mradermacher/Eduardo-27b-i1-GGUF
Reinforcement Learning • 27B • Updated • 1.96k
problem stringlengths 20 1.22k | answer stringlengths 1 17 | llama8b_solve_rate float64 0.02 0.59 | transfer_problem stringlengths 20 1.18k ⌀ | transfer_answer stringlengths 1 21 ⌀ | pre_solve_rate float64 0 1 | pre_solve_rate_n_scored int64 16 16 | pre_solve_rate_transfer_problem float64 0 1 ⌀ | pre_solve_rate_transfer_n_scored int64 0 16 |
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Four positive integers $a, b, c, d$ satisfy the condition: $a < b < c < d$ . For what smallest possible value of $d$ could the following condition be true: the arithmetic mean of numbers $a, b, c$ is twice smaller than the arithmetic mean of numbers $a, b, c, d$ ? | 10 | 0.15625 | Four positive integers $p, q, r, s$ are strictly increasing, such that $p < q < r < s$. What is the minimum possible value of $s$ if the arithmetic mean of $p, q,$ and $r$ is exactly one-third of the arithmetic mean of all four numbers $p, q, r,$ and $s$? | 18 | 0.1875 | 16 | 0.375 | 16 |
Among the 100 natural numbers 1, 3, 5, 7, …, 199, select several numbers so that none of the selected numbers is an integer multiple of another. The maximum number of such numbers that can be selected is . | 67 | 0.015625 | Among the 150 natural numbers 1, 3, 5, 7, …, 299, select several numbers so that none of the selected numbers is an integer multiple of another. The maximum number of such numbers that can be selected is ____. | 100 | 0 | 16 | 0 | 16 |
If the line $l_1: ax+3y+1=0$ is parallel to the line $l_2: 2x+(a+1)y+1=0$, the value of $a$ is ___. | -3 | 0.171875 | If the line $l_1: kx+2y+1=0$ is parallel to the line $l_2: 4x+(k-2)y+1=0$, the value of $k$ is ___. | -2 | 0.1875 | 16 | 0.1875 | 16 |
Let $f(x)$ be an odd function defined on $\mathbb{R}$, and $f(x+2)=-f(x)$. If $0 \leq x \leq 1$, $f(x)=x$, then $f(7.5)=$ ? | -0.5 | 0.109375 | Let $f(x)$ be an odd function defined on $\mathbb{R}$, and $f(x+3)=-f(x)$. If $0 \leq x \leq 1.5$, $f(x)=2x$, then find the value of $f(17)$. | -2 | 0 | 16 | 0.125 | 16 |
The radius of the circle $\rho=2\cos\theta$ is. | 1 | 0.265625 | The radius of the circle $\rho=4\sin\theta$ is. | 2 | 0.1875 | 16 | 0.1875 | 16 |
Let $T$ be the set of ordered triples $(x,y,z)$, where $x,y,z$ are integers with $0\leq x,y,z\leq9$. Players $A$ and $B$ play the following guessing game. Player $A$ chooses a triple $(x,y,z)$ in $T$, and Player $B$ has to discover $A$[i]'s[/i] triple in as few moves as possible. A [i]move[/i] consists of the followin... | 3 | 0.28125 | Let $S$ be the set of ordered triples $(p,q,r)$ of integers such that $1 \leq p,q,r \leq 10$. Alice and Bob play a game where Alice secretly chooses a triple $(p,q,r)$ from $S$. Bob wants to determine Alice's triple in the minimum number of guesses. In each guess, Bob chooses a triple $(u,v,w)$ from $S$ and Alice tells... | 3 | 0.3125 | 16 | 0.25 | 16 |
Find the smallest positive integer $n$ such that $\frac{5^{n+1}+2^{n+1}}{5^{n}+2^{n}}>4.99$. | 7 | 0.078125 | Find the smallest positive integer $n$ such that $\frac{6^{n+1}+2^{n+1}}{6^{n}+2^{n}}>5.99$. | 6 | 0.125 | 16 | 0.125 | 16 |
If $x$ is a real number and $\lfloor x \rfloor = -9,$ how many possible values are there for $\lfloor 5x \rfloor$? | 5 | 0.421875 | If $x$ is a real number and $\lfloor x \rfloor = -5,$ how many possible values are there for $\lfloor 8x \rfloor$? | 8 | 0.1875 | 16 | 0.5 | 16 |
Given that $2$, $x$, $y$, $z$, $18$ form a geometric sequence, find the value of $y$. | 6 | 0.328125 | Given that $3$, $p$, $q$, $r$, $48$ form a geometric sequence, find the value of $q$. | 12 | 0.1875 | 16 | 0.375 | 16 |
Given the sequence $\{a_n\}$ that satisfies $a_{n+1} = \frac{4a_n + 3}{4}$ ($n \in \mathbb{N}^*$), and $a_1 = 1$, find $a_{17}$. | 13 | 0.1875 | Consider a sequence $\{a_n\}$ defined by $a_1 = 2$ and the recurrence relation $a_{n+1} = \frac{5a_n + 2}{5}$ for all positive integers $n$. Calculate the value of $a_{21}$. | 10 | 0.25 | 16 | 0.4375 | 16 |
If a line is perpendicular to a plane, then this line and the plane form a "perpendicular line-plane pair". In a cube, the number of "perpendicular line-plane pairs" formed by a line determined by two vertices and a plane containing four vertices is _________. | 36 | 0.015625 | In a 3D modeling program, a designer creates 4 independent, non-intersecting cubes. A "normal pair" is defined as a line determined by two vertices of a single cube and a plane containing four vertices of that same cube, such that the line is perpendicular to the plane. What is the total number of "normal pairs" that c... | 144 | 0 | 16 | 0.125 | 16 |
Two children at a time can play pairball. For 90 minutes, with only two children playing at a time, five children take turns so that each one plays the same amount of time. Calculate the number of minutes each child plays. Express your answer as a whole number. | 36 | 0.0625 | Three children at a time can play a multiplayer racing game. For 120 minutes, with only three children playing at a time, eight children take turns so that each one plays the same amount of time. Calculate the number of minutes each child plays. Express your answer as a whole number. | 45 | 0 | 16 | 0 | 16 |
Let \( S_{n}=1-2+3-4+\cdots+(-1)^{n-1} n \), where \( n=1,2,\cdots \). Calculate the value of \( S_{17}+S_{33}+S_{50} \). Express your answer as a single integer. | 1 | 0.0625 | Consider the sequence defined by \( A_{n}=1-2+3-4+\cdots+(-1)^{n-1} n \) for all positive integers \( n \). Determine the value of \( A_{21}+A_{45}+A_{64} \). Express your answer as a single integer. | 2 | 0.0625 | 16 | 0.125 | 16 |
Given functions f(x) = ae^x and g(x) = 2x + b, if f(x) >= g(x) always holds, find the maximum value of b/a. Express your answer as a single number. | 1 | 0.25 | Given functions f(x) = ae^x and g(x) = 5x + b, if f(x) >= g(x) always holds, find the maximum value of b/a. Express your answer as a single number. | 1 | 0.1875 | 16 | 0.25 | 16 |
In the ancient Chinese mathematical classic "Jiuzhang Suanshu," there is a problem called "Rice and Wheat Grains Division": When the granary opens to receive grains, someone delivered 1536 dan of rice. Upon inspection, it was found that there were wheat grains mixed in. A handful of rice was taken for sampling, and it ... | 192 | 0.34375 | A merchant receives a large shipment of 2160 kilograms of coffee beans. Upon inspection, it is found that there are some defective beans mixed in. A random handful of beans is taken for sampling, and it is counted that among 240 beans, there are 30 defective beans. Estimate the amount of defective beans mixed in this s... | 270 | 0.5625 | 16 | 0.75 | 16 |
Given the polar equation
$$
\rho = \frac{1}{1 - C_{m}^{n} \cos \theta}
$$
where \( 1 \leq n \leq m \leq 5 \), how many different hyperbolas are represented by this equation when \( C_{m}^{n} > 1 \)? Express your answer as a whole number. | 10 | 0.21875 | Given the polar equation
$$
\rho = \frac{5}{1 - \binom{m}{n} \sin \theta}
$$
where $m$ and $n$ are integers such that $1 \leq n \leq m \leq 6$. For how many ordered pairs $(m, n)$ does this equation represent a hyperbola? Express your answer as a whole number. | 15 | 0.3125 | 16 | 0.3125 | 16 |
On a true-false test of 100 items, every question that is a multiple of 4 is true, and all others are false. If a student marks every item that is a multiple of 3 false and all others true, how many of the 100 items will be correctly answered? | 42 | 0.09375 | On a true-false test of 150 questions, the correct answer to every question that is a multiple of 5 is true, and all others are false. If a student marks every question that is a multiple of 4 false and all others true, how many of the 150 questions will be answered correctly? | 53 | 0.0625 | 16 | 0.125 | 16 |
Given that the sequence ${a_n}$ is an arithmetic sequence, and the sequence ${b_n}$ is a geometric sequence, where $({a_{2017}}+{a_{2018}}=π )$ and $({{b_{20}}}^2=4 )$, find the value of $\tan \frac{{a_{2}}+{a_{4033}}}{{b_{1}}{b_{39}}}$. Express your answer as a single number. | 1 | 0.28125 | Given that the sequence $u_n$ is an arithmetic sequence, and the sequence $v_n$ is a geometric sequence, where $u_{50} + u_{51} = \pi$ and $v_{15}^2 = 4$, find the value of $\tan \frac{u_{12} + u_{89}}{v_3 v_{27}}$. Express your answer as a single number. | 1 | 0.25 | 16 | 0.3125 | 16 |
Compute
\[\sum_{n = 1}^\infty \frac{1}{F_n F_{n + 2}},\]where $F_n$ denotes the $n$th Fibonacci number, so $F_0 = 0$ and $F_1 = 1.$ | 1 | 0.390625 | Compute
\[\sum_{n = 1}^\infty \frac{1}{L_n L_{n + 2}},\]where $L_n$ denotes the $n$th Lucas number, defined by the recurrence $L_n = L_{n-1} + L_{n-2}$ for $n \ge 2,$ with initial values $L_0 = 2$ and $L_1 = 1.$ | \frac{1}{3} | 0.375 | 16 | 0.1875 | 16 |
Given $\overrightarrow{a}=(2,-1,3),\overrightarrow{b}=(-1,4,2),\overrightarrow{c}=(-3,5,\lambda)$, if the three vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ lie in the same plane, then the real number $\lambda =$____. | -1 | 0.015625 | Given $\overrightarrow{u}=(3,-2,1)$, $\overrightarrow{v}=(-2,3,4)$, and $\overrightarrow{w}=(-1,4,\lambda)$, if the three vectors $\overrightarrow{u}$, $\overrightarrow{v}$, and $\overrightarrow{w}$ lie in the same plane, then the real number $\lambda =$____. | 9 | 0 | 16 | 0.25 | 16 |
Given that F is the focus of the parabola $C: y^2 = 8x$, and let M be a point on C. The extension line of FM intersects the y-axis at point N. If M is the midpoint of FN, then $|FN| =$ __________ . | 6 | 0.015625 | Given that $F$ is the focus of the parabola $C: y^2 = 12x$, and let $P$ be a point on $C$. The line extending from $F$ through $P$ intersects the $y$-axis at point $Q$. If $P$ is the midpoint of the segment $FQ$, then what is the length of $FQ$? | 9 | 0 | 16 | 0.0625 | 16 |
In triangle $\triangle ABC$, if ${a^2}+{b^2}={c^2}+\sqrt{3}ab$, then find the measure of angle $C$ in degrees. Express your answer as a whole number. | 30 | 0.4375 | In triangle $\triangle XYZ$, if ${x^2}+{y^2}={z^2}+xy$, then find the measure of angle $Z$ in degrees. Express your answer as a whole number. | 60 | 0.25 | 16 | 0.5 | 16 |
Observe the following set of equations:
$S_{1}=1$,
$S_{2}=2+3+4=9$,
$S_{3}=3+4+5+6+7=25$,
$S_{4}=4+5+6+7+8+9+10=49$,
...
Based on the equations above, guess that $S_{2n-1}=(4n-3)(an+b)$, then $a^{2}+b^{2}=$_______. | 25 | 0.03125 | Observe the following set of equations:
$S_{1}=1$,
$S_{2}=2+3+4=9$,
$S_{3}=3+4+5+6+7=25$,
$S_{4}=4+5+6+7+8+9+10=49$,
...
Based on the equations above, guess that $S_{3n-1}=(6n-3)(pn+q)$, then $p^{2}+q^{2}=$_______. | 45 | 0.25 | 16 | 0.0625 | 16 |
Given that points A, B, and C move on the circle x²+y²=1, and AB is perpendicular to BC. If the coordinates of point P are (2,0), find the maximum value of $|$$\overrightarrow {PA}$$+$$\overrightarrow {PB}$$+$$\overrightarrow {PC}$$|$. | 7 | 0.015625 | Given that points $K$, $L$, and $M$ move on the circle $x^2+y^2=9$, and $KL$ is perpendicular to $LM$. If the coordinates of point $T$ are $(0,4)$, find the maximum value of $|\overrightarrow{TK} + \overrightarrow{TL} + \overrightarrow{TM}|$. | 15 | 0 | 16 | 0.0625 | 16 |
Find the number of complex numbers z that satisfy the conditions z^24 = 1 and z^6 is a real number. Express your answer as a whole number. | 12 | 0.359375 | Find the number of complex numbers $z$ that satisfy the conditions $z^{36} = 1$ and $z^9$ is a real number. Express your answer as a whole number. | 18 | 0.3125 | 16 | 0.3125 | 16 |
What is the product of all the coordinates of all the points of intersection of the two circles defined by $x^2-2x +y^2-10y+25=0$ and $x^2-8x+y^2-10y+37=0$? | 10 | 0.234375 | What is the product of all the coordinates of all the points of intersection of the two circles defined by $x^2-6x+y^2-12y+41=0$ and $x^2-16x+y^2-12y+91=0$? | 30 | 0.25 | 16 | 0.0625 | 16 |
Calculate the value of the expression: ${C}_{2}^{2}+{C}_{3}^{2}+{C}_{4}^{2}+\ldots +{C}_{10}^{2}$. Express your answer as a single integer. | 165 | 0.140625 | Calculate the value of the expression: ${C}_{2}^{2}+{C}_{3}^{2}+{C}_{4}^{2}+\ldots +{C}_{11}^{2}$. Express your answer as a single integer. | 220 | 0.125 | 16 | 0.3125 | 16 |
One angle of a triangle is twice another, and the sides opposite these angles have lengths 15 and 9. Compute the length of the third side of the triangle. | 16 | 0.03125 | One angle of a triangle is twice another, and the sides opposite these angles have lengths 12 and 8. Compute the length of the third side of the triangle. | 10 | 0 | 16 | 0 | 16 |
Given $f(x) = ax^2 + bx + 3a + b$ is an even function, and its domain is $[a-3, 2a]$, find the value of $a+b$. | 1 | 0.25 | Given $f(x) = px^2 + qx + 4p - 2q$ is an even function, and its domain is $[p-6, 2p]$, find the value of $p+q$. | 2 | 0.25 | 16 | 0.25 | 16 |
Given points A(0, 1) and B(4, a), if the intercepts of line AB on the x-axis and the y-axis are equal, then the real number $a=$ \_\_\_\_\_. | -3 | 0.546875 | Given points $P(0, 3)$ and $Q(7, m)$, if the intercepts of line $PQ$ on the $x$-axis and the $y$-axis are equal, then the real number $m=$ _____. | -4 | 0.6875 | 16 | 0.3125 | 16 |
Let $a, b, c, x$ be reals with $(a+b)(b+c)(c+a) \neq 0$ that satisfy $$\frac{a^{2}}{a+b}=\frac{a^{2}}{a+c}+20, \quad \frac{b^{2}}{b+c}=\frac{b^{2}}{b+a}+14, \quad \text { and } \quad \frac{c^{2}}{c+a}=\frac{c^{2}}{c+b}+x$$ Compute $x$. | -34 | 0.03125 | Let $u, v, w, k$ be real numbers with $(u+v)(v+w)(w+u) \neq 0$ that satisfy
$$\frac{u^{2}}{u+v}=\frac{u^{2}}{u+w}+30, \quad \frac{v^{2}}{v+w}=\frac{v^{2}}{v+u}+45, \quad \text { and } \quad \frac{w^{2}}{w+u}=\frac{w^{2}}{w+v}+k.$$
Compute $k$. | -75 | 0 | 16 | 0.0625 | 16 |
In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\left( \begin{array}{cc} 1 & 2 3 & 4 \end... | 17 | 0.015625 | In the galaxy of Matrixia, sectors are labeled with $2 \times 2$ arrays of positive reals. One can hyper-jump from sector $M$ to sector $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can hyper-jump from $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ t... | 21 | 0 | 16 | 0 | 16 |
Given two circles $x^{2}+y^{2}+2ax+a^{2}-4=0$ and $x^{2}+y^{2}-4by-1+4b^{2}=0$ with exactly three common tangents, find the minimum value of $\dfrac{1}{a^{2}}+\dfrac{1}{b^{2}}$, where $a\in R$, $b\in R$, and $ab\neq 0$. Express your answer as a single numerical value. | 1 | 0.03125 | Given two circles $x^{2}+y^{2}+2mx+m^{2}-9=0$ and $x^{2}+y^{2}-8ny+16n^{2}-4=0$ with exactly three common tangents, find the minimum value of $\dfrac{1}{m^{2}}+\dfrac{1}{n^{2}}$, where $m\in \mathbb{R}$, $n\in \mathbb{R}$, and $mn\neq 0$. Express your answer as a single numerical value. | 1 | 0 | 16 | 0 | 16 |
Twenty-seven solid gold spherical beads each of radius 3 are melted down and recast into a larger, solid gold sphere. How many units are in the radius of this larger gold sphere? | 9 | 0.59375 | Sixty-four solid silver spherical marbles, each with a radius of 2, are melted down and recast into a single larger, solid silver sphere. How many units are in the radius of this larger silver sphere? | 8 | 0.9375 | 16 | 0.6875 | 16 |
Each integer in $\{1, 2, 3, . . . , 2020\}$ is coloured in such a way that, for all positive integers $a$ and $b$ such that $a + b \leq 2020$ , the numbers $a$ , $b$ and $a + b$ are not coloured with three different colours.
Determine the maximum number of colours that can be used. | 11 | 0.015625 | Each integer in the set $\{1, 2, 3, \dots, 5000\}$ is assigned a color in such a way that, for any pair of positive integers $x$ and $y$ satisfying $x + y \le 5000$, the three numbers $x$, $y$, and $x + y$ are never assigned three distinct colors.
Determine the maximum number of colors that can be used in such a colo... | 13 | 0 | 16 | 0 | 16 |
Given that $a$ is a real number, and the complex number $z=\frac{{1+2i}}{{a-i}}$ is purely imaginary (where $i$ is the imaginary unit), find the value of $a$. | 2 | 0.546875 | Let $c$ be a real number. If the complex number $w = \frac{2+5i}{c-2i}$ is purely imaginary (where $i$ is the imaginary unit), what is the value of $c$? | 5 | 0.8125 | 16 | 0.9375 | 16 |
Evaluate: $$27^{ \frac {2}{3}}-2^{log_{2}3}×log_{2} \frac {1}{8}+2lg( \sqrt {3+ \sqrt {5}}+ \sqrt {3- \sqrt {5}})$$ | 19 | 0.28125 | Evaluate: $$64^{ \frac {2}{3}}-3^{\log_{3}5} \times \log_{3} \frac {1}{27}+2\lg( \sqrt {4+ \sqrt {15}}+ \sqrt {4- \sqrt {15}})$$ | 32 | 0.125 | 16 | 0.1875 | 16 |
A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$. | 106 | 0.0625 | A car dealership lines up two red cars, five blue cars, and four silver cars in a single row. The cars are parked in a random order, with each arrangement being equally likely. Let $\frac{m}{n}$ in lowest terms be the probability that no two silver cars are parked next to one another. Find $m+n$. | 40 | 0.3125 | 16 | 0.0625 | 16 |
Given that the positive numbers $x$ and $y$ satisfy the equation $x + y = 1$, find the minimum value of $\frac{1}{x} + \frac{4}{y}$. | 9 | 0.171875 | Given that the positive numbers $m$ and $n$ satisfy the equation $m + n = 1$, find the minimum value of $\frac{4}{m} + \frac{9}{n}$. | 25 | 0.125 | 16 | 0.4375 | 16 |
Given the formula for converting Celsius to Fahrenheit: F = (9/5)C + 32, and a temperature of 10 degrees Celsius, calculate the temperature in degrees Fahrenheit. Express your answer as a decimal value with one digit after the decimal point, and include the unit (degrees Fahrenheit). | 50.0 | 0.203125 | Use the conversion formula F = (9/5)C + 32 to find the Fahrenheit equivalent of a temperature measuring 35 degrees Celsius. Write your final answer as a decimal with a single digit following the decimal point. | 95.0 | 0.875 | 16 | 0.75 | 16 |
There are 100 positive integers written on a board. At each step, Alex composes 50 fractions using each number written on the board exactly once, brings these fractions to their irreducible form, and then replaces the 100 numbers on the board with the new numerators and denominators to create 100 new numbers.
Find th... | 99 | 0.0625 | null | null | 0 | 16 | null | 0 |
Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is $3$ jumps east and $10$ jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at Frist, located $2$ jumps east and $4$ jumps north of Fine Hall. How many ways ca... | 181 | 0.109375 | A delivery robot navigates a city grid by traveling only north and east. The robot starts at the charging station and needs to reach the repair bay, which is $4$ blocks east and $8$ blocks north of the charging station. However, the robot must avoid a construction zone located $2$ blocks east and $3$ blocks north of th... | 285 | 0.25 | 16 | 0.3125 | 16 |
Calculate the area of the closed region formed by the function y = |||x|-1|-1| and the x-axis. Express your answer as a single number. | 2 | 0.15625 | Calculate the area of the closed region formed by the function y = |||x|-2|-2| and the x-axis. Express your answer as a single number. | 8 | 0.25 | 16 | 0.0625 | 16 |
Given that the lengths of two sides of a triangle are $3$ and $5$, and the third side is a root of the equation $x^{2}-3x=x-3$, then the perimeter of this triangle is ______. | 11 | 0.28125 | A triangle has two sides of lengths $4$ and $7$. Its third side length is a solution to the equation $x^2 - 5x = 2x - 10$. What is the perimeter of the triangle? | 16 | 0.3125 | 16 | 0.5625 | 16 |
Given that point A $(a, 4)$ is symmetric to point B across the y-axis, and the coordinates of point B are $(-2, b)$, find the value of $a+b$. | 2 | 0.03125 | Given that point P $(c, 9)$ is symmetric to point Q across the x-axis, and the coordinates of point Q are $(-4, d)$, find the value of $c+d$. | -13 | 0 | 16 | 0.8125 | 16 |
Given that the random variable ξ follows a normal distribution N(μ, σ^2), and P(ξ < 2) = P(ξ > 6) = 0.15, calculate the probability P(2 ≤ ξ < 4). Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places if necessary. | 0.35 | 0.1875 | Given that the random variable $X$ follows a normal distribution $N(\mu, \sigma^2)$, and $P(X < 14) = P(X > 26) = 0.18$, calculate the probability $P(14 \le X < 20)$. Express your answer as a decimal value between 0 and 1. | 0.32 | 0.5 | 16 | 0.0625 | 16 |
Find the length of the chord that is perpendicular to the axis of symmetry and passes through the focus of the parabola $y^2 = 4x$. | 4 | 0.390625 | Find the length of the chord that is perpendicular to the axis of symmetry and passes through the focus of the parabola $x^2 = 12y$. | 12 | 0.5 | 16 | 0.5 | 16 |
There are 148 students whose scores range from 100 to 120 (with scores being positive integers), and students with the same score are given the same rank. What is the smallest possible number x such that each rank can have at most x students? Express your answer as a single integer. | 8 | 0.25 | There are 130 participants whose scores range from 80 to 99 (with scores being positive integers), and participants with the same score are given the same rank. What is the smallest possible number $x$ such that each rank can have at most $x$ participants? Express your answer as a single integer. | 7 | 0.4375 | 16 | 0.3125 | 16 |
Po writes down five consecutive integers and then erases one of them. The four remaining integers sum to 153. Compute the integer that Po erased. | 37 | 0.125 | Maya writes down five consecutive integers and then erases one of them. The four remaining integers sum to 207. Compute the integer that Maya erased. | 53 | 0 | 16 | 0 | 16 |
In a rectangle, the perimeter of quadrilateral $PQRS$ is given. If the horizontal distance between adjacent dots in the same row is 1 and the vertical distance between adjacent dots in the same column is 1, what is the perimeter of quadrilateral $PQRS$? | 14 | 0.109375 | Quadrilateral $WXYZ$ is drawn on a coordinate plane with its vertices at $W(2, 1)$, $X(8, 1)$, $Y(8, 6)$, and $Z(2, 6)$. If the horizontal and vertical distances between adjacent grid lines are both 1 unit, what is the perimeter of quadrilateral $WXYZ$? | 22 | 0.5625 | 16 | 0.9375 | 16 |
There are several pairs of integers $ (a, b) $ satisfying $ a^2 - 4a + b^2 - 8b = 30 $ . Find the sum of the sum of the coordinates of all such points. | 60 | 0.03125 | There are several pairs of integers $ (m, n) $ satisfying $ m^2 - 2m + n^2 - 6n = 15 $. Find the sum of the sum of the coordinates of all such points. | 48 | 0 | 16 | 0 | 16 |
Back in 1930, Tillie had to memorize her multiplication facts from $0 \times 0$ to $12 \times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. Calculate the fraction of the numbers in the body of the table that are odd. Express you... | 0.21 | 0.109375 | Arthur is studying his extended multiplication facts from $0 \times 0$ to $14 \times 14$. The chart he uses has rows and columns labeled with the numbers $0$ through $14$, and the products fill the main grid. Find the fraction of the products in the grid that are odd numbers. Express your answer as a decimal rounded to... | 0.22 | 0.25 | 16 | 0.125 | 16 |
Given that the terms of the geometric sequence $\\{a\_n\\}$ are all positive, the common ratio $q=2$, and $a\_3 \cdot a\_{13}=16$, find $a\_9$. | 8 | 0.53125 | Let $\{b_n\}$ be a geometric sequence consisting entirely of positive terms. If the common ratio is $q=3$ and $b_4 \cdot b_{10} = 81$, what is the value of $b_8$? | 27 | 0.8125 | 16 | 0.8125 | 16 |
Find the number of solutions to the equation $$C_{16}^{x^{2}-x} = C_{16}^{5x-5},$$ considering the constraints on the values of x that satisfy the conditions of the binomial coefficient. Express your answer as a single integer. | 2 | 0.421875 | Find the number of solutions to the equation $$C_{8}^{x^{2}-2x} = C_{8}^{5x-10},$$ considering the constraints on the values of $x$ that satisfy the conditions of the binomial coefficient. Express your answer as a single integer. | 2 | 0.5 | 16 | 0.5625 | 16 |
Given that $i$ is the imaginary unit, solve for $z$ in the equation $1+i=z(1-i)$ and find the imaginary part of $z$. Express your answer as a numerical value. | 1 | 0.390625 | Given that $i$ is the imaginary unit, solve for $z$ in the equation $11 + 13i = z(3 - i)$ and find the imaginary part of $z$. Express your answer as a numerical value. | 5 | 0.3125 | 16 | 0.3125 | 16 |
Let $S_{n}$ be the sum of the first $n$ terms of the geometric sequence ${a_{n}}$. Given that $3S_{2}=a_{3}-2$, $3S_{1}=a_{2}-2$, find the common ratio $q=$ _____. | 4 | 0.0625 | Let $S_{n}$ be the sum of the first $n$ terms of a geometric sequence $\{a_{n}\}$. Given that $5S_{2}=a_{3}-3$ and $5S_{1}=a_{2}-3$, find the common ratio $q$. | 6 | 0 | 16 | 0 | 16 |
There are 2012 backgammon checkers with one side black and the other side white, arranged in a line such that no two consecutive checkers are the same color. At each move, two checkers are chosen, and they and all checkers between them are flipped. What is the minimum number of moves required to make all checkers the s... | 1006 | 0.4375 | There are 2024 coins with one side heads and the other side tails, arranged in a straight line such that no two consecutive coins show the same face. In a single move, you may choose any two coins and flip them along with all the coins between them. What is the minimum number of moves required to make all the coins sho... | 1012 | 0 | 16 | 0.1875 | 16 |
Given the quadratic function $y=a(a+1)x^{2}-(2a+1)x+1$, when $a$ takes the values of $1, 2, 3, \dots, n$, the lengths of the line segments intercepted by the parabola on the $x$-axis are $d_{1}, d_{2}, d_{3}, \dots, d_{n}$ respectively. Find the limit $\lim\limits_{n→∞}(d_{1}+d_{2}+⋯+d_{n})$. | 1 | 0.34375 | Given the quadratic function $y=k(k+1)x^{2}-2(2k+1)x+4$, when $k$ takes the values of $1, 2, 3, \dots, n$, the lengths of the line segments intercepted by the parabola on the $x$-axis are $L_{1}, L_{2}, L_{3}, \dots, L_{n}$ respectively. Find the limit $\lim\limits_{n\to\infty}(L_{1}+L_{2}+\dots+L_{n})$. | 2 | 0.25 | 16 | 0.0625 | 16 |
Given four natural numbers $a$, $b$, $c$, and $d$, where the sum of every three numbers is respectively $180$, $197$, $208$, and $222$, find the largest number among $a$, $b$, $c$, and $d$. | 89 | 0.296875 | Given four positive integers $w$, $x$, $y$, and $z$, where the sum of every three numbers is respectively $145$, $150$, $165$, and $185$, find the largest number among $w$, $x$, $y$, and $z$. | 70 | 0.375 | 16 | 0.25 | 16 |
Given the set $A = \{x \,|\, |x| \geq 3\}$, and $B = \{x \,|\, x \geq a\}$. If $A \subseteq B$, then the range of the real number $a$ is $(-\infty, c]$, where $c = \_\_\_\_\_\_$. | -3 | 0.359375 | Given the set $P = \{x \,|\, |x| \le 4\}$, and $Q = \{x \,|\, x \ge k\}$. If $P \subseteq Q$, then the range of the real number $k$ is $(-\infty, d]$, where $d = \_\_\_\_\_\_$. | -4 | 0.5 | 16 | 0.375 | 16 |
Among the numbers $\dfrac{1}{6}$, $\sqrt {5}$, $0$, $\sqrt [3] {9}$, $-\dfrac{\pi }{3}$, how many are irrational? Express your answer as a whole number. | 3 | 0.421875 | Among the numbers $-\dfrac{4}{9}$, $\sqrt{7}$, $-2$, $\sqrt[3]{16}$, $\dfrac{3\pi}{2}$, how many are irrational? Express your answer as a whole number. | 3 | 0.125 | 16 | 0.5 | 16 |
Let $[n] = \{1, 2, 3, ... ,n\}$ and for any set $S$ , let $ P(S)$ be the set of non-empty subsets of $S$ . What is the last digit of $|P(P([2013]))|$ ? | 7 | 0.234375 | Let $S_{2024} = \{1, 2, 3, \dots, 2024\}$. For any set $A$, let $N(A)$ denote the set of all non-empty subsets of $A$. What is the units digit of $|N(N(S_{2024}))|$? | 7 | 0.0625 | 16 | 0.25 | 16 |
A lottery ticket has $50$ cells into which one must put a permutation of $1, 2, 3, ... , 50$ . Any ticket with at least one cell matching the winning permutation wins a prize. How many tickets are needed to be sure of winning a prize? | 26 | 0.015625 | A secret code consists of a permutation of the integers $1, 2, 3, \ldots, 40$. A computer program can test multiple permutations. The program will report a success if at least one of the tested permutations has at least one integer in the exact same position as the secret code. What is the minimum number of permutation... | 21 | 0 | 16 | 0 | 16 |
Construct a rectangular open-top pool with a volume of $18m^3$ and a depth of $2m$. If the cost per square meter for the pool bottom is $200$ yuan and for the pool walls is $150$ yuan, then the minimum cost of constructing the pool is _______ yuan. | 5400 | 0.109375 | A rectangular open-top storage tank is to be constructed with a volume of $32\text{ m}^3$ and a depth of $2\text{ m}$. If the material cost per square meter for the tank's base is $300$ dollars and for the tank's walls is $100$ dollars, then the minimum cost of materials to construct the tank is _______ dollars. | 8000 | 0.1875 | 16 | 0.1875 | 16 |
Mattis is hosting a badminton tournament for $40$ players on $20$ courts numbered from $1$ to $20$. The players are distributed with $2$ players on each court. In each round a winner is determined on each court. Afterwards, the player who lost on court $1$, and the player who won on court $20$ stay in place. For the re... | 39 | 0.03125 | Sarah is organizing a ping-pong tournament for $64$ players on $32$ tables numbered from $1$ to $32$. The players are distributed with $2$ players on each table. In each round, a winner is determined on each table. Afterwards, the player who lost on table $1$ and the player who won on table $32$ stay in place. For the ... | 63 | 0.0625 | 16 | 0 | 16 |
Given point $M$ is on the parabola $y^{2}=4x$, $F$ is the focus of the parabola, and $A$ is on the circle $C$: $(x-4)^{2}+(y-1)^{2}=1$, the minimum value of $|MA|+|MF|$ is ______. | 4 | 0.015625 | Given point $M$ is on the parabola $y^{2}=8x$, $F$ is the focus of the parabola, and $A$ is on the circle $C$: $(x-5)^{2}+(y-3)^{2}=4$, find the minimum value of $|MA|+|MF|$. | 5 | 0 | 16 | 0 | 16 |
Given $(1+x)+(1+x)^{2}+\ldots+(1+x)^{n}=a_{0}+a_{1}x+\ldots+a_{n}x^{n}$, if $a_{1}+a_{2}+a_{3}+\ldots+a_{n-1}=29-n$, calculate the value of the natural number $n$. Express your answer as a natural number. | 4 | 0.078125 | Given $(1+x)+(1+x)^{2}+\ldots+(1+x)^{n}=c_{0}+c_{1}x+\ldots+c_{n}x^{n}$, if $c_{1}+c_{2}+c_{3}+\ldots+c_{n-1}=125-n$, determine the value of the positive integer $n$. | 6 | 0 | 16 | 0 | 16 |
Alexis imagines a $2008\times 2008$ grid of integers arranged sequentially in the following way:
\[\begin{array}{r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r}1,&2,&3,&\ldots,&20082009,&2010,&2011,&\ldots,&40264017,&4018,&4019,&\ldots,&6024\vdots&&&&\vdots2008^2-2008+1,&2008^2-2008+2,&2008^2-... | 1004 | 0.0625 | Beatrice imagines a $100 \times 100$ grid of integers arranged sequentially in the following way:
\[\begin{array}{r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r}
1,&2,&3,&\ldots,&100\\
101,&102,&103,&\ldots,&200\\
201,&202,&203,&\ldots,&300\\
\vdots&&&&\vdots\\
9901,&9902,&9903,&\ldots,&10000
\e... | 50 | 0 | 16 | 0.0625 | 16 |
Given that $a > 2$ and $b > 2$, and the equation $\frac{1}{2}\log_{2}(a+b)+\log_{2} \frac{ \sqrt {2}}{a}= \frac{1}{2}\log_{2} \frac {1}{a+b}+\log_{2} \frac {b}{ \sqrt {2}}$ holds, find the value of $\log_{2}(a-2)+\log_{2}(b-2)$. Express your answer as a single numerical value. | 2 | 0.09375 | Given that $a > 9$ and $b > 9$, and the equation $\frac{1}{2}\log_{3}(a+b)+\log_{3} \frac{ 3}{a}= \frac{1}{2}\log_{3} \frac {1}{a+b}+\log_{3} \frac {b}{ 3}$ holds, find the value of $\log_{3}(a-9)+\log_{3}(b-9)$. Express your answer as a single numerical value. | 4 | 0 | 16 | 0 | 16 |
Given natural numbers \( a, b, \) and \( c \) with the sum \( S \), satisfying \( a + b = 1014 \), \( c - b = 497 \), and \( a > b \). Calculate the maximum value of \( S \). Express your answer as a single integer. | 2017 | 0.125 | Given natural numbers \( x, y, \) and \( z \) with the sum \( T \), satisfying \( x + y = 820 \), \( z - y = 315 \), and \( x > y \). Calculate the maximum value of \( T \). Express your answer as a single integer. | 1544 | 0.375 | 16 | 0.1875 | 16 |
If the value of the algebraic expression $x^{2}-x$ is $1$, then the value of the algebraic expression $1+2x-2x^{2}$ is ______. | -1 | 0.328125 | If the value of the algebraic expression $y^{2}-3y$ is $2$, then the value of the algebraic expression $4+9y-3y^{2}$ is ______. | -2 | 0.5 | 16 | 0.3125 | 16 |
Given sets A = {1, 2, 3,..., 10} and B = {1, 2, 3, 4}, where C is a subset of A such that C ∩ B ≠ ∅, calculate the number of subsets C that satisfy this condition. | 960 | 0.4375 | Given sets $X = \{1, 2, 3, \dots, 12\}$ and $Y = \{2, 4, 6, 8, 10\}$, where $Z$ is a subset of $X$ such that $Z \cap Y \neq \emptyset$, calculate the number of subsets $Z$ that satisfy this condition. | 3968 | 0 | 16 | 0.3125 | 16 |
If $AB=2$, what is the perimeter of parallelogram ABCD? | 6 | 0.015625 | Three consecutive vertices of a parallelogram $PQRS$ are given by the coordinates $P(0, 0)$, $Q(6, 0)$, and $R(11, 12)$. What is the perimeter of parallelogram $PQRS$? | 38 | 0 | 16 | 0.4375 | 16 |
Anastasia's tablecloth has a regular pattern. The tablecloth can be divided into 25 equal squares, with half of each of the 16 outer squares colored black. Find the percentage of Anastasia's tablecloth that is black. Express your answer as a percentage. | 32 | 0.15625 | Beatrice's rectangular quilt has a regular pattern. The quilt can be divided into 50 equal squares arranged in a 5 by 10 grid, with half of each of the 26 outer squares colored purple. Find the percentage of Beatrice's quilt that is purple. Express your answer as a percentage. | 26 | 0.3125 | 16 | 0.0625 | 16 |
Simplify first, then evaluate: $a\left(1-2a\right)+2\left(a+1\right)\left(a-1\right)$, where $a=2023$. | 2021 | 0.46875 | Simplify first, then evaluate: $m\left(5-2m\right)+2\left(m+3\right)\left(m-3\right)$, where $m=400$. | 1982 | 0.6875 | 16 | 0.75 | 16 |
Mary and Jenna are playing a game in which each girl rolls a twenty-sided die with numbers from 1 to 20 on the faces. If the number on the die that Mary rolls is a factor of the number on the die that Jenna rolls, Mary wins. If the number on the die that Jenna rolls is a factor of the number on the die that Mary roll... | 20 | 0.140625 | Leo and Maya are playing a game in which each player rolls a thirty-sided die with numbers from 1 to 30 on the faces. If the number on the die that Leo rolls is a factor of the number on the die that Maya rolls, Leo wins. If the number on the die that Maya rolls is a factor of the number on the die that Leo rolls, Maya... | 30 | 0.125 | 16 | 0.0625 | 16 |
It is known that all the vertices of a regular quadrilateral prism are on the surface of a sphere, and the height of the prism is 2. The surface area of the sphere is $12\pi$. Then, the volume of this regular quadrilateral prism is ___. | 12 | 0.28125 | It is known that all the vertices of a regular quadrilateral prism lie on the surface of a sphere, and the height of the prism is $6$. If the surface area of the sphere is $68\pi$, then find the volume of this regular quadrilateral prism. | 96 | 0.375 | 16 | 0 | 16 |
Let the complex number $z$ satisfy $zi=(1-i)^{2}$. Calculate the conjugate of the complex number $z$, denoted as $\overline{z}$. Express your answer as a complex number in the form $a + bi$. | -2 | 0.40625 | Let the complex number $z$ satisfy $zi=(3-3i)^{2}$. Calculate the conjugate of the complex number $z$, denoted as $\overline{z}$. Express your answer as a complex number in the form $a + bi$. | -18 | 0.75 | 16 | 0.625 | 16 |
Find the number of integral terms in the binomial expansion of $\left( \frac {y}{ \sqrt {x}} - \frac {x}{ \sqrt {y}} \right)^{16}$. Express your answer as a whole number. | 3 | 0.015625 | Find the number of integral terms in the binomial expansion of $\left( \frac {v}{ \sqrt {u}} + \frac {u}{ \sqrt {v}} \right)^{20}$. Express your answer as a whole number. | 3 | 0 | 16 | 0 | 16 |
Given $\triangle ABC$, where $BC=4$, $AC=8$, and $\angle C=60^{\circ}$, find the dot product of $\overrightarrow{BC}$ and $\overrightarrow{CA}$. | -16 | 0.09375 | Given $\triangle XYZ$, where $YZ=10$, $XZ=14$, and $\angle Z=60^{\circ}$, find the dot product of $\overrightarrow{YZ}$ and $\overrightarrow{ZX}$. | -70 | 0.0625 | 16 | 0.3125 | 16 |
As shown in the diagram, circles \( \odot O_{1} \) and \( \odot O_{2} \) are externally tangent. The line segment \( O_{1}O_{2} \) intersects \( \odot O_{1} \) at points \( A \) and \( B \), and intersects \( \odot O_{2} \) at points \( C \) and \( D \). Circle \( \odot O_{3} \) is internally tangent to \( \odot O_{1} ... | 1.2 | 0.21875 | As shown in the diagram, circles $\omega_1$ and $\omega_2$ are externally tangent. The line connecting their centers intersects $\omega_1$ at points $P$ and $Q$, and intersects $\omega_2$ at points $R$ and $S$, such that the points lie in the order $P, Q, R, S$ on the line (note that $Q$ and $R$ coincide at the point o... | 2.5 | 0 | 16 | 0 | 16 |
If the graph of the function $f(x)=\sin 2x+a\cos 2x$ is symmetrical about the line $x= \frac {\pi}{8}$, then the real number $a=$ _____ . | 1 | 0.328125 | If the graph of the function $f(x)=\sqrt{3}\sin 2x+a\cos 2x$ is symmetrical about the line $x= \frac {\pi}{12}$, then the real number $a=$ _____ . | 3 | 0.125 | 16 | 0.125 | 16 |
Given the polynomial p = x^4 + 6x^3 + 11x^2 + 3x + 31, how many integer values of x make this polynomial a perfect square? Express your answer as a single integer. | 1 | 0.109375 | Given the polynomial $P(x) = x^4 + 2x^3 + 3x^2 - 3x + 41$, how many integer values of $x$ make this polynomial a perfect square? Express your answer as a single integer. | 1 | 0.0625 | 16 | 0 | 16 |
Given that the angle between the plane vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ is $\frac{π}{6}$ and $|\overrightarrow{m}|=\sqrt{3}, |\overrightarrow{n}|=2$, in $\Delta ABC$, $\overrightarrow{AB}=2\overrightarrow{m}+2\overrightarrow{n}$, $\overrightarrow{AC}=2\overrightarrow{m}-6\overrightarrow{n}$, $D$ is... | 2 | 0.125 | Given that the angle between the plane vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ is $\frac{2\pi}{3}$ and $|\overrightarrow{u}|=2, |\overrightarrow{v}|=3$. In $\Delta PQR$, $\overrightarrow{PQ}=4\overrightarrow{u}+5\overrightarrow{v}$ and $\overrightarrow{PR}=2\overrightarrow{u}-\overrightarrow{v}$. If $M$ i... | 6 | 0.5 | 16 | 0.1875 | 16 |
Andy and Bob went to the canteen to buy snacks. They spent the same amount. Andy bought a can of soda at $1 and two hamburgers at $2 each. Bob ordered two sandwiches for $3 and a can of fruit drink. How much did Bob's fruit drink cost? | 2 | 0.421875 | Alice and Clara went to the stationery store to buy supplies. They spent the same amount. Alice bought a notebook for $3 and three pens at $2 each. Clara ordered two folders for $5 and a pack of stickers. How much did Clara's pack of stickers cost? | 4 | 0.3125 | 16 | 0.5 | 16 |
Let $ ABC$ be an equilateral triangle. Let $ \Omega$ be its incircle (circle inscribed in the triangle) and let $ \omega$ be a circle tangent externally to $ \Omega$ as well as to sides $ AB$ and $ AC$ . Determine the ratio of the radius of $ \Omega$ to the radius of $ \omega$ . | 2 | 0.078125 | Let $ABC$ be an equilateral triangle. Let $\Omega$ be its incircle (the circle inscribed in the triangle) and let $\omega$ be a smaller circle tangent externally to $\Omega$ as well as to sides $AB$ and $AC$. If the radius of $\Omega$ is 24, determine the radius of $\omega$. | 8 | 0.0625 | 16 | 0.0625 | 16 |
Given that $a$ is a root of the equation $2x^{2}+3x-4=0$, the value of the algebraic expression $2a^{2}+3a$ is equal to \_\_\_\_\_\_. | 4 | 0.328125 | Given that $k$ is a root of the equation $4x^2-2x-9=0$, the value of the algebraic expression $4k^2-2k$ is equal to ______. | 9 | 0.5625 | 16 | 0.5625 | 16 |
Among the coordinates $(x,y)$ $(1\leq x,y\leq 101)$ , choose some points such that there does not exist $4$ points which form a isoceles trapezium with its base parallel to either the $x$ or $y$ axis(including rectangles). Find the maximum number of coordinate points that can be chosen. | 201 | 0.015625 | null | null | 0 | 16 | null | 0 |
A library has six identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least one book is in the library and at least one is checked out? ... | 5 | 0.578125 | A cafe has eight identical mugs. At any given time, some of these mugs are on the display shelf and some are in the wash bin. How many different ways are there for some of the mugs to be on the display shelf and the rest to be in the wash bin if at least one mug is on the shelf and at least one is in the wash bin? (The... | 7 | 0.8125 | 16 | 0.5 | 16 |
The point $A$ $(3,4)$ is reflected over the $x$-axis to $B$. Then $B$ is reflected over the line $y=x$ to $C$. What is the area of triangle $ABC$? | 28 | 0.5 | Point $P$ at $(4,6)$ is reflected across the $x$-axis to form point $Q$. Point $Q$ is then reflected across the line $y=x$ to form point $R$. What is the area of triangle $PQR$? | 60 | 0.5625 | 16 | 0.5 | 16 |
In triangle ABC with AB = AC = 3.6, a point D is taken on AB at a distance 1.2 from A. Point D is joined to E in the prolongation of AC so that triangle AED is equal in area to ABC. Calculate the length of AE. | 10.8 | 0.015625 | In an isosceles triangle PQR with PQ = PR = 6.0, a point S is chosen on PQ such that the distance from P to S is 2.4. Point S is connected to a point T located on the extension of PR past R, such that the area of triangle PST is exactly equal to the area of triangle PQR. Determine the length of PT. | 15 | 0.0625 | 16 | 0 | 16 |
If the function $f\left(x\right)=\frac{1}{2}\left(m-2\right){x}^{2}+\left(n-8\right)x+1\left(m\geqslant 0,n\geqslant 0\right)$ is monotonically decreasing in the interval $\left[\frac{1}{2},2\right]$, find the maximum value of $mn$. | 18 | 0.015625 | If the function $g(x)=\frac{1}{2}(a-2)x^2+(b-12)x+5$ ($a \ge 0, b \ge 0$) is monotonically decreasing in the interval $\left[\frac{1}{2}, 2\right]$, find the maximum value of $ab$. | 32 | 0 | 16 | 0 | 16 |
Find the greatest common divisor of $10! + 2$ and $11! + 8$. | 2 | 0.46875 | Find the greatest common divisor of $12! + 5$ and $13! + 30$. | 5 | 0.3125 | 16 | 0.3125 | 16 |
Given the hyperbola $C$: $\frac{x^2}{2} - \frac{y^2}{4} = 1$ with asymptote $l$: $y = \sqrt{2}x$, and foci $F_1(-\sqrt{6}, 0)$ and $F_2(\sqrt{6}, 0)$, find the distance from point $P$ on $l$ to the x-axis, given that $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 0$. | 2 | 0.21875 | Given the hyperbola $C$: $\frac{x^2}{3} - \frac{y^2}{9} = 1$ with asymptote $l$: $y = \sqrt{3}x$, and foci $F_1(-2\sqrt{3}, 0)$ and $F_2(2\sqrt{3}, 0)$, find the distance from point $P$ on $l$ to the x-axis, given that $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 0$. | 3 | 0.1875 | 16 | 0.1875 | 16 |
Find the number of functions $f(x),$ taking positive real numbers to real numbers, such that
\[f(xy) = f(x) f(y) - 2\]for all positive real numbers $x$ and $y.$ | 2 | 0.421875 | Find the number of functions $g(x),$ taking positive real numbers to real numbers, such that
\[g(xy) = g(x) g(y) - 12\]for all positive real numbers $x$ and $y.$ | 2 | 0.4375 | 16 | 0.5625 | 16 |
Observe the following equations
$$(1+x+x^{2})^{1} = 1+x+x^{2},$$
$$(1+x+x^{2})^{2} = 1+2x+3x^{2}+2x^{3}+x^{4},$$
$$(1+x+x^{2})^{3} = 1+3x+6x^{2}+7x^{3}+6x^{4}+3x^{5}+x^{6},$$
$$(1+x+x^{2})^{4} = 1+4x+10x^{2}+16x^{3}+19x^{4}+16x^{5}+10x^{6}+4x^{7}+x^{8},$$
...
If $(1+x+x^{2})^{6} = a_{0}+a_{1}x+a_{2}x^{2}+\l... | 21 | 0.171875 | Consider the following sequence of polynomial expansions:
$$(1+x+x^{2})^{1} = 1+x+x^{2},$$
$$(1+x+x^{2})^{2} = 1+2x+3x^{2}+2x^{3}+x^{4},$$
$$(1+x+x^{2})^{3} = 1+3x+6x^{2}+7x^{3}+6x^{4}+3x^{5}+x^{6},$$
$$(1+x+x^{2})^{4} = 1+4x+10x^{2}+16x^{3}+19x^{4}+16x^{5}+10x^{6}+4x^{7}+x^{8},$$
...
If $(1+x+x^{2})^{8} = ... | 36 | 0.3125 | 16 | 0.125 | 16 |
Given that $i$ is the imaginary unit, if the complex number $(a+i)^{2}i$ is a positive real number, find the value of the real number $a$. Express your answer as a single real number. | -1 | 0.359375 | Suppose $i$ represents the imaginary unit. If the complex number $(a-2i)^2 i$ evaluates to a strictly positive real number, determine the value of the real number $a$. Express your answer as a single real number. | 2 | 0.625 | 16 | 0.5 | 16 |
There are 7 parking spaces arranged in a row. Now, there are 3 different models of cars that need to be parked. If it is required that the remaining 4 parking spaces must be connected, find the number of distinct parking arrangements. Express your answer as a whole number. | 24 | 0.015625 | There are 9 display pedestals arranged in a row. Now, there are 4 different trophies that need to be placed on them. If it is required that the remaining 5 empty pedestals must be adjacent to one another, find the number of distinct display arrangements. Express your answer as a whole number. | 120 | 0 | 16 | 0.0625 | 16 |
Find the value of $\tan \left( -\frac{13}{4} \pi \right)$. Express your answer as a single number. | -1 | 0.5625 | Find the value of $\tan \left( -\frac{15}{4} \pi \right)$. Express your answer as a single number. | 1 | 0.625 | 16 | 0.3125 | 16 |
If $800670 = 8 \times 10^{x} + 6 \times 10^{y} + 7 \times 10^{z}$, where $x$, $y$, and $z$ are whole numbers, calculate the sum of $x$, $y$, and $z$. Express your answer as a whole number. | 8 | 0.25 | If $307020 = 3 \times 10^{a} + 7 \times 10^{b} + 2 \times 10^{c}$, where $a$, $b$, and $c$ are whole numbers, calculate the sum of $a$, $b$, and $c$. Express your answer as a whole number. | 9 | 0.1875 | 16 | 0 | 16 |
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