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 |
|---|---|---|---|---|---|---|
Given the hyperbola $mx^{2}+y^{2}=1$ and one of its asymptotes has a slope of $2$, find the value of $m$. | -4 | 0.875 | 3,599.6875 | 3,051.857143 | 7,434.5 | |
The sum of two numbers is $30$. Their difference is $4$. What is the larger of the two numbers? | 17 | 1 | 1,893.125 | 1,893.125 | -1 | |
When \( q(x) = Dx^4 + Ex^2 + Fx + 6 \) is divided by \( x - 2 \), the remainder is 14. Find the remainder when \( q(x) \) is divided by \( x + 2 \). | 14 | 0.125 | 8,192 | 8,192 | 8,192 | |
Three times Dick's age plus Tom's age equals twice Harry's age.
Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age.
Their respective ages are relatively prime to each other. The sum of the squares of their ages is | 42 | We are given two equations involving the ages of Dick, Tom, and Harry, denoted as $d$, $t$, and $h$ respectively:
1. \(3d + t = 2h\)
2. \(2h^3 = 3d^3 + t^3\)
We start by expressing $t$ in terms of $d$ and $h$ from the first equation:
\[ t = 2h - 3d \]
Substitute this expression for $t$ into the second equation:
\[ 2h... | 0.625 | 4,738.8125 | 4,017.4 | 5,941.166667 |
Using the distinct digits \( a, b, \) and \( c \), Araceli wrote the number \( abc \), and Luana wrote the numbers \( ab, bc, \) and \( ca \). Find the digits \( a, b, \) and \( c \), knowing that the sum of the numbers written by Luana is equal to the number written by Araceli. | 198 | 0.0625 | 3,043.125 | 8,192 | 2,699.866667 | |
The function $f(x)$ satisfies
\[f(2^x) + xf(2^{-x}) = 1\]for all real numbers $x.$ Find $f(2).$ | 0 | 1 | 3,947.5 | 3,947.5 | -1 | |
For $i = 1, 2, \cdots, n$, if $\left|x_{i}\right| < 1$ and $\left|x_{1}\right| + \left|x_{2}\right| + \cdots + \left|x_{n}\right| = 2005 + \left|x_{1} + x_{2} + \cdots + x_{n} \right|$, find the minimum value of the positive integer $n$. | 2006 | 0.375 | 7,367.5625 | 5,993.5 | 8,192 | |
In the tetrahedron A-BCD inscribed within sphere O, we have AB=6, AC=10, $\angle ABC = \frac{\pi}{2}$, and the maximum volume of the tetrahedron A-BCD is 200. Find the radius of sphere O. | 13 | 0.375 | 7,145.6875 | 5,475.666667 | 8,147.7 | |
A line contains the points $(-1, 6)$, $(6, k)$ and $(20, 3)$. What is the value of $k$? | 5 | 1 | 1,972.6875 | 1,972.6875 | -1 | |
Suppose that \( ABCDEF \) is a regular hexagon with sides of length 6. Each interior angle of \( ABCDEF \) is equal to \( 120^{\circ} \).
(a) A circular arc with center \( D \) and radius 6 is drawn from \( C \) to \( E \). Determine the area of the shaded sector.
(b) A circular arc with center \( D \) and radius 6 i... | 18\pi - 27\sqrt{3} | 0.0625 | 8,087.6875 | 6,523 | 8,192 | |
My grandpa has 12 pieces of art, including 4 prints by Escher and 3 by Picasso. What is the probability that all four Escher prints and all three Picasso prints will be placed consecutively? | \dfrac{1}{660} | 0.4375 | 6,883.25 | 5,705.714286 | 7,799.111111 | |
Calculate the square of 1007 without using a calculator. | 1014049 | 0.5625 | 526.5625 | 504.888889 | 554.428571 | |
A school wants to understand the psychological state of its senior high school students regarding their studies. They decide to use a systematic sampling method to select 40 students out of 800 for a certain test. The students are randomly numbered from 1 to 800. After grouping, the first group is selected through simp... | 12 | 0.625 | 5,660.875 | 4,142.2 | 8,192 | |
A total of 1000 senior high school students from a certain school participated in a mathematics exam. The scores in this exam follow a normal distribution N(90, σ²). If the probability of a score being within the interval (70, 110] is 0.7, estimate the number of students with scores not exceeding 70. | 150 | 0.9375 | 3,927.3125 | 3,643 | 8,192 | |
Given $DC = 7$, $CB = 8$, $AB = \frac{1}{4}AD$, and $ED = \frac{4}{5}AD$, find $FC$. Express your answer as a decimal. [asy]
draw((0,0)--(-20,0)--(-20,16)--cycle);
draw((-13,0)--(-13,10.4));
draw((-5,0)--(-5,4));
draw((-5,0.5)--(-5+0.5,0.5)--(-5+0.5,0));
draw((-13,0.5)--(-13+0.5,0.5)--(-13+0.5,0));
draw((-20,0.5)--(-... | 10.4 | 0.875 | 5,627.875 | 5,261.571429 | 8,192 | |
A ticket contains six digits \(a, b, c, d, e, f\). This ticket is said to be "lucky" if \(a + b + c = d + e + f\). How many lucky tickets are there (including the ticket 000000)? | 55252 | 0 | 8,192 | -1 | 8,192 | |
From 1 to 100, take a pair of integers (repetitions allowed) so that their sum is greater than 100. How many ways are there to pick such pairs? | 5050 | 0.3125 | 7,432.125 | 6,341.8 | 7,927.727273 | |
Order the numbers $3$, $\frac{5}{2}$, and $\sqrt{10}$ from smallest to largest. | \frac{5}{2}, 3, \sqrt{10} | Since $3 = \frac{6}{2}$ and $\frac{5}{2} < \frac{6}{2}$, then $\frac{5}{2} < 3$. Since $3 = \sqrt{9}$ and $\sqrt{9} < \sqrt{10}$, then $3 < \sqrt{10}$. Thus, $\frac{5}{2} < 3 < \sqrt{10}$, and so the list of the three numbers in order from smallest to largest is $\frac{5}{2}, 3, \sqrt{10}$. | 0.375 | 394.8125 | 418.166667 | 380.8 |
What digits should replace the asterisks to make the number 454** divisible by 2, 7, and 9? | 45486 | 0.25 | 5,308.6875 | 4,979.25 | 5,418.5 | |
The railway between Station A and Station B is 840 kilometers long. Two trains start simultaneously from the two stations towards each other, with Train A traveling at 68.5 kilometers per hour and Train B traveling at 71.5 kilometers per hour. After how many hours will the two trains be 210 kilometers apart? | 7.5 | 0 | 650.375 | -1 | 650.375 | |
Simplify $$\frac{13!}{11! + 3 \cdot 9!}$$ | \frac{17160}{113} | 0.875 | 5,623.9375 | 5,257.071429 | 8,192 | |
What is the product of the two largest one-digit primes and the largest two-digit prime? | 3395 | 1 | 1,739.9375 | 1,739.9375 | -1 | |
Jeff has a 50 point quiz at 11 am . He wakes up at a random time between 10 am and noon, then arrives at class 15 minutes later. If he arrives on time, he will get a perfect score, but if he arrives more than 30 minutes after the quiz starts, he will get a 0 , but otherwise, he loses a point for each minute he's late (... | \frac{55}{2} | If he wakes up between 10:00 and 10:45, he will arrive on time and get a perfect score of 50. If he wakes up between 10:45 and 11:15, he will arrive late and lose points. If he wakes up $k$ minutes after $10: 45$, then he gets $50-k$ points. Finally, if he wakes up between 11:15 and 12:00 he gets 0 points. So he has a ... | 0.0625 | 7,093.3125 | 4,646 | 7,256.466667 |
A bag contains 4 tan, 3 pink, 5 violet, and 2 green chips. If all 14 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the 4 tan chips, the 3 pink chips, and the 5 violet chips are each drawn consecutively, and there is at least one green chip placed between any ... | \frac{1440}{14!} | 0 | 8,192 | -1 | 8,192 | |
Given the sequences $\{a_{n}\}$ and $\{b_{n}\}$ satisfying $2a_{n+1}+a_{n}=3$ for $n\geqslant 1$, $a_{1}=10$, and $b_{n}=a_{n}-1$. Find the smallest integer $n$ that satisfies the inequality $|{{S_n}-6}|＜\frac{1}{{170}}$. | 10 | 0.0625 | 8,192 | 8,192 | 8,192 | |
An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $20$, and one of the base angles is $\arcsin(0.6)$. Find the area of the trapezoid given that the height of the trapezoid is $9$. | 100 | 0 | 7,546.625 | -1 | 7,546.625 | |
A permutation of \{1,2, \ldots, 7\} is chosen uniformly at random. A partition of the permutation into contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes sorted. For example, the permutation $(3,4,2,1,6,5,7)$ can be partitioned correctly into the blocks $[3,4,2,1]$ ... | \frac{151}{105} | Let $\sigma$ be a permutation on \{1, \ldots, n\}. Call $m \in\{1, \ldots, n\}$ a breakpoint of $\sigma$ if $\{\sigma(1), \ldots, \sigma(m)\}=$ $\{1, \ldots, m\}$. Notice that the maximum partition is into $k$ blocks, where $k$ is the number of breakpoints: if our breakpoints are $m_{1}, \ldots, m_{k}$, then we take $\... | 0 | 8,084.9375 | -1 | 8,084.9375 |
Given that a child builds towers with $2$ red cubes, $3$ blue cubes, and $4$ green cubes, determine the number of different towers with a height of $8$ cubes that can be built, with one cube left out. | 1,260 | 0 | 5,586.875 | -1 | 5,586.875 | |
Given the polar equation of curve $C$ is $\rho-4\sin \theta=0$. With the pole as the origin and the positive half-axis of the $x$-axis as the polar axis, a Cartesian coordinate system is established. Line $l$ passes through point $M(1,0)$ with an inclination angle of $\dfrac{3\pi}{4}$.
$(1)$ Find the Cartesian equation... | 3\sqrt{2} | 0.9375 | 5,136.0625 | 5,050.933333 | 6,413 | |
Rationalize the denominator of $\frac{3}{2\sqrt[3]{5}}$. The answer can be written in the form of $\frac{A\sqrt[3]{B}}{C}$, where $A$, $B$, and $C$ are integers, $C$ is positive, and $B$ is not divisible by the cube of any prime. Find $A+B+C$. | 38 | 0.9375 | 2,136.4375 | 1,732.733333 | 8,192 | |
A cashier, upon checking the account before leaving work, finds that the cash is 153 yuan less than the account book. She knows the actual amount collected cannot be wrong, so it must be due to a decimal point error during bookkeeping. What is the actual amount of the cash that was recorded incorrectly? | 17 | 0.1875 | 6,005.75 | 5,932.666667 | 6,022.615385 | |
Given $m+n=2$ and $mn=-2$. Find the value of:
1. $2^{m}\cdot 2^{n}-(2^{m})^{n}$
2. $(m-4)(n-4)$
3. $(m-n)^{2}$. | 12 | 1 | 1,957.875 | 1,957.875 | -1 | |
A square is divided into $25$ unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property? | 12 | 0.0625 | 8,178.875 | 7,982 | 8,192 | |
Given that \( a \) and \( b \) are real numbers, and the following system of inequalities in terms of \( x \):
\[
\left\{\begin{array}{l}
20x + a > 0, \\
15x - b \leq 0
\end{array}\right.
\]
has integer solutions of only 2, 3, and 4, find the maximum value of \( ab \). | -1200 | 0.125 | 8,161.75 | 8,044.5 | 8,178.5 | |
Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$ . Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all
numbers $N$ with such property. Find the sum of all possible values ... | 135 | 0 | 8,156.625 | -1 | 8,156.625 | |
The value of \(0.001 + 1.01 + 0.11\) is | 1.121 | 0.9375 | 3,729.5 | 3,432 | 8,192 | |
The functions $p(x),$ $q(x),$ and $r(x)$ are all invertible. We set
\[f = q \circ p \circ r.\]Which is the correct expression for $f^{-1}$?
A. $r^{-1} \circ q^{-1} \circ p^{-1}$
B. $p^{-1} \circ q^{-1} \circ r^{-1}$
C. $r^{-1} \circ p^{-1} \circ q^{-1}$
D. $q^{-1} \circ p^{-1} \circ r^{-1}$
E. $q^{-1} \circ r^{-1... | \text{C} | 0 | 3,476.125 | -1 | 3,476.125 | |
Given that the point $(1, \frac{1}{3})$ lies on the graph of the function $f(x)=a^{x}$ ($a > 0$ and $a \neq 1$), and the sum of the first $n$ terms of the geometric sequence $\{a_n\}$ is $f(n)-c$, the first term and the sum $S_n$ of the sequence $\{b_n\}$ ($b_n > 0$) satisfy $S_n-S_{n-1}= \sqrt{S_n}+ \sqrt{S_{n+1}}$ ($... | 112 | 0 | 8,192 | -1 | 8,192 | |
A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction. | \frac{9}{2} | 0.4375 | 6,842.8125 | 5,108.142857 | 8,192 | |
Consider the two hands of an analog clock, each of which moves with constant angular velocity. Certain positions of these hands are possible (e.g. the hour hand halfway between the 5 and 6 and the minute hand exactly at the 6), while others are impossible (e.g. the hour hand exactly at the 5 and the minute hand exactly... | 143 | 143 We can look at the twelve-hour cycle beginning at midnight and ending just before noon, since during this time, the clock goes through each possible position exactly once. The minute hand has twelve times the angular velocity of the hour hand, so if the hour hand has made $t$ revolutions from its initial position $... | 0 | 8,100.5 | -1 | 8,100.5 |
Using the numbers $1$, $2$, $3$, $4$ to form a four-digit number without repeating digits, the number of four-digit numbers larger than $2134$ is _____. (Answer in digits) | 17 | 0.6875 | 6,596.4375 | 6,357 | 7,123.2 | |
The midpoint of a line segment is located at $(1, -2)$. If one of the endpoints is $(4, 5)$, what is the other endpoint? Express your answer as an ordered pair. | (-2,-9) | 1 | 1,375.125 | 1,375.125 | -1 | |
Given an arithmetic sequence $\{a_n\}$, it is known that $\frac{a_{11}}{a_{10}} + 1 < 0$. Determine the maximum value of $n$ for which $S_n > 0$ holds. | 19 | 0.375 | 7,484 | 6,304 | 8,192 | |
Given that point $A(1,\sqrt{5})$ lies on the parabola $C:y^{2}=2px$, the distance from $A$ to the focus of $C$ is ______. | \frac{9}{4} | 1 | 2,150.8125 | 2,150.8125 | -1 | |
Two angles of a triangle measure 30 and 45 degrees. If the side of the triangle opposite the 30-degree angle measures $6\sqrt2$ units, what is the sum of the lengths of the two remaining sides? Express your answer as a decimal to the nearest tenth. | 28.4 | 0.875 | 5,870.375 | 5,538.714286 | 8,192 | |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5 \times 5$ square array of dots? | 100 | 0.4375 | 7,267.0625 | 6,339.714286 | 7,988.333333 | |
( Elgin Johnston ) Legs $L_1, L_2, L_3, L_4$ of a square table each have length $n$ , where $n$ is a positive integer. For how many ordered 4-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers can we cut a piece of length $k_i$ from the end of leg $L_i \; (i = 1,2,3,4)$ and still have a stable table?
(The table ... | \[
\binom{n+3}{3}
\] |
The problem involves determining how many ways we can cut legs of a square table such that the table remains stable. Specifically, we aim to find the number of ordered 4-tuples \((k_1, k_2, k_3, k_4)\) where the lengths of the cuts \(k_i\) are non-negative integers, and the legs of the table after cutting are stable, ... | 0 | 7,957.25 | -1 | 7,957.25 |
Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$ s that may occur among the $100$ numbers. | 95 | 0 | 8,192 | -1 | 8,192 | |
Consider all non-empty subsets of the set \( S = \{1, 2, \cdots, 10\} \). A subset is called a "good subset" if the number of even numbers in the subset is not less than the number of odd numbers. How many "good subsets" are there? | 637 | 0.125 | 7,662.1875 | 8,192 | 7,586.5 | |
Find the maximum real number $\lambda$ such that for the real coefficient polynomial $f(x)=x^{3}+a x^{2}+b x+c$ having all non-negative real roots, it holds that $f(x) \geqslant \lambda(x-a)^{3}$ for all $x \geqslant 0$. Additionally, determine when equality holds in the given expression. | -1/27 | 0 | 8,192 | -1 | 8,192 | |
The set $S=\{1,2,3, \ldots, 49,50\}$ contains the first 50 positive integers. After the multiples of 2 and the multiples of 3 are removed, how many integers remain in the set $S$? | 17 | The set $S$ contains 25 multiples of 2 (that is, even numbers). When these are removed, the set $S$ is left with only the odd integers from 1 to 49. At this point, there are $50-25=25$ integers in $S$. We still need to remove the multiples of 3 from $S$. Since $S$ only contains odd integers at this point, then we must ... | 0.9375 | 3,072.125 | 2,730.8 | 8,192 |
Given that $C$ is an interior angle of $\triangle ABC$, and the vectors $\overrightarrow{m}=(2\cos C-1,-2)$, $\overrightarrow{n}=(\cos C,\cos C+1)$. If $\overrightarrow{m}\perp \overrightarrow{n}$, calculate the value of $\angle C$. | \dfrac{2\pi}{3} | 0 | 1,679.625 | -1 | 1,679.625 | |
Billy Bones has two coins — one gold and one silver. One of these coins is fair, while the other is biased. It is unknown which coin is biased but it is known that the biased coin lands heads with a probability of $p=0.6$.
Billy Bones tossed the gold coin and it landed heads immediately. Then, he started tossing the s... | 5/9 | 0.1875 | 6,425.5 | 6,987 | 6,295.923077 | |
Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$? | 3+\sqrt3 | 1. **Identify the Configuration**: We start by noting that $\triangle ABC$ is equilateral with side length $1$. Squares $ABDE$, $BCHI$, and $CAFG$ are constructed outside the triangle. We need to find the area of hexagon $DEFGHI$.
2. **Understanding the Hexagon**: Each side of the hexagon is a side of one of the squar... | 0 | 8,133.75 | -1 | 8,133.75 |
The movie "Thirty Thousand Miles in Chang'an" allows the audience to experience the unique charm of Tang poetry that has been passed down for thousands of years and the beauty of traditional Chinese culture. In the film, Li Bai was born in the year $701$ AD. If we represent this as $+701$ years, then Confucius was born... | -551 | 0.0625 | 373.4375 | 465 | 367.333333 | |
Simplify
\[\tan x + 2 \tan 2x + 4 \tan 4x + 8 \cot 8x.\]The answer will be a trigonometric function of some simple function of $x,$ like "$\cos 2x$" or "$\sin (x^3)$". | \cot x | 0.1875 | 7,621.75 | 5,150.666667 | 8,192 | |
The sequence $2, 7, 12, a, b, 27$ is arithmetic. What is the value of $a + b$? | 39 | 1 | 1,707.0625 | 1,707.0625 | -1 | |
A number is divided by \(7, 11, 13\). The sum of the quotients is 21, and the sum of the remainders is 21. What is the number? | 74 | 0.0625 | 8,155.0625 | 7,601 | 8,192 | |
If Fang Fang cuts a piece of paper into 9 pieces, then selects one of the resulting pieces to cut into 9 pieces again, and so on, determine the number of cuts made to achieve a total of 2009 paper pieces. | 251 | 1 | 1,905.0625 | 1,905.0625 | -1 | |
Equilateral triangle $DEF$ has each side equal to $9$. A circle centered at $Q$ is tangent to side $DE$ at $D$ and passes through $F$. Another circle, centered at $R$, is tangent to side $DF$ at $F$ and passes through $E$. Find the magnitude of segment $QR$.
A) $12\sqrt{3}$
B) $9\sqrt{3}$
C) $15$
D) $18$
E) $9$ | 9\sqrt{3} | 0 | 7,348.875 | -1 | 7,348.875 | |
Let \( S = \{(x, y) \mid x, y \in \mathbb{Z}, 0 \leq x, y \leq 2016\} \). Given points \( A = (x_1, y_1), B = (x_2, y_2) \) in \( S \), define
\[ d_{2017}(A, B) = (x_1 - x_2)^2 + (y_1 - y_2)^2 \pmod{2017} \]
The points \( A = (5, 5) \), \( B = (2, 6) \), and \( C = (7, 11) \) all lie in \( S \). There is also a point ... | 1021 | 0.3125 | 7,654.6875 | 6,472.6 | 8,192 | |
For a positive integer $n$, let $\theta(n)$ denote the number of integers $0 \leq x<2010$ such that $x^{2}-n$ is divisible by 2010. Determine the remainder when $\sum_{n=0}^{2009} n \cdot \theta(n)$ is divided by 2010. | 335 | Let us consider the $\operatorname{sum} \sum_{n=0}^{2009} n \cdot \theta(n)(\bmod 2010)$ in a another way. Consider the sum $0^{2}+1^{2}+2^{2}+\cdots+2007^{2}(\bmod 2010)$. For each $0 \leq n<2010$, in the latter sum, the term $n$ appears $\theta(n)$ times, so the sum is congruent to $\sum_{n=0}^{2009} n \cdot \theta(n... | 0.1875 | 7,732.125 | 6,830.666667 | 7,940.153846 |
For positive integers $m, n$, let \operatorname{gcd}(m, n) denote the largest positive integer that is a factor of both $m$ and $n$. Compute $$\sum_{n=1}^{91} \operatorname{gcd}(n, 91)$$ | 325 | Since $91=7 \times 13$, we see that the possible values of \operatorname{gcd}(n, 91) are 1, 7, 13, 91. For $1 \leq n \leq 91$, there is only one value of $n$ such that \operatorname{gcd}(n, 91)=91. Then, we see that there are 12 values of $n$ for which \operatorname{gcd}(n, 91)=7 (namely, multiples of 7 other than 91 )... | 0.875 | 4,668.125 | 4,164.714286 | 8,192 |
Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading... | 61 | Let P, Q, and R be the east-west distance of the field, the north-south distance, and the distance from the southeast corner to point D, respectively.
Ana's distance to point D = $P + Q + (P - R) = 2P + Q - R$
Bob's distance to point D = $Q + R$
Cao's distance to point D = $\sqrt{Q^2 + R^2}$
Since they arrive at th... | 0 | 8,192 | -1 | 8,192 |
In a regular dodecagon $A B C D E F G H I J K L$ inscribed in a circle with a radius of $6 \mathrm{~cm}$, determine the perimeter of the pentagon $A C F H K$. | 18 + 12\sqrt{2} | 0.375 | 6,373.5 | 5,205.5 | 7,074.3 | |
Call an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n$. How many oddly powerful integers are less than $2010$? | 16 | 0.0625 | 8,061.875 | 6,110 | 8,192 | |
In how many different ways can 7 different prizes be awarded to 5 students such that each student has at least one prize? | 16800 | 0.875 | 4,273.1875 | 3,713.357143 | 8,192 | |
Write the digits from 0 to 9 in a line, in any order you choose. On the line below, combine the neighboring digits to form nine new numbers, and sum these numbers as in the example below:
| 2 | | 1 | | 3 | | 7 | | 4 | | 9 | | 5 | | 8 | | 0 | | 6 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :---... | 494 | 0 | 8,085.25 | -1 | 8,085.25 | |
In $\triangle ABC$ in the adjoining figure, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD$, $DE$ and $EC$ are $2$, $3$, and $6$, respectively. The length of the shortest side of $\triangle ABC$ is | 2\sqrt{10} | 1. **Assign Variables:**
Let $AC = b$, $AB = c$, $AD = d$, and $AE = e$.
2. **Use the Angle Bisector Theorem:**
Since $AD$ and $AE$ trisect $\angle BAC$, we have:
\[
\frac{BD}{DA} = \frac{BE}{EA} \quad \text{and} \quad \frac{DA}{AE} = \frac{EA}{EC}.
\]
Given $BD = 2$, $DE = 3$, and $EC = 6$, we find... | 0.1875 | 7,757.25 | 5,873.333333 | 8,192 |
A right pyramid with a square base has total surface area 432 square units. The area of each triangular face is half the area of the square face. What is the volume of the pyramid in cubic units? | 288\sqrt{3} | 1 | 2,325.875 | 2,325.875 | -1 | |
A right triangle has side lengths $a, b$, and $\sqrt{2016}$ in some order, where $a$ and $b$ are positive integers. Determine the smallest possible perimeter of the triangle. | 48+\sqrt{2016} | There are no integer solutions to $a^{2}+b^{2}=2016$ due to the presence of the prime 7 on the right-hand side (by Fermat's Christmas Theorem). Assuming $a<b$, the minimal solution $(a, b)=(3,45)$ which gives the answer above. | 0 | 8,192 | -1 | 8,192 |
There are four positive integers that are divisors of each number in the list $$36, 72, -12, 114, 96.$$Find the sum of these four positive integers. | 12 | 1 | 2,316.25 | 2,316.25 | -1 | |
Find the maximum value of $m$ for a sequence $P_{0}, P_{1}, \cdots, P_{m+1}$ of points on a grid satisfying certain conditions. | n(n-1) | We will show that the desired maximum value for $m$ is $n(n-1)$. First, let us show that $m \leq n(n-1)$ always holds for any sequence $P_{0}, P_{1}, \cdots, P_{m+1}$ satisfying the conditions of the problem. Call a point a turning point if it coincides with $P_{i}$ for some $i$ with $1 \leq i \leq m$. Let us say also ... | 0 | 7,461.6875 | -1 | 7,461.6875 |
Triangle $\triangle ABC$ has a right angle at $C$, $\angle A = 45^\circ$, and $AC=12$. Find the radius of the incircle of $\triangle ABC$. | 6 - 3\sqrt{2} | 0 | 4,274 | -1 | 4,274 | |
The distance between points A and B is 1200 meters. Dachen starts from point A, and 6 minutes later, Xiaogong starts from point B. After another 12 minutes, they meet. Dachen walks 20 meters more per minute than Xiaogong. How many meters does Xiaogong walk per minute? | 28 | 0.6875 | 2,693.9375 | 2,098.545455 | 4,003.8 | |
Find $a$ if the remainder is constant when $10x^3-7x^2+ax+6$ is divided by $2x^2-3x+1$. | -7 | 1 | 3,108.9375 | 3,108.9375 | -1 | |
Determine the value of $\sin {15}^{{}^\circ }+\cos {15}^{{}^\circ }$. | \frac{\sqrt{6}}{2} | 0 | 2,499.6875 | -1 | 2,499.6875 | |
Consider the 800-digit integer
$$
234523452345 \cdots 2345 .
$$
The first \( m \) digits and the last \( n \) digits of the above integer are crossed out so that the sum of the remaining digits is 2345. Find the value of \( m+n \). | 130 | 0 | 7,953.5625 | -1 | 7,953.5625 | |
$(1)$ Given the function $f(x) = |x+1| + |2x-4|$, find the solution to $f(x) \geq 6$;<br/>$(2)$ Given positive real numbers $a$, $b$, $c$ satisfying $a+2b+4c=8$, find the minimum value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$. | \frac{11+6\sqrt{2}}{8} | 0 | 7,208.5625 | -1 | 7,208.5625 | |
Given the geometric sequence with 8 inserted numbers between 1 and 3, find the product of these 8 inserted numbers. | 81 | 0.9375 | 2,801.375 | 2,442 | 8,192 | |
What is the sum of all positive integer divisors of 77? | 96 | 1 | 1,487.4375 | 1,487.4375 | -1 | |
A regular dodecagon \(Q_1 Q_2 \ldots Q_{12}\) is drawn in the coordinate plane with \(Q_1\) at \((2,0)\) and \(Q_7\) at \((4,0)\). If \(Q_n\) is the point \((x_n, y_n)\), compute the numerical value of the product:
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \ldots (x_{12} + y_{12} i).
\] | 531440 | 0.625 | 6,297.0625 | 5,395.3 | 7,800 | |
Choose one of the three conditions in $①$ $ac=\sqrt{3}$, $②$ $c\sin A=3$, $③$ $c=\sqrt{3}b$, and supplement it in the following question. If the triangle in the question exists, find the value of $c$; if the triangle in the question does not exist, explain the reason.<br/>Question: Does there exist a $\triangle ABC$ wh... | 2\sqrt{3} | 0.125 | 7,282.6875 | 6,932 | 7,332.785714 | |
Convert $427_8$ to base 5. | 2104_5 | 0.9375 | 4,504.1875 | 4,258.333333 | 8,192 | |
Given the equation $3x^{2}-4=-2x$, find the quadratic coefficient, linear coefficient, and constant term. | -4 | 0.0625 | 490.375 | 523 | 488.2 | |
Six IMO competitions are hosted sequentially by two Asian countries, two European countries, and two African countries, where each country hosts once but no continent can host consecutively. How many such arrangements are possible? | 240 | 0.125 | 7,897.0625 | 7,519.5 | 7,951 | |
Given the new operation $n\heartsuit m=n^{3+m}m^{2+n}$, evaluate $\frac{2\heartsuit 4}{4\heartsuit 2}$. | \frac{1}{2} | 0.875 | 2,827.0625 | 2,878 | 2,470.5 | |
Solve the equation \(\frac{15}{x\left(\sqrt[3]{35-8 x^{3}}\right)}=2x+\sqrt[3]{35-8 x^{3}}\). Write the sum of all obtained solutions as the answer. | 2.5 | 0 | 5,665.125 | -1 | 5,665.125 | |
Find [the decimal form of] the largest prime divisor of $100111011_6$.
| 181 | 0.1875 | 7,865.8125 | 7,479.333333 | 7,955 | |
If $n$ is a multiple of $4$, the sum $s=1+2i+3i^2+\cdots+(n+1)i^n$, where $i=\sqrt{-1}$, equals: | \frac{1}{2}(n+2-ni) | To solve the problem, we need to evaluate the sum $s = 1 + 2i + 3i^2 + \cdots + (n+1)i^n$ where $i = \sqrt{-1}$ and $n$ is a multiple of $4$. We start by analyzing the powers of $i$:
1. **Powers of $i$:**
- $i^0 = 1$
- $i^1 = i$
- $i^2 = -1$
- $i^3 = -i$
- $i^4 = 1$ (and the cycle repeats every four ter... | 0 | 7,613.75 | -1 | 7,613.75 |
Determine the required distance between the pins and the length of the string in order to draw an ellipse with a length of 12 cm and a width of 8 cm. Find a simple rule that allows for constructing an ellipse of predetermined dimensions. | 24 | 0 | 3,111.3125 | -1 | 3,111.3125 | |
If $x$ is a real number, find $(x+1)^2+2(x+1)(3-x)+(3-x)^2$. | 16 | 1 | 1,943.9375 | 1,943.9375 | -1 | |
Using the same relationships between ball weights, how many blue balls are needed to balance $5$ green, $3$ yellow, and $3$ white balls? | 22 | 0 | 7,185.3125 | -1 | 7,185.3125 | |
It is known that the 3 sides of a triangle are consecutive positive integers and the largest angle is twice the smallest angle. Find the perimeter of this triangle. | 15 | 0.875 | 5,354.4375 | 4,949.071429 | 8,192 | |
A set of $n$ numbers has the sum $s$. Each number of the set is increased by $20$, then multiplied by $5$, and then decreased by $20$. The sum of the numbers in the new set thus obtained is: | $5s + 80n$ | Let the original set of numbers be $\{x_1, x_2, \ldots, x_n\}$, and the sum of these numbers is given by $s = x_1 + x_2 + \ldots + x_n$.
Each number in the set undergoes the following transformations:
1. Increased by $20$: $x_i + 20$
2. Multiplied by $5$: $5(x_i + 20) = 5x_i + 100$
3. Decreased by $20$: $(5x_i + 100) ... | 0 | 3,507 | -1 | 3,507 |
Let \( PROBLEMZ \) be a regular octagon inscribed in a circle of unit radius. Diagonals \( MR \) and \( OZ \) meet at \( I \). Compute \( LI \). | \sqrt{2} | 0.75 | 6,367.5625 | 5,759.416667 | 8,192 | |
Given a regular decagon, the probability that exactly one of the sides of the triangle formed by connecting three randomly chosen vertices of the decagon is also a side of the decagon. | \frac{1}{2} | 0.25 | 7,859.125 | 7,047.25 | 8,129.75 | |
If circle $$C_{1}: x^{2}+y^{2}+ax=0$$ and circle $$C_{2}: x^{2}+y^{2}+2ax+ytanθ=0$$ are both symmetric about the line $2x-y-1=0$, then $sinθcosθ=$ \_\_\_\_\_\_ . | -\frac{2}{5} | 0.0625 | 5,250.0625 | 3,168 | 5,388.866667 | |
The first four terms of a sequence are $1,4,2$, and 3. Beginning with the fifth term in the sequence, each term is the sum of the previous four terms. What is the eighth term? | 66 | The first four terms of the sequence are $1,4,2,3$. Since each term starting with the fifth is the sum of the previous four terms, then the fifth term is $1+4+2+3=10$. Also, the sixth term is $4+2+3+10=19$, the seventh term is $2+3+10+19=34$, and the eighth term is $3+10+19+34=66$. | 1 | 1,719 | 1,719 | -1 |
If \( a < b < c < d \) are distinct positive integers such that \( a+b+c+d \) is a square, what is the minimum value of \( c+d \)? | 11 | 0.625 | 6,634.8125 | 5,972.9 | 7,738 |
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