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 |
|---|---|---|---|---|---|---|
Five friends — Sarah, Lily, Emma, Nora, and Kate — performed in a theater as quartets, with one friend sitting out each time. Nora performed in 10 performances, which was the most among all, and Sarah performed in 6 performances, which was the fewest among all. Calculate the total number of performances. | 10 | 0.0625 | 7,859.625 | 7,004 | 7,916.666667 | |
Juca is a scout exploring the vicinity of his camp. After collecting fruits and wood, he needs to fetch water from the river and return to his tent. Represent Juca by the letter $J$, the river by the letter $r$, and his tent by the letter $B$. The distance from the feet of the perpendiculars $C$ and $E$ on $r$ from poi... | 180\sqrt{2} | 0.0625 | 7,231.25 | 6,641 | 7,270.6 | |
Let $S$ be the sum of all positive integers $n$ such that $n^2+12n-2007$ is a perfect square. Find the remainder when $S$ is divided by $1000.$
| 464 | 1 | 4,122.75 | 4,122.75 | -1 | |
Given that Bill's age in two years will be three times his current age, and the digits of both Jack's and Bill's ages are reversed, find the current age difference between Jack and Bill. | 18 | 0 | 6,266.6875 | -1 | 6,266.6875 | |
If a function $f(x)$ satisfies both (1) for any $x$ in the domain, $f(x) + f(-x) = 0$ always holds; and (2) for any $x_1, x_2$ in the domain where $x_1 \neq x_2$, the inequality $\frac{f(x_1) - f(x_2)}{x_1 - x_2} < 0$ always holds, then the function $f(x)$ is called an "ideal function." Among the following three functi... | (3) | 0 | 5,570.5 | -1 | 5,570.5 | |
Given the graph of $y = mx + 2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$, find the maximum possible value of $a$. | \frac{50}{99} | 0.1875 | 7,860.25 | 6,558 | 8,160.769231 | |
Given a regular tetrahedron with four vertices \(A, B, C, D\) and each edge of length 1 meter, a bug starts at point \(A\) and moves according to the following rule: at each vertex, it randomly chooses one of the three edges connected to that vertex with equal probability and crawls to the other end of the chosen edge.... | \frac{182}{729} | 0.3125 | 7,450.5625 | 5,819.4 | 8,192 | |
Calculate the value of $v_2$ when $x = 2$ for $f(x) = 3x^4 + x^3 + 2x^2 + x + 4$ using Horner's method. | 16 | 0.9375 | 3,026.125 | 3,071.266667 | 2,349 | |
Point \( P \) is located on the side \( AB \) of the square \( ABCD \) such that \( AP: PB = 2:3 \). Point \( Q \) lies on the side \( BC \) of the square and divides it in the ratio \( BQ: QC = 3 \). Lines \( DP \) and \( AQ \) intersect at point \( E \). Find the ratio of lengths \( AE: EQ \). | 4:9 | 0.4375 | 5,385.25 | 5,943 | 4,951.444444 | |
In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 30 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. Calculate the area of trapezoid $EFCD$ if the height from $E$ to line $CD$ is 10 units. | 250 | 0.125 | 8,124.75 | 7,654 | 8,192 | |
For a $5 \times 5$ chessboard colored as shown below, place 5 different rooks on black squares such that no two rooks can attack each other (rooks attack if they are in the same row or column). How many different ways are there to do this? | 1440 | 0.0625 | 7,685.4375 | 8,192 | 7,651.666667 | |
One million bucks (i.e. one million male deer) are in different cells of a $1000 \times 1000$ grid. The left and right edges of the grid are then glued together, and the top and bottom edges of the grid are glued together, so that the grid forms a doughnut-shaped torus. Furthermore, some of the bucks are honest bucks, ... | 1200000 | Note that each honest buck has at most one honest neighbor, and each dishonest buck has at least two honest neighbors. The connected components of honest bucks are singles and pairs. Then if there are $K$ honest bucks and $B$ buckaroo pairs, we get $B \geq 3 K$. From the dishonest buck condition we get $B \geq 2(100000... | 0 | 8,108.9375 | -1 | 8,108.9375 |
Given that the sequence $\{a_n\}$ is a geometric sequence, and $a_4 = e$, if $a_2$ and $a_7$ are the two real roots of the equation $$ex^2 + kx + 1 = 0, (k > 2\sqrt{e})$$ (where $e$ is the base of the natural logarithm),
1. Find the general formula for $\{a_n\}$.
2. Let $b_n = \ln a_n$, and $S_n$ be the sum of the firs... | n = 4 | 0.0625 | 8,059.25 | 6,068 | 8,192 | |
Use the six digits 0, 1, 2, 3, 4, 5 to form four-digit even numbers without repeating any digit. Calculate the total number of such numbers that can be formed. | 156 | 0.625 | 5,967.625 | 5,150.4 | 7,329.666667 | |
Given that $a > 0$, $b > 0$, and $a + b = 1$, find the minimum value of $\frac{2}{a} + \frac{3}{b}$. | 5 + 2\sqrt{6} | 0.9375 | 4,773 | 4,545.066667 | 8,192 | |
Read the following material and then answer the following questions: When simplifying algebraic expressions, sometimes we encounter expressions like $\frac{5}{\sqrt{3}}$, $\frac{2}{\sqrt{3}+1}$, in fact, we can further simplify them:<br/>$($1) $\frac{5}{\sqrt{3}}=\frac{5×\sqrt{3}}{\sqrt{3}×\sqrt{3}}=\frac{5}{3}\sqrt{3}... | \frac{3\sqrt{11}-1}{2} | 0 | 4,569.4375 | -1 | 4,569.4375 | |
Two distinct similar rhombi share a diagonal. The smaller rhombus has area 1, and the larger rhombus has area 9. Compute the side length of the larger rhombus. | \sqrt{15} | Let $d$ be the length of the smaller diagonal of the smaller rhombus. Since the ratio of the areas is $9: 1$, the ratio of the lengths is $3: 1$. This means that the smaller diagonal of the larger rhombus (which is also the longer diagonal of the smaller rhombus) has length $3 d$. Therefore, the smaller rhombus has dia... | 0.4375 | 7,608.75 | 6,858.857143 | 8,192 |
$\frac{10^7}{5\times 10^4}=$ | 200 | 1. **Simplify the expression**: We start by simplifying the fraction given in the problem:
\[
\frac{10^7}{5 \times 10^4}
\]
We can separate the powers of ten and the constant:
\[
\frac{10^7}{5 \times 10^4} = \frac{10^7}{10^4} \times \frac{1}{5}
\]
2. **Simplify the powers of ten**: Using the laws ... | 1 | 2,879.3125 | 2,879.3125 | -1 |
There are 6 blue, 7 red, and 9 white light bulbs. In how many ways can you arrange them (using all the light bulbs) in a garland so that no two white light bulbs are consecutive? | 3435432 | 0.125 | 7,300.125 | 4,061 | 7,762.857143 | |
Let $a = \pi/2008$. Find the smallest positive integer $n$ such that \[2[\cos(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)]\] is an integer. | 251 | By the product-to-sum identities, we have that $2\cos a \sin b = \sin (a+b) - \sin (a-b)$. Therefore, this reduces to a telescoping series: \begin{align*} \sum_{k=1}^{n} 2\cos(k^2a)\sin(ka) &= \sum_{k=1}^{n} [\sin(k(k+1)a) - \sin((k-1)ka)]\\ &= -\sin(0) + \sin(2a)- \sin(2a) + \sin(6a) - \cdots - \sin((n-1)na) + \sin(n(... | 0.3125 | 7,880.625 | 7,613.6 | 8,002 |
Let $\mathrm {P}$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have a positive imaginary part, and suppose that $\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})$, where $0<r$ and $0\leq \theta <360$. Find $\theta$. | 276 | 0.0625 | 7,845.8125 | 5,291 | 8,016.133333 | |
Given the function $f(x)= \sqrt{3}\sin ωx−2{(\sin \dfrac{ωx}{2})}^{2}(ω > 0)$ with the smallest positive period of $3π$.
(1) Find the maximum and minimum values of the function $f(x)$ in the interval $[-{\dfrac{3π }{4}},π]$.
(2) Given $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively in an ... | \dfrac {3+ \sqrt {3}}{3} | 0 | 8,021.0625 | -1 | 8,021.0625 | |
The gardener Fedya has a miracle tree with seven branches in his garden. On each branch, there can either grow 6 apples, 5 pears, or 3 oranges. Fedya discovered that the tree has fruit of all types, with the most pears and the fewest apples.
How many fruits in total grew on the miracle tree? | 30 | 0.125 | 6,360.6875 | 3,052.5 | 6,833.285714 | |
Calculate the volumes of the bodies bounded by the surfaces.
$$
z = 2x^2 + 18y^2, \quad z = 6
$$ | 6\pi | 0 | 7,357 | -1 | 7,357 | |
We draw 6 circles of equal radius on the surface of a unit sphere such that the circles do not intersect. What is the maximum possible radius of these circles?
| \frac{\sqrt{2}}{2} | 0 | 7,184.6875 | -1 | 7,184.6875 | |
In $\triangle{ABC}$ with $AB = 12$, $BC = 13$, and $AC = 15$, let $M$ be a point on $\overline{AC}$ such that the incircles of $\triangle{ABM}$ and $\triangle{BCM}$ have equal radii. Then $\frac{AM}{CM} = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | 45 | Let $CM=x, AM=rx, BM=d$. $x+rx=15\Rightarrow x=\frac{15}{1+r}$.
Similar to Solution 1, we have \[r=\frac{[AMB]}{[CMB]}=\frac{12+rx+d}{13+x+d} \Rightarrow d=\frac{13r-12}{1-r}\] as well as \[12^2\cdot x + 13^2 rx=15x\cdot rx+15d^2 (\text{via Stewart's Theorem})\] \[\frac{(12^2 + 13^2r) \cdot 15}{1+r} - \frac{15r\cdot 1... | 0.0625 | 8,192 | 8,192 | 8,192 |
How many of the positive divisors of 3240 are multiples of 3? | 32 | 0.9375 | 3,594.125 | 3,287.6 | 8,192 | |
For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the gre... | 18 | 0 | 7,775.75 | -1 | 7,775.75 | |
Let $f(x)$ be an odd function defined on $\mathbb{R}$. When $x > 0$, $f(x)=x^{2}+2x-1$.
(1) Find $f(-2)$;
(2) Find the expression of $f(x)$. | -7 | 0.375 | 4,320.625 | 5,019.666667 | 3,901.2 | |
Find the smallest positive integer $Y$ such that a number $U$, made only of digits 0s and 1s, is divisible by 15, and $U = 15Y$. | 74 | 0.4375 | 6,359.375 | 4,433.285714 | 7,857.444444 | |
What is the value of $\displaystyle\frac{235^2-221^2}{14}$? | 456 | 0.875 | 2,698.25 | 2,328 | 5,290 | |
The average of the numbers 23 and $x$ is 27. What is the positive difference between 23 and $x$? (Recall that the average of two numbers is their sum divided by 2). | 8 | 1 | 1,015.125 | 1,015.125 | -1 | |
$5y$ varies inversely as the square of $x$. When $y=16$, $x=1$. When $x=8$, $y$ equals: | \frac{1}{4} | 1. **Understanding the relationship**: Given that $5y$ varies inversely as the square of $x$, we can express this relationship mathematically as:
\[
5y = \frac{k}{x^2}
\]
where $k$ is a constant.
2. **Finding the constant $k$**: We know that when $y = 16$ and $x = 1$, the equation holds. Plugging these val... | 1 | 1,655.0625 | 1,655.0625 | -1 |
In a plane, 100 points are marked. It turns out that 40 marked points lie on each of two different lines \( a \) and \( b \). What is the maximum number of marked points that can lie on a line that does not coincide with \( a \) or \( b \)? | 23 | 0 | 7,852.875 | -1 | 7,852.875 | |
Al and Bert must arrive at a town 22.5 km away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at 8 km/h, leaves the bicycle, and then walks at 5 km/h. Al walks at 4 km/h, reaches the bicycle, and rides at 10 km/h. For how many minutes was the bicycle not in motion? | 75 | 0.125 | 7,030.625 | 6,071.5 | 7,167.642857 | |
Let set \( A = \{a \mid a = 3k + 2, k \leqslant 2000, k \in \mathbf{N}_{+}\} \) and set \( B = \{b \mid b = 4k - 1, k \leqslant 2000, k \in \mathbf{N}_{+}\} \). How many elements are in \( A \cap B \)? | 500 | 0.75 | 4,291.875 | 3,802 | 5,761.5 | |
Let \(D\) be a point on side \(BC\) of \(\triangle ABC\). Points \(E\) and \(F\) are the centroids of \(\triangle ABD\) and \(\triangle ACD\), respectively. The line segment \(EF\) intersects \(AD\) at point \(G\). Find the value of \(\frac{DG}{GA}\). | 1/2 | 0.6875 | 6,220.875 | 5,324.909091 | 8,192 | |
The lateral surface of a cylinder unfolds into a square. What is the ratio of its lateral surface area to the base area. | 4\pi | 0.875 | 2,548.5 | 2,690.714286 | 1,553 | |
Given a triangle $ABC$ with internal angles $A$, $B$, and $C$ opposite to the sides $a$, $b$, and $c$ respectively. If $(2a-c)\cos B=b\cos C$, and the dot product $\vec{AB}\cdot \vec{BC} = -3$,
1. Find the area of $\triangle ABC$;
2. Find the minimum value of side $AC$. | \sqrt{6} | 0.875 | 5,372.875 | 4,970.142857 | 8,192 | |
Eric is taking a biology class. His problem sets are worth 100 points in total, his three midterms are worth 100 points each, and his final is worth 300 points. If he gets a perfect score on his problem sets and scores $60 \%, 70 \%$, and $80 \%$ on his midterms respectively, what is the minimum possible percentage he ... | 60 \% | We see there are a total of $100+3 \times 100+300=700$ points, and he needs $70 \% \times 700=490$ of them. He has $100+60+70+80=310$ points before the final, so he needs 180 points out of 300 on the final, which is $60 \%$. | 0.4375 | 914.75 | 852.428571 | 963.222222 |
How many polynomials of degree exactly 5 with real coefficients send the set \{1,2,3,4,5,6\} to a permutation of itself? | 714 | For every permutation $\sigma$ of \{1,2,3,4,5,6\}, Lagrange Interpolation gives a polynomial of degree at most 5 with $p(x)=\sigma(x)$ for every $x=1,2,3,4,5,6$. Additionally, this polynomial is unique: assume that there exist two polynomials $p, q$ of degree \leq 5 such that they map \{1,2,3,4,5,6\} to the same permut... | 0 | 7,929.1875 | -1 | 7,929.1875 |
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the side... | 2(w+h)^2 | 1. **Understanding the Problem Setup**: We have a square sheet of wrapping paper and a box with a square base of side $w$ and height $h$. The box is placed such that its base vertices lie on the midlines of the wrapping paper. The wrapping paper is folded up to meet at a point $A$ at the center of the top of the box.
... | 0 | 8,192 | -1 | 8,192 |
Given a geometric sequence $\{a_n\}$ with a common ratio of $2$ and the sum of the first $n$ terms denoted by $S_n$. If $a_2= \frac{1}{2}$, find the expression for $a_n$ and the value of $S_5$. | \frac{31}{16} | 0 | 2,269.9375 | -1 | 2,269.9375 | |
From the set of three-digit numbers that do not contain the digits $0,1,2,3,4,5$, several numbers were written down in such a way that no two numbers could be obtained from each other by swapping two adjacent digits. What is the maximum number of such numbers that could have been written? | 40 | 0 | 7,671.625 | -1 | 7,671.625 | |
Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, \ldots, 8,9,10\}$ that satisfy $|S|+\min (S)$. $\max (S)=0$. | 335 | Since $\min (S) \cdot \max (S)<0$, we must have $\min (S)=-a$ and $\max (S)=b$ for some positive integers $a$ and $b$. Given $a$ and $b$, there are $|S|-2=a b-2$ elements left to choose, which must come from the set $\{-a+1,-a+2, \ldots, b-2, b-1\}$, which has size $a+b-1$. Therefore the number of possibilities for a g... | 0 | 8,192 | -1 | 8,192 |
The orthocenter of triangle $ABC$ divides altitude $\overline{CF}$ into segments with lengths $HF = 6$ and $HC = 15.$ Calculate $\tan A \tan B.$
[asy]
unitsize (1 cm);
pair A, B, C, D, E, F, H;
A = (0,0);
B = (5,0);
C = (4,4);
D = (A + reflect(B,C)*(A))/2;
E = (B + reflect(C,A)*(B))/2;
F = (C + reflect(A,B)*(C))/2;... | \frac{7}{2} | 0.25 | 7,644.0625 | 6,371.5 | 8,068.25 | |
For each positive integer $n$, define $s(n)$ to equal the sum of the digits of $n$. The number of integers $n$ with $100 \leq n \leq 999$ and $7 \leq s(n) \leq 11$ is $S$. What is the integer formed by the rightmost two digits of $S$? | 24 | We write an integer $n$ with $100 \leq n \leq 999$ as $n=100a+10b+c$ for some digits $a, b$ and $c$. That is, $n$ has hundreds digit $a$, tens digit $b$, and ones digit $c$. For each such integer $n$, we have $s(n)=a+b+c$. We want to count the number of such integers $n$ with $7 \leq a+b+c \leq 11$. When $100 \leq n \l... | 0.0625 | 7,077.8125 | 4,170 | 7,271.666667 |
In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks needs to be replaced by any digit from $0, 1, 2, 3, 4, 5, 6, 7, 8$ (digits can repeat) so that the resulting 10-digit number is divisible by 45. In how many ways can this be done? | 1458 | 0.5 | 6,074.0625 | 5,297.375 | 6,850.75 | |
A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples.
What is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples? | 6 | 1. **Identify the range of possible apple counts per crate**: Each crate contains between $120$ and $144$ apples, inclusive. Therefore, the number of different possible apple counts per crate is calculated as:
\[
144 - 120 + 1 = 25
\]
This means there are 25 different possible numbers of apples that each cr... | 1 | 2,741.0625 | 2,741.0625 | -1 |
Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$ | (3, 3) |
To solve the problem, we need to identify all pairs of primes \( p \) and \( q \) such that the expression \( 3p^{q-1} + 1 \) divides \( 11^p + 17^p \). The reference answer indicates that the only solution is the pair \((3, 3)\). Let's go through the process of verifying this.
Given the division condition:
\[
3p^{... | 0 | 8,192 | -1 | 8,192 |
How many distinct equilateral triangles can be constructed by connecting three different vertices of a regular dodecahedron? | 60 | 0 | 8,184.3125 | -1 | 8,184.3125 | |
Let $a$ , $b$ , $c$ be positive integers such that $abc + bc + c = 2014$ . Find the minimum possible value of $a + b + c$ . | 40 | 0.3125 | 7,684.625 | 6,619.4 | 8,168.818182 | |
Find the smallest $n$ such that $n!$ ends with 10 zeroes. | 45 | The number of zeroes that $n!$ ends with is the largest power of 10 dividing $n!$. The exponent of 5 dividing $n!$ exceeds the exponent of 2 dividing $n!$, so we simply seek the exponent of 5 dividing $n!$. For a number less than 125, this exponent is just the number of multiples of 5, but not 25, less than $n$ plus tw... | 1 | 4,004.875 | 4,004.875 | -1 |
Eight identical cubes with of size $1 \times 1 \times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$ , number $3$ written on the face opposite number $5$ , and number $4$ written on the face opposite number $6$ . The eight cubes... | 24 | 0 | 7,581.0625 | -1 | 7,581.0625 | |
Calculate the remainder when $1 + 11 + 11^2 + \cdots + 11^{1024}$ is divided by $500$. | 25 | 0.25 | 7,685.375 | 6,702.5 | 8,013 | |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and vectors $\overrightarrow{m}=(1-\cos (A+B),\cos \frac {A-B}{2})$ and $\overrightarrow{n}=( \frac {5}{8},\cos \frac {A-B}{2})$ with $\overrightarrow{m}\cdot \overrightarrow{n}= \frac {9}{8}$,
(1) Find the ... | -\frac {3}{8} | 0 | 8,192 | -1 | 8,192 | |
Given a $4 \times 4$ grid with 16 unit squares, each painted white or black independently and with equal probability, find the probability that the entire grid becomes black after a 90° clockwise rotation, where any white square landing on a place previously occupied by a black square is repainted black. | \frac{1}{65536} | 0.0625 | 8,172.75 | 7,884 | 8,192 | |
Regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are all equal. In how many ways can this be done?
[asy]
p... | 1152 | 0.25 | 8,037.3125 | 7,779.5 | 8,123.25 | |
Given the plane rectangular coordinate system $(xOy)$, with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system. The polar coordinate equation of the curve $C_{1}$ is $\rho = 4\cos \theta$, and the line $l$: $\begin{cases} x = 1 - \frac{2\sqrt{5}}{5}t \\ y = 1 + \fra... | \frac{\sqrt{10}}{5} | 0 | 5,255.25 | -1 | 5,255.25 | |
A lattice point in the plane is a point of the form $(n, m)$, where $n$ and $m$ are integers. Consider a set $S$ of lattice points. We construct the transform of $S$, denoted by $S^{\prime}$, by the following rule: the pair $(n, m)$ is in $S^{\prime}$ if and only if any of $(n, m-1),(n, m+1),(n-1, m)$, $(n+1, m)$, and ... | 421 | Transforming it $k \geq 1$ times yields the 'diamond' of points $(n, m)$ such that $|n|+|m| \leq k$. The diamond contains $(k+1)^{2}+k^{2}$ lattice points (this can be seen by rotating the plane 45 degrees and noticing the lattice points in the transforms form two squares, one of which is contained in the other), so th... | 0.3125 | 7,373.375 | 5,572.4 | 8,192 |
If \( 3^x = 5 \), what is the value of \( 3^{x+2} \)? | 45 | Using exponent laws, \( 3^{x+2} = 3^x \cdot 3^2 = 3^x \cdot 9 \). Since \( 3^x = 5 \), then \( 3^{x+2} = 3^x \cdot 9 = 5 \cdot 9 = 45 \). | 1 | 1,595.125 | 1,595.125 | -1 |
Given a sequence $\{a_n\}$ where $a_n = n$, for each positive integer $k$, in between $a_k$ and $a_{k+1}$, insert $3^{k-1}$ twos (for example, between $a_1$ and $a_2$, insert three twos, between $a_2$ and $a_3$, insert $3^1$ twos, between $a_3$ and $a_4$, insert $3^2$ twos, etc.), to form a new sequence $\{d_n\}$. Let ... | 245 | 0 | 7,722.25 | -1 | 7,722.25 | |
There are 7 line segments with integer lengths in centimeters: $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$, satisfying $a_1 < a_2 < a_3 < a_4 < a_5 < a_6 < a_7$, and any three of these line segments cannot form a triangle. If $a_1 = 1$ cm and $a_7 = 21$ cm, then $a_6 = \ $. | 13 | 0.5625 | 6,099.375 | 4,471.777778 | 8,192 | |
Reading material: In class, the teacher explained the following two problems on the blackboard:
Problem 1: Calculate: $77.7\times 11-77.7\times 5+77.7\times 4$.
Solution: $77.7\times 11-77.7\times 5+77.7\times 4=77.7\times \left(11-5+4\right)=777$.
This problem uses the distributive property of multiplication.
Problem... | -\frac{1}{3} | 0.25 | 7,604.125 | 5,840.5 | 8,192 | |
John now has 15 marbles of different colors, including two red, two green, and one blue marble. In how many ways can he choose 5 marbles, if exactly two of the chosen marbles are red and one is green? | 110 | 0.375 | 3,429 | 2,269.833333 | 4,124.5 | |
Two arithmetic sequences $A$ and $B$ both begin with 30 and have common differences of absolute value 10, with sequence $A$ increasing and sequence $B$ decreasing. What is the absolute value of the difference between the 51st term of sequence $A$ and the 51st term of sequence $B$? | 1000 | 1 | 1,882.9375 | 1,882.9375 | -1 | |
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_{n+1}=2S_n+1$, where $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, and $n\in\mathbb{N}^*$.
(1) Find $a_n$;
(2) If the sequence $\{b_n\}$ satisfies $b_n=\dfrac{1}{(1+\log_{3}a_n)(3+\log_{3}a_n)}$, and the sum of the first $n$ terms of $\{b_n\}$ is $T_n$, ... | \dfrac{3}{4} | 0.9375 | 4,698.3125 | 4,735.333333 | 4,143 | |
If $f(x)=\log \left(\frac{1+x}{1-x}\right)$ for $-1<x<1$, then $f\left(\frac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$ is | 3f(x) | Given the function \( f(x) = \log \left(\frac{1+x}{1-x}\right) \) for \( -1 < x < 1 \), we need to find \( f\left(\frac{3x+x^3}{1+3x^2}\right) \) in terms of \( f(x) \).
1. **Substitute \( y = \frac{3x+x^3}{1+3x^2} \) into \( f(y) \):**
\[
f(y) = \log \left(\frac{1+y}{1-y}\right)
\]
2. **Simplify \( y \):**
... | 0.75 | 6,191.625 | 5,524.833333 | 8,192 |
$100_{10}$ in base $b$ has exactly $5$ digits. What is the value of $b$? | 3 | 0.875 | 4,068.5 | 3,479.428571 | 8,192 | |
A square with side length 1 is rotated about one vertex by an angle of $\alpha,$ where $0^\circ < \alpha < 90^\circ$ and $\cos \alpha = \frac{4}{5}.$ Find the area of the shaded region that is common to both squares.
[asy]
unitsize(3 cm);
pair A, B, C, D, Bp, Cp, Dp, P;
A = (0,0);
B = (-1,0);
C = (-1,-1);
D = (0,-1... | \frac{1}{2} | 0.125 | 7,765.4375 | 7,337 | 7,826.642857 | |
Say that an integer $A$ is yummy if there exist several consecutive integers, including $A$, that add up to 2014. What is the smallest yummy integer? | -2013 | 0.375 | 7,698.4375 | 6,991.333333 | 8,122.7 | |
An entrepreneur invested \$12,000 in a three-month savings certificate that paid a simple annual interest rate of $8\%$. After three months, she invested the total value of her investment in another three-month certificate. After three more months, the investment was worth \$12,980. If the annual interest rate of the s... | 24\% | 0 | 7,866.375 | -1 | 7,866.375 | |
Determine the value of
\[3003 + \frac{1}{3} \left( 3002 + \frac{1}{3} \left( 3001 + \dots + \frac{1}{3} \left( 4 + \frac{1}{3} \cdot 3 \right) \right) \dotsb \right).\] | 9006 | 0 | 8,141.8125 | -1 | 8,141.8125 | |
Two cars, A and B, start from points A and B respectively and travel towards each other at the same time. They meet at point C after 6 hours. If car A maintains its speed and car B increases its speed by 5 km/h, they will meet 12 km away from point C. If car B maintains its speed and car A increases its speed by 5 km/h... | 30 | 0 | 8,192 | -1 | 8,192 | |
A store prices an item in dollars and cents so that when 4% sales tax is added, no rounding is necessary because the result is exactly $n$ dollars where $n$ is a positive integer. The smallest value of $n$ is | 13 | 1. **Define the variables**: Let the price of the item in cents be $x$. Since the sales tax is 4%, the total cost including tax is $x + 0.04x = 1.04x$ cents.
2. **Convert the total cost to dollars**: Since $1$ dollar = $100$ cents, the total cost in dollars is $\frac{1.04x}{100} = 0.0104x$ dollars.
3. **Set up the eq... | 0.875 | 4,983 | 4,524.571429 | 8,192 |
Given an obtuse triangle \(ABC\) with obtuse angle \(C\). Points \(P\) and \(Q\) are marked on its sides \(AB\) and \(BC\) respectively, such that \(\angle ACP = CPQ = 90^\circ\). Find the length of segment \(PQ\) if it is known that \(AC = 25\), \(CP = 20\), and \(\angle APC = \angle A + \angle B\). | 16 | 0 | 8,192 | -1 | 8,192 | |
In triangle $∆ABC$, the angles $A$, $B$, $C$ correspond to the sides $a$, $b$, $c$ respectively. The vector $\overrightarrow{m}\left(a, \sqrt{3b}\right)$ is parallel to $\overrightarrow{n}=\left(\cos A,\sin B\right)$.
(1) Find $A$;
(2) If $a= \sqrt{7},b=2$, find the area of $∆ABC$. | \dfrac{3 \sqrt{3}}{2} | 0 | 7,546.75 | -1 | 7,546.75 | |
Let the function \( f(x) = 3 \sin x + 2 \cos x + 1 \). If real numbers \( a, b, c \) satisfy \( a f(x) + b f(x-c) = 1 \) for all real numbers \( x \), find the value of \( \frac{b \cos c}{a} \). | -1 | 0.75 | 6,272.0625 | 5,640.416667 | 8,167 | |
Given the vectors $\overrightarrow{a} = (\cos 25^\circ, \sin 25^\circ)$, $\overrightarrow{b} = (\sin 20^\circ, \cos 20^\circ)$, and $\overrightarrow{u} = \overrightarrow{a} + t\overrightarrow{b}$, where $t\in\mathbb{R}$, find the minimum value of $|\overrightarrow{u}|$. | \frac{\sqrt{2}}{2} | 0 | 4,819.625 | -1 | 4,819.625 | |
For a nonnegative integer \(n\), let \(r_{11}(7n)\) stand for the remainder left when \(n\) is divided by \(11.\) For example, \(r_{11}(7 \cdot 3) = 10.\)
What is the \(15^{\text{th}}\) entry in an ordered list of all nonnegative integers \(n\) that satisfy $$r_{11}(7n) \leq 5~?$$ (Note that the first entry in this li... | 29 | 0 | 7,221.8125 | -1 | 7,221.8125 | |
One dimension of a cube is increased by $1$, another is decreased by $1$, and the third is left unchanged. The volume of the new rectangular solid is $5$ less than that of the cube. What was the volume of the cube? | 125 | 1. **Define the dimensions and volume of the original cube:**
Let the edge length of the original cube be $a$. Therefore, the volume of the cube is given by:
\[
V_{\text{cube}} = a^3
\]
2. **Define the dimensions and volume of the new rectangular solid:**
One dimension is increased by $1$, another is de... | 1 | 1,273.3125 | 1,273.3125 | -1 |
Given $$(5x- \frac {1}{ \sqrt {x}})^{n}$$, the sum of the binomial coefficients in its expansion is 64. Find the constant term in the expansion. | 375 | 0.375 | 2,779.6875 | 2,986.666667 | 2,655.5 | |
In Geometrytown, all streets are $30$ feet wide and the blocks they enclose are rectangles with side lengths of $300$ feet and $500$ feet. Anne runs around the block on the $300$-foot side of the street, while Bob runs on the opposite side of the street. How many more feet than Anne does Bob run for every lap around th... | 240 | 0.1875 | 6,596.6875 | 4,756.333333 | 7,021.384615 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\cos A= \frac{4}{5}$.
(1) Find the value of $\sin ^{2} \frac{B+C}{2}+\cos 2A$;
(2) If $b=2$, the area of $\triangle ABC$ is $S=3$, find $a$. | \sqrt{13} | 0.8125 | 4,602.75 | 4,181.615385 | 6,427.666667 | |
A straight one-way city street has 8 consecutive traffic lights. Every light remains green for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are synchronized so that each light turns red 10 seconds after the preceding one turns red. Calculate the longest interval of time, in seconds, during whi... | 20 | 0.125 | 8,074.0625 | 7,248.5 | 8,192 | |
What is the units digit in the product of all natural numbers from 1 to 99, inclusive? | 0 | 1 | 2,702.875 | 2,702.875 | -1 | |
(1) Simplify: $\dfrac{\sin(\pi -\alpha)\cos(\pi +\alpha)\sin(\dfrac{\pi}{2}+\alpha)}{\sin(-\alpha)\sin(\dfrac{3\pi}{2}+\alpha)}$.
(2) Given $\alpha \in (\dfrac{\pi}{2}, \pi)$, and $\sin(\pi -\alpha) + \cos \alpha = \dfrac{7}{13}$, find $\tan \alpha$. | -\dfrac{12}{5} | 0.9375 | 5,427.1875 | 5,242.866667 | 8,192 | |
Stacy has $d$ dollars. She enters a mall with 10 shops and a lottery stall. First she goes to the lottery and her money is doubled, then she goes into the first shop and spends 1024 dollars. After that she alternates playing the lottery and getting her money doubled (Stacy always wins) then going into a new shop and sp... | 1023 | Work backwards. Before going into the last shop she had $\$ 1024$, before the lottery she had $\$ 512$, then $\$ 1536, \$ 768, \ldots$. We can easily prove by induction that if she ran out of money after $n$ shops, $0 \leq n \leq 10$, she must have started with $1024-2^{10-n}$ dollars. Therefore $d$ is $\mathbf{1023}$. | 0.4375 | 7,072.8125 | 5,658.428571 | 8,172.888889 |
What is $(3^{12} \times 9^{-3})^2$? Write your answer as an integer. | 531441 | 0.9375 | 3,072.25 | 2,964.333333 | 4,691 | |
All triangles have the same value, and all circles have the same value. What is the sum of three circles? \begin{align*}
\Delta + \bigcirc + \Delta + \bigcirc + \Delta&= 21\\
\bigcirc + \Delta+\bigcirc+\Delta+\bigcirc &= 19\\
\bigcirc + \bigcirc + \bigcirc &= \ ?
\end{align*} | 9 | 1 | 1,723.75 | 1,723.75 | -1 | |
A piece of platinum, which has a density of $2.15 \cdot 10^{4} \mathrm{kg} / \mathrm{m}^{3}$, is connected to a piece of cork wood (density $2.4 \cdot 10^{2} \mathrm{kg} / \mathrm{m}^{3}$). The density of the combined system is $4.8 \cdot 10^{2} \mathrm{kg} / \mathrm{m}^{3}$. What is the mass of the piece of wood, if t... | 85 | 0.875 | 6,125.6875 | 5,830.5 | 8,192 | |
There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? | 80 | 1 | 3,326.3125 | 3,326.3125 | -1 | |
Given four points \( K, L, M, N \) that are not coplanar. A sphere touches the planes \( K L M \) and \( K L N \) at points \( M \) and \( N \) respectively. Find the surface area of the sphere, knowing that \( M L = 1 \), \( K M = 2 \), \( \angle M N L = 60^\circ \), and \( \angle K M L = 90^\circ \). | \frac{64\pi}{11} | 0 | 8,108.3125 | -1 | 8,108.3125 | |
Find the smallest positive integer \( n \) such that
\[
\sqrt{5 n}-\sqrt{5 n-4}<0.01
\] | 8001 | 0 | 8,192 | -1 | 8,192 | |
Let the function $f(x)=\ln x- \frac{1}{2}ax^{2}-bx$.
$(1)$ When $a=b= \frac{1}{2}$, find the maximum value of the function $f(x)$;
$(2)$ Let $F(x)=f(x)+ \frac{1}{2}ax^{2}+bx+ \frac{a}{x}$, $(0 < x\leqslant 3)$, the slope of the tangent line at any point $P(x_{0},y_{0})$ on its graph is $k\leqslant \frac{1}{2}$ always... | \frac{1}{2} | 0.75 | 6,803 | 6,340 | 8,192 | |
A random sample of $10$ households was taken from a residential area, and the monthly income data ${x_i}$ (in units of thousand yuan) of the $i$-th household was obtained. The following statistical values were calculated: $\sum_{i=1}^{10} x_i = 80$, $\sum_{i=1}^{10} y_i = 20$, $\sum_{i=1}^{10} x_i y_i = 184$, and $\sum... | 1.7 | 1 | 2,715.3125 | 2,715.3125 | -1 | |
Let $x$, $y$, and $z$ be distinct real numbers that sum to $0$. Compute \[ \dfrac {xy+yz+zx}{x^2+y^2+z^2}. \] | -\frac12 | 1 | 1,952.125 | 1,952.125 | -1 | |
Two distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction. | \frac{1}{2} | 0.9375 | 4,147.8125 | 3,878.2 | 8,192 | |
Among the six-digit numbers formed by the digits 0, 1, 2, 3, 4, 5 without repetition, calculate the number of the numbers that are divisible by 2. | 312 | 0.625 | 5,962.6875 | 5,142.4 | 7,329.833333 | |
Consider the graphs $y=Ax^2$ and $y^2+3=x^2+4y$, where $A$ is a positive constant and $x$ and $y$ are real variables. In how many points do the two graphs intersect? | 4 | #### Step-by-step Algebraic Approach:
1. **Substitute $y = Ax^2$ into the second equation:**
\[
y^2 + 3 = x^2 + 4y \implies (Ax^2)^2 + 3 = x^2 + 4Ax^2
\]
Simplifying, we get:
\[
A^2x^4 + 3 = x^2 + 4Ax^2
\]
Rearranging terms:
\[
A^2x^4 - (4A+1)x^2 + 3 = 0
\]
2. **Let $u = x^2$, then th... | 0.1875 | 7,849.5625 | 6,365.666667 | 8,192 |
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