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 |
|---|---|---|---|---|---|---|
Lord Moneybag said to his grandson, "Bill, listen carefully! Christmas is almost here. I have taken an amount between 300 and 500 pounds, which is a multiple of 6. You will receive 5 pounds in 1-pound coins. When I give you each pound, the remaining amount will first be divisible by 5, then by 4, then by 3, then by 2, ... | 426 | 0.3125 | 7,166.3125 | 4,909.8 | 8,192 | |
Two cross sections of a right hexagonal pyramid are obtained by cutting the pyramid with planes parallel to the hexagonal base. The areas of the cross sections are $216\sqrt{3}$ square feet and $486\sqrt{3}$ square feet. The two planes are $8$ feet apart. How far from the apex of the pyramid is the larger cross section... | 24 | 0.8125 | 3,844.75 | 3,563.153846 | 5,065 | |
Consider a triangle with vertices at points \( (0, 0), (30, 0), \) and \( (18, 26) \). The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the original triangle along the sides of its midpoint triangle. What is the volume of this pyramid? | 3380 | 0 | 8,192 | -1 | 8,192 | |
A square $WXYZ$ with side length 8 units is divided into four smaller squares by drawing lines from the midpoints of one side to the midpoints of the opposite sides. The top right square of each iteration is shaded. If this dividing and shading process is done 100 times, what is the total area of the shaded squares?
A)... | \frac{64}{3} | 0 | 4,387.75 | -1 | 4,387.75 | |
If "For all $x \in \mathbb{R}, (a-2)x+1>0$" is a true statement, then the set of values for the real number $a$ is. | \{2\} | 0.1875 | 2,670.125 | 2,807 | 2,638.538462 | |
In the rectangular parallelepiped shown, $AB = 3$, $BC = 1$, and $CG = 2$. Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$? | \frac{4}{3} | 1. **Identify the dimensions of the rectangular parallelepiped**: Given $AB = 3$, $BC = 1$, and $CG = 2$.
2. **Determine the length of $EB$**: Since $EB$ is the diagonal of the rectangle $ABCE$ on the base of the parallelepiped, we use the Pythagorean theorem in the plane:
\[
EB = \sqrt{AB^2 + BC^2 + AC^2} = \s... | 0 | 6,749.1875 | -1 | 6,749.1875 |
In the diagram, the rectangular wire grid contains 15 identical squares. The length of the rectangular grid is 10. What is the length of wire needed to construct the grid? | 76 | 0.1875 | 5,605.3125 | 5,964.333333 | 5,522.461538 | |
A sequence of length 15 consisting of the letters $A$ and $B$ satisfies the following conditions:
For any two consecutive letters, $AA$ appears 5 times, $AB$, $BA$, and $BB$ each appear 3 times. How many such sequences are there?
For example, in $AA B B A A A A B A A B B B B$, $AA$ appears 5 times, $AB$ appears 3 tim... | 560 | 0 | 8,192 | -1 | 8,192 | |
How many of the smallest 2401 positive integers written in base 7 include the digits 4, 5, or 6? | 2146 | 0 | 7,965.625 | -1 | 7,965.625 | |
For any real numbers \( x \) and \( y \), the operation is defined as
\[ x \oplus y = x + 2y + 3. \]
Given that real numbers \( a \) and \( b \) satisfy
\[ \left(a^3 \oplus a^2\right) \oplus a = a^3 \oplus \left(a^2 \oplus a\right) = b, \]
find the value of \( a + b \). | \frac{21}{8} | 0.9375 | 4,058.875 | 3,783.333333 | 8,192 | |
Amanda, Ben, and Carlos share a sum of money. Their portions are in the ratio of 1:2:7, respectively. If Amanda's portion is $\$$20, what is the total amount of money shared? | 200 | 1 | 1,176.4375 | 1,176.4375 | -1 | |
There are 10 street lamps, and to save electricity, three of them are turned off. However, the two at the ends must not be turned off, and no two consecutive lamps should be off. Calculate the number of ways to do this. | 20 | 0.625 | 6,069.9375 | 4,959.1 | 7,921.333333 | |
Given that the chord common to circle C: x²+(y-4)²=18 and circle D: (x-1)²+(y-1)²=R² has a length of $6\sqrt {2}$, find the radius of circle D. | 2\sqrt {7} | 0 | 7,115.8125 | -1 | 7,115.8125 | |
Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is $132.$ | 7 |
#### Solution 1 - Detailed Analysis
1. **Represent the number and its reverse:**
Let the two-digit number be represented as $10a + b$, where $a$ is the tens digit and $b$ is the units digit. The reverse of this number would be $10b + a$.
2. **Formulate the equation:**
According to the problem, the sum of t... | 0.875 | 3,361.125 | 2,671 | 8,192 |
In triangle $ABC$, $AB=20$ and $AC=11$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive ... | 51 | Assign mass points as follows: by Angle-Bisector Theorem, $BD / DC = 20/11$, so we assign $m(B) = 11, m(C) = 20, m(D) = 31$. Since $AM = MD$, then $m(A) = 31$, and $\frac{CP}{PA} = \frac{m(A) }{ m(C)} = \frac{31}{20}$, so $m+n = \boxed{51}$. | 0.625 | 5,721.5625 | 4,687.7 | 7,444.666667 |
Determine the minimum possible value of the sum
\[\frac{a}{2b} + \frac{b}{4c} + \frac{c}{8a},\]where $a,$ $b,$ and $c$ are positive real numbers. | \frac{3}{4} | 0.9375 | 4,748.0625 | 4,518.466667 | 8,192 | |
Let $ a,\ b$ be real constants. Find the minimum value of the definite integral:
$ I(a,\ b)\equal{}\int_0^{\pi} (1\minus{}a\sin x \minus{}b\sin 2x)^2 dx.$ | \pi - \frac{8}{\pi} | 1 | 5,097.125 | 5,097.125 | -1 | |
The bacteria in a lab dish double in number every four hours. If 500 bacteria cells are in the dish now, in how many hours will there be exactly 32,000 bacteria? | 24 | 1 | 1,876.0625 | 1,876.0625 | -1 | |
Define mutually externally tangent circles $\omega_1$ , $\omega_2$ , and $\omega_3$ . Let $\omega_1$ and $\omega_2$ be tangent at $P$ . The common external tangents of $\omega_1$ and $\omega_2$ meet at $Q$ . Let $O$ be the center of $\omega_3$ . If $QP = 420$ and $QO = 427$ , find the radius of $\om... | 77 | 0 | 8,192 | -1 | 8,192 | |
The number $7.21\times 10^{11}$ has how many digits in the original number. | 12 | 0.8125 | 509.3125 | 520.769231 | 459.666667 | |
Let \( K, L, \) and \( M \) be the midpoints of the edges \( AD \), \( A_1B_1 \), and \( CC_1 \) respectively of a rectangular parallelepiped \( ABCDA_1B_1C_1D_1 \), where \( AB = a \), \( AA_1 = b \), and \( AD = c \). Find the ratio of the sum of the squares of the sides of triangle \( KLM \) to the square of the dia... | \frac{3}{2} | 0.75 | 5,476.625 | 4,886.666667 | 7,246.5 | |
In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group? Note that identically sized groups are indistinguishable. | 36 | 0 | 7,979.5625 | -1 | 7,979.5625 | |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, ... | 20\sqrt{5} | 0.0625 | 8,114.5625 | 7,535 | 8,153.2 | |
A car is averaging 50 miles per hour. If the car maintains this speed, how many minutes less would a 450-mile trip take than a 475-mile trip? | 30 | 1 | 1,664.0625 | 1,664.0625 | -1 | |
Given that both $m$ and $n$ are non-negative integers, find the number of "simple" ordered pairs $(m, n)$ with a value of 2019. | 60 | 0 | 4,959.8125 | -1 | 4,959.8125 | |
What is the value of $\sqrt{64 \times \sqrt{49}}$? | 8\sqrt{7} | 0.9375 | 2,633.25 | 2,262.666667 | 8,192 | |
Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$, and a point $P(1,\frac{{\sqrt{2}}}{2})$ on the ellipse, satisfying $|PF_{1}|+|PF_{2}|=2\sqrt{2}$.<br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ A line $l$ passing through $F_{2}$ intersects t... | \frac{\sqrt{2}}{2} | 0 | 8,192 | -1 | 8,192 | |
Find the sum of all possible positive integer values of $b$ such that the quadratic equation $2x^2 + 5x + b = 0$ has rational roots. | 5 | 1 | 4,018.3125 | 4,018.3125 | -1 | |
Find the least positive integer \( x \) that satisfies both \( x + 7219 \equiv 5305 \pmod{17} \) and \( x \equiv 4 \pmod{7} \). | 109 | 1 | 2,861.5625 | 2,861.5625 | -1 | |
A factory makes chocolate bars. Five boxes, labelled $V, W, X, Y, Z$, are each packed with 20 bars. Each of the bars in three of the boxes has a mass of 100 g. Each of the bars in the other two boxes has a mass of 90 g. One bar is taken from box $V$, two bars are taken from box $W$, four bars are taken from box $X$, ei... | W \text{ and } Z | The number of bars taken from the boxes is $1+2+4+8+16=31$. If these bars all had mass 100 g, their total mass would be 3100 g. Since their total mass is 2920 g, they are $3100 \mathrm{~g}-2920 \mathrm{~g}=180 \mathrm{~g}$ lighter. Since all of the bars have a mass of 100 g or of 90 g, then it must be the case that 18 ... | 0 | 7,846.5 | -1 | 7,846.5 |
The lines $x=\frac{1}{4}y+a$ and $y=\frac{1}{4}x+b$ intersect at the point $(1,2)$. What is $a+b$? | \frac{9}{4} | 1. **Substitute $(1,2)$ into the first equation:**
Given the equation $x = \frac{1}{4}y + a$, substitute $x = 1$ and $y = 2$:
\[
1 = \frac{1}{4} \cdot 2 + a
\]
Simplify the equation:
\[
1 = \frac{1}{2} + a
\]
Solve for $a$:
\[
a = 1 - \frac{1}{2} = \frac{1}{2}
\]
2. **Substitute $... | 1 | 1,964.3125 | 1,964.3125 | -1 |
Given the function $f(x)=x^{3}+ax^{2}+bx+c$, it reaches a maximum value of $7$ when $x=-1$, and it reaches a minimum value when $x=3$. Find the values of $a$, $b$, $c$, and this minimum value. | -25 | 1 | 2,449.375 | 2,449.375 | -1 | |
Suppose that $a$ and $b$ are real numbers such that the line $y=a x+b$ intersects the graph of $y=x^{2}$ at two distinct points $A$ and $B$. If the coordinates of the midpoint of $A B$ are $(5,101)$, compute $a+b$. | 61 | Solution 1: Let $A=\left(r, r^{2}\right)$ and $B=\left(s, s^{2}\right)$. Since $r$ and $s$ are roots of $x^{2}-a x-b$ with midpoint 5, $r+s=10=a$ (where the last equality follows by Vieta's formula). Now, as $-r s=b$ (Vieta's formula), observe that $$202=r^{2}+s^{2}=(r+s)^{2}-2 r s=100+2 b$$ This means $b=51$, so the a... | 1 | 2,273.3125 | 2,273.3125 | -1 |
Alloy $A$ of two metals has a mass of 6 kg, with the first metal being twice as abundant as the second metal. When placed in a container of water, it exerts a force of $30\ \mathrm{N}$ on the bottom. Alloy $B$ of the same metals has a mass of 3 kg, with the first metal being five times less abundant than the second met... | 40 | 0 | 8,192 | -1 | 8,192 | |
Given $x_1$ and $x_2$ are the two real roots of the quadratic equation in $x$: $x^2 - 2(m+2)x + m^2 = 0$.
(1) When $m=0$, find the roots of the equation;
(2) If $(x_1 - 2)(x_2 - 2) = 41$, find the value of $m$;
(3) Given an isosceles triangle $ABC$ with one side length of 9, if $x_1$ and $x_2$ happen to be the len... | 19 | 0.625 | 6,318.8125 | 5,687.3 | 7,371.333333 | |
Lynne chooses four distinct digits from 1 to 9 and arranges them to form the 24 possible four-digit numbers. These 24 numbers are added together giving the result \(N\). For all possible choices of the four distinct digits, what is the largest sum of the distinct prime factors of \(N\)? | 146 | 0.4375 | 7,234.1875 | 6,828.428571 | 7,549.777778 | |
Using the digits 0, 1, 2, 3, and 4, how many even numbers can be formed without repeating any digits? | 163 | 0 | 7,469.4375 | -1 | 7,469.4375 | |
The value of \( 444 - 44 - 4 \) is | 396 | 0.625 | 5,218.75 | 3,468.8 | 8,135.333333 | |
Determine the area of the region bounded by the graph of \[x^2+y^2 = 6|x-y| + 6|x+y|\].
A) 54
B) 63
C) 72
D) 81
E) 90 | 72 | 0 | 8,192 | -1 | 8,192 | |
The first tourist travels for 1.5 hours on a bicycle at a speed of 16 km/h, then takes a break for 1.5 hours, and then continues at the initial speed. Four hours after the first tourist starts, a second tourist starts chasing the first tourist on a motorcycle at a speed of 56 km/h. What distance will they each have tra... | 56 | 0.375 | 6,746.6875 | 5,095.333333 | 7,737.5 | |
When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even? | \frac{5}{8} | 1. **Determine the probability of rolling an even or odd number:** Given that an even number is 3 times as likely to appear as an odd number, let the probability of rolling an odd number be $p$. Then, the probability of rolling an even number is $3p$. Since the total probability must sum to 1, we have:
\[
p + 3p ... | 0.6875 | 4,977.5 | 4,078.818182 | 6,954.6 |
Given a number $\overline{abcd}$ , where $a$ , $b$ , $c$ , and $d$ , represent the digits of $\overline{abcd}$ , find the minimum value of
\[\frac{\overline{abcd}}{a+b+c+d}\]
where $a$ , $b$ , $c$ , and $d$ are distinct
<details><summary>Answer</summary>$\overline{abcd}=1089$ , minimum value of $\dfrac{\ove... | 60.5 | 0.375 | 7,589.25 | 6,584.666667 | 8,192 | |
A four-digit number with digits in the thousands, hundreds, tens, and units places respectively denoted as \(a, b, c, d\) is formed by \(10 \cdot 23\). The sum of these digits is 26. The tens digit of the product of \(b\) and \(d\) equals \((a+c)\). Additionally, \(( b d - c^2 )\) is an integer power of 2. Find the fou... | 1979 | 0.125 | 7,969 | 6,408 | 8,192 | |
Let $S$ be the set of all positive factors of 6000. What is the probability of a random quadruple $(a, b, c, d) \in S^{4}$ satisfies $$\operatorname{lcm}(\operatorname{gcd}(a, b), \operatorname{gcd}(c, d))=\operatorname{gcd}(\operatorname{lcm}(a, b), \operatorname{lcm}(c, d)) ?$$ | \frac{41}{512} | For each prime factor, let the greatest power that divides $a, b, c, d$ be $p, q, r, s$. WLOG assume that $p \leq q$ and $r \leq s$, and further WLOG assume that $p \leq r$. Then we need $r=\min (q, s)$. If $q=r$ then we have $p \leq q=r \leq s$, and if $r=s$ then we have $p \leq r=s \leq q$, and in either case the con... | 0 | 8,192 | -1 | 8,192 |
a and b are real numbers for which the equation \(x^4 + ax^3 + bx^2 + ax + 1 = 0\) has at least one real solution. Find the least possible value of \(a^2 + b^2\). | 4/5 | 0.1875 | 7,918.125 | 7,038 | 8,121.230769 | |
The square quilt block shown is made from 16 unit squares, four of which have been divided in half to form triangles. Additionally, two squares are completely filled while others are empty. What fraction of the square quilt is shaded? Express your answer as a common fraction. | \frac{1}{4} | 0.6875 | 4,957.5 | 5,449.272727 | 3,875.6 | |
Find the smallest positive integer $N$ such that among the four numbers $N$, $N+1$, $N+2$, and $N+3$, one is divisible by $3^2$, one by $5^2$, one by $7^2$, and one by $11^2$. | 363 | 0 | 8,192 | -1 | 8,192 | |
Let $a, b$ and $c$ be positive real numbers such that $$\begin{aligned} a^{2}+a b+b^{2} & =9 \\ b^{2}+b c+c^{2} & =52 \\ c^{2}+c a+a^{2} & =49 \end{aligned}$$ Compute the value of $\frac{49 b^{2}-33 b c+9 c^{2}}{a^{2}}$. | 52 | Consider a triangle $A B C$ with Fermat point $P$ such that $A P=a, B P=b, C P=c$. Then $$A B^{2}=A P^{2}+B P^{2}-2 A P \cdot B P \cos \left(120^{\circ}\right)$$ by the Law of Cosines, which becomes $$A B^{2}=a^{2}+a b+b^{2}$$ and hence $A B=3$. Similarly, $B C=\sqrt{52}$ and $A C=7$. Furthermore, we have $$\begin{alig... | 0.3125 | 7,513.5625 | 6,477.2 | 7,984.636364 |
Given that \( x_{i}=\frac{i}{101} \), find the value of \( S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \). | 51 | 0.0625 | 8,178.125 | 7,970 | 8,192 | |
Points \( D \) and \( E \) are located on side \( AC \) of triangle \( ABC \). Lines \( BD \) and \( BE \) divide the median \( AM \) of triangle \( ABC \) into three equal segments.
Find the area of triangle \( BDE \) if the area of triangle \( ABC \) is 1. | 0.3 | 0 | 6,817.5 | -1 | 6,817.5 | |
Chicks hatch on the night from Sunday to Monday. For two weeks, a chick sits with its beak open, during the third week it silently grows feathers, and during the fourth week it flies out of the nest. Last week, there were 20 chicks in the nest sitting with their beaks open, and 14 growing feathers, while this week 15 ... | 165 | 0 | 654.1875 | -1 | 654.1875 | |
Given that $({x-1})^4({x+2})^5=a_0+a_1x+a_2x^2+⋯+a_9x^9$, find the value of $a_{2}+a_{4}+a_{6}+a_{8}$. | -24 | 0.125 | 5,690.1875 | 3,919 | 5,943.214286 | |
A game board is constructed by shading three of the regions formed by the diagonals of a regular pentagon. What is the probability that the tip of the spinner will come to rest in a shaded region? Assume the spinner can land in any region with equal likelihood. | \frac{3}{10} | 0 | 5,553.8125 | -1 | 5,553.8125 | |
Rationalize the denominator: $$\frac{1}{\sqrt[3]{3}+\sqrt[3]{27}}$$ | \frac{\sqrt[3]{9}}{12} | 0 | 5,255.625 | -1 | 5,255.625 | |
Find the distance between the foci of the ellipse
\[\frac{x^2}{36} + \frac{y^2}{16} = 8.\] | 8\sqrt{10} | 0.8125 | 4,078.1875 | 4,030 | 4,287 | |
The ratio of the length to the width of a rectangle is $4$ : $3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$? | \frac{12}{25} | 1. **Assign Variables to Dimensions:**
Let the length of the rectangle be $4x$ and the width be $3x$. This assignment is based on the given ratio of length to width, which is $4:3$.
2. **Use the Pythagorean Theorem:**
The diagonal $d$ of the rectangle forms a right triangle with the length and width. According t... | 1 | 1,319.25 | 1,319.25 | -1 |
In triangle $XYZ$, where $\angle X = 90^\circ$, the hypotenuse $YZ = 13$, and $\tan Z = 3\cos Y$. What is the length of side $XY$? | \frac{2\sqrt{338}}{3} | 0 | 5,180.0625 | -1 | 5,180.0625 | |
Given a quadrilateral formed by the two foci and the two endpoints of the conjugate axis of a hyperbola $C$, one of its internal angles is $60^{\circ}$. Determine the eccentricity of the hyperbola $C$. | \frac{\sqrt{6}}{2} | 0 | 6,585.375 | -1 | 6,585.375 | |
Define the operation: $\begin{vmatrix} a_{1} & a_{2} \\ a_{3} & a_{4}\end{vmatrix} =a_{1}a_{4}-a_{2}a_{3}$, and consider the function $f(x)= \begin{vmatrix} \sqrt {3} & \sin \omega x \\ 1 & \cos \omega x\end{vmatrix} (\omega > 0)$. If the graph of $f(x)$ is shifted to the left by $\dfrac {2\pi}{3}$ units, and the resul... | \dfrac{5}{4} | 0.5625 | 7,355.1875 | 6,704.333333 | 8,192 | |
The product underwent a price reduction from 25 yuan to 16 yuan. Calculate the average percentage reduction for each price reduction. | 20\% | 0 | 404.375 | -1 | 404.375 | |
8. Shortening. There is a sequence of 2015 digits. All digits are chosen randomly from the set {0, 9} independently of each other. The following operation is performed on the resulting sequence. If several identical digits go in a row, they are replaced by one such digit. For example, if there was a fragment ...0445666... | 1813.6 | 0 | 7,665.125 | -1 | 7,665.125 | |
Let (a,b,c,d) be an ordered quadruple of not necessarily distinct integers, each one of them in the set {0,1,2,3,4}. Determine the number of such quadruples that make the expression $a \cdot d - b \cdot c + 1$ even. | 136 | 0 | 7,463.8125 | -1 | 7,463.8125 | |
Given that $$cos(α- \frac {π}{6})-sinα= \frac {2 \sqrt {3}}{5}$$, find the value of $$cos(α+ \frac {7π}{6})$$. | - \frac {2 \sqrt {3}}{5} | 0 | 5,395.375 | -1 | 5,395.375 | |
George now has an unfair eight-sided die. The probabilities of rolling each number from 1 to 5 are each $\frac{1}{15}$, the probability of rolling a 6 or a 7 is $\frac{1}{6}$ each, and the probability of rolling an 8 is $\frac{1}{5}$. What is the expected value of the number shown when this die is rolled? Express your ... | 4.7667 | 0.0625 | 7,897.0625 | 8,192 | 7,877.4 | |
Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$ . Find the value of $2^{-(1+\log_23)x}$ | 216 | 0 | 5,004.25 | -1 | 5,004.25 | |
In the figure, the area of square $WXYZ$ is $25 \text{ cm}^2$. The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In $\triangle
ABC$, $AB = AC$, and when $\triangle ABC$ is folded over side $\overline{BC}$, point $A$ coincides with $O$, the center of sq... | \frac{27}{4} | 0 | 8,113.5 | -1 | 8,113.5 | |
A circle touches the extensions of two sides \(AB\) and \(AD\) of square \(ABCD\), and the point of tangency cuts off a segment of length \(2 + \sqrt{5 - \sqrt{5}}\) cm from vertex \(A\). From point \(C\), two tangents are drawn to this circle. Find the side length of the square, given that the angle between the tange... | \frac{\sqrt{\sqrt{5} - 1} \cdot \sqrt[4]{125}}{5} | 0 | 8,192 | -1 | 8,192 | |
$(1)$ Calculate: $(\frac{1}{2})^{-1}+(\sqrt{2})^{2}-4\times |-\frac{1}{2}|$.
$(2)$ Simplify first, then find the value: $(1+\frac{4}{a-1})÷\frac{{a}^{2}+6a+9}{{a}^{2}-a}$, where $a=2$. | \frac{2}{5} | 1 | 2,249.25 | 2,249.25 | -1 | |
Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value? | \frac{113}{648} | 0.25 | 6,811.375 | 6,144.25 | 7,033.75 | |
Ancient astronaut theorist Nutter B. Butter claims that the Caloprians from planet Calop, 30 light years away and at rest with respect to the Earth, wiped out the dinosaurs. The iridium layer in the crust, he claims, indicates spaceships with the fuel necessary to travel at 30% of the speed of light here and back, and ... | 111 | 0.5 | 5,276.3125 | 4,317.875 | 6,234.75 | |
Find distinct digits to replace the letters \(A, B, C, D\) such that the following division in the decimal system holds:
$$
\frac{ABC}{BBBB} = 0,\overline{BCDB \, BCDB \, \ldots}
$$
(in other words, the quotient should be a repeating decimal). | 219 | 0 | 8,192 | -1 | 8,192 | |
In a kindergarten, each child was given three cards, each of which has either "MA" or "NY" written on it. It turned out that 20 children can arrange their cards to spell the word "MAMA", 30 children can arrange their cards to spell the word "NYANYA", and 40 children can arrange their cards to spell the word "MANYA". Ho... | 10 | 0 | 8,055.4375 | -1 | 8,055.4375 | |
A line $l$ in the coordinate plane has the equation $2x - 3y + 30 = 0$. This line is rotated $90^\circ$ counterclockwise about the point $(15,10)$ to obtain line $k'$. Find the $x$-coordinate of the $x$-intercept of $k'$. | \frac{65}{3} | 0.0625 | 7,560.75 | 6,563 | 7,627.266667 | |
In the $2013\cdot Jining$ test, if the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n = n^2 - 4n + 2$, then $|a_1| + |a_2| + \ldots + |a_{10}| = \_\_\_\_\_\_\_$. | 66 | 0.75 | 4,800.4375 | 4,495.583333 | 5,715 | |
Given $\sin\left(\frac{\pi}{3}+\frac{\alpha}{6}\right)=-\frac{3}{5}$, $\cos\left(\frac{\pi}{12}-\frac{\beta}{2}\right)=-\frac{12}{13}$, $-5\pi < \alpha < -2\pi$, $-\frac{11\pi}{6} < \beta < \frac{\pi}{6}$,
Find the value of $\sin \left(\frac{\alpha }{6}+\frac{\beta }{2}+\frac{\pi }{4}\right)$. | \frac{16}{65} | 0.5 | 6,423.25 | 4,762.75 | 8,083.75 | |
In triangle \( A B C \), the base of the height \( C D \) lies on side \( A B \), and the median \( A E \) is equal to 5. The height \( C D \) is equal to 6.
Find the area of triangle \( A B C \), given that the area of triangle \( A D C \) is three times the area of triangle \( B C D \). | 96/7 | 0.8125 | 5,063.0625 | 4,341 | 8,192 | |
Let \( k=-\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i} \). In the complex plane, the vertices of \(\triangle ABC\) correspond to the complex numbers \( z_{1}, z_{2}, z_{3} \) which satisfy the equation
\[ z_{1}+k z_{2}+k^{2}\left(2 z_{3}-z_{1}\right)=0 \text {. } \]
Find the radian measure of the smallest interior angle ... | \frac{\pi}{6} | 0 | 8,192 | -1 | 8,192 | |
What is the sum of all integer values $n$ for which $\binom{26}{13}+\binom{26}{n}=\binom{27}{14}$? | 26 | 0.8125 | 4,277.125 | 3,373.692308 | 8,192 | |
Given the function $f(x)=3\sin ( \frac {1}{2}x+ \frac {π}{4})-1$, where $x\in R$, find:
1) The minimum value of the function $f(x)$ and the set of values of the independent variable $x$ at this time;
2) How is the graph of the function $y=\sin x$ transformed to obtain the graph of the function $f(x)=3\sin ( \frac {1}{2... | (4) | 0 | 4,491.8125 | -1 | 4,491.8125 | |
A random variable \(X\) is given by the probability density function \(f(x) = \frac{1}{2} \sin x\) within the interval \((0, \pi)\); outside this interval, \(f(x) = 0\). Find the variance of the function \(Y = \varphi(X) = X^2\) using the probability density function \(g(y)\). | \frac{\pi^4 - 16\pi^2 + 80}{4} | 0.1875 | 6,121.6875 | 4,972 | 6,387 | |
$1,000,000,000,000-777,777,777,777=$ | $222,222,222,223$ | To solve the problem $1,000,000,000,000 - 777,777,777,777$, we can perform the subtraction directly:
1. **Align the numbers for subtraction:**
\[
\begin{array}{r}
1,000,000,000,000 \\
-777,777,777,777 \\
\end{array}
\]
2. **Subtract each corresponding set of digits starting from the rightmost digit:... | 0 | 4,641.4375 | -1 | 4,641.4375 |
I bought a lottery ticket, the sum of the digits of its five-digit number turned out to be equal to the age of my neighbor. Determine the number of the ticket, given that my neighbor easily solved this problem. | 99999 | 0 | 6,725.25 | -1 | 6,725.25 | |
A line is expressed in the form
\[\begin{pmatrix} -2 \\ -5 \end{pmatrix} \cdot \left( \begin{pmatrix} x \\ y \end{pmatrix} - \begin{pmatrix} 1 \\ 11 \end{pmatrix} \right) = 0.\]The equation of the line can be expressed in the form $y = mx + b.$ Enter the ordered pair $(m,b).$ | \left( -\frac{2}{5}, \frac{57}{5} \right) | 1 | 2,662.8125 | 2,662.8125 | -1 | |
A student has five different physics questions numbered 1, 2, 3, 4, and 5, and four different chemistry questions numbered 6, 7, 8, and 9. The student randomly selects two questions, each with an equal probability of being chosen. Let the event `(x, y)` represent "the two questions with numbers x and y are chosen, wher... | \frac{5}{12} | 0.4375 | 5,399.1875 | 4,508.285714 | 6,092.111111 | |
How many zeros are there in the last digits of the following number $P = 11\times12\times ...\times 88\times 89$ ? | 18 | 0.875 | 4,167.6875 | 3,945.5 | 5,723 | |
In $\triangle XYZ$, we have $\angle X = 90^\circ$ and $\tan Z = 3$. What is $\cos Z$? | \frac{\sqrt{10}}{10} | 0 | 1,653.5 | -1 | 1,653.5 | |
Given that $i^2=-1$, for how many integers $n$ is $(n+i)^4$ an integer? | 3 | 1. **Expand the expression**: We start by expanding $(n+i)^4$ using the binomial theorem:
\[
(n+i)^4 = \binom{4}{0}n^4i^0 + \binom{4}{1}n^3i^1 + \binom{4}{2}n^2i^2 + \binom{4}{3}n^1i^3 + \binom{4}{4}n^0i^4.
\]
Simplifying each term, we get:
\[
(n+i)^4 = n^4 + 4n^3i - 6n^2 - 4ni + i^4.
\]
Since $... | 1 | 2,671.75 | 2,671.75 | -1 |
Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is ... | 8748 | 0.5 | 6,515.5 | 4,839 | 8,192 | |
Compute the product of all positive integers $b \geq 2$ for which the base $b$ number $111111_{b}$ has exactly $b$ distinct prime divisors. | 24 | Notice that this value, in base $b$, is $$\frac{b^{6}-1}{b-1}=(b+1)\left(b^{2}-b+1\right)\left(b^{2}+b+1\right)$$ This means that, if $b$ satisfies the problem condition, $(b+1)\left(b^{2}-b+1\right)\left(b^{2}+b+1\right)>p_{1} \ldots p_{b}$, where $p_{i}$ is the $i$ th smallest prime. We claim that, if $b \geq 7$, the... | 0.0625 | 8,171.25 | 7,860 | 8,192 |
Given the quadratic equation \( ax^2 + bx + c \) and the table of values \( 6300, 6481, 6664, 6851, 7040, 7231, 7424, 7619, 7816 \) for a sequence of equally spaced increasing values of \( x \), determine the function value that does not belong to the table. | 6851 | 0 | 8,192 | -1 | 8,192 | |
In the expression \(5 * 4 * 3 * 2 * 1 = 0\), replace the asterisks with arithmetic operators \(+, -, \times, \div\), using each operator exactly once, so that the equality holds true (note: \(2 + 2 \times 2 = 6\)). | 5 - 4 \times 3 : 2 + 1 | 0 | 7,328 | -1 | 7,328 | |
Bob's Rice ID number has six digits, each a number from 1 to 9, and any digit can be used any number of times. The ID number satisfies the following property: the first two digits is a number divisible by 2, the first three digits is a number divisible by 3, etc. so that the ID number itself is divisible by 6. One ID n... | 324 | We will count the number of possibilities for each digit in Bob's ID number, then multiply them to find the total number of possibilities for Bob's ID number. There are 3 possibilities for the first digit given any last 5 digits, because the entire number must be divisible by 3, so the sum of the digits must be divisib... | 0 | 7,847.5 | -1 | 7,847.5 |
Solve for $p$: $\frac 56 = \frac n{72} = \frac {m+n}{84}= \frac {p - m}{120}$. | 110 | 1 | 1,688.3125 | 1,688.3125 | -1 | |
Inside a non-isosceles acute triangle \(ABC\) with \(\angle ABC = 60^\circ\), point \(T\) is marked such that \(\angle ATB = \angle BTC = \angle ATC = 120^\circ\). The medians of the triangle intersect at point \(M\). The line \(TM\) intersects the circumcircle of triangle \(ATC\) at point \(K\) for the second time. Fi... | 1/2 | 0 | 8,192 | -1 | 8,192 | |
Points \( M \) and \( N \) are located on side \( BC \) of triangle \( ABC \), and point \( K \) is on side \( AC \), with \( BM : MN : NC = 1 : 1 : 2 \) and \( CK : AK = 1 : 4 \). Given that the area of triangle \( ABC \) is 1, find the area of quadrilateral \( AMNK \). | 13/20 | 0.1875 | 7,250.5625 | 5,796.333333 | 7,586.153846 | |
For arbitrary real numbers \(a\) and \(b\) (\(a \neq 0\)), find the minimum value of the expression \(\frac{1}{a^{2}} + 2a^{2} + 3b^{2} + 4ab\). | \sqrt{\frac{8}{3}} | 0 | 7,122.125 | -1 | 7,122.125 | |
How many of the first 1000 positive integers can be written as the sum of finitely many distinct numbers from the sequence $3^{0}, 3^{1}, 3^{2}, \ldots$? | 105 | We want to find which integers have only 0 's and 1 's in their base 3 representation. Note that $1000_{10}=1101001_{3}$. We can construct a bijection from all such numbers to the binary strings, by mapping $x_{3} \leftrightarrow x_{2}$. Since $1101001_{2}=105_{10}$, we conclude that the answer is 105. | 0 | 8,192 | -1 | 8,192 |
Given two points $A(-2,0)$ and $B(0,2)$, and point $C$ is any point on the circle $x^{2}+y^{2}-2x=0$, find the minimum area of $\triangle ABC$. | 3 - \sqrt{2} | 0.75 | 6,848.9375 | 6,401.25 | 8,192 | |
What is the ratio of the volume of a cube with edge length six inches to the volume of a cube with edge length one foot? Express your answer as a common fraction. | \frac{1}{8} | 1 | 1,684 | 1,684 | -1 | |
Given that \( a, b, \) and \( c \) are all positive integers, and the parabola \( y = ax^2 + bx + c \) intersects the x-axis at two distinct points \( A \) and \( B \). If both \( A \) and \( B \) are less than 1 unit away from the origin, what is the minimum value of \( a + b + c \)? | 11 | 0 | 8,192 | -1 | 8,192 |
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