0.75 | 4,118.5625 | 3,678.333333 | 5,439.25 | |
Steph Curry is playing the following game and he wins if he has exactly 5 points at some time. Flip a fair coin. If heads, shoot a 3-point shot which is worth 3 points. If tails, shoot a free throw which is worth 1 point. He makes \frac{1}{2} of his 3-point shots and all of his free throws. Find the probability he will... | \frac{140}{243} | If he misses the shot, then the state of the game is the same as before he flipped the coin. Since the probability of making a free throw is \frac{1}{2} and the probability of making a 3-point shot is \frac{1}{4}. Therefore, given that he earns some point, the probability it is a 3-point shot is \frac{1}{3}. The possib... | 0 | 7,135.125 | -1 | 7,135.125 |
What is the least positive integer with exactly $12$ positive factors? | 72 | 0 | 3,682.1875 | -1 | 3,682.1875 | |
If $f(x)=2x^3+4$, find $f^{-1}(58)$. | 3 | 1 | 1,900.375 | 1,900.375 | -1 | |
There are 5 balls numbered $(1)$, $(2)$, $(3)$, $(4)$, $(5)$ and 5 boxes numbered $(1)$, $(2)$, $(3)$, $(4)$, $(5)$. Each box contains one ball. The number of ways in which at most two balls have the same number as their respective boxes is $\_\_\_\_\_\_$. | 109 | 1 | 3,794.5 | 3,794.5 | -1 | |
Simplify the expression \[\sqrt{37-20\sqrt3}.\] | 5-2\sqrt3 | 1 | 1,900.8125 | 1,900.8125 | -1 | |
In the following diagram (not to scale), $A$ , $B$ , $C$ , $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$ . Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$ . Find $\angle OPQ$ in degrees.
[asy]
pathpen = rgb(0,0,0.6)+linewidth(0.7); pointpen = black+linew... | 30 | 0.1875 | 8,172.75 | 8,089.333333 | 8,192 | |
Suppose $x$ and $y$ are inversely proportional and positive. If $x$ increases by $p\%$, then $y$ decreases by | \frac{100p}{100+p}\%$ | 1. **Understanding Inverse Proportionality**: Given that $x$ and $y$ are inversely proportional, we can express this relationship as:
\[
xy = k
\]
for some constant $k$.
2. **Change in $x$**: If $x$ increases by $p\%$, the new value of $x$, denoted as $x'$, can be calculated as:
\[
x' = x \left(1 + \... | 0 | 4,065.625 | -1 | 4,065.625 |
Let $ABCDE$ be a convex pentagon such that: $\angle ABC=90,\angle BCD=135,\angle DEA=60$ and $AB=BC=CD=DE$ . Find angle $\angle DAE$ . | 30 | 0 | 8,192 | -1 | 8,192 | |
A class participates in a tree-planting event, divided into three groups. The first group plants 5 trees per person, the second group plants 4 trees per person, and the third group plants 3 trees per person. It is known that the number of people in the second group is one-third of the sum of the number of people in the... | 32 | 1 | 4,035.25 | 4,035.25 | -1 | |
In a triangle configuration, each row consists of increasing multiples of 3 unit rods. The number of connectors in a triangle always forms an additional row than the rods, with connectors enclosing each by doubling the requirements of connecting joints from the previous triangle. How many total pieces are required to b... | 60 | 0 | 7,663.5 | -1 | 7,663.5 | |
Let $a$ and $b$ be real numbers, and let $r, s$, and $t$ be the roots of $f(x)=x^{3}+a x^{2}+b x-1$. Also, $g(x)=x^{3}+m x^{2}+n x+p$ has roots $r^{2}, s^{2}$, and $t^{2}$. If $g(-1)=-5$, find the maximum possible value of $b$. | 1+\sqrt{5} | By Vieta's Formulae, $m=-\left(r^{2}+s^{2}+t^{2}\right)=-a^{2}+2 b, n=r^{2} s^{2}+s^{2} t^{2}+t^{2} r^{2}= b^{2}+2 a$, and $p=-1$. Therefore, $g(-1)=-1-a^{2}+2 b-b^{2}-2 a-1=-5 \Leftrightarrow(a+1)^{2}+(b-1)^{2}=5$. This is an equation of a circle, so $b$ reaches its maximum when $a+1=0 \Rightarrow a=-1$. When $a=-1$, ... | 0.6875 | 5,699.4375 | 5,053 | 7,121.6 |
Given that the terminal side of angle $\alpha$ passes through point $P(m, 2\sqrt{2})$, $\sin{\alpha} = \frac{2\sqrt{2}}{3}$, and $\alpha$ is in the second quadrant.
(1) Find the value of $m$;
(2) If $\tan{\beta} = \sqrt{2}$, find the value of $\frac{\sin{\alpha}\cos{\beta} + 3\sin({\frac{\pi}{2} + \alpha})\sin{\beta}... | \frac{\sqrt{2}}{11} | 0 | 4,785.125 | -1 | 4,785.125 | |
The area of the larger base of a truncated pyramid is $T$, the area of the smaller base is $t$, and the height is $m$. The volume is $V = kTm$, where $\frac{1}{3} \leq k \leq 1$. What is the proportionality ratio $\lambda(<1)$ between the two bases? (The proportionality ratio is the ratio of corresponding distances in ... | \frac{2 - \sqrt{3}}{2} | 0 | 8,028.5 | -1 | 8,028.5 | |
Let \( a \) and \( b \) be positive integers such that \( 79 \mid (a + 77b) \) and \( 77 \mid (a + 79b) \). Find the smallest possible value of \( a + b \). | 193 | 0 | 8,192 | -1 | 8,192 | |
Add the square of the smallest area from squares of size $1 \times 1, 2 \times 2,$ and $3 \times 3,$ such that the number of squares of each size is the same. | 14 | 0.5 | 4,435.625 | 4,803.375 | 4,067.875 | |
Find the number of positive integer solutions to $n^{x}+n^{y}=n^{z}$ with $n^{z}<2001$. | 10 | If $n=1$, the relation can not hold, so assume otherwise. If $x>y$, the left hand side factors as $n^{y}\left(n^{x-y}+1\right)$ so $n^{x-y}+1$ is a power of $n$. But it leaves a remainder of 1 when divided by $n$ and is greater than 1, a contradiction. We reach a similar contradiction if $y>x$. So $y=x$ and $2 n^{x}=n^... | 0 | 7,767.1875 | -1 | 7,767.1875 |
The area of the triangle formed by the tangent to the curve $y=\ln(x)-2x$ at the point $(1, -2)$ and the coordinate axes. | \frac{1}{2} | 1 | 3,515.625 | 3,515.625 | -1 | |
(1) Find the domain of the function $y= \sqrt {\sin x}+ \sqrt { \frac{1}{2}-\cos x}$.
(2) Find the range of the function $y=\cos ^{2}x-\sin x$, where $x\in\left[-\frac {\pi}{4}, \frac {\pi}{4}\right]$. | \frac{1-\sqrt{2}}{2} | 0 | 5,522.25 | -1 | 5,522.25 | |
Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 17x\]for all integers $x.$ Find the smallest possible value of $a.$ | 17 | 0.5625 | 7,585.875 | 7,114.444444 | 8,192 | |
Out of 1500 people surveyed, 25% do not like television, and 15% of those who do not like television also do not like games. How many people surveyed do not like both television and games? | 56 | 0.375 | 2,649.8125 | 388.5 | 4,006.6 | |
The mode and median of the data $9.30$, $9.05$, $9.10$, $9.40$, $9.20$, $9.10$ are ______ and ______, respectively. | 9.15 | 0.5 | 710.25 | 748.375 | 672.125 | |
In the five-digit number abcde, \(a, b, c, d, e\) respectively represent the digits in the ten-thousands, thousands, hundreds, tens, and units places. Given that \(d > e\), \(c > d + e\), \(b > c + d + e\), and \(a > b + c + d + e\), what is the largest five-digit number that satisfies these conditions? | 95210 | 0 | 8,186.6875 | -1 | 8,186.6875 | |
For what positive value of $t$ is $|6+ti| = 10$? | 8 | 1 | 1,181.8125 | 1,181.8125 | -1 | |
Find the sum of all prime numbers between $1$ and $120$ that are simultaneously $1$ greater than a multiple of $3$ and $1$ less than a multiple of $5$. | 207 | 1 | 3,664.4375 | 3,664.4375 | -1 | |
Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then the... | C |
We are given that players \( A \), \( B \), and \( C \) each receive one card per game, and the points received correspond to the numbers written on their respective cards. After several games, the total points are as follows: \( A \) has 20 points, \( B \) has 10 points, and \( C \) has 9 points. In the last game, \(... | 0 | 8,192 | -1 | 8,192 |
The ferry boat begins transporting tourists to an island every hour starting at 9 AM until its last trip, which starts at 4 PM. On the first trip at 9 AM, there were 120 tourists, and on each successive trip, there were 2 fewer tourists than on the previous trip. Determine the total number of tourists the ferry transpo... | 904 | 0.625 | 1,745.875 | 1,874.1 | 1,532.166667 | |
What is the minimum number of children required in a school to be sure that at least 3 of them have their birthday on the same day? (Keep in mind that some people are born on February 29.) | 733 | 0.9375 | 3,218.5625 | 2,887 | 8,192 | |
Given an ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ passes through points $M(2,0)$ and $N(0,1)$.
$(1)$ Find the equation of ellipse $C$ and its eccentricity;
$(2)$ A line $y=kx (k \in \mathbb{R}, k \neq 0)$ intersects ellipse $C$ at points $A$ and $B$, point $D$ is a moving point on ellipse $C$, a... | \dfrac{8}{5} | 0.0625 | 7,696.5 | 7,492 | 7,710.133333 | |
If the digit $1$ is placed after a two digit number whose tens' digit is $t$, and units' digit is $u$, the new number is: | 100t+10u+1 | 1. **Identify the original number**: A two-digit number with tens' digit $t$ and units' digit $u$ can be expressed as $10t + u$.
2. **Placing the digit 1 after the number**: When the digit $1$ is placed after this two-digit number, it effectively shifts the original number one place to the left in the decimal system a... | 0.9375 | 3,401.9375 | 3,082.6 | 8,192 |
Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $... | 160 | 0.375 | 4,967.875 | 3,454.5 | 5,875.9 | |
In the arithmetic sequence $\{a_n\}$, it is known that $a_1=10$, and the sum of the first $n$ terms is $S_n$. If $S_9=S_{12}$, find the maximum value of $S_n$ and the corresponding value of $n$. | 55 | 0.3125 | 7,477.6875 | 7,451.6 | 7,489.545455 | |
China Unicom charges for mobile phone calls with two types of packages: Package $A$ (monthly fee of $15$ yuan, call fee of $0.1 yuan per minute) and Package $B$ (monthly fee of $0$ yuan, call fee of $0.15 yuan per minute). Let $y_{1}$ represent the monthly bill for Package $A$ (in yuan), $y_{2}$ represent the monthly b... | 300 | 1 | 1,685.9375 | 1,685.9375 | -1 | |
Let $ABC$ be a triangle whose angles measure $A$ , $B$ , $C$ , respectively. Suppose $\tan A$ , $\tan B$ , $\tan C$ form a geometric sequence in that order. If $1\le \tan A+\tan B+\tan C\le 2015$ , find the number of possible integer values for $\tan B$ . (The values of $\tan A$ and $\tan C$ need not b... | 11 | 0.125 | 8,060.5625 | 7,140.5 | 8,192 | |
There are 22 people standing in a circle, and each of them is either a knight (who always tells the truth) or a liar (who always lies). Each person says: "The next 10 people clockwise from me are liars." How many liars are there among these 22 people? | 20 | 0.125 | 7,856.3125 | 6,211 | 8,091.357143 | |
A store sells a type of notebook. The retail price for each notebook is 0.30 yuan, a dozen (12 notebooks) is priced at 3.00 yuan, and for purchases of more than 10 dozen, each dozen can be paid for at 2.70 yuan.
(1) There are 57 students in the ninth grade class 1, and each student needs one notebook of this type. Wh... | 51.30 | 0.4375 | 6,763.9375 | 5,219.714286 | 7,965 | |
Given that $f(x)$ is an odd function on $\mathbb{R}$ and satisfies $f(x+4)=f(x)$, when $x \in (0,2)$, $f(x)=2x^2$. Evaluate $f(2015)$. | -2 | 0.9375 | 3,460.25 | 3,144.8 | 8,192 | |
A tetrahedron with each edge length equal to $\sqrt{2}$ has all its vertices on the same sphere. Calculate the surface area of this sphere. | 3\pi | 0.9375 | 3,493.9375 | 3,180.733333 | 8,192 | |
Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x, y, \) and \( z \) such that \( x \mid y^{3} \), \( y \mid z^{3} \), and \( z \mid x^{3} \), it always holds that \( x y z \mid (x+y+z)^{n} \). | 13 | 0.125 | 8,097.3125 | 7,434.5 | 8,192 | |
A hexagonal prism has a height of 165 cm. Its two hexagonal faces are regular hexagons with sides of length 30 cm. Its other six faces are rectangles. A fly and an ant start at point \(X\) on the bottom face and travel to point \(Y\) on the top face. The fly flies directly along the shortest route through the prism. Th... | 19 | Throughout this solution, we remove the units (cm) as each length is in these same units. First, we calculate the distance flown by the fly, which we call \(f\). Let \(Z\) be the point on the base on the prism directly underneath \(Y\). Since the hexagonal base has side length 30, then \(XZ = 60\). This is because a he... | 0.125 | 7,995.875 | 7,657 | 8,044.285714 |
A positive integer \( m \) has the property that when multiplied by 12, the result is a four-digit number \( n \) of the form \( 20A2 \) for some digit \( A \). What is the four-digit number \( n \)? | 2052 | 0.875 | 3,959.25 | 3,354.571429 | 8,192 | |
A regular polygon has sides of length 5 units and an exterior angle of 120 degrees. What is the perimeter of the polygon, in units? | 15 | 1 | 1,717.4375 | 1,717.4375 | -1 | |
A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_n$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10} = .4$ and $a_n\le.4$ for all $n$ such that $1\le n\le9$ is given to be $p^aq^br/\left(s^c\right)$ where $p... | 660 | We graph the $10$ shots on a grid. Suppose that a made shot is represented by a step of $(0,1)$, and a missed shot is represented by $(1,0)$. Then the basketball player's shots can be represented by the number of paths from $(0,0)$ to $(6,4)$ that always stay below the line $y=\frac{2x}{3}$. We can find the number of s... | 0 | 8,192 | -1 | 8,192 |
A box contains 5 white balls and 6 black balls. A ball is drawn out of the box at random. What is the probability that the ball is white? | \dfrac{5}{11} | 1 | 988.0625 | 988.0625 | -1 | |
How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | 152 | 0 | 7,596.6875 | -1 | 7,596.6875 | |
Consider the points $A(0,12), B(10,9), C(8,0),$ and $D(-4,7).$ There is a unique square $S$ such that each of the four points is on a different side of $S.$ Let $K$ be the area of $S.$ Find the remainder when $10K$ is divided by $1000$. | 936 | Let $(a,b)$ denote a normal vector of the side containing $A$. Note that $\overline{AC}, \overline{BD}$ intersect and hence must be opposite vertices of the square. The lines containing the sides of the square have the form $ax+by=12b$, $ax+by=8a$, $bx-ay=10b-9a$, and $bx-ay=-4b-7a$. The lines form a square, so the dis... | 0 | 8,192 | -1 | 8,192 |
In a regular tetrahedron O-ABC with each edge length equal to 1, if point P satisfies $$\overrightarrow {OP}=x \overrightarrow {OA}+y \overrightarrow {OB}+z \overrightarrow {OC}(x+y+z=1)$$, find the minimum value of $$| \overrightarrow {OP}|$$. | \frac{\sqrt{6}}{3} | 0 | 7,803 | -1 | 7,803 | |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles? Express your answer in fully expanded form in terms of $\pi$. | \frac{9\pi}{2} - 9 | 0.6875 | 4,791.375 | 4,171.181818 | 6,155.8 | |
Given $\sin x_{1}=\sin x_{2}=\frac{1}{3}$ and $0 \lt x_{1} \lt x_{2} \lt 2\pi$, find $\cos |\overrightarrow{a}|$. | -\frac{7}{9} | 0.625 | 5,685.9375 | 4,540.2 | 7,595.5 | |
Each of the lateral edges of a pyramid is equal to 269/32. The base of the pyramid is a triangle with sides 13, 14, 15. Find the volume of the pyramid. | 483/8 | 0.9375 | 5,515.375 | 5,336.933333 | 8,192 | |
The bottoms of two vertical poles are 20 feet apart on a flat ground. One pole is 8 feet tall and the other is 18 feet tall. Simultaneously, the ground between the poles is sloped, with the base of the taller pole being 2 feet higher than the base of the shorter pole due to the slope. Calculate the length in feet of a ... | \sqrt{544} | 0 | 6,260.6875 | -1 | 6,260.6875 | |
Petya invented four distinct natural numbers and wrote down all their pairwise sums on the board. Below those, he wrote all their sums taken three at a time. It turned out that the sum of the two largest pairwise sums and the two smallest sums from those taken three at a time (a total of four sums) is 2017. Find the la... | 1006 | 0 | 8,192 | -1 | 8,192 | |
Given point $P(2,2)$, and circle $C$: $x^{2}+y^{2}-8y=0$. A moving line $l$ passing through point $P$ intersects circle $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $M$, and $O$ being the origin.
$(1)$ Find the equation of the trajectory of point $M$;
$(2)$ When $|OP|=|OM|$, find the equation of... | \frac{16}{5} | 0.1875 | 7,836.1875 | 6,294.333333 | 8,192 | |
Consider a sequence $\{a_n\}$ whose sum of the first $n$ terms $S_n = n^2 - 4n + 2$. Find the sum of the absolute values of the first ten terms: $|a_1| + |a_2| + \cdots + |a_{10}|$. | 68 | 0.4375 | 4,956.625 | 3,771.285714 | 5,878.555556 | |
Let \( x \) be a positive integer, and write \( a = \left\lfloor \log_{10} x \right\rfloor \) and \( b = \left\lfloor \log_{10} \frac{100}{x} \right\rfloor \). Here \( \lfloor c \rfloor \) denotes the greatest integer less than or equal to \( c \). Find the largest possible value of \( 2a^2 - 3b^2 \). | 24 | 0 | 8,192 | -1 | 8,192 | |
Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in $10$ , $9$ , $11$ , and $12$ days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer... | 12 | 0 | 8,177.875 | -1 | 8,177.875 | |
How many pairs of integers $(a, b)$, with $1 \leq a \leq b \leq 60$, have the property that $b$ is divisible by $a$ and $b+1$ is divisible by $a+1$? | 106 | The divisibility condition is equivalent to $b-a$ being divisible by both $a$ and $a+1$, or, equivalently (since these are relatively prime), by $a(a+1)$. Any $b$ satisfying the condition is automatically $\geq a$, so it suffices to count the number of values $b-a \in$ $\{1-a, 2-a, \ldots, 60-a\}$ that are divisible by... | 0 | 8,169.4375 | -1 | 8,169.4375 |
Given non-zero complex numbers \( x \) and \( y \) satisfying
\[ y^{2}(x^{2}-xy+y^{2})+x^{3}(x-y)=0, \]
find the value of
\[ \sum_{m=0}^{29} \sum_{n=0}^{29} x^{18mn} y^{-18mn}. \] | 180 | 0.0625 | 7,707.0625 | 7,852 | 7,697.4 | |
A professor is assigning grades to a class of 10 students. As a very kind professor, he only gives out A's, B's, and C's. How many ways can the professor assign grades to all his students? | 59049 | 0.875 | 1,298.9375 | 1,428.071429 | 395 | |
A wholesaler bought 500 kilograms of a certain type of fruit at a price of 40 yuan per kilogram. According to market forecasts, the selling price $y$ (yuan per kilogram) of this fruit is a function of the storage time $x$ (days), given by $y=60+2x$. However, an average of 10 kilograms of this fruit will be lost each da... | 11600 | 0.875 | 3,845.875 | 3,930.071429 | 3,256.5 | |
What is the least three-digit positive integer that has 2, 5 and 7 as factors? | 140 | 1 | 2,056.8125 | 2,056.8125 | -1 | |
In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. If $a^{2}=b^{2}+4bc\sin A$ and $\tan A \cdot \tan B=2$, then $\tan B-\tan A=$ ______. | -8 | 0.25 | 7,806.75 | 6,996.25 | 8,076.916667 | |
How many positive two-digit integers leave a remainder of 2 when divided by 8? | 12 | 1 | 2,358.5625 | 2,358.5625 | -1 | |
Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations $\begin{cases} & x=2\sqrt{3}\cos \alpha \\ & y=2\sin \alpha \end{cases}$, where $\alpha$ is the parameter, $\alpha \in (0,\pi)$. In the polar coordinate system with the ... | 6 \sqrt{2} | 0.6875 | 6,893.625 | 6,303.454545 | 8,192 | |
A cube with an edge length of 4 units has the same volume as a square-based pyramid with base edge lengths of 8 units and a height of $h$ units. What is the value of $h$? | 3 | 1 | 868.5625 | 868.5625 | -1 | |
Four fair coins are tossed once. For every head that appears, two six-sided dice are rolled. What is the probability that the sum of all dice rolled is exactly ten?
A) $\frac{1} {48}$
B) $\frac{1} {20}$
C) $\frac{1} {16}$
D) $\frac{1} {30}$ | \frac{1} {20} | 0 | 8,192 | -1 | 8,192 | |
Given $x$, $y \in \mathbb{R}^{+}$ and $2x+3y=1$, find the minimum value of $\frac{1}{x}+ \frac{1}{y}$. | 5+2 \sqrt{6} | 0.8125 | 6,209.8125 | 5,752.384615 | 8,192 | |
In a bag, there are 2 black balls labeled $1$ and $2$, and 3 white balls labeled $3$, $4$, and $5$. These 5 balls are identical except for their labels and colors.
$(1)$ If two balls are randomly drawn from the bag with replacement, one at a time, what is the probability of drawing a black ball first and then a white... | \frac{3}{10} | 0.1875 | 7,238.25 | 7,529.333333 | 7,171.076923 | |
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. What is $c$? | \frac{2}{3} |
We are given a configuration of five unit squares in the coordinate plane, and a line extending from $(c,0)$ to $(3,3)$ that divides the entire region into two regions of equal area. We need to find the value of $c$.
#### Step-by-step Analysis:
1. **Total Area of the Squares**: The total area of the five unit square... | 0 | 8,186.125 | -1 | 8,186.125 |
Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 8x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$ | 27 | 0.1875 | 7,980.5 | 7,064 | 8,192 | |
Ivan rents a car for $\$$25 a day and $\$$0.20 a mile. If he rents it for 4 days and drives it 400 miles, how many dollars does he pay? | \$180 | 1 | 1,237 | 1,237 | -1 | |
Calculate the volumes of solids formed by the rotation of regions bounded by the graphs of the functions around the y-axis.
$$
y=\arcsin \frac{x}{5}, y=\arcsin x, y=\frac{\pi}{2}
$$ | 6 \pi^2 | 0.125 | 8,154.625 | 7,893 | 8,192 | |
In rectangle $ABCD$, side $AB$ measures $6$ units and side $BC$ measures $3$ units, as shown. Points $F$ and $G$ are on side $CD$ with segment $DF$ measuring $1$ unit and segment $GC$ measuring $2$ units, and lines $AF$ and $BG$ intersect at $E$. What is the area of triangle $AEB$? [asy]
draw((0,0)--(6,0)--(6,3)--(0,3)... | 18 | 0.8125 | 5,673.5625 | 5,092.384615 | 8,192 | |
In the number sequence $1,1,2,3,5,8,x,21,34,55$, what is the value of $x$? | 13 | 0.9375 | 1,339.9375 | 1,399.466667 | 447 | |
Express $7.\overline{316}$ as a common fraction in lowest terms. | \frac{7309}{999} | 1 | 3,906.25 | 3,906.25 | -1 | |
Define the *hotel elevator cubic*as the unique cubic polynomial $P$ for which $P(11) = 11$ , $P(12) = 12$ , $P(13) = 14$ , $P(14) = 15$ . What is $P(15)$ ?
*Proposed by Evan Chen* | 13 | 0.5625 | 6,629.8125 | 5,414.777778 | 8,192 | |
Let $A$ , $M$ , and $C$ be nonnegative integers such that $A+M+C=10$ . What is the maximum value of $A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A$ ? | 69 | 0.875 | 6,126.625 | 5,831.571429 | 8,192 | |
The consignment shop received for sale cameras, clocks, pens, and receivers totaling 240 rubles. The sum of the prices of the receiver and one clock is 4 rubles more than the sum of the prices of the camera and the pen, and the sum of the prices of one clock and the pen is 24 rubles less than the sum of the prices of t... | 18 | 0.375 | 7,566.875 | 6,540.5 | 8,182.7 | |
In triangle \( \triangle ABC \), \( BD \) is a median, \( CF \) intersects \( BD \) at \( E \), and \( BE = ED \). Point \( F \) is on \( AB \), and if \( BF = 5 \), then the length of \( BA \) is: | 15 | 1 | 4,449.0625 | 4,449.0625 | -1 | |
Given \( x \in (0,1) \) and \( \frac{1}{x} \notin \mathbf{Z} \), define
\[
a_{n}=\frac{n x}{(1-x)(1-2 x) \cdots(1-n x)} \quad \text{for} \quad n=1,2, \cdots
\]
A number \( x \) is called a "good number" if and only if it makes the sequence \(\{a_{n}\}\) satisfy
\[
a_{1}+a_{2}+\cdots+a_{10} > -1 \quad \text{and} \q... | 61/210 | 0 | 7,265.1875 | -1 | 7,265.1875 | |
Given that $(a_n)_{n \equal{} 1}^\infty$ is defined on real numbers with $a_n \not \equal{} 0$, $a_na_{n \plus{} 3} = a_{n \plus{} 2}a_{n \plus{} 5}$, and $a_1a_2 + a_3a_4 + a_5a_6 = 6$. Find the value of $a_1a_2 + a_3a_4 + \cdots + a_{41}a_{42}$. | 42 | 0.125 | 7,825.3125 | 5,258.5 | 8,192 | |
A survey of $150$ teachers determined the following:
- $90$ had high blood pressure
- $60$ had heart trouble
- $50$ had diabetes
- $30$ had both high blood pressure and heart trouble
- $20$ had both high blood pressure and diabetes
- $10$ had both heart trouble and diabetes
- $5$ had all three conditions
What percent ... | 3.33\% | 0.9375 | 4,099.375 | 3,826.533333 | 8,192 | |
Every day from Monday to Friday, an old man went to the blue sea and cast his net. Each day, the number of fish caught in the net was not greater than the number caught the previous day. Over the five days, the old man caught exactly 100 fish. What is the minimum total number of fish he could have caught over three day... | 50 | 0.0625 | 8,031.25 | 5,620 | 8,192 |
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