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 |
|---|---|---|---|---|---|---|
Let \(a\), \(b\), \(c\) be distinct complex numbers such that
\[
\frac{a+1}{2 - b} = \frac{b+1}{2 - c} = \frac{c+1}{2 - a} = k.
\]
Find the sum of all possible values of \(k\). | 1.5 | 0 | 7,793.375 | -1 | 7,793.375 | |
Given that acute angles $\alpha$ and $\beta$ satisfy $\alpha+2\beta=\frac{2\pi}{3}$ and $\tan\frac{\alpha}{2}\tan\beta=2-\sqrt{3}$, find the value of $\alpha +\beta$. | \frac{5\pi}{12} | 0.8125 | 4,936.625 | 4,185.384615 | 8,192 | |
For real numbers $t,$ consider the point of intersection of the triplet of lines $3x - 2y = 8t - 5$, $2x + 3y = 6t + 9$, and $x + y = 2t + 1$. All the plotted points lie on a line. Find the slope of this line. | -\frac{1}{6} | 0.125 | 6,827.5625 | 3,891.5 | 7,247 | |
Assuming $a \neq 3$, $b \neq 4$, and $c \neq 5$, what is the value in simplest form of the following expression?
\[\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}\] | -1 | 1. **Identify the pattern in the expression**: We are given the expression:
\[
\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}
\]
We notice that each fraction has a numerator and a denominator that are differences of variables and constants.
2. **Apply the property of differences**: Recall the ... | 0.9375 | 3,100.1875 | 2,760.733333 | 8,192 |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 499 | 0 | 8,191.5 | -1 | 8,191.5 | |
Given that $P$ is a moving point on the line $l: x-2y+4=0$, two tangents are drawn from point $P$ to the circle $C: x^{2}+y^{2}-2x=0$, with tangents intersecting at points $A$ and $B$. Find the minimum area of the circumcircle of quadrilateral $PACB$. | \frac{5\pi}{4} | 0 | 8,192 | -1 | 8,192 | |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of
\[\frac{x + y}{xyz}.\] | 16 | 0.75 | 6,278.1875 | 5,640.25 | 8,192 | |
Rectangle $ABCD$ has an area of $32$, and side $\overline{AB}$ is parallel to the x-axis. Side $AB$ measures $8$ units. Vertices $A,$ $B$, and $C$ are located on the graphs of $y = \log_a x$, $y = 2\log_a x$, and $y = 4\log_a x$, respectively. Determine the value of $a$.
A) $\sqrt[3]{\frac{1 + \sqrt{33}}{2} + 8}$
B) $\... | \sqrt[4]{\frac{1 + \sqrt{33}}{2} + 8} | 0 | 8,064.25 | -1 | 8,064.25 | |
Of the following complex numbers $z$, which one has the property that $z^5$ has the greatest real part? | -\sqrt{3} + i | We evaluate the fifth power of each answer choice and determine the real part of each result.
1. **For $\textbf{(A)}$:**
- $z = -2$
- $z^5 = (-2)^5 = -32$
- $\operatorname{Re}(z^5) = -32$
2. **For $\textbf{(E)}$:**
- $z = 2i$
- $z^5 = (2i)^5 = 32i$
- $\operatorname{Re}(z^5) = 0$
3. **For $\textbf{(... | 0 | 7,184.125 | -1 | 7,184.125 |
Given the sequence of even counting numbers starting from $2$, find the sum of the first $3000$ terms and the sequence of odd counting numbers starting from $3$, find the sum of the first $3000$ terms, and then calculate their difference. | -3000 | 0.3125 | 4,163.125 | 4,216.6 | 4,138.818182 | |
A factory has a fixed daily cost of 20,000 yuan, and the maximum daily production capacity is 360 units. The cost increases by 100 yuan for each unit produced. The revenue function for producing $x$ units of product per day is $R(x) = -\frac{1}{2}x^2 + 400x$. Let $L(x)$ and $P(x)$ represent the daily profit and average... | 95 | 0.6875 | 5,780.875 | 5,455.818182 | 6,496 | |
Given that when 81849, 106392, and 124374 are divided by an integer \( n \), the remainders are equal. If \( a \) is the maximum value of \( n \), find \( a \). | 243 | 0.75 | 4,934.625 | 4,067.166667 | 7,537 | |
For real numbers $a,$ $b,$ and $c,$ the matrix
\[\begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix}\]is not invertible. List all possible values of
\[\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}.\] | -3 | 0.0625 | 3,978.375 | 4,572 | 3,938.8 | |
A sphere is inscribed in a right cone with base radius \(15\) cm and height \(30\) cm. Find the radius \(r\) of the sphere, which can be expressed as \(b\sqrt{d} - b\) cm. What is the value of \(b + d\)? | 12.5 | 0 | 8,178.375 | -1 | 8,178.375 | |
Let \( ABC \) be an equilateral triangle with side length 16. Three circles of the same radius \( r \) are mutually tangent to each other, and each circle is also tangent to two sides of the triangle. The radius \( r \) can be expressed as \( r = \sqrt{a} - b \), where \( a \) and \( b \) are integers. Determine \( a +... | 52 | 0.3125 | 7,492.75 | 5,954.4 | 8,192 | |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | \sqrt{40} | 0 | 3,085.25 | -1 | 3,085.25 | |
On the edge \(AD\) and the diagonal \(A_1C\) of the parallelepiped \(ABCDA_1B_1C_1D_1\), points \(M\) and \(N\) are taken respectively, such that the line \(MN\) is parallel to the plane \(BDC_1\) and \(AM:AD = 1:5\). Find the ratio \(CN:CA_1\). | 3/5 | 0.0625 | 6,952.625 | 4,584 | 7,110.533333 | |
What is the value of \((-1)^{3}+(-1)^{2}+(-1)\)? | -1 | Since -1 raised to an even exponent equals 1 and -1 raised to an odd exponent equals -1, then \((-1)^{3}+(-1)^{2}+(-1)=-1+1-1=-1\). | 1 | 1,036.75 | 1,036.75 | -1 |
Three vertices of parallelogram $PQRS$ are $P(-3,-2)$, $Q(1,-5)$, $R(9,1)$ with $P$ and $R$ diagonally opposite. The sum of the coordinates of vertex $S$ is: | 9 | To find the coordinates of vertex $S$ of parallelogram $PQRS$, where $P$ and $R$ are diagonally opposite, we can use the property that the diagonals of a parallelogram bisect each other. This means that the midpoint of diagonal $PR$ is also the midpoint of diagonal $QS$.
1. **Calculate the midpoint of diagonal $PR$:**... | 1 | 3,178.6875 | 3,178.6875 | -1 |
Rohan wants to cut a piece of string into nine pieces of equal length. He marks his cutting points on the string. Jai wants to cut the same piece of string into only eight pieces of equal length. He marks his cutting points on the string. Yuvraj then cuts the string at all the cutting points that are marked. How many p... | 16 | 0.0625 | 4,433 | 440 | 4,699.2 | |
In $\triangle ABC$, if $bc=3$, $a=2$, then the minimum value of the area of the circumcircle of $\triangle ABC$ is $\_\_\_\_\_\_$. | \frac{9\pi}{8} | 0.6875 | 7,110.875 | 7,010.181818 | 7,332.4 | |
Let $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2 +y^2-10x-24y+153=0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_2$ and internally tangent to $w_1.$ Given that $m^2=\frac pq,$ where $p$ and ... | 169 | We use the same reflection as in Solution 2. As $OF_1'=OF_2=13$, we know that $\triangle OF_1'F_2$ is isosceles. Hence $\angle F_2F_1'O=\angle F_1'F_2O$. But by symmetry, we also know that $\angle OF_1T=\angle F_2F_1'O$. Hence $\angle OF_1T=\angle F_1'F_2O$. In particular, as $\angle OF_1T=\angle OF_2T$, this implies t... | 0.375 | 7,649.4375 | 7,121.833333 | 7,966 |
The moisture content of freshly cut grass is $70\%$, while the moisture content of hay is $16\%. How much grass needs to be cut to obtain 1 ton of hay? | 2800 | 0.125 | 5,301.375 | 5,202.5 | 5,315.5 | |
A triangle with side lengths in the ratio 2:3:4 is inscribed in a circle of radius 4. What is the area of the triangle? | 3\sqrt{15} | 0 | 7,019.1875 | -1 | 7,019.1875 | |
The area of rectangle \(ABCD\) is 48, and the diagonal is 10. On the plane where the rectangle is located, a point \(O\) is chosen such that \(OB = OD = 13\). Find the distance from point \(O\) to the vertex of the rectangle farthest from it. | 17 | 0 | 8,192 | -1 | 8,192 | |
The Student council has 24 members: 12 boys and 12 girls. A 5-person committee is selected at random. What is the probability that the committee includes at least one boy and at least one girl? | \frac{455}{472} | 0 | 5,243.375 | -1 | 5,243.375 | |
Find \(\lim _{x \rightarrow -1} \frac{3 x^{4} + 2 x^{3} - x^{2} + 5 x + 5}{x^{3} + 1}\). | -\frac{1}{3} | 0 | 3,100.375 | -1 | 3,100.375 | |
When any two numbers are taken from the set {0, 1, 2, 3, 4, 5} to perform division, calculate the number of different sine values that can be obtained. | 10 | 0 | 8,162.3125 | -1 | 8,162.3125 | |
Evaluate $\sum_{n=2}^{17} \frac{n^{2}+n+1}{n^{4}+2 n^{3}-n^{2}-2 n}$. | \frac{592}{969} | Observe that the denominator $n^{4}+2 n^{3}-n^{2}-2 n=n(n-1)(n+1)(n+2)$. Thus we can rewrite the fraction as $\frac{n^{2}-n+1}{n^{4}+2 n^{3}-n^{2}-2 n}=\frac{a}{n-1}+\frac{b}{n}+\frac{c}{n+1}+\frac{d}{n+2}$ for some real numbers $a, b, c$, and $d$. This method is called partial fractions. Condensing the right hand side... | 0 | 8,015.125 | -1 | 8,015.125 |
Given that the sum of the first three terms of a geometric sequence $\{a_n\}$ is $3$ and the sum of the first nine terms is $39$, calculate the value of the sum of the first six terms. | 12 | 0.125 | 8,140.75 | 7,782 | 8,192 | |
(This question is worth 14 points.)
A newspaper stand in a city buys the "Evening News" from the newspaper office at a price of 0.20 yuan per copy and sells it at 0.30 yuan per copy. The unsold newspapers can be returned to the newspaper office at a price of 0.05 yuan per copy. In a month (calculated as 30 days), ther... | 825 | 0 | 8,019.5625 | -1 | 8,019.5625 | |
The solution of $8x+1\equiv 5 \pmod{12}$ is $x\equiv a\pmod{m}$ for some positive integers $m\geq 2$ and $a<m$. Find $a+m$. | 5 | 1 | 2,439.8125 | 2,439.8125 | -1 | |
Find the number of 10-digit numbers $\overline{a_{1} a_{2} \cdots a_{10}}$ which are multiples of 11 such that the digits are non-increasing from left to right, i.e. $a_{i} \geq a_{i+1}$ for each $1 \leq i \leq 9$. | 2001 | It is well known that $\overline{a_{1} a_{2} \cdots a_{10}}$ is divisible by 11 if and only if $S=a_{1}-a_{2}+a_{3}-\cdots-a_{10}$ is. By the non-increasing condition, we deduce that $$S=\left(a_{1}-a_{2}\right)+\left(a_{3}-a_{4}\right)+\cdots+\left(a_{9}-a_{10}\right) \geq 0$$ Also, $$S=a_{1}-\left(a_{2}-a_{3}\right)-... | 0 | 8,182 | -1 | 8,182 |
Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid? | 750 | 1. **Identify the Components of the Trapezoid**:
Given that $ABCD$ is a trapezoid with $AB$ and $CD$ as the parallel sides, and the altitude (height) from $AB$ to $CD$ is $12$. The lengths of the sides are $AD = 15$, $AB = 50$, $BC = 20$.
2. **Draw Altitudes and Form Right Triangles**:
By drawing altitudes from ... | 0.3125 | 4,888 | 5,160.8 | 4,764 |
Let the set \( S = \{1, 2, \cdots, 15\} \). Define \( A = \{a_{1}, a_{2}, a_{3}\} \) as a subset of \( S \), such that \( (a_{1}, a_{2}, a_{3}) \) satisfies \( 1 \leq a_{1} < a_{2} < a_{3} \leq 15 \) and \( a_{3} - a_{2} \leq 6 \). Find the number of such subsets that satisfy these conditions. | 371 | 0.25 | 7,858.75 | 6,859 | 8,192 | |
In the diagram, $AB = 25 \mbox{ cm},$ $AC = 20 \mbox{ cm},$ and $\angle A = 90^\circ.$ What is the area of triangle $ABC?$
[asy]
draw((0,0)--(25,0)--(0,20)--cycle,black+linewidth(1));
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black+linewidth(1));
label("$A$",(0,0),SW);
label("$B$",(25,0),SE);
label("$C$",(0,20),NW);
[/as... | 250 | 0.875 | 1,770.125 | 1,956.428571 | 466 | |
Given that the lines $l_{1}$: $ax+y+3=0$ and $l_{2}$: $2x+\left(a-1\right)y+a+1=0$ are parallel, find the value of $a$. | -1 | 0.1875 | 6,420.875 | 4,292 | 6,912.153846 | |
For positive integers $n,$ let $s(n)$ be the sum of the digits of $n.$ Over all four-digit positive integers $n,$ which value of $n$ maximizes the ratio $\frac{s(n)}{n}$ ?
*Proposed by Michael Tang* | 1099 | 0 | 8,192 | -1 | 8,192 | |
The domain of the function $y=\sin x$ is $[a,b]$, and its range is $\left[-1, \frac{1}{2}\right]$. Calculate the maximum value of $b-a$. | \frac{4\pi}{3} | 0.125 | 7,714.75 | 5,837 | 7,983 | |
An equilateral triangle $ABC$ shares a side with a square $BCDE$ . If the resulting pentagon has a perimeter of $20$ , what is the area of the pentagon? (The triangle and square do not overlap). | 16 + 4\sqrt{3} | 0.125 | 7,735.0625 | 4,536.5 | 8,192 | |
Given a tetrahedron P-ABC, if PA, PB, and PC are mutually perpendicular, and PA=2, PB=PC=1, then the radius of the inscribed sphere of the tetrahedron P-ABC is \_\_\_\_\_\_. | \frac {1}{4} | 0.6875 | 6,246 | 5,361.454545 | 8,192 | |
Inside a right triangle \(ABC\) with hypotenuse \(AC\), a point \(M\) is chosen such that the areas of triangles \(ABM\) and \(BCM\) are one-third and one-quarter of the area of triangle \(ABC\) respectively. Find \(BM\) if \(AM = 60\) and \(CM = 70\). If the answer is not an integer, round it to the nearest whole numb... | 38 | 0.125 | 7,876.3125 | 5,666.5 | 8,192 | |
A and B began riding bicycles from point A to point C, passing through point B on the way. After a while, A asked B, "How many kilometers have we ridden?" B responded, "We have ridden a distance equivalent to one-third of the distance from here to point B." After riding another 10 kilometers, A asked again, "How many k... | \frac{40}{3} | 0 | 8,037.5625 | -1 | 8,037.5625 | |
In \(\triangle ABC\), \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively. Given \(a+c=2b\) and \(A-C=\frac{\pi}{3}\), find the value of \(\sin B\). | \frac{\sqrt{39}}{8} | 0 | 8,016.6875 | -1 | 8,016.6875 | |
Points $A, B$, and $C$ lie in that order on line $\ell$, such that $A B=3$ and $B C=2$. Point $H$ is such that $C H$ is perpendicular to $\ell$. Determine the length $C H$ such that $\angle A H B$ is as large as possible. | \sqrt{10} | Let $\omega$ denote the circumcircle of triangle $A B H$. Since $A B$ is fixed, the smaller the radius of $\omega$, the bigger the angle $A H B$. If $\omega$ crosses the line $C H$ in more than one point, then there exists a smaller circle that goes through $A$ and $B$ that crosses $C H$ at a point $H^{\prime}$. But an... | 0.6875 | 6,345.4375 | 5,506.090909 | 8,192 |
In a psychiatric hospital, there is a chief doctor and many madmen. During the week, each madman bit someone once a day (possibly themselves). At the end of the week, it was found that each patient has two bites, and the chief doctor has one hundred bites. How many madmen are there in the hospital? | 20 | 0.1875 | 6,540.1875 | 2,667 | 7,434 | |
Simplify $(2^5+7^3)(2^3-(-2)^2)^8$. | 24576000 | 0.9375 | 3,664.3125 | 3,647.4 | 3,918 | |
Triangle $A B C$ has perimeter 1. Its three altitudes form the side lengths of a triangle. Find the set of all possible values of $\min (A B, B C, C A)$. | \left(\frac{3-\sqrt{5}}{4}, \frac{1}{3}\right] | Let $a, b, c$ denote the side lengths $B C, C A$, and $A B$, respectively. Without loss of generality, assume $a \leq b \leq c$; we are looking for the possible range of $a$. First, note that the maximum possible value of $a$ is $\frac{1}{3}$, which occurs when $A B C$ is equilateral. It remains to find a lower bound f... | 0 | 7,848.3125 | -1 | 7,848.3125 |
Let $a,$ $b,$ $c,$ and $d$ be the roots of \[x^4 + 8x^3 + 9x^2 + 5x + 4 = 0.\]Find the value of \[\frac{1}{ab} + \frac{1}{ac} + \frac{1}{ad} + \frac{1}{bc} + \frac{1}{bd} + \frac{1}{cd}.\] | \tfrac 94 | 0.9375 | 2,766.875 | 2,405.2 | 8,192 | |
In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$. | 306 | Refer to the diagram in solution 2; let $a^2=[E'EP]$, $b^2=[D'DP]$, and $c^2=[F'FP]$. Now, note that $[E'BD]$, $[D'DP]$, and $[E'EP]$ are similar, so through some similarities we find that $\frac{E'P}{PD}=\frac{a}{b}\implies\frac{E'D}{PD}=\frac{a+b}{b}\implies[E'BD]=b^2\left(\frac{a+b}{b}\right)^2=(a+b)^2$. Similarly, ... | 0 | 7,215.3125 | -1 | 7,215.3125 |
Thirty identical toothpicks were used to create the figure below. There are over 25 triangles in the figure. What is the fewest number of toothpicks that could be removed so that no triangles remain?
[asy]
draw((0,0)--(8,0), linewidth(1));
draw(2dir(60)--(2dir(60)+(6,0)), linewidth(1));
draw(4dir(60)--(4dir(60)+(4,0))... | 10 | 0.1875 | 8,001.1875 | 7,888.333333 | 8,027.230769 | |
Given vectors $\overrightarrow{a}=(\sin x,\frac{3}{2})$ and $\overrightarrow{b}=(\cos x,-1)$.
$(1)$ When $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $\sin 2x$.
$(2)$ Find the minimum value of $f(x)=(\overrightarrow{a}+\overrightarrow{b}) \cdot \overrightarrow{b}$ for $x \in [-\frac{\pi}{2},... | -\frac{\sqrt{2}}{2} | 0 | 5,454.25 | -1 | 5,454.25 | |
A, B, and C start from the same point on a circular track with a circumference of 360 meters: A starts first and runs in the counterclockwise direction; before A completes a lap, B and C start simultaneously and run in the clockwise direction; when A and B meet for the first time, C is exactly half a lap behind them; a... | 90 | 0 | 8,192 | -1 | 8,192 | |
Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle?
*Auth... | 256 | 0.375 | 7,208.25 | 5,568.666667 | 8,192 | |
Evaluate \(\left(a^a - a(a-2)^a\right)^a\) when \( a = 4 \). | 1358954496 | 0.75 | 3,314.9375 | 4,248.166667 | 515.25 | |
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$ , where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$ , $2+2$ , $2+1+1$ , $1+2+1$ , $1+1+2$ , and $1+1+1+1$ . Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd. | \[ 2047 \] | First of all, note that $f(n)$ = $\sum_{i=0}^{k} f(n-2^{i})$ where $k$ is the largest integer such that $2^k \le n$ . We let $f(0) = 1$ for convenience.
From here, we proceed by induction, with our claim being that the only $n$ such that $f(n)$ is odd are $n$ representable of the form $2^{a} - 1, a \in \mathbb{Z}$
We... | 0 | 8,192 | -1 | 8,192 |
Each time you click a toggle switch, the switch either turns from *off* to *on* or from *on* to *off*. Suppose that you start with three toggle switches with one of them *on* and two of them *off*. On each move you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive ... | 61 | 0 | 7,977.6875 | -1 | 7,977.6875 | |
Find the positive value of $x$ that satisfies the equation:
\[\log_2 (x + 2) + \log_{4} (x^2 - 2) + \log_{\frac{1}{2}} (x + 2) = 5.\] | \sqrt{1026} | 0.875 | 3,203.3125 | 3,222 | 3,072.5 | |
An equilateral triangle is drawn with a side of length $a$. A new equilateral triangle is formed by joining the midpoints of the sides of the first one. Then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. The limit of the sum of the perimeters of all the t... | 6a | 1. **Identify the sequence of perimeters**:
- The perimeter of the first equilateral triangle is $3a$ since each side is $a$ and there are three sides.
- Each subsequent triangle is formed by joining the midpoints of the sides of the previous triangle, thus each side of the new triangle is half the length of the... | 1 | 2,238.3125 | 2,238.3125 | -1 |
The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is | \{x \mid 2 < x < 3\} | 1. **Understanding the Problem:**
The problem asks us to find the set of values for $x$ such that the sum of the greatest integer less than or equal to $x$ (denoted $\lfloor x \rfloor$) and the least integer greater than or equal to $x$ (denoted $\lceil x \rceil$) equals 5.
2. **Analyzing the Floor and Ceiling Func... | 0 | 2,662.875 | -1 | 2,662.875 |
Compute $$\sum_{n=0}^{\infty} \frac{n}{n^{4}+n^{2}+1}$$ | 1/2 | Note that $$n^{4}+n^{2}+1=\left(n^{4}+2 n^{2}+1\right)-n^{2}=\left(n^{2}+1\right)^{2}-n^{2}=\left(n^{2}+n+1\right)\left(n^{2}-n+1\right)$$ Decomposing into partial fractions, we find that $$\frac{n}{n^{4}+n^{2}+1}=\frac{1}{2}\left(\frac{1}{n^{2}-n+1}-\frac{1}{n^{2}+n+1}\right)$$ Now, note that if $f(n)=\frac{1}{n^{2}-n... | 0.6875 | 6,542.3125 | 6,109 | 7,495.6 |
Suppose $a$ and $b$ are the points obtained by throwing a dice in order, and the function is $f(x)=\frac{1}{2}ax^{2}+bx+1$.
(1) Find the probability that $f(x)$ is a decreasing function in the interval $(-\infty,-1]$;
(2) Find the probability that the function $f(x)$ has zero points. | \frac{2}{3} | 0.0625 | 4,418.1875 | 6,579 | 4,274.133333 | |
Let \( S = \{1, 2, \cdots, 2005\} \). If in any set of \( n \) pairwise coprime numbers in \( S \) there is at least one prime number, find the minimum value of \( n \). | 16 | 0.0625 | 8,115.125 | 8,192 | 8,110 | |
In SHORT BINGO, a $5\times5$ card is filled by marking the middle square as WILD and placing 24 other numbers in the remaining 24 squares.
Specifically a card is made by placing 5 distinct numbers from the set $1-10$ in the first column, 5 distinct numbers from $11-20$ in the second column, 4 distinct numbers $21-30$... | 30240 | 1 | 1,154.9375 | 1,154.9375 | -1 | |
The length of rectangle $ABCD$ is 5 inches and its width is 3 inches. Diagonal $AC$ is divided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is: | \frac{5}{2} | 1. **Draw the rectangle and identify key components**: Let rectangle $ABCD$ have $AB = 5$ inches (length) and $AD = 3$ inches (width). Diagonal $AC$ is drawn, which divides the rectangle into two right triangles, $ABC$ and $ADC$.
2. **Calculate the length of diagonal $AC$ using the Pythagorean Theorem**:
\[
AC =... | 1 | 5,115.8125 | 5,115.8125 | -1 |
A right cone has a base with a circumference of $20\pi$ inches and a height of 40 inches. The height of the cone is reduced while the circumference stays the same. After reduction, the volume of the cone is $400\pi$ cubic inches. What is the ratio of the new height to the original height, and what is the new volume? | 400\pi | 1 | 2,399.8125 | 2,399.8125 | -1 | |
If the universal set $U = \{-1, 0, 1, 2\}$, and $P = \{x \in \mathbb{Z} \,|\, -\sqrt{2} < x < \sqrt{2}\}$, determine the complement of $P$ in $U$. | \{2\} | 1 | 1,407.25 | 1,407.25 | -1 | |
Consider the set of 5-tuples of positive integers at most 5. We say the tuple $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$ is perfect if for any distinct indices $i, j, k$, the three numbers $a_{i}, a_{j}, a_{k}$ do not form an arithmetic progression (in any order). Find the number of perfect 5-tuples. | 780 | There are two situations. 1. The multiset is aabbc; the only condition here is $c \neq \frac{1}{2}(a+b)$, for $\left(\binom{5}{3}-|S|\right) \cdot\binom{3}{1}=18$ such triples, where $S$ is the set of unordered triples $(a, b, c)$ which do not satisfy the condition, and $S=\{(1,2,3),(2,3,4),(3,4,5),(1,3,5)\}$. Each one... | 0 | 8,192 | -1 | 8,192 |
Given a triangle \( \triangle ABC \) with interior angles \( \angle A, \angle B, \angle C \) and opposite sides \( a, b, c \) respectively, where \( \angle A - \angle C = \frac{\pi}{2} \) and \( a, b, c \) are in arithmetic progression, find the value of \( \cos B \). | \frac{3}{4} | 0.5 | 7,164.3125 | 6,136.625 | 8,192 | |
Given a [rational number](https://artofproblemsolving.com/wiki/index.php/Rational_number), write it as a [fraction](https://artofproblemsolving.com/wiki/index.php/Fraction) in lowest terms and calculate the product of the resulting [numerator](https://artofproblemsolving.com/wiki/index.php/Numerator) and [denominator](... | 128 | 0.6875 | 6,019.5 | 5,032 | 8,192 | |
Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$. | 108 | Let $M$ be midpoint $BC, BM = x, AB = y, \angle IBM = \alpha.$
$BI$ is the bisector of $\angle ABM$ in $\triangle ABM.$ $BI = \frac {2 xy \cos \alpha}{x+y} = 8, \cos \alpha = \frac {x}{8} \implies \frac {x^2 y}{x+y} = 32.$ \[y = \frac {32 x} {x^2 - 32}.\] $BC = 2x$ is integer, $5.5^2 < 32 \implies x \ge 6.$ $BM < BI \i... | 0 | 8,192 | -1 | 8,192 |
In terms of $k$, for $k>0$, how likely is it that after $k$ minutes Sherry is at the vertex opposite the vertex where she started? | \frac{1}{6}+\frac{1}{3(-2)^{k}} | The probability that she ends up on the original vertex is equal to the probability that she ends up on the top vertex, and both are equal to $\frac{1-p(n)}{2}$ for $n \geq 1$. From the last problem, $$\begin{aligned} p(n+1) & =1-\frac{p(n)}{2} \\ p(n+1)-\frac{2}{3} & =-\frac{1}{2}\left(p(n)-\frac{2}{3}\right) \end{ali... | 0 | 8,192 | -1 | 8,192 |
$JKLM$ is a square and $PQRS$ is a rectangle. If $JK$ is parallel to $PQ$, $JK = 8$ and $PS = 2$, then the total area of the shaded regions is: | 48 | 0.1875 | 6,630.1875 | 5,628 | 6,861.461538 | |
The quadrilateral \(ABCD\) is circumscribed around a circle with a radius of \(1\). Find the greatest possible value of \(\left| \frac{1}{AC^2} + \frac{1}{BD^2} \right|\). | 1/4 | 0 | 8,192 | -1 | 8,192 | |
Integers from 1 to 100 are placed in a row in some order. Let us call a number *large-right*, if it is greater than each number to the right of it; let us call a number *large-left*, is it is greater than each number to the left of it. It appears that in the row there are exactly $k$ large-right numbers and exactly ... | 50 | 0.3125 | 7,832.875 | 7,042.8 | 8,192 | |
If the scores for innovation capability, innovation value, and innovation impact are $8$ points, $9$ points, and $7$ points, respectively, and the total score is calculated based on the ratio of $5:3:2$ for the three scores, calculate the total score of the company. | 8.1 | 0.6875 | 432.3125 | 462.909091 | 365 | |
The ratio of the areas of two squares is $\frac{300}{147}$. Find the simplified form of the ratio of their side lengths, expressed as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. Additionally, if the perimeter of the larger square is 60 units, determine the side length of the smaller square. | 10.5 | 0 | 3,458.9375 | -1 | 3,458.9375 | |
Find the smallest number composed exclusively of ones that is divisible by 333...33 (where there are 100 threes in the number). | 300 | 0.0625 | 7,675.0625 | 4,689 | 7,874.133333 | |
Let $x$ and $y$ be complex numbers such that
\[\frac{x + y}{x - y} + \frac{x - y}{x + y} = 1.\]Find
\[\frac{x^4 + y^4}{x^4 - y^4} + \frac{x^4 - y^4}{x^4 + y^4}.\] | \frac{41}{20} | 0.9375 | 4,347.625 | 4,091.333333 | 8,192 | |
Find the number of positive integers \( n \) that satisfy
\[
(n - 1)(n - 3)(n - 5) \dotsm (n - 99) < 0.
\] | 25 | 0.1875 | 7,467.9375 | 5,914.333333 | 7,826.461538 | |
Distribute 5 students into two dormitories, A and B, with each dormitory accommodating at least 2 students. Find the number of distinct arrangements. | 20 | 0.375 | 7,087.5625 | 5,522 | 8,026.9 | |
In the trapezoid \(ABCD\) (\(AD \parallel BC\)), a perpendicular \(EF\) is drawn from point \(E\) (the midpoint of \(CD\)) to line \(AB\). Find the area of the trapezoid if \(AB = 5\) and \(EF = 4\). | 20 | 0.3125 | 7,401.625 | 6,884.8 | 7,636.545455 | |
In an acute triangle $\triangle ABC$, altitudes $\overline{AD}$ and $\overline{BE}$ intersect at point $H$. Given that $HD=6$ and $HE=3$, calculate $(BD)(DC)-(AE)(EC)$. | 27 | 0 | 8,053.25 | -1 | 8,053.25 | |
The increasing sequence \(1, 3, 4, 9, 10, 12, 13, \cdots\) consists of some positive integers that are either powers of 3 or sums of distinct powers of 3. Find the value of the 2014th term. | 88329 | 0.625 | 6,515.375 | 5,699.8 | 7,874.666667 | |
Consider the non-decreasing sequence of positive integers
\[1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,\cdots\]
in which the $n^{th}$ positive integer appears $n$ times. The remainder when the $1993^{rd}$ term is divided by $5$ is | 3 | 1. **Identify the sequence pattern**: The given sequence is such that each integer $n$ appears $n$ times. This means the sequence looks like:
\[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots\]
2. **Determine the position of each integer**: The position where each integer $n$ ends can be calculated by summing up the first $n$... | 0.875 | 3,701.375 | 3,345.357143 | 6,193.5 |
In the diagram, \( AB \) and \( CD \) intersect at \( E \). If \(\triangle BCE\) is equilateral and \(\triangle ADE\) is a right-angled triangle, what is the value of \( x \)? | 30 | 0.0625 | 7,885.1875 | 5,840 | 8,021.533333 | |
If $x = 202$ and $x^3y - 4x^2y + 2xy = 808080$, what is the value of $y$? | \frac{1}{10} | 0 | 8,192 | -1 | 8,192 | |
In triangle \( \triangle ABC \), it is given that \( \angle C=90^\circ \), \( \angle A=60^\circ \), and \( AC=1 \). Points \( D \) and \( E \) are on sides \( BC \) and \( AB \) respectively such that triangle \( \triangle ADE \) is an isosceles right triangle with \( \angle ADE=90^\circ \). Find the length of \( BE \)... | 4-2\sqrt{3} | 0.6875 | 6,918.75 | 6,724.272727 | 7,346.6 | |
The operation $\dagger$ is defined as $\frac{m}{n}\dagger\frac{p}{q} = (m)(p)(\frac{q}{n}).$ What is the simplified value of $\frac{7}{12}\dagger\frac{8}{3}$? | 14 | 1 | 2,638.5 | 2,638.5 | -1 | |
The perimeter of one square is $3$ times the perimeter of another square. The area of the larger square is how many times the area of the smaller square? | 9 | 1. **Define Variables:**
Let $a$ be the side length of the larger square and $b$ be the side length of the smaller square.
2. **Relate Perimeters:**
The perimeter of a square is given by $4$ times the side length. If the perimeter of the larger square is $3$ times the perimeter of the smaller square, we have:
... | 1 | 979.75 | 979.75 | -1 |
(1) Given $\sin\left( \alpha +\frac{\pi }{4} \right)=\frac{\sqrt{2}}{10}$, with $\alpha\in(0,\pi)$, find the value of $\cos \alpha$;
(2) Evaluate: $\left( \tan {10^{\circ} }-\sqrt{3} \right)\sin {40^{\circ} }$. | -1 | 0.9375 | 5,359.5625 | 5,170.733333 | 8,192 | |
A $4 \times 4$ square is divided into $16$ unit squares. Each unit square is painted either white or black, each with a probability of $\frac{1}{2}$, independently. The square is then rotated $180^\circ$ about its center. After rotation, any white square that occupies a position previously held by a black square is rep... | \frac{6561}{65536} | 0.3125 | 6,400.25 | 5,055.8 | 7,011.363636 | |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 such that its bases are parallel to the base of the hemisphere and the top of the cylinder touches the top of the hemisphere. What is the height of the cylinder? | 2\sqrt{10} | 0.625 | 5,267.25 | 3,532.6 | 8,158.333333 | |
Let \( a_{k} \) be the coefficient of \( x^{k} \) in the expansion of \( (1+2x)^{100} \), where \( 0 \leq k \leq 100 \). Find the number of integers \( r \) such that \( 0 \leq r \leq 99 \) and \( a_{r} < a_{r+1} \). | 67 | 0.5625 | 7,168.3125 | 6,372.111111 | 8,192 | |
Compute: $\left(\frac{1}{2} \right)^{3} \cdot \left(\frac{1}{7} \right)$. | \frac{1}{56} | 1 | 367.125 | 367.125 | -1 | |
How many unordered pairs of edges in a regular tetrahedron determine a plane? | 12 | 0.9375 | 4,951.6875 | 4,735.666667 | 8,192 | |
When $x^{2}$ was added to the quadratic polynomial $f(x)$, its minimum value increased by 1. When $x^{2}$ was subtracted from it, its minimum value decreased by 3. How will the minimum value of $f(x)$ change if $2x^{2}$ is added to it? | \frac{3}{2} | 0.75 | 5,979.375 | 5,587.666667 | 7,154.5 | |
The slope of a line is $-2$ and its $x$-intercept is $(5,0).$ What is the $y$-intercept point of the line? Express your answer as an ordered pair. | (0,10) | 1 | 1,537.5625 | 1,537.5625 | -1 | |
Given the set of positive odd numbers $\{1, 3, 5, \ldots\}$, we are grouping them in order such that the $n$-th group contains $(2n-1)$ odd numbers, determine which group the number 2009 belongs to. | 31 | 0 | 7,458.9375 | -1 | 7,458.9375 | |
Egorov decided to open a savings account to buy a car worth 900,000 rubles. The initial deposit is 300,000 rubles. Every month, Egorov plans to add 15,000 rubles to his account. The bank offers a monthly interest rate of $12\%$ per annum. The interest earned each month is added to the account balance, and the interest... | 29 | 0.5625 | 7,243.75 | 6,706 | 7,935.142857 |
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