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 |
|---|---|---|---|---|---|---|
The maximum value of the function $y=4^x+2^{x+1}+5$, where $x\in[1,2]$, is to be found. | 29 | 1 | 2,449.75 | 2,449.75 | -1 | |
Which of the following numbers is closest to 1: $ rac{11}{10}$, $ rac{111}{100}$, 1.101, $ rac{1111}{1000}$, 1.011? | 1.011 | When we convert each of the possible answers to a decimal, we obtain 1.1, 1.11, 1.101, 1.111, and 1.011. Since the last of these is the only one greater than 1 and less than 1.1, it is closest to 1. | 0.9375 | 637 | 637.6 | 628 |
If the square roots of a positive number are $a+2$ and $2a-11$, find the positive number. | 225 | 0.125 | 3,044.5 | 2,024.5 | 3,190.214286 | |
If $2 x^{2}=9 x-4$ and $x
eq 4$, what is the value of $2 x$? | 1 | Since $2 x^{2}=9 x-4$, then $2 x^{2}-9 x+4=0$. Factoring, we obtain $(2 x-1)(x-4)=0$. Thus, $2 x=1$ or $x=4$. Since $x
eq 4$, then $2 x=1$. | 1 | 629.9375 | 629.9375 | -1 |
In the country of Francisca, there are 2010 cities, some of which are connected by roads. Between any two cities, there is a unique path which runs along the roads and which does not pass through any city twice. What is the maximum possible number of cities in Francisca which have at least 3 roads running out of them? | 1004 | The restrictions on how roads connect cities directly imply that the graph of the cities of Francisca with the roads as edges is a tree. Therefore the sum of the degrees of all the vertices is $2009 \cdot 2=4018$. Suppose that $b$ vertices have degree \geq 3. The other $2010-b$ vertices must have a degree of at least 1... | 0.375 | 7,063.4375 | 5,182.5 | 8,192 |
As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\omega$ and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be bisector p... | 60^\circ |
To approach this problem, we will analyze the geometric properties and symmetrical nature of the heart shape and the properties of the cyclic quadrilateral \(XYZT\).
1. **Understanding the Geometry of the Heart Shape:**
- The heart shape is constructed from three semicircles: with diameters \(AB\), \(BC\), and \(A... | 0 | 8,192 | -1 | 8,192 |
Find the largest positive integer $n$ such that $\sigma(n) = 28$ , where $\sigma(n)$ is the sum of the divisors of $n$ , including $n$ . | 12 | 0.0625 | 8,152.875 | 7,566 | 8,192 | |
Find $\begin{pmatrix} 3 \\ -7 \end{pmatrix} + \begin{pmatrix} -6 \\ 11 \end{pmatrix}.$ | \begin{pmatrix} -3 \\ 4 \end{pmatrix} | 1 | 1,701.1875 | 1,701.1875 | -1 | |
Calculate the result of the expression:
\[ 2013 \times \frac{5.7 \times 4.2 + \frac{21}{5} \times 4.3}{\frac{14}{73} \times 15 + \frac{5}{73} \times 177 + 656} \] | 126 | 0.8125 | 4,619.625 | 4,079.076923 | 6,962 | |
In the middle of the school year, $40\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how many students does the Poolesville magnet ha... | 170 | 0.25 | 2,072.1875 | 3,251 | 1,679.25 | |
In a bag, there are 70 balls which differ only in color: 20 red, 20 blue, 20 yellow, and the rest are black and white.
What is the minimum number of balls that must be drawn from the bag, without seeing them, to ensure that there are at least 10 balls of one color among them? | 38 | 0.375 | 5,653.1875 | 4,709.5 | 6,219.4 | |
Given that $\sin(\theta + 3\pi) = -\frac{2}{3}$, find the value of $\frac{\tan(-5\pi - \theta) \cdot \cos(\theta - 2\pi) \cdot \sin(-3\pi - \theta)}{\tan(\frac{7\pi}{2} + \theta) \cdot \sin(-4\pi + \theta) \cdot \cot(-\theta - \frac{\pi}{2})} + 2 \tan(6\pi - \theta) \cdot \cos(-\pi + \theta)$. | \frac{2}{3} | 0.3125 | 5,352.125 | 4,236.8 | 5,859.090909 | |
Given the function $f(x)= \frac {1}{2}x^{2}-2\ln x+a(a\in\mathbb{R})$, $g(x)=-x^{2}+3x-4$.
$(1)$ Find the intervals of monotonicity for $f(x)$;
$(2)$ Let $a=0$, the line $x=t$ intersects the graphs of $f(x)$ and $g(x)$ at points $M$ and $N$ respectively. When $|MN|$ reaches its minimum value, find the value of $t$;... | \frac {3+ \sqrt {33}}{6} | 0 | 8,192 | -1 | 8,192 | |
Three merchants - Sosipatra Titovna, Olympiada Karpovna, and Poliksena Uvarovna - sat down to drink tea. Olympiada Karpovna and Sosipatra Titovna together drank 11 cups, Poliksena Uvarovna and Olympiada Karpovna drank 15 cups, and Sosipatra Titovna and Poliksena Uvarovna drank 14 cups. How many cups of tea did all thre... | 20 | 0.9375 | 2,238.125 | 1,841.2 | 8,192 | |
A dot is marked at each vertex of a triangle $A B C$. Then, 2,3 , and 7 more dots are marked on the sides $A B, B C$, and $C A$, respectively. How many triangles have their vertices at these dots? | 357 | Altogether there are $3+2+3+7=15$ dots, and thus $\binom{15}{3}=455$ combinations of 3 dots. Of these combinations, $\binom{2+2}{3}+\binom{2+3}{3}+\binom{2+7}{3}=4+10+84=98$ do not give triangles because they are collinear (the rest do give triangles). Thus $455-98=357$ different triangles can be formed. | 0.25 | 6,178.8125 | 4,588 | 6,709.083333 |
Rectangle $EFGH$ has area $4032$. An ellipse with area $4032\pi$ passes through points $E$ and $G$ and has foci at $F$ and $H$. Determine the perimeter of the rectangle $EFGH$. | 8\sqrt{2016} | 0 | 6,607.375 | -1 | 6,607.375 | |
Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$. | 441 | Let $A,B,C$ be the weights of the respective vertices. We see that the weights of the feet of the cevians are $A+B,B+C,C+A$. By mass points, we have that: \[\dfrac{a}{3}=\dfrac{B+C}{A}\] \[\dfrac{b}{3}=\dfrac{C+A}{B}\] \[\dfrac{c}{3}=\dfrac{A+B}{C}\]
If we add the equations together, we get $\frac{a+b+c}{3}=\frac{A^2B... | 0 | 7,478.625 | -1 | 7,478.625 |
Into how many regions do the x-axis and the graphs of \( y = 2 - x^2 \) and \( y = x^2 - 1 \) split the plane? | 10 | 0 | 8,124.0625 | -1 | 8,124.0625 | |
Let $a_{1}=3$, and for $n>1$, let $a_{n}$ be the largest real number such that $$4\left(a_{n-1}^{2}+a_{n}^{2}\right)=10 a_{n-1} a_{n}-9$$ What is the largest positive integer less than $a_{8}$ ? | 335 | Let $t_{n}$ be the larger real such that $a_{n}=t_{n}+\frac{1}{t_{n}}$. Then $t_{1}=\frac{3+\sqrt{5}}{2}$. We claim that $t_{n}=2 t_{n-1}$. Writing the recurrence as a quadratic polynomial in $a_{n}$, we have: $$4 a_{n}^{2}-10 a_{n-1} a_{n}+4 a_{n-1}^{2}+9=0$$ Using the quadratic formula, we see that $a_{n}=\frac{5}{4}... | 0.3125 | 7,829.1875 | 7,031 | 8,192 |
Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=18,$ find the smallest possible value of $|w^3+z^3|.$ | 50 | 0.4375 | 7,023.0625 | 6,013.285714 | 7,808.444444 | |
Three distinct vertices of a regular 2020-gon are chosen uniformly at random. The probability that the triangle they form is isosceles can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a+b$. | 773 | The number of isosceles triangles that share vertices with the 2020-gon is $2020 \cdot 1009$, since there are 2020 ways to choose the apex of the triangle and then 1009 ways to choose the other two vertices. (Since 2020 is not divisible by 3, there are no equilateral triangles, so no triangle is overcounted.) Therefore... | 0.125 | 7,899 | 5,848 | 8,192 |
Given that the random variable $\xi$ follows a normal distribution $N(4, 6^2)$, and $P(\xi \leq 5) = 0.89$, find the probability $P(\xi \leq 3)$. | 0.11 | 0.6875 | 5,730.875 | 5,025.909091 | 7,281.8 | |
Find the sum of all prime numbers whose representation in base 14 has the form $101010...101$ (alternating ones and zeros). | 197 | 0.0625 | 8,090.875 | 6,574 | 8,192 | |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to $\angle A$, $\angle B$, $\angle C$ respectively. If $\cos 2B + \cos B + \cos (A-C) = 1$ and $b = \sqrt{7}$, find the minimum value of $a^2 + c^2$. | 14 | 0.25 | 7,765.625 | 6,807 | 8,085.166667 | |
Two people, A and B, are working together to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they have completed half of the document, A's typing speed triples, while B takes a 5-minute break and then continues typing at his original speed. By the time the docu... | 18000 | 0.125 | 7,805.6875 | 5,101.5 | 8,192 | |
Find the largest positive integer $m$ such that an $m \times m$ square can be exactly divided into 7 rectangles with pairwise disjoint interiors, and the lengths of the 14 sides of these 7 rectangles are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14$. | 22 | 0 | 8,146.4375 | -1 | 8,146.4375 | |
A square has a side length of $4$. Within this square, two equilateral triangles are placed such that one has its base along the bottom side of the square, and the other is rotated such that its vertex touches the midpoint of the top side of the square, and its base is parallel to the bottom of the square. Find the are... | 4\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Given an isosceles trapezoid \(ABCD\), where \(AD \parallel BC\), \(BC = 2AD = 4\), \(\angle ABC = 60^\circ\), and \(\overrightarrow{CE} = \frac{1}{3} \overrightarrow{CD}\), find \(\overrightarrow{CA} \cdot \overrightarrow{BE}\). | -10 | 0.625 | 5,489.75 | 4,728 | 6,759.333333 | |
Determine the residue of $-998\pmod{28}$. Your answer should be an integer in the range $0,1,2,\ldots,25,26,27$. | 10 | 1 | 3,005.1875 | 3,005.1875 | -1 | |
A right isosceles triangle is inscribed in a triangle with a base of 30 and a height of 10 such that its hypotenuse is parallel to the base of the given triangle, and the vertex of the right angle lies on this base. Find the hypotenuse. | 12 | 0.25 | 7,435.6875 | 5,166.75 | 8,192 | |
Calculate the values of:
(1) $8^{\frac{2}{3}} - (0.5)^{-3} + \left(\frac{1}{\sqrt{3}}\right)^{-2} \times \left(\frac{81}{16}\right)^{-\frac{1}{4}}$;
(2) $\log 5 \cdot \log 8000 + (\log 2^{\sqrt{3}})^2 + e^{\ln 1} + \ln(e \sqrt{e})$. | \frac{11}{2} | 0.75 | 5,916.5625 | 5,158.083333 | 8,192 | |
Given that $x$ is a multiple of $12600$, what is the greatest common divisor of $g(x) = (5x + 7)(11x + 3)(17x + 8)(4x + 5)$ and $x$? | 840 | 0.5 | 6,634.625 | 5,077.25 | 8,192 | |
The numbers from 1 to 150, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is not a perfect power (integers that can be expressed as $x^{y}$ where $x$ is an integer and $y$ is an integer greater than 1. For example, $2^{4}=16$ is a perfect power, while $2\times3... | \frac{133}{150} | 0.5625 | 6,576.3125 | 6,473.888889 | 6,708 | |
Given that the perimeter of triangle \( \triangle ABC \) is 20, the radius of the inscribed circle is \( \sqrt{3} \), and \( BC = 7 \). Find the value of \( \tan A \). | \sqrt{3} | 0.9375 | 4,335.625 | 4,078.533333 | 8,192 | |
Determine the share of the Japanese yen in the currency structure of the National Wealth Fund (NWF) as of 01.12.2022 using one of the following methods:
First method:
a) Find the total amount of NWF funds placed in Japanese yen as of 01.12.2022:
\[ J P Y_{22} = 1388.01 - 41.89 - 2.77 - 309.72 - 554.91 - 0.24 = 478.4... | -12.6 | 0.3125 | 4,122.9375 | 4,165.2 | 4,103.727273 | |
If $x, y$ and $2x + \frac{y}{2}$ are not zero, then
$\left( 2x + \frac{y}{2} \right)^{-1} \left[(2x)^{-1} + \left( \frac{y}{2} \right)^{-1} \right]$ equals | \frac{1}{xy} | 1. **Rewrite the expression**:
Given the expression $\left( 2x + \frac{y}{2} \right)^{-1} \left[(2x)^{-1} + \left( \frac{y}{2} \right)^{-1} \right]$, we start by simplifying each component:
\[
\left( 2x + \frac{y}{2} \right)^{-1} = \left( \frac{4x+y}{2} \right)^{-1} = \frac{2}{4x+y}
\]
and
\[
(2x)^... | 1 | 2,440.75 | 2,440.75 | -1 |
From the numbers 0, 1, 2, 3, and 4, select three different digits to form a three-digit number and calculate the total number of such numbers that are odd. | 18 | 0.5625 | 6,470.1875 | 5,284 | 7,995.285714 | |
Given $\overrightarrow{a}=(-3,4)$, $\overrightarrow{b}=(5,2)$, find $|\overrightarrow{a}|$, $|\overrightarrow{b}|$, and $\overrightarrow{a}\cdot \overrightarrow{b}$. | -7 | 1 | 1,529.6875 | 1,529.6875 | -1 | |
Let $N = 34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$? | 1 : 14 | 1. **Prime Factorization of \(N\):**
Given \(N = 34 \cdot 34 \cdot 63 \cdot 270\), we start by factorizing each number:
- \(34 = 2 \cdot 17\)
- \(63 = 3^2 \cdot 7\)
- \(270 = 2 \cdot 3^3 \cdot 5\)
Therefore, \(N = (2 \cdot 17)^2 \cdot (3^2 \cdot 7) \cdot (2 \cdot 3^3 \cdot 5) = 2^3 \cdot 3^5 \cdot 5 \cd... | 0 | 6,020 | -1 | 6,020 |
Points $A,B,C$ and $D$ lie on a line in that order, with $AB = CD$ and $BC = 16$. Point $E$ is not on the line, and $BE = CE = 13$. The perimeter of $\triangle AED$ is three times the perimeter of $\triangle BEC$. Find $AB$.
A) $\frac{32}{3}$
B) $\frac{34}{3}$
C) $\frac{36}{3}$
D) $\frac{38}{3}$ | \frac{34}{3} | 0 | 8,192 | -1 | 8,192 | |
Given that the parabola $y=x^2-5x+2$ is symmetric about the point $(3,2)$ with $y=ax^2+bx+c$, find the value of $3a+3c+b$. | -8 | 0.9375 | 3,395.625 | 3,075.866667 | 8,192 | |
In a right triangle \(ABC\) with a right angle at \(B\) and \(\angle A = 30^\circ\), a height \(BD\) is drawn. Then, in triangle \(BDC\), a median \(DE\) is drawn, and in triangle \(DEC\), an angle bisector \(EF\) is drawn. Find the ratio \( \frac{FC}{AC} \). | 1/8 | 0.875 | 5,999.875 | 5,686.714286 | 8,192 | |
If the sum of all the angles except one of a convex polygon is $2190^{\circ}$, then the number of sides of the polygon must be | 15 | 1. **Identify the formula for the sum of interior angles of a polygon**: The sum of the interior angles of a polygon with $n$ sides is given by the formula:
\[
180^\circ \times (n-2)
\]
This formula arises from the fact that a polygon can be divided into $(n-2)$ triangles, and the sum of angles in each tria... | 1 | 2,216.75 | 2,216.75 | -1 |
Calculate the value of the expression given by: $$2 + \cfrac{3}{4 + \cfrac{5}{6}}.$$ | \frac{76}{29} | 0.75 | 1,436.625 | 1,660.5 | 765 | |
Side $AB$ of triangle $ABC$ was divided into $n$ equal parts (dividing points $B_0 = A, B_1, B_2, ..., B_n = B$ ), and side $AC$ of this triangle was divided into $(n + 1)$ equal parts (dividing points $C_0 = A, C_1, C_2, ..., C_{n+1} = C$ ). Colored are the triangles $C_iB_iC_{i+1}$ (where $i = 1,2, ...... | \frac{1}{2} | 0.1875 | 7,534.0625 | 5,138.333333 | 8,086.923077 | |
Simplify the product \[\frac{8}{4}\cdot\frac{12}{8}\cdot\frac{16}{12} \dotsm \frac{4n+4}{4n} \dotsm \frac{2008}{2004}.\] | 502 | 1 | 1,787.125 | 1,787.125 | -1 | |
Twelve chess players played a round-robin tournament. Each player then wrote 12 lists. In the first list, only the player himself was included, and in the $(k+1)$-th list, the players included those who were in the $k$-th list as well as those whom they defeated. It turned out that each player's 12th list differed from... | 54 | 0 | 7,868.625 | -1 | 7,868.625 | |
The quadratic \( x^2 + 1800x + 1800 \) can be written in the form \( (x+b)^2 + c \), where \( b \) and \( c \) are constants. What is \( \frac{c}{b} \)? | -898 | 1 | 2,307.75 | 2,307.75 | -1 | |
Find the number of triples $(x,y,z)$ of real numbers that satisfy
\begin{align*}
x &= 2018 - 2019 \operatorname{sign}(y + z), \\
y &= 2018 - 2019 \operatorname{sign}(x + z), \\
z &= 2018 - 2019 \operatorname{sign}(x + y).
\end{align*}Note: For a real number $a,$
\[\operatorname{sign} (a) = \left\{
\begin{array}{cl}
1 &... | 3 | 0.0625 | 7,996.5625 | 8,192 | 7,983.533333 | |
A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$ . Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger? | 672 | 0.375 | 6,804.5625 | 4,842.333333 | 7,981.9 | |
At Beaumont High School, there are 20 players on the basketball team. All 20 players are taking at least one of biology or chemistry. (Biology and chemistry are two different science courses at the school.) If there are 8 players taking biology and 4 players are taking both sciences, how many players are taking chem... | 16 | 1 | 1,450.0625 | 1,450.0625 | -1 | |
What is half of the absolute value of the difference of the squares of 21 and 15 added to the absolute value of the difference of their cubes? | 3051 | 0.75 | 2,935 | 2,478.833333 | 4,303.5 | |
Arrange 6 volunteers for 3 different tasks, each task requires 2 people. Due to the work requirements, A and B must work on the same task, and C and D cannot work on the same task. How many different arrangements are there? | 12 | 0.1875 | 7,387.375 | 6,447.666667 | 7,604.230769 | |
A pentagon is inscribed around a circle, with the lengths of its sides being whole numbers, and the lengths of the first and third sides equal to 1. Into what segments does the point of tangency divide the second side? | \frac{1}{2} | 0.125 | 8,016.3125 | 6,786.5 | 8,192 | |
Solve
\[(x^3 + 3x^2 \sqrt{2} + 6x + 2 \sqrt{2}) + (x + \sqrt{2}) = 0.\]Enter all the solutions, separated by commas. | -\sqrt{2}, -\sqrt{2} + i, -\sqrt{2} - i | 0 | 5,210.6875 | -1 | 5,210.6875 | |
Determine the number of all positive integers which cannot be written in the form $80k + 3m$ , where $k,m \in N = \{0,1,2,...,\}$ | 79 | 0.6875 | 5,869.4375 | 4,813.727273 | 8,192 | |
There are several balls of the same shape and size in a bag, including $a+1$ red balls, $a$ yellow balls, and $1$ blue ball. Now, randomly draw a ball from the bag, with the rule that drawing a red ball earns $1$ point, a yellow ball earns $2$ points, and a blue ball earns $3$ points. If the expected value of the score... | \frac{3}{10} | 0.6875 | 5,174.75 | 3,850.272727 | 8,088.6 | |
The cube shown is divided into 64 small cubes. Exactly one of the cubes is grey, as shown in the diagram. Two cubes are said to be 'neighbours' if they have a common face.
On the first day, the white neighbours of the grey cube are changed to grey. On the second day, the white neighbours of all the grey cubes are chan... | 17 | 0 | 8,035.875 | -1 | 8,035.875 | |
\( A, B, C \) are positive integers. It is known that \( A \) has 7 divisors, \( B \) has 6 divisors, \( C \) has 3 divisors, \( A \times B \) has 24 divisors, and \( B \times C \) has 10 divisors. What is the minimum value of \( A + B + C \)? | 91 | 0.0625 | 8,147 | 7,472 | 8,192 | |
Given that $\binom{24}{3}=2024$, $\binom{24}{4}=10626$, and $\binom{24}{5}=42504$, find $\binom{26}{6}$. | 230230 | 0.4375 | 7,366.3125 | 6,304.714286 | 8,192 | |
In quadrilateral $ABCD$, sides $\overline{AB}$ and $\overline{BC}$ both have length 10, sides $\overline{CD}$ and $\overline{DA}$ both have length 17, and the measure of angle $ADC$ is $60^\circ$. What is the length of diagonal $\overline{AC}$? [asy]
draw((0,0)--(17,0));
draw(rotate(301, (17,0))*(0,0)--(17,0));
picture... | 17 | 0.8125 | 5,925.8125 | 5,402.846154 | 8,192 | |
Compute $i^{600} + i^{599} + \cdots + i + 1$, where $i^2=-1$. | 1 | 0.625 | 6,118.5 | 4,874.4 | 8,192 | |
Given that $\frac{\cos \alpha + \sin \alpha}{\cos \alpha - \sin \alpha} = 2$, find the value of $\frac{1 + \sin 4\alpha - \cos 4\alpha}{1 + \sin 4\alpha + \cos 4\alpha}$. | \frac{3}{4} | 0.875 | 5,139.75 | 4,703.714286 | 8,192 | |
In the quadrilateral $MARE$ inscribed in a unit circle $\omega,$ $AM$ is a diameter of $\omega,$ and $E$ lies on the angle bisector of $\angle RAM.$ Given that triangles $RAM$ and $REM$ have the same area, find the area of quadrilateral $MARE.$ | \frac{8\sqrt{2}}{9} | 0 | 7,954.125 | -1 | 7,954.125 | |
Jenny wants to create all the six-letter words where the first two letters are the same as the last two letters. How many combinations of letters satisfy this property? | 17576 | 0.25 | 6,591.8125 | 5,416.25 | 6,983.666667 | |
The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$. Given that $S_{10}=0$ and $S_{15}=25$, find the minimum value of $nS_n$. | -49 | 0.5625 | 5,605.8125 | 5,481.111111 | 5,766.142857 | |
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^{2}}{16}{+}\frac{y^{2}}{b^{2}}{=}1$, and the line $l$ passing through $F_1$ intersects the ellipse at points $A$ and $B$. If the maximum value of $|AF_2|+|BF_2|$ is $10$, find the eccentricity of the ellipse. | \frac{1}{2} | 0.4375 | 7,161.875 | 5,837.428571 | 8,192 | |
What are the last three digits of \(2003^N\), where \(N = 2002^{2001}\)? | 241 | 0.625 | 6,571.375 | 6,054.3 | 7,433.166667 | |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a King and the second card is a $\heartsuit$? | \dfrac{1}{52} | 0.5625 | 6,565.1875 | 5,299.888889 | 8,192 | |
An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct? | \frac{56}{225} | To solve this problem, we need to calculate the total number of integers between $1000$ and $9999$ and then find how many of these integers meet the given conditions (odd and all digits distinct).
1. **Total number of integers between $1000$ and $9999$:**
- These integers are all the four-digit integers.
- The s... | 0.25 | 5,784.4375 | 5,233.75 | 5,968 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and they satisfy the equation $$\frac {2c-b}{a} = \frac {\cos{B}}{\cos{A}}$$. If $a = 2\sqrt {5}$, find the maximum value of $b + c$. | 4\sqrt{5} | 0.625 | 6,508.0625 | 5,497.7 | 8,192 | |
Let $ABCDEF$ be a regular hexagon, and let $G$ , $H$ , $I$ , $J$ , $K$ , and $L$ be the midpoints of sides $AB$ , $BC$ , $CD$ , $DE$ , $EF$ , and $FA$ , respectively. The intersection of lines $\overline{AH}$ , $\overline{BI}$ , $\overline{CJ}$ , $\overline{DK}$ , $\overline{EL}$ , and $\overline{F... | 4/7 | 0.0625 | 8,073.4375 | 6,295 | 8,192 | |
Given that the positive numbers $x$ and $y$ satisfy the equation $$3x+y+ \frac {1}{x}+ \frac {2}{y}= \frac {13}{2}$$, find the minimum value of $$x- \frac {1}{y}$$. | - \frac {1}{2} | 0 | 8,192 | -1 | 8,192 | |
Consider a large square divided into a grid of \(5 \times 5\) smaller squares, each with side length \(1\) unit. A shaded region within the large square is formed by connecting the centers of four smaller squares, creating a smaller square inside. Calculate the ratio of the area of the shaded smaller square to the area... | \frac{2}{25} | 0 | 7,858.5 | -1 | 7,858.5 | |
If the complex number \( z \) satisfies \( |z| = 2 \), then the maximum value of \( \frac{\left|z^{2}-z+1\right|}{|2z-1-\sqrt{3}i|} \) is _______. | \frac{3}{2} | 0.1875 | 8,032.75 | 7,342.666667 | 8,192 | |
Xiaoming started adding the page numbers of a book from page 1, sequentially until the end, and obtained a sum of 4979. Later, he discovered that one sheet (which includes two consecutive pages) was missing from the book. How many pages did the book originally have? | 100 | 0.9375 | 3,287.75 | 2,960.8 | 8,192 | |
Find the integer $n,$ $-180 \le n \le 180,$ such that $\cos n^\circ = \cos 430^\circ.$ | -70 | 0.125 | 5,652.0625 | 4,930 | 5,755.214286 | |
Given that x > 0, y > 0, and x + 2y = 4, find the minimum value of $$\frac {(x+1)(2y+1)}{xy}$$. | \frac {9}{2} | 0.75 | 5,870.75 | 5,481.583333 | 7,038.25 | |
Given the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), with the circle centered at the right focus $F$ of $C$ ($c$, $0$) and with radius $a$ intersects one of the asymptotes of $C$ at points $A$ and $B$. If $|AB| = \frac{2}{3}c$, determine the eccentricity of the hyperbola $C$. | \frac{3\sqrt{5}}{5} | 0 | 5,736.5 | -1 | 5,736.5 | |
Given the function $f(x)=\frac{1}{2}x^{2}-a\ln x+b$ where $a\in R$.
(I) If the equation of the tangent line to the curve $y=f(x)$ at $x=1$ is $3x-y-3=0$, find the values of the real numbers $a$ and $b$.
(II) If $x=1$ is the extreme point of the function $f(x)$, find the value of the real number $a$.
(III) If $-2\leqsla... | 12 | 0.3125 | 7,302.125 | 5,656 | 8,050.363636 | |
Define the determinant operation $\begin{vmatrix} a_{1} & a_{2} \\ b_{1} & b_{2}\end{vmatrix} =a_{1}b_{2}-a_{2}b_{1}$, and consider the function $f(x)= \begin{vmatrix} \sqrt {3} & \sin x \\ 1 & \cos x\end{vmatrix}$. If the graph of this function is translated to the left by $t(t > 0)$ units, and the resulting graph cor... | \dfrac{5\pi}{6} | 0.6875 | 6,788.75 | 6,320 | 7,820 | |
In $\triangle ABC$, given $A(1,4)$, $B(4,1)$, $C(0,-4)$, find the minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB} + \overrightarrow{PB} \cdot \overrightarrow{PC} + \overrightarrow{PC} \cdot \overrightarrow{PA}$. | - \dfrac {62}{3} | 0.6875 | 5,713.3125 | 5,232.818182 | 6,770.4 | |
In 2023, a special international mathematical conference is held. Let $A$, $B$, and $C$ be distinct positive integers such that the product $A \cdot B \cdot C = 2023$. What is the largest possible value of the sum $A+B+C$? | 297 | 1 | 5,435.8125 | 5,435.8125 | -1 | |
A point has rectangular coordinates $(-5,-7,4)$ and spherical coordinates $(\rho, \theta, \phi).$ Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, -\phi).$ | (5,7,4) | 0.75 | 5,680.5625 | 4,843.416667 | 8,192 | |
The expression $\dfrac{\sqrt[3]{5}}{\sqrt[5]{5}}$ equals 5 raised to what power? | 2/15 | 0.9375 | 2,322.5 | 1,931.2 | 8,192 | |
The shortest distance from a point on the curve $y=\ln x$ to the line $y=x+2$ is what value? | \frac{3\sqrt{2}}{2} | 0 | 5,395.25 | -1 | 5,395.25 | |
The diagram shows a shape made from ten squares of side-length \(1 \mathrm{~cm}\), joined edge to edge. What is the length of its perimeter, in centimetres?
A) 14
B) 18
C) 30
D) 32
E) 40 | 18 | 0 | 7,606 | -1 | 7,606 | |
Given $f(x) = \frac {\log_{2}x-1}{2\log_{2}x+1}$ (where $x > 2$), and $f(x_1) + f(2x_2) = \frac {1}{2}$, find the minimum value of $f(x_1x_2)$. | \frac {1}{3} | 0.5 | 6,712 | 5,565.125 | 7,858.875 | |
Let \( f : \mathbb{C} \to \mathbb{C} \) be defined by \( f(z) = z^2 - 2iz + 2 \). Determine how many complex numbers \( z \) exist such that \( \text{Im}(z) > 0 \) and both the real and the imaginary parts of \( f(z) \) are integers within \( |a|, |b| \leq 5 \). | 110 | 0 | 8,192 | -1 | 8,192 | |
Given that \( F_1 \) and \( F_2 \) are the left and right foci of the ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) where \( a > b > 0 \), and \( P \) is a point on the ellipse \( C \). The incenter of triangle \( \triangle F_{1}PF_{2} \) is \( I \). If there exists a real number \( \lambda \) such that:
$... | \frac{1}{2} | 0.625 | 6,927.25 | 6,168.4 | 8,192 | |
Let $a_{10} = 10$, and for each positive integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least positive $n > 10$ such that $a_n$ is a multiple of $99$.
| 45 | 0.25 | 7,978.25 | 7,337 | 8,192 | |
What is the sum of the reciprocals of the natural-number factors of 6? | 2 | 1 | 2,638.9375 | 2,638.9375 | -1 | |
In the Cartesian coordinate plane $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. Let $M$ be a moving point on the parabola, then the maximum value of $\frac{MO}{MF}$ is _______. | \frac{2\sqrt{3}}{3} | 0 | 6,810 | -1 | 6,810 | |
For a positive number $x$, define $f(x)=\frac{x}{x+1}$. For example, $f(1)=\frac{1}{1+1}=\frac{1}{2}$, $f(2)=\frac{2}{2+1}=\frac{2}{3}$, $f(\frac{1}{2})=\frac{\frac{1}{2}}{\frac{1}{2}+1}=\frac{1}{3}$.
$(1)$ Find the value of: $f(3)+f(\frac{1}{3})=$______; $f(4)+f(\frac{1}{4})=$______.
$(2)$ Conjecture: $f(x)+f(\fra... | 2022.5 | 0 | 6,626.5 | -1 | 6,626.5 | |
Given the function $f(x)$ and its derivative $f''(x)$ on $\mathbb{R}$, and for any real number $x$, it satisfies $f(x)+f(-x)=2x^{2}$, and for $x < 0$, $f''(x)+1 < 2x$, find the minimum value of the real number $a$ such that $f(a+1) \leqslant f(-a)+2a+1$. | -\dfrac {1}{2} | 0 | 8,192 | -1 | 8,192 | |
Find all pairs of distinct rational numbers $(a,b)$ such that $a^a=b^b$. | \left(\left(\frac{u}{v}\right)^{\frac{u}{v-u}}, \left(\frac{u}{v}\right)^{\frac{v}{v-u}}\right) |
To find all pairs of distinct rational numbers \((a, b)\) such that \(a^a = b^b\), we start by setting up the equation:
\[
a^a = b^b.
\]
This can be rewritten using logarithms as:
\[
a \ln a = b \ln b.
\]
Consider \(a = \left(\frac{u}{v}\right)^{\frac{u}{v-u}}\) and \(b = \left(\frac{u}{v}\right)^{\frac{v}{v-u}}\)... | 0 | 8,189.125 | -1 | 8,189.125 |
Onkon wants to cover his room's floor with his favourite red carpet. How many square yards of red carpet are required to cover a rectangular floor that is $12$ feet long and $9$ feet wide? (There are 3 feet in a yard.) | 12 | 1. **Convert dimensions from feet to yards**:
Given that there are 3 feet in a yard, we convert the dimensions of the room from feet to yards by dividing each dimension by 3:
- Length in yards: \( \frac{12 \text{ feet}}{3 \text{ feet/yard}} = 4 \text{ yards} \)
- Width in yards: \( \frac{9 \text{ feet}}{3 \te... | 1 | 1,116.8125 | 1,116.8125 | -1 |
In triangle $ABC$, $a=3$, $\angle C = \frac{2\pi}{3}$, and the area of $ABC$ is $\frac{3\sqrt{3}}{4}$. Find the lengths of sides $b$ and $c$. | \sqrt{13} | 0.75 | 4,994 | 3,928 | 8,192 | |
"The Nine Chapters on the Mathematical Art" is an ancient Chinese mathematical text, which records: "If it can be halved, then halve it; if not, juxtapose the numerator and denominator, subtract the lesser from the greater, continue to subtract in turn, seeking their equality. Use the equal number to reduce them." This... | 273 | 0.1875 | 5,802.5625 | 4,200.666667 | 6,172.230769 | |
Ella adds up all the odd integers from 1 to 499, inclusive. Mike adds up all the integers from 1 to 500, inclusive. What is Ella's sum divided by Mike's sum? | \frac{500}{1001} | 0 | 5,263.25 | -1 | 5,263.25 |
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