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 |
|---|---|---|---|---|---|---|
A natural number plus 13 is a multiple of 5, and its difference with 13 is a multiple of 6. What is the smallest natural number that satisfies these conditions? | 37 | 0 | 3,328.0625 | -1 | 3,328.0625 | |
Let $N = 123456789101112\dots505152$ be the number obtained by writing out the integers from 1 to 52 consecutively. Compute the remainder when $N$ is divided by 45. | 37 | 0.6875 | 4,880.1875 | 4,313.181818 | 6,127.6 | |
Compute the sum of all integers $1 \leq a \leq 10$ with the following property: there exist integers $p$ and $q$ such that $p, q, p^{2}+a$ and $q^{2}+a$ are all distinct prime numbers. | 20 | Odd $a$ fail for parity reasons and $a \equiv 2(\bmod 3)$ fail for $\bmod 3$ reasons. This leaves $a \in\{4,6,10\}$. It is easy to construct $p$ and $q$ for each of these, take $(p, q)=(3,5),(5,11),(3,7)$, respectively. | 0.5625 | 7,354.5 | 6,909.555556 | 7,926.571429 |
Two chords \(AB\) and \(CD\) of a circle with center \(O\) each have a length of 10. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) respectively intersect at point \(P\), with \(DP = 3\). The line \(PO\) intersects segment \(AC\) at point \(L\). Find the ratio \(AL : LC\). | 3/13 | 0 | 8,020.5625 | -1 | 8,020.5625 | |
Positive integers $a$ and $b$ satisfy $a b=2010$. If $a>b$, what is the smallest possible value of $a-b$? | 37 | Note that $2010=10(201)=2(5)(3)(67)$ and that 67 is prime. Therefore, the positive divisors of 2010 are $1,2,3,5,6,10,15,30,67,134,201,335,402,670$, $1005,2010$. Thus, the possible pairs $(a, b)$ with $a b=2010$ and $a>b$ are $(2010,1),(1005,2),(670,3)$, $(402,5),(335,6),(201,10),(134,15),(67,30)$. Of these pairs, the ... | 0.8125 | 995.125 | 884.769231 | 1,473.333333 |
Find the value of the product \( \cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{3 \pi}{15} \cdots \cos \frac{7 \pi}{15} \). | \frac{1}{128} | 0 | 8,192 | -1 | 8,192 | |
$(2x-1)^{10} = a_0 + a_1x + a_2x^2 + \ldots + a_9x^9 + a_{10}x^{10}$, then $a_2 + a_3 + \ldots + a_9 + a_{10} =$ \_\_\_\_\_\_. | 20 | 0.625 | 5,205.4375 | 3,878.1 | 7,417.666667 | |
After walking so much that his feet get really tired, the beaver staggers so that, at each step, his coordinates change by either $(+1,+1)$ or $(+1,-1)$. Now he walks from $(0,0)$ to $(8,0)$ without ever going below the $x$-axis. How many such paths are there? | 14 | $C(4)=14$. | 0.875 | 5,250 | 4,829.714286 | 8,192 |
Let $u$ and $v$ be integers satisfying $0 < v < u$. Let $A = (u,v)$, let $B$ be the reflection of $A$ across the line $y = x$, let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $ABCDE$ is $... | 21 | We find the coordinates like in the solution above: $A = (u,v)$, $B = (v,u)$, $C = (-v,u)$, $D = (-v,-u)$, $E = (v,-u)$. Then we apply the Shoelace Theorem. \[A = \frac{1}{2}[(u^2 + vu + vu + vu + v^2) - (v^2 - uv - uv - uv -u^2)] = 451\] \[\frac{1}{2}(2u^2 + 6uv) = 451\] \[u(u + 3v) = 451\]
This means that $(u,v) = (... | 0.875 | 4,658.5 | 4,153.714286 | 8,192 |
Let $a,$ $b,$ $c,$ $d$ be nonzero integers such that
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^2 = \begin{pmatrix} 7 & 0 \\ 0 & 7 \end{pmatrix}.\]Find the smallest possible value of $|a| + |b| + |c| + |d|.$ | 7 | 0.6875 | 6,206 | 5,303.272727 | 8,192 | |
Given that a rectangular room is 15 feet long and 108 inches wide, calculate the area of the new extended room after adding a 3 feet wide walkway along the entire length of one side, in square yards, where 1 yard equals 3 feet and 1 foot equals 12 inches. | 20 | 0.0625 | 595.9375 | 535 | 600 | |
The quadratic equation $ax^2+20x+c=0$ has exactly one solution. If $a+c=29$, and $a<c$ find the ordered pair $(a,c)$. | (4,25) | 1 | 1,641.0625 | 1,641.0625 | -1 | |
The average lifespan of a motor is 4 years. Estimate from below the probability that this motor will not last more than 20 years. | 0.8 | 0.6875 | 4,154.3125 | 3,539.909091 | 5,506 | |
When \( n \) is a positive integer, the function \( f \) satisfies \( f(n+3)=\frac{f(n)-1}{f(n)+1} \), with \( f(1) \neq 0 \) and \( f(1) \neq \pm 1 \). Find the value of \( f(8) \cdot f(2018) \). | -1 | 0.5 | 7,129.0625 | 6,066.125 | 8,192 | |
When Alia was young, she could cycle 18 miles in 2 hours. Now, as an older adult, she walks 8 kilometers in 3 hours. Given that 1 mile is approximately 1.609 kilometers, determine how many minutes longer it takes for her to walk a kilometer now compared to when she was young. | 18 | 0 | 958 | -1 | 958 | |
What is $88 \div 4 \div 2$? | 11 | 1 | 175.0625 | 175.0625 | -1 | |
A biologist found a pond with frogs. When classifying them by their mass, he noticed the following:
*The $50$ lightest frogs represented $30\%$ of the total mass of all the frogs in the pond, while the $44$ heaviest frogs represented $27\%$ of the total mass.*As fate would have it, the frogs escaped and the bi... | 165 | 0.6875 | 5,645.0625 | 4,487.363636 | 8,192 | |
Simplify first, then evaluate: $(1-\frac{m}{{m-3}})\div \frac{{{m^2}-3m}}{{{m^2}-6m+9}}$, where $m=4\sqrt{3}$. | -\frac{\sqrt{3}}{4} | 0 | 2,980 | -1 | 2,980 | |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a\cos B - b\cos A = c$, and $C = \frac{π}{5}$, calculate the measure of $\angle B$. | \frac{3\pi}{10} | 0.8125 | 4,928 | 4,174.769231 | 8,192 | |
Given that a normal vector of line $l$ is $\overrightarrow{n}=(\sqrt{3}, -1)$, find the size of the slope angle of line $l$. | \frac{\pi}{3} | 0 | 3,671.375 | -1 | 3,671.375 | |
Given the expression $2-(-3)-4\times(-5)-6-(-7)-8\times(-9)+10$, evaluate this expression. | 108 | 0.6875 | 1,794.4375 | 2,219.636364 | 859 | |
The school plans to set up two computer labs, each equipped with one teacher's computer and several student computers. In a standard lab, the teacher's computer costs 8000 yuan, and each student computer costs 3500 yuan; in an advanced lab, the teacher's computer costs 11500 yuan, and each student computer costs 7000 y... | 27 | 0.0625 | 5,598.75 | 8,192 | 5,425.866667 | |
Let \( a, b, c, d \) be real numbers defined by
$$
a=\sqrt{4-\sqrt{5-a}}, \quad b=\sqrt{4+\sqrt{5-b}}, \quad c=\sqrt{4-\sqrt{5+c}}, \quad d=\sqrt{4+\sqrt{5+d}}
$$
Calculate their product. | 11 | 0.1875 | 7,785 | 6,814.333333 | 8,009 | |
Let $\mathcal{S}$ be the set $\lbrace1,2,3,\ldots,10\rbrace$ Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when $n$ is divided by $1000$.
| 501 | 0.875 | 5,094.125 | 4,651.571429 | 8,192 | |
In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's *score* is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n\times n$ grid with probability $k$ ; he notices t... | 51 | 0.625 | 6,079.75 | 5,246.2 | 7,469 | |
Given the function $f(x) = 2\sin x\cos x - 2\sin^2 x + 1$, determine the smallest positive value of $\varphi$ such that the graph of $f(x)$ shifted to the right by $\varphi$ units is symmetric about the y-axis. | \frac{3\pi}{8} | 0.8125 | 6,172.5 | 5,706.461538 | 8,192 | |
In quadrilateral $ABCD$ , $AB \parallel CD$ and $BC \perp AB$ . Lines $AC$ and $BD$ intersect at $E$ . If $AB = 20$ , $BC = 2016$ , and $CD = 16$ , find the area of $\triangle BCE$ .
*Proposed by Harrison Wang* | 8960 | 0.75 | 6,375.5625 | 6,074.916667 | 7,277.5 | |
What percent of square $ABCD$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.7));
xaxis(0,5,Ticks(1.0,NoZero));
yaxis(0,5,Ticks(1.0,NoZero));
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle);
fill((2,0)--(3,0)--(3,3)--(0,3)--(0,2)--(2,2)--cycle);
fill((4,0)--(5,0)--(5,5)--(0... | 60 | 0.75 | 5,624.5 | 5,108.166667 | 7,173.5 | |
Suppose $ABCD$ is a rectangle whose diagonals meet at $E$ . The perimeter of triangle $ABE$ is $10\pi$ and the perimeter of triangle $ADE$ is $n$ . Compute the number of possible integer values of $n$ . | 47 | 0.625 | 6,474.125 | 5,443.4 | 8,192 | |
Find the area of the parallelogram generated by $\begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ -3 \\ 4 \end{pmatrix}.$
[asy]
unitsize(0.4 cm);
pair A, B, C, D;
A = (0,0);
B = (7,2);
C = (1,3);
D = B + C;
draw(A--B,Arrow(6));
draw(A--C,Arrow(6));
draw(B--D--C);
[/asy] | 10 \sqrt{3} | 1 | 3,276 | 3,276 | -1 | |
A group of 101 Dalmathians participate in an election, where they each vote independently on either candidate \(A\) or \(B\) with equal probability. If \(X\) Dalmathians voted for the winning candidate, the expected value of \(X^{2}\) can be expressed as \(\frac{a}{b}\) for positive integers \(a, b\) with \(\operatorna... | 51 | Claim: with 101 replaced with \(2k+1\), the expectation of \(X^{2}\) is \(\frac{\binom{2k}{k}}{2^{2k+1}}(2k+1)^{2}+\frac{(2k+1)(2k+2)}{4}\). The answer is this value taken modulo 103, which can be calculated by noting that the integers modulo 103 form a finite field. Note that the multiplicative inverse of 4 is 26, the... | 0 | 5,748.6875 | -1 | 5,748.6875 |
What is ${-\frac{1}{2} \choose 100} \div {\frac{1}{2} \choose 100}$? | -199 | 0.25 | 7,525.3125 | 5,525.25 | 8,192 | |
Let $X_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$, \[(*)\quad\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}\] for $(m,n)=(2,3),(3,2),(2,5)$, or $(5,2)$. Determine [i]all[/i] other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $x+y+z=0$. | (2, 3), (3, 2), (2, 5), (5, 2) |
Let's start by understanding the problem statement correctly. We have a sequence defined by
\[ S_r = x^r + y^r + z^r \]
where \( x, y, \) and \( z \) are real numbers. We are informed that if \( S_1 = x + y + z = 0 \), then the following relationship holds:
\[
(*)\quad \frac{S_{m+n}}{m+n} = \frac{S_m}{m} \cdot \frac{... | 0 | 8,192 | -1 | 8,192 |
Given $f(x)= \sqrt {3}\sin x\cos (x+ \dfrac {π}{6})+\cos x\sin (x+ \dfrac {π}{3})+ \sqrt {3}\cos ^{2}x- \dfrac { \sqrt {3}}{2}$.
(I) Find the range of $f(x)$ when $x\in(0, \dfrac {π}{2})$;
(II) Given $\dfrac {π}{12} < α < \dfrac {π}{3}$, $f(α)= \dfrac {6}{5}$, $- \dfrac {π}{6} < β < \dfrac {π}{12}$, $f(β)= \dfrac {10}{... | -\dfrac{33}{65} | 0.0625 | 8,037.125 | 5,714 | 8,192 | |
With all angles measured in degrees, calculate the product $\prod_{k=1}^{30} \csc^2(3k)^\circ$, and express the result as $m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$. | 31 | 0 | 8,192 | -1 | 8,192 | |
An equilateral triangle with a side length of 1 is cut along a line parallel to one of its sides, resulting in a trapezoid. Let $S = \frac{\text{(perimeter of the trapezoid)}^2}{\text{area of the trapezoid}}$. Find the minimum value of $S$. | \frac{32\sqrt{3}}{3} | 0 | 7,987.5 | -1 | 7,987.5 | |
Given $2^x = 8^{y+1}$ and $9^y = 3^{x-9}$, find the value of $x+y$ | 27 | 1. **Rewrite the equations with the same base:**
Given $2^x = 8^{y+1}$, we know $8 = 2^3$, so:
\[
2^x = (2^3)^{y+1} = 2^{3(y+1)}
\]
Similarly, given $9^y = 3^{x-9}$, and knowing $9 = 3^2$, we have:
\[
9^y = (3^2)^y = 3^{2y}
\]
Therefore, the equations become:
\[
2^x = 2^{3(y+1)} \quad... | 1 | 1,585.4375 | 1,585.4375 | -1 |
We denote by gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd $(4, 6, 8)=2$ and gcd $(12, 15)=3$ .) Suppose that positive integers $a, b, c$ satisfy the following four conditions: $\bullet$ gcd $(a, b, c)=1$ , $\bullet$ gcd $(a, b + c)>1$ , $\bullet$ gcd $(b, c + a)>1$ , $\bul... | 30 | 0 | 8,192 | -1 | 8,192 | |
The graph of the function $f(x)=\frac{x}{x+a}$ is symmetric about the point $(1,1)$, and the function $g(x)=\log_{10}(10^x+1)+bx$ is even. Find the value of $a+b$. | -\frac{3}{2} | 0.3125 | 7,328.8125 | 6,388.8 | 7,756.090909 | |
In recent years, live streaming e-commerce has gradually become an emerging marketing model, bringing new growth points to the e-commerce industry. At the beginning of the first year, a certain live streaming platform had an initial capital of 5 million yuan. Due to the participation of some well-known hosts, the platf... | 46.8 | 0 | 8,192 | -1 | 8,192 | |
The inclination angle of the line $x+ \sqrt {3}y+c=0$ is \_\_\_\_\_\_. | \frac{5\pi}{6} | 0.375 | 2,984.3125 | 3,386.333333 | 2,743.1 | |
The base of a prism is an equilateral triangle $ABC$. The lateral edges of the prism $AA_1$, $BB_1$, and $CC_1$ are perpendicular to the base. A sphere, whose radius is equal to the edge of the base of the prism, touches the plane $A_1B_1C_1$ and the extensions of the segments $AB_1$, $BC_1$, and $CA_1$ beyond the poin... | \sqrt{44} - 6 | 0 | 8,192 | -1 | 8,192 | |
A, B, C, and D obtained the top 4 positions in the school (no ties). They made the following statements:
- A: "I am neither first nor second."
- B: "I am neither second nor third."
- C: "My position is adjacent to B."
- D: "My position is adjacent to C."
Given that A, B, C, and D are all honest students, determine the... | 4123 | 0.9375 | 3,479.25 | 3,467.466667 | 3,656 | |
Hawkins, Dustin, and Lucas start playing a game where each begins with $\$2$. A bell rings every $10$ seconds, and each player with money independently chooses one of the other two players at random to give $\$1$. If a player has only $\$1$ left, there is a $\frac{1}{3}$ probability they will keep their money and not g... | \frac{1}{4} | 0 | 7,852.375 | -1 | 7,852.375 | |
The sequence is defined as \( a_{0}=134, a_{1}=150, a_{k+1}=a_{k-1}-\frac{k}{a_{k}} \) for \( k=1,2, \cdots, n-1 \). Determine the value of \( n \) for which \( a_{n}=0 \). | 201 | 0.375 | 6,909.9375 | 5,425.333333 | 7,800.7 | |
Rational Man and Irrational Man both buy new cars, and they decide to drive around two racetracks from time $t = 0$ to $t = \infty.$ Rational Man drives along the path parameterized by
\begin{align*}
x &= \cos t, \\
y &= \sin t,
\end{align*}and Irrational Man drives along the path parameterized by
\begin{align*}
x &= ... | \frac{\sqrt{33} - 3}{3} | 0 | 8,192 | -1 | 8,192 | |
Let point $P$ be on the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$. A line through $P$ intersects the asymptotes at $P_{1}$ and $P_{2}$, and $\overrightarrow{P_{1} P} \overrightarrow{P P_{2}}$ $=3$. Let $O$ be the origin. Find the area of $\triangle O P_{1} P_{2}$. | 16 | 0 | 8,192 | -1 | 8,192 | |
In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0, 2, 4, 7, 8, 9\) (digits can repeat) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done? | 1728 | 0.0625 | 7,715.0625 | 5,820 | 7,841.4 | |
Equilateral triangle $ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = \sqrt{11}$. Find $\sum_{k=1}^4(CE_k)^2$. | 677 | 0 | 8,192 | -1 | 8,192 | |
When the square root of $x$ is cubed, the answer is 64. What is the value of $x$? | 16 | 1 | 1,265.1875 | 1,265.1875 | -1 | |
A positive integer \( n \) with \( n \) digits is called an "auspicious number" if, when appended to the end of any two positive integers, the product of these two new numbers ends in \( x \). For example, 6 is an "auspicious number," but 16 is not, because \( 116 \times 216 = 25056 \), which does not end in 16. What i... | 1114 | 0.125 | 7,800 | 7,110 | 7,898.571429 | |
The equation $x^2 - kx - 24 = 0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$? | 40 | 0.9375 | 3,732.6875 | 3,441.066667 | 8,107 | |
A shooter fires at a target until they hit it for the first time. The probability of hitting the target each time is 0.6. If the shooter has 4 bullets, the expected number of remaining bullets after stopping the shooting is \_\_\_\_\_\_\_\_. | 2.376 | 0 | 7,910.25 | -1 | 7,910.25 | |
(The full score for this question is 8 points) Arrange 3 male students and 2 female students in a row,
(1) The number of all different arrangements;
(2) The number of arrangements where exactly two male students are adjacent;
(3) The number of arrangements where male students are of different heights and are ar... | 20 | 0.75 | 6,081.625 | 5,378.166667 | 8,192 | |
For every dollar Ben spent on pastries, David spent $50$ cents less. Ben paid $20$ more than David. Calculate the amount Ben and David spent together in the pastry shop. | 60 | 0.9375 | 2,528.625 | 2,151.066667 | 8,192 | |
A computer can apply three operations to a number: "increase by 2," "increase by 3," "multiply by 2." The computer starts with the number 1 and is made to go through all possible combinations of 6 operations (each combination is applied to the initial number 1). After how many of these combinations will the computer en... | 486 | 0.1875 | 6,764.1875 | 5,370.333333 | 7,085.846154 | |
Let $f$ be a mapping from set $A = \{a, b, c, d\}$ to set $B = \{0, 1, 2\}$.
(1) How many different mappings $f$ are there?
(2) If it is required that $f(a) + f(b) + f(c) + f(d) = 4$, how many different mappings $f$ are there? | 19 | 0.625 | 6,294.625 | 5,156.2 | 8,192 | |
What is the largest possible rational root of the equation $ax^2 + bx + c = 0{}$ where $a, b$ and $c{}$ are positive integers that do not exceed $100{}$? | \frac{1}{99} |
To determine the largest possible rational root of the quadratic equation \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are positive integers not exceeding 100, we use the Rational Root Theorem. This theorem states that any rational root, expressed as \(\frac{p}{q}\), must have \( p \) as a divisor of the con... | 0 | 8,192 | -1 | 8,192 |
Given that bricklayer Alice can build a wall alone in 8 hours, and bricklayer Bob can build it alone in 12 hours, and they complete the wall in 6 hours when working together with a 15-brick-per-hour decrease in productivity, determine the number of bricks in the wall. | 360 | 1 | 1,991.25 | 1,991.25 | -1 | |
Convert the point $(\sqrt{2},-\sqrt{2})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | \left( 2, \frac{7 \pi}{4} \right) | 1 | 1,571.0625 | 1,571.0625 | -1 | |
Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. | 110.8333 | 0 | 4,974.6875 | -1 | 4,974.6875 | |
What is the sum of all odd integers between $400$ and $600$? | 50000 | 0.875 | 3,659.8125 | 3,012.357143 | 8,192 | |
Please write an irrational number that is smaller than $3$. | \sqrt{2} | 0.875 | 1,602.625 | 661.285714 | 8,192 | |
Consider a square in the coordinate plane with vertices at $(2, 1)$, $(5, 1)$, $(5, 4)$, and $(2, 4)$. A line joining $(2, 3)$ and $(5, 1)$ divides the square shown into two parts. Determine the fraction of the area of the square that is above this line. | \frac{5}{6} | 0 | 7,103.0625 | -1 | 7,103.0625 | |
Many calculators have a reciprocal key $\boxed{\frac{1}{x}}$ that replaces the current number displayed with its reciprocal. For example, if the display is $\boxed{00004}$ and the $\boxed{\frac{1}{x}}$ key is depressed, then the display becomes $\boxed{000.25}$. If $\boxed{00032}$ is currently displayed, what is the ... | 2 | 1. **Define the function**: Let $f(x) = \frac{1}{x}$ represent the operation of the reciprocal key on the calculator.
2. **Apply the function**: We need to determine how many times we must apply $f(x)$ to return to the original number. Start by applying $f(x)$ to the number 32:
\[
f(32) = \frac{1}{32}
\]
3. ... | 0.75 | 5,503.0625 | 4,683.666667 | 7,961.25 |
There are $4$ distinct codes used in an intelligence station, one of them applied in each week. No two codes used in two adjacent weeks are the same code. Knowing that code $A$ is used in the first week, find the probability that code $A$ is used in the seventh week. | 61/243 | 0.4375 | 7,613.5 | 6,897.428571 | 8,170.444444 | |
Given that a travel agency plans to arrange a trip for 900 passengers using two types of buses, A and B, with capacities 36 and 60 passengers respectively, rental costs 1600 yuan and 2400 yuan per bus respectively, and the total number of buses rented does not exceed 21, and the number of type B buses cannot exceed the... | 36800 | 0.625 | 6,747.9375 | 6,150.1 | 7,744.333333 | |
A 24-hour digital clock shows times $h: m: s$, where $h, m$, and $s$ are integers with $0 \leq h \leq 23$, $0 \leq m \leq 59$, and $0 \leq s \leq 59$. How many times $h: m: s$ satisfy $h+m=s$? | 1164 | We are solving $h+m=s$ in $0 \leq s \leq 59,0 \leq m \leq 59$, and $0 \leq h \leq 23$. If $s \geq 24$, each $h$ corresponds to exactly 1 solution, so we get $24(59-23)=24(36)$ in this case. If $s \leq 23$, we want the number of nonnegative integer solutions to $h+m \leq 23$, which by lattice point counting (or balls an... | 0.875 | 6,237.875 | 5,958.714286 | 8,192 |
Determine the value of $a + b$ if the points $(2,a,b),$ $(a,3,b),$ and $(a,b,4)$ are collinear. | -2 | 0 | 4,067.5625 | -1 | 4,067.5625 | |
There are 2012 distinct points in the plane, each of which is to be coloured using one of \( n \) colours so that the number of points of each colour are distinct. A set of \( n \) points is said to be multi-coloured if their colours are distinct. Determine \( n \) that maximizes the number of multi-coloured sets. | 61 | 0 | 8,192 | -1 | 8,192 | |
In the arithmetic sequence $\{a_n\}$, $S_{10} = 10$, $S_{20} = 30$, then $S_{30} = \ ?$ | 60 | 0.6875 | 5,276.625 | 3,951.454545 | 8,192 | |
Jose is $4$ years younger than Zack. Zack is $3$ years older than Inez. Inez is $15$ years old. How old is Jose? | 14 | 1. **Determine Inez's Age:**
Given that Inez is $15$ years old.
2. **Calculate Zack's Age:**
Zack is $3$ years older than Inez. Therefore, we calculate Zack's age as follows:
\[
\text{Zack's age} = \text{Inez's age} + 3 = 15 + 3 = 18 \text{ years}
\]
3. **Calculate Jose's Age:**
Jose is $4$ years yo... | 1 | 932.0625 | 932.0625 | -1 |
What is the value of $x$ if a cube's volume is $5x$ cubic units and its surface area is $x$ square units? | 5400 | 1 | 1,681.4375 | 1,681.4375 | -1 | |
The front view of a cone is an equilateral triangle with a side length of 4. Find the surface area of the cone. | 12\pi | 0.9375 | 2,169.0625 | 2,154.066667 | 2,394 | |
A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find th... | 57 | 0.5 | 6,315.125 | 5,444.125 | 7,186.125 | |
What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction. | 11 | 0 | 7,289.5625 | -1 | 7,289.5625 | |
If the coefficient of the $x^2$ term in the expansion of $(1-ax)(1+2x)^4$ is $4$, then $\int_{\frac{e}{2}}^{a}{\frac{1}{x}}dx =$ . | \ln(5) - 1 | 0 | 3,535.4375 | -1 | 3,535.4375 | |
In $\triangle A B C, \omega$ is the circumcircle, $I$ is the incenter and $I_{A}$ is the $A$-excenter. Let $M$ be the midpoint of arc $\widehat{B A C}$ on $\omega$, and suppose that $X, Y$ are the projections of $I$ onto $M I_{A}$ and $I_{A}$ onto $M I$, respectively. If $\triangle X Y I_{A}$ is an equilateral triangle... | \frac{\sqrt{6}}{7} | Using Fact 5, we know that $I I_{A}$ intersects the circle $(A B C)$ at $M_{A}$, which is the center of $(I I_{A} B C X Y)$. Let $R$ be the radius of the latter circle. We have $R=\frac{1}{\sqrt{3}}$. We have $\angle A I M=\angle Y I I_{A}=\angle Y I X=\frac{\pi}{3}$. Also, $\angle I I_{A} M=\angle I M I_{A}$ by calcul... | 0 | 8,192 | -1 | 8,192 |
For how many integers $n$ with $1 \le n \le 2016$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 2 + e^{4 \pi i k / n} \right)^n - 1 \right)
\]
equal to zero? | 504 | 0.0625 | 7,718.4375 | 6,095 | 7,826.666667 | |
$1000 \times 1993 \times 0.1993 \times 10 =$ | $(1993)^2$ | 1. **Break down the expression**: We start by simplifying the expression $1000 \times 1993 \times 0.1993 \times 10$. We can rearrange the terms for easier computation:
\[
1000 \times 10 \times 1993 \times 0.1993
\]
2. **Simplify the powers of 10**: Calculate $1000 \times 10$:
\[
1000 \times 10 = 10^4
... | 0 | 3,680.875 | -1 | 3,680.875 |
What is the largest number of positive, consecutive integers whose sum is 105? | 14 | 0.9375 | 4,651.6875 | 4,415.666667 | 8,192 | |
A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles? | 4 \times 8 | 1. **Identify the original dimensions of the square**: The square has a side length of 8 units. Therefore, each side of the square is 8 units long.
2. **Understand the effect of cutting the square in half**: When the square is cut in half, one dimension is halved while the other remains the same. This is because the c... | 0.0625 | 2,616.5625 | 1,309 | 2,703.733333 |
Let \( a_1, a_2, \dots \) be a sequence of positive real numbers such that
\[ a_n = 7a_{n-1} - 2n \] for all \( n > 1 \). Find the smallest possible value of \( a_1 \). | \frac{13}{18} | 0.25 | 7,574 | 5,720 | 8,192 | |
Find all natural numbers \( n \) such that, when writing the numbers \( n^3 \) and \( n^4 \) side by side in decimal notation, each of the ten digits appears exactly once.
(Former Yugoslavia Mathematical Olympiad, 1983) | 18 | 0.4375 | 7,504.75 | 6,621.142857 | 8,192 | |
Let $f(n) = \frac{x_1 + x_2 + \cdots + x_n}{n}$, where $n$ is a positive integer. If $x_k = (-1)^k, k = 1, 2, \cdots, n$, the set of possible values of $f(n)$ is: | $\{0, -\frac{1}{n}\}$ | To find the set of possible values of $f(n)$, we first need to evaluate the sum $x_1 + x_2 + \cdots + x_n$ where $x_k = (-1)^k$ for $k = 1, 2, \ldots, n$.
1. **Expression for $x_k$:**
- $x_k = (-1)^k$ means that $x_k$ alternates between $-1$ and $1$ starting with $-1$ when $k$ is odd and $1$ when $k$ is even.
2. *... | 0 | 6,144.4375 | -1 | 6,144.4375 |
If $a*b=a^2+ab-b^2$, find $3*2$. | 11 | 1 | 701.875 | 701.875 | -1 | |
A line $l$ passes through two points $P(-1,2)$ and $Q(2,-2)$, and intersects the hyperbola $(y-2)^{2}-x^{2}=1$ at two points $A$ and $B$.
$(1)$ Write the parametric equation of $l$ as required by the question;
$(2)$ Find the distance between the midpoint $M$ of $AB$ and point $P$. | 5 \sqrt {65} | 0 | 3,690.5625 | -1 | 3,690.5625 | |
Several people completed the task of planting 2013 trees, with each person planting the same number of trees. If 5 people do not participate in the planting, the remaining people each need to plant 2 more trees but still cannot complete the task. However, if each person plants 3 more trees, they can exceed the task. Ho... | 61 | 0.375 | 6,485.6875 | 4,540.333333 | 7,652.9 | |
The scores on Trisha's first three tests were 88, 73 and 70. After two more tests, the mean score for all five tests was 81. Each test score was less than 90, and all of Trisha's test scores were different integer values. List Trisha's five test scores from the greatest to the least, separated by commas. | 89, 88, 85, 73, 70 | 0.1875 | 4,362.625 | 4,261.666667 | 4,385.923077 | |
Find the smallest positive integer $n$ such that there exists a sequence of $n+1$ terms $a_{0}, a_{1}, \cdots, a_{n}$ satisfying $a_{0}=0, a_{n}=2008$, and $\left|a_{i}-a_{i-1}\right|=i^{2}$ for $i=1,2, \cdots, n$. | 19 | 0.1875 | 8,005.4375 | 7,197 | 8,192 | |
In a pot, there are 6 sesame-filled dumplings, 5 peanut-filled dumplings, and 4 red bean paste-filled dumplings. These three types of dumplings look exactly the same from the outside. If 4 dumplings are randomly scooped out, the probability that at least one dumpling of each type is scooped out is ______. | \dfrac{48}{91} | 0.25 | 7,145.375 | 4,702 | 7,959.833333 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. It is given that $\angle B=30^{\circ}$, the area of $\triangle ABC$ is $\frac{3}{2}$, and $\sin A + \sin C = 2\sin B$. Calculate the value of $b$. | \sqrt{3}+1 | 0 | 7,406.5 | -1 | 7,406.5 | |
Around a circular table, there are 18 girls seated, 11 dressed in blue and 7 dressed in red. Each girl is asked if the girl to her right is dressed in blue, and each one responds with either yes or no. It is known that a girl tells the truth only when both of her neighbors, the one to her right and the one to her left,... | 11 | 0 | 8,192 | -1 | 8,192 | |
Perform the calculations.
$(54+38) \times 15$
$1500-32 \times 45$
$157 \times (70 \div 35)$ | 314 | 1 | 490.5 | 490.5 | -1 | |
Let $a,$ $b,$ $c$ be three distinct positive real numbers such that $a,$ $b,$ $c$ form a geometric sequence, and
\[\log_c a, \ \log_b c, \ \log_a b\]form an arithmetic sequence. Find the common difference of the arithmetic sequence. | \frac{3}{2} | 0.5 | 7,336.0625 | 6,480.125 | 8,192 | |
In triangle \( ABC \), \( AC = 3 AB \). Let \( AD \) bisect angle \( A \) with \( D \) lying on \( BC \), and let \( E \) be the foot of the perpendicular from \( C \) to \( AD \). Find \( \frac{[ABD]}{[CDE]} \). (Here, \([XYZ]\) denotes the area of triangle \( XYZ \)). | 1/3 | 0.4375 | 7,575.5625 | 6,783 | 8,192 | |
The Grunters play the Screamers 6 times. The Grunters have a 60% chance of winning any given game. If a game goes to overtime, the probability of the Grunters winning changes to 50%. There is a 10% chance that any game will go into overtime. What is the probability that the Grunters will win all 6 games, considering th... | \frac{823543}{10000000} | 0 | 5,765.375 | -1 | 5,765.375 | |
Call the pentominoes found in the last problem square pentominoes. Just like dominos and ominos can be used to tile regions of the plane, so can square pentominoes. In particular, a square pentomino tiling of a region of the plane is a way of covering it (and only it) completely by nonoverlapping square pentominoes. Ho... | 0 | Since 5 does not divide 144, there are 0. | 0.875 | 2,821.875 | 2,173.857143 | 7,358 |
The number of distinct points common to the graphs of $x^2+y^2=9$ and $y^2=9$ is: | 2 | 1. **Identify the equations**: We are given two equations:
- Circle: \(x^2 + y^2 = 9\)
- Horizontal lines: \(y^2 = 9\)
2. **Solve the second equation**: To find the values of \(y\), we solve \(y^2 = 9\):
\[
y = \pm 3
\]
3. **Substitute \(y\) values into the first equation**: We substitute \(y = 3\) and... | 1 | 1,824.3125 | 1,824.3125 | -1 |
An icosahedron is a regular polyhedron with twenty faces, all of which are equilateral triangles. If an icosahedron is rotated by $\theta$ degrees around an axis that passes through two opposite vertices so that it occupies exactly the same region of space as before, what is the smallest possible positive value of $\th... | 72^{\circ} | Because this polyhedron is regular, all vertices must look the same. Let's consider just one vertex. Each triangle has a vertex angle of $60^{\circ}$, so we must have fewer than 6 triangles; if we had 6 , there would be $360^{\circ}$ at each vertex and you wouldn't be able to "fold" the polyhedron up (that is, it would... | 0.9375 | 4,342.6875 | 4,288.133333 | 5,161 |
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