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 $x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.$ Find $(x+1)^{48}$. | 125 | Like Solution $2$, let $z=\sqrt[16]{5}$ Then, the expression becomes
$x=\frac{4}{(z+1)(z^2+1)(z^4+1)(z^8+1)}$ Now, multiplying by the conjugate of each binomial in the denominator, we obtain...
$x=\frac{4(z-1)(z^2-1)(z^4-1)(z^8-1)}{(z^2-1)(z^4-1)(z^8-1)(z^{16}-1)}\implies x=\frac{4(z-1)}{z^{16}-1}$ Plugging back in,
$x... | 0.5625 | 5,752.5625 | 3,948.555556 | 8,072 |
Po is trying to solve the following equation by completing the square: $$49x^2+56x-64 = 0.$$He successfully rewrites the above equation in the following form: $$(ax + b)^2 = c,$$where $a$, $b$, and $c$ are integers and $a > 0$. What is the value of $a + b + c$? | 91 | 0.9375 | 4,137.0625 | 3,866.733333 | 8,192 | |
A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2007. ... | 372 | 0.0625 | 7,918.4375 | 7,666 | 7,935.266667 | |
What is the slope of a line parallel to the line $2x - 4y = 9$? Express your answer as a common fraction. | \frac{1}{2} | 1 | 1,183.25 | 1,183.25 | -1 | |
Determine the exact value of
\[\sqrt{\left( 2 - \sin^2 \frac{\pi}{7} \right) \left( 2 - \sin^2 \frac{2 \pi}{7} \right) \left( 2 - \sin^2 \frac{3 \pi}{7} \right)}.\] | \frac{13}{8} | 0.375 | 6,894.1875 | 4,828.333333 | 8,133.7 | |
What value should the real number $m$ take so that the point representing the complex number $z=(m^2-8m+15)+(m^2-5m-14)i$ in the complex plane
(Ⅰ) lies in the fourth quadrant;
(Ⅱ) lies on the line $y=x$. | \frac{29}{3} | 1 | 3,391 | 3,391 | -1 | |
In the $xy$-plane, how many lines whose $x$-intercept is a positive prime number and whose $y$-intercept is a positive integer pass through the point $(4,3)$? | 2 | 1. **Identify the general form of the line**: A line with $x$-intercept $a$ and $y$-intercept $b$ can be represented by the equation:
\[
\frac{x}{a} + \frac{y}{b} = 1
\]
where $a$ and $b$ are the intercepts on the $x$-axis and $y$-axis respectively.
2. **Substitute the point $(4,3)$ into the equation**: Si... | 0.9375 | 4,691.4375 | 4,458.066667 | 8,192 |
Solve the equations:
(1) $3x^2 -32x -48=0$
(2) $4x^2 +x -3=0$
(3) $(3x+1)^2 -4=0$
(4) $9(x-2)^2 =4(x+1)^2.$ | \frac{4}{5} | 0.5625 | 3,260.8125 | 3,122 | 3,439.285714 | |
Let $\triangle PQR$ be a right triangle such that $Q$ is a right angle. A circle with diameter $QR$ meets side $PR$ at point $S$. If $PS = 3$ and $QS = 9$, then what is $RS$? | 27 | 0.5625 | 6,488.625 | 5,582.888889 | 7,653.142857 | |
Given a regular triangular pyramid \(P-ABC\), where points \(P\), \(A\), \(B\), and \(C\) all lie on the surface of a sphere with radius \(\sqrt{3}\), and \(PA\), \(PB\), and \(PC\) are mutually perpendicular, find the distance from the center of the sphere to the cross-section \(ABC\). | \frac{\sqrt{3}}{3} | 0 | 7,991.3125 | -1 | 7,991.3125 | |
Sam spends his days walking around the following $2 \times 2$ grid of squares. Say that two squares are adjacent if they share a side. He starts at the square labeled 1 and every second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers on every square he visits in his path is equal... | 167 | Note that on the first step, Sam can either step on 2 or 4. On the second step, Sam can either step on 1 or 3, regardless of whether he is on 2 or 4. Now, for example, say that Sam takes 8 steps. His total sum will be $2+1+2+1+2+1+2+1+2 a$, where $a$ is the number of times that he decides to step on the larger number o... | 0 | 8,192 | -1 | 8,192 |
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$? | 20 | 1. **Calculate the total amount spent and the individual shares:**
- Total amount spent by Tom, Dorothy, and Sammy:
\[
105 + 125 + 175 = 405 \text{ dollars}
\]
- Since they agreed to split the costs evenly, each person's share is:
\[
\frac{405}{3} = 135 \text{ dollars}
\]
2. **Deter... | 1 | 1,467 | 1,467 | -1 |
According to the latest revision of the "Regulations on the Application and Use of Motor Vehicle Driving Licenses" by the Ministry of Public Security: each driving license applicant must pass the "Subject One" (theoretical subject) and "Comprehensive Subject" (combination of driving skills and some theoretical knowledg... | 0.9988 | 0.375 | 6,956.625 | 5,804.5 | 7,647.9 | |
The real roots of the equations \( x^{5} + x + 1 = 0 \) and \( x + \sqrt[5]{x} + 1 = 0 \) are \(\alpha\) and \(\beta\), respectively. What is the value of \(\alpha + \beta\)? | -1 | 0.625 | 5,772.8125 | 4,321.3 | 8,192 | |
Given that $a$, $b$, $c$ are all non-zero, and the maximum value of $\dfrac{a}{|a|} + \dfrac{b}{|b|} + \dfrac{c}{|c|} - \dfrac{abc}{|abc|}$ is $m$, and the minimum value is $n$, find the value of $\dfrac{n^{m}}{mn}$. | -16 | 0 | 4,087.375 | -1 | 4,087.375 | |
Given three vertices of a rectangle are located at $(2, 5)$, $(2, -4)$ and $(10, 5)$. Calculate the area of the intersection of this rectangle with the region inside the graph of the equation $(x - 10)^2 + (y - 5)^2 = 16$. | 4\pi | 0.5625 | 6,733.6875 | 5,648.444444 | 8,129 | |
In their base $10$ representations, the integer $a$ consists of a sequence of $1985$ eights and the integer $b$ consists of a sequence of $1985$ fives. What is the sum of the digits of the base $10$ representation of $9ab$? | 17865 | 1. **Express $a$ and $b$ in terms of geometric series**:
The integer $a$ consists of $1985$ eights, which can be written as:
\[
a = 8 \cdot 10^0 + 8 \cdot 10^1 + \cdots + 8 \cdot 10^{1984}
\]
Using the formula for the sum of a geometric series, $S = a \frac{r^n - 1}{r - 1}$ where $a$ is the first term,... | 0.25 | 7,774.25 | 6,521 | 8,192 |
What is the least positive integer with exactly $12$ positive factors? | 72 | 0 | 4,174.125 | -1 | 4,174.125 | |
Given the function $f\left(x\right)=x^{2}-2$, find $\lim_{{Δx→0}}\frac{{f(3)-f({3-2Δx})}}{{Δx}}$. | 12 | 0.875 | 4,353.8125 | 3,972.857143 | 7,020.5 | |
Compute
$$\sum_{k=1}^{2000} k(\lceil \log_{\sqrt{3}}{k}\rceil - \lfloor\log_{\sqrt{3}}{k} \rfloor).$$ | 1999907 | 0.25 | 4,570.0625 | 4,461.5 | 4,606.25 | |
Each of the digits 3, 5, 6, 7, and 8 is placed one to a box in the diagram. If the two-digit number is subtracted from the three-digit number, what is the smallest difference? | 269 | 0.25 | 7,975.9375 | 7,327.75 | 8,192 | |
Maya lists all the positive divisors of $2010^2$. She then randomly selects two distinct divisors from this list. Let $p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $p$ can be expressed in the form $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive int... | 107 | $2010^2 = 2^2\cdot3^2\cdot5^2\cdot67^2$. Thus there are $(2+1)^4$ divisors, $(1+1)^4$ of which are squares (the exponent of each prime factor must either be $0$ or $2$). Therefore the probability is \[\frac {2\cdot2^4\cdot(3^4 - 2^4)}{3^4(3^4 - 1)} = \frac {26}{81} \Longrightarrow 26+ 81 = \boxed{107}.\] | 1 | 3,115.3125 | 3,115.3125 | -1 |
Given a regular triangular prism \(ABC-A_1B_1C_1\) with side edges and base edges all equal to 1, find the volume of the common part of the tetrahedra \(A_1ABC\), \(B_1ABC\), and \(C_1ABC\). | \frac{\sqrt{3}}{36} | 0 | 8,192 | -1 | 8,192 | |
Find \(AX\) in the diagram where \(AC = 27\) units, \(BC = 36\) units, and \(BX = 30\) units. | 22.5 | 0 | 6,829.5 | -1 | 6,829.5 | |
If triangle $PQR$ has sides of length $PQ = 8,$ $PR = 7,$ and $QR = 5,$ then calculate
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\] | \frac{5}{7} | 0 | 5,443.375 | -1 | 5,443.375 | |
Let $S=\{1,2, \ldots, 2014\}$. For each non-empty subset $T \subseteq S$, one of its members is chosen as its representative. Find the number of ways to assign representatives to all non-empty subsets of $S$ so that if a subset $D \subseteq S$ is a disjoint union of non-empty subsets $A, B, C \subseteq S$, then the rep... | \[ 108 \cdot 2014! \] | Answer: $108 \cdot 2014$ !. For any subset $X$ let $r(X)$ denotes the representative of $X$. Suppose that $x_{1}=r(S)$. First, we prove the following fact: $$ \text { If } x_{1} \in X \text { and } X \subseteq S \text {, then } x_{1}=r(X) $$ If $|X| \leq 2012$, then we can write $S$ as a disjoint union of $X$ and two o... | 0 | 8,192 | -1 | 8,192 |
The sequence \((b_n)\) satisfies
\[kb_1 + kb_2 + kb_3 + \dots + kb_n = n^2 kb_n\] for all \(n \ge 2.\) If \(b_{70} = 2\) and \(k=3\), find \(b_1.\) | 4970 | 0.9375 | 4,694.875 | 4,461.733333 | 8,192 | |
Consider a $6 \times 6$ grid of squares. Edmond chooses four of these squares uniformly at random. What is the probability that the centers of these four squares form a square? | \frac{1}{561} | Firstly, there are $\binom{36}{4}$ possible combinations of points. Call a square proper if its sides are parallel to the coordinate axes and improper otherwise. Note that every improper square can be inscribed in a unique proper square. Hence, an $n \times n$ proper square represents a total of $n$ squares: 1 proper a... | 0 | 8,192 | -1 | 8,192 |
On the side of a triangle, a point is taken such that an angle equals another angle. What is the smallest possible distance between the centers of the circles circumscribed around triangles, if \( BC = 1 \)? | 1/2 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Given $\tan \alpha = 3$, evaluate the expression $2\sin ^{2}\alpha + 4\sin \alpha \cos \alpha - 9\cos ^{2}\alpha$. | \dfrac{21}{10} | 0.6875 | 5,517.75 | 4,828.727273 | 7,033.6 | |
Two different natural numbers are selected from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | \frac{3}{4} | 0.25 | 7,838.625 | 6,778.5 | 8,192 | |
Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sum of the first $n$ terms of each sequence is denoted as $S_n$ and $T_n$ respectively. If $$\frac {S_{n}}{T_{n}}= \frac {7n+45}{n+3}$$, and $$\frac {a_{n}}{b_{2n}}$$ is an integer, then the value of $n$ is \_\_\_\_\_\_. | 15 | 0.6875 | 6,178.0625 | 5,262.636364 | 8,192 | |
Given the function $f(x)=x^{2}-3x$. If for any $x_{1}$, $x_{2}$ in the interval $[-3,2]$, we have $|f(x_{1})-f(x_{2})| \leqslant m$, then the minimum value of the real number $m$ is _______. | \frac{81}{4} | 0 | 8,192 | -1 | 8,192 | |
In the cells of an $80 \times 80$ table, pairwise distinct natural numbers are placed. Each number is either prime or the product of two prime numbers (possibly the same). It is known that for any number $a$ in the table, there is a number $b$ in the same row or column such that $a$ and $b$ are not coprime. What is the... | 4266 | 0.0625 | 8,084.9375 | 6,479 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $C= \frac{\pi}{3}$, $b=8$, and the area of $\triangle ABC$ is $10\sqrt{3}$.
$(1).$ Find the value of $c$; $(2).$ Find the value of $\cos(B-C)$. | \frac{13}{14} | 0.8125 | 4,171.3125 | 3,656.384615 | 6,402.666667 | |
A soccer ball kicked vertically upward reaches, after $t$ seconds, a height of $s$ meters where $s = 180t - 20t^2$. Find the maximum height reached by the ball. | 405 | 1 | 4,100.1875 | 4,100.1875 | -1 | |
Let \( g(x) = \log_{\frac{1}{3}}\left(\log_9\left(\log_{\frac{1}{9}}\left(\log_{81}\left(\log_{\frac{1}{81}}x\right)\right)\right)\right) \). Determine the length of the interval that forms the domain of \( g(x) \), and express it in the form \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive in... | 82 | 0 | 8,192 | -1 | 8,192 | |
Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\overline{AC}$ and points $D$ and $F$ are on $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have the same perimeter. What is $DE+FG$... | \frac{21}{13} | 1. **Assign Variables:**
Let $AD = x$ and $AG = y$. Since $DE$ and $FG$ are parallel to $BC$, and $ABC$ is equilateral, $DE$ and $FG$ are also equal to $x$ and $y$ respectively. We need to find $DE + FG = x + y$.
2. **Use Perimeter Conditions:**
Given that $\triangle ADE$, trapezoid $DFGE$, and trapezoid $FBCG$ ... | 0.25 | 6,857 | 4,558 | 7,623.333333 |
The total corn yield in centners, harvested from a certain field area, is expressed as a four-digit number composed of the digits 0, 2, 3, and 5. When the average yield per hectare was calculated, it was found to be the same number of centners as the number of hectares of the field area. Determine the total corn yield. | 3025 | 0.75 | 5,676.5 | 4,838 | 8,192 | |
What is the maximum possible area of a quadrilateral with side lengths 1, 4, 7, and 8? | 18 | 0.6875 | 5,781.1875 | 4,685.363636 | 8,192 | |
Two lines defined by the equations $y = mx + 4$ and $y = 3x + b$, where $m$ and $b$ are constants, intersect at the point $(6, 10)$. What is the value of $b + m$? | -7 | 1 | 1,252.5625 | 1,252.5625 | -1 | |
Find the values of the real number \( a \) such that all the roots of the polynomial in the variable \( x \),
\[ x^{3}-2x^{2}-25x+a \]
are integers. | -50 | 0 | 8,055.8125 | -1 | 8,055.8125 | |
On the lateral side \(CD\) of trapezoid \(ABCD (AD \parallel BC)\), point \(M\) is marked. A perpendicular \(AH\) is dropped from vertex \(A\) to segment \(BM\). It turns out that \(AD = HD\). Find the length of segment \(AD\), given that \(BC = 16\), \(CM = 8\), and \(MD = 9\). | 18 | 0 | 8,192 | -1 | 8,192 | |
$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD.$ The crease is $EF,$ where $E$ is on $AB$ and $F$ is on $CD.$ The dimensions $AE=8, BE=17,$ and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n,$ where $m$ and $n$ are relatively prime posi... | 293 | Use the prepared diagram for this solution.
Call the intersection of DF and B'C' G. AB'E is an 8-15-17 right triangle, and so are B'DG and C'FG. Since C'F is 3, then using the properties of similar triangles GF is 51/8. DF is 22, so DG is 125/8. Finally, DB can to calculated to be 25/3. Add all the sides together to g... | 0.875 | 4,233.8125 | 3,668.357143 | 8,192 |
For every four points $P_{1},P_{2},P_{3},P_{4}$ on the plane, find the minimum value of $\frac{\sum_{1\le\ i<j\le\ 4}P_{i}P_{j}}{\min_{1\le\ i<j\le\ 4}(P_{i}P_{j})}$ . | 4 + 2\sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and finds himself in a room with 4 doors. This room is the first in a sequence of 6 indistinguishable rooms. In each room, 1 door leads to the next room in the sequence (or, for the last room, Bowser's level), while the other 3 doors lea... | 5460 | This problem works in the same general way as the last problem, but it can be more succinctly solved using the general formula, which is provided below in the solution to the next problem. | 0 | 7,971.0625 | -1 | 7,971.0625 |
The traffic police brigade of our county is carrying out a comprehensive road traffic safety rectification campaign "Hundred-Day Battle" throughout the county, which strictly requires riders of electric bicycles and motorcycles to comply with the rule of "one helmet, one belt". A certain dealer purchased a type of helm... | 50 | 0.5625 | 3,541.375 | 2,697.222222 | 4,626.714286 | |
In a right triangle, instead of having one $90^{\circ}$ angle and two small angles sum to $90^{\circ}$, consider now the acute angles are $x^{\circ}$, $y^{\circ}$, and a smaller angle $z^{\circ}$ where $x$, $y$, and $z$ are all prime numbers, and $x^{\circ} + y^{\circ} + z^{\circ} = 90^{\circ}$. Determine the largest p... | 47 | 0 | 7,241.4375 | -1 | 7,241.4375 | |
You want to arrange the numbers $1,2,3, \ldots, 25$ in a sequence with the following property: if $n$ is divisible by $m$, then the $n$th number is divisible by the $m$ th number. How many such sequences are there? | 24 | Let the rearranged numbers be $a_{1}, \ldots, a_{25}$. The number of pairs $(n, m)$ with $n \mid m$ must equal the number of pairs with $a_{n} \mid a_{m}$, but since each pair of the former type is also of the latter type, the converse must be true as well. Thus, $n \mid m$ if and only if $a_{n} \mid a_{m}$. Now for ea... | 0 | 8,192 | -1 | 8,192 |
A train starts its journey, then stops after 1 hour due to an incident and remains halted for half an hour. After that, it continues at $\frac{3}{4}$ of its original speed, resulting in a delay of $3 \frac{1}{2}$ hours upon reaching its destination. If the incident had occurred 90 miles further ahead, the train would h... | 600 | 0.125 | 5,490.8125 | 2,928.5 | 5,856.857143 | |
$A B C D$ is a cyclic quadrilateral with sides $A B=10, B C=8, C D=25$, and $D A=12$. A circle $\omega$ is tangent to segments $D A, A B$, and $B C$. Find the radius of $\omega$. | \sqrt{\frac{1209}{7}} \text{ OR } \frac{\sqrt{8463}}{7} | Denote $E$ an intersection point of $A D$ and $B C$. Let $x=E A$ and $y=E B$. Because $A B C D$ is a cyclic quadrilateral, $\triangle E A B$ is similar to $\triangle E C D$. Therefore, $\frac{y+8}{x}=\frac{25}{10}$ and $\frac{x+12}{y}=\frac{25}{10}$. We get $x=\frac{128}{21}$ and $y=\frac{152}{21}$. Note that $\omega$ ... | 0 | 8,192 | -1 | 8,192 |
Compute the square of 1023 without a calculator. | 1046529 | 0.5 | 1,608.375 | 1,662.875 | 1,553.875 | |
Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$ with distinct terms, given that $a_{3}a_{8}=3a_{11}$, $S_{3}=9$.
1. Find the general term formula for the sequence $\{a_{n}\}$.
2. If $b_{n}= \frac {1}{ \sqrt {a_{n}}+ \sqrt {a_{n+1}}}$, and the sum of the first $n$ terms of the sequenc... | 4 \sqrt {2} | 0 | 6,081.3125 | -1 | 6,081.3125 | |
Find all natural numbers which are divisible by $30$ and which have exactly $30$ different divisors.
(M Levin) | 11250, 4050, 7500, 1620, 1200, 720 |
To find all natural numbers divisible by 30 with exactly 30 different divisors, we first explore the properties of divisors and the structure of such numbers.
A natural number \( n \) has exactly 30 divisors if its prime factorization can be expressed to fit the divisors' formula:
\[
(n = p_1^{a_1} \cdot p_2^{a_2} \c... | 0 | 6,159.125 | -1 | 6,159.125 |
Let $x$ and $y$ be real numbers satisfying $3 \leqslant xy^2 \leqslant 8$ and $4 \leqslant \frac{x^2}{y} \leqslant 9$. Find the maximum value of $\frac{x^3}{y^4}$. | 27 | 0.625 | 5,994.25 | 5,135.8 | 7,425 | |
Binbin's height is 1.46 meters, his father is 0.32 meters taller than Binbin, and his mother's height is 1.5 meters.
(1) How tall is Binbin's father?
(2) How much shorter is Binbin's mother than his father? | 0.28 | 0.375 | 432.375 | 439 | 428.4 | |
Define a modified Ackermann function \( A(m, n) \) with the same recursive relationships as the original problem:
\[ A(m,n) = \left\{
\begin{aligned}
&n+1& \text{ if } m = 0 \\
&A(m-1, 1) & \text{ if } m > 0 \text{ and } n = 0 \\
&A(m-1, A(m, n-1))&\text{ if } m > 0 \text{ and } n > 0.
\end{aligned}
\right.\]
Compu... | 29 | 0.4375 | 7,276.6875 | 6,407.428571 | 7,952.777778 | |
Given real numbers $x$ and $y$ satisfying $x^2+4y^2=4$, find the maximum value of $\frac {xy}{x+2y-2}$. | \frac {1+ \sqrt {2}}{2} | 0 | 8,052.5 | -1 | 8,052.5 | |
Bully Dima constructed a rectangle from 38 wooden toothpicks in the shape of a $3 \times 5$ grid. Then he simultaneously ignited two adjacent corners of the rectangle, as marked on the diagram.
It is known that one toothpick burns in 10 seconds. How many seconds will it take for the entire structure to burn?
(The fir... | 65 | 0 | 6,842.4375 | -1 | 6,842.4375 | |
Given the function $f(x)=a\ln x + x - \frac{1}{x}$, where $a$ is a real constant.
(I) If $x=\frac{1}{2}$ is a local maximum point of $f(x)$, find the local minimum value of $f(x)$.
(II) If the inequality $a\ln x - \frac{1}{x} \leqslant b - x$ holds for any $-\frac{5}{2} \leqslant a \leqslant 0$ and $\frac{1}{2} \leqsla... | \frac{3}{2} | 0.875 | 5,789.75 | 5,446.571429 | 8,192 | |
A lady made 3 round doilies with radii of 2, 3, and 10 inches, respectively. She placed them on a round table so that each doily touches the two others and the edge of the table. What is the radius of the table? | 15 | 0.375 | 7,412.6875 | 6,113.833333 | 8,192 | |
For $t = 1, 2, 3, 4$, define $S_t = \sum_{i = 1}^{350}a_i^t$, where $a_i \in \{1,2,3,4\}$. If $S_1 = 513$ and $S_4 = 4745$, find the minimum possible value for $S_2$. | 905 | Because the order of the $a$'s doesn't matter, we simply need to find the number of $1$s $2$s $3$s and $4$s that minimize $S_2$. So let $w, x, y,$ and $z$ represent the number of $1$s, $2$s, $3$s, and $4$s respectively. Then we can write three equations based on these variables. Since there are a total of $350$ $a$s, w... | 0.8125 | 5,471.5 | 5,142.384615 | 6,897.666667 |
If set $A=\{-4, 2a-1, a^2\}$, $B=\{a-5, 1-a, 9\}$, and $A \cap B = \{9\}$, then the value of $a$ is. | -3 | 0.5 | 7,023.5625 | 7,016.625 | 7,030.5 | |
Arrange the letters a, a, b, b, c, c into three rows and two columns, such that in each row and each column, the letters are different. How many different arrangements are there? | 12 | 0.0625 | 8,120.9375 | 7,055 | 8,192 | |
Let \(C\) be a cube with side length 4 and center \(O\). Let \(S\) be the sphere centered at \(O\) with radius 2. Let \(A\) be one of the vertices of the cube. Let \(R\) be the set of points in \(C\) but not in \(S\), which are closer to \(A\) than to any other vertex of \(C\). Find the volume of \(R\). | 8 - \frac{4\pi}{3} | 0 | 8,192 | -1 | 8,192 | |
The edge of cube \( ABCD A_1 B_1 C_1 D_1 \) is 12. Point \( K \) lies on the extension of edge \( BC \) at a distance of 9 from vertex \( C \). Point \( L \) on edge \( AB \) is at a distance of 5 from \( A \). Point \( M \) divides segment \( A_1 C_1 \) in a ratio of 1:3, starting from \( A_1 \). Find the area of the ... | 156 | 0.0625 | 8,048 | 8,192 | 8,038.4 | |
Given that the right focus of ellipse $I$: $\frac{{x}^{2}}{{a}^{2}}+ \frac{{y}^{2}}{{b}^{2}}=1 (a > 0,b > 0)$ is $(2 \sqrt{2},0)$, and ellipse $I$ passes through point $(3,1)$.
(1) Find the equation of ellipse $I$;
(2) Let line $l$ with slope $1$ intersect ellipse $I$ at two distinct points $A$ and $B$. Construct an is... | \frac {9}{2} | 1 | 5,799.5625 | 5,799.5625 | -1 | |
What is the greatest integer less than or equal to \[\frac{5^{80} + 3^{80}}{5^{75} + 3^{75}}?\] | 3124 | 0.5625 | 7,735.3125 | 7,380.111111 | 8,192 | |
In Pascal's Triangle, we know each number is the combination of two numbers just above it. What is the sum of the middle three numbers in each of Rows 5, 6, and 7? | 157 | 0 | 5,612.4375 | -1 | 5,612.4375 | |
For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\sqrt{p}| < \frac{1}{2}$. For example, $b(6) = 2$ and $b(23) = 5$. If $S = \sum_{p=1}^{2007} b(p),$ find the remainder when $S$ is divided by 1000. | 955 | $\left(k- \frac 12\right)^2=k^2-k+\frac 14$ and $\left(k+ \frac 12\right)^2=k^2+k+ \frac 14$. Therefore $b(p)=k$ if and only if $p$ is in this range, or $k^2-k<p\leq k^2+k$. There are $2k$ numbers in this range, so the sum of $b(p)$ over this range is $(2k)k=2k^2$. $44<\sqrt{2007}<45$, so all numbers $1$ to $44$ have t... | 0.5625 | 7,145.25 | 6,690.666667 | 7,729.714286 |
The number $15$ is written on a blackboard. A move consists of erasing the number $x$ and replacing it with $x+y$ where $y$ is a randomly chosen number between $1$ and $5$ (inclusive). The game ends when the number on the blackboard exceeds $51$ . Which number is most likely to be on the blackboard at the ... | 54 | 0 | 8,192 | -1 | 8,192 | |
Convert the point $\left( 8, \frac{\pi}{4}, \sqrt{3} \right)$ in cylindrical coordinates to rectangular coordinates. | (4 \sqrt{2}, 4 \sqrt{2}, \sqrt{3}) | 1 | 1,464.5625 | 1,464.5625 | -1 | |
Given that quadrilateral \(ABCD\) is a right trapezoid with the upper base \(AD = 8\) cm, the lower base \(BC = 10\) cm, and the right leg \(CD = 6\) cm. Point \(E\) is the midpoint of \(AD\), and point \(F\) is on \(BC\) such that \(BF = \frac{2}{3} BC\). Point \(G\) is on \(DC\) such that the area of triangle \(DEG\)... | 24 | 0 | 7,080.875 | -1 | 7,080.875 | |
Triangle $AHI$ is equilateral. We know $\overline{BC}$, $\overline{DE}$ and $\overline{FG}$ are all parallel to $\overline{HI}$ and $AB = BD = DF = FH$. What is the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$? Express your answer as a common fraction.
[asy]
unitsize(0.2inch);
defaultpen(linewid... | \frac{7}{16} | 0.1875 | 7,800.25 | 6,102.666667 | 8,192 | |
If $x=2$, what is the value of $4x^2 - 3x^2$? | 4 | Simplifying, $4 x^{2}-3 x^{2}=x^{2}$. When $x=2$, this expression equals 4 . Alternatively, when $x=2$, we have $4 x^{2}-3 x^{2}=4 \cdot 2^{2}-3 \cdot 2^{2}=16-12=4$. | 1 | 1,018.875 | 1,018.875 | -1 |
Let $r_{1}, r_{2}, \ldots, r_{7}$ be the distinct complex roots of the polynomial $P(x)=x^{7}-7$. Let $$K=\prod_{1 \leq i<j \leq 7}\left(r_{i}+r_{j}\right)$$ that is, the product of all numbers of the form $r_{i}+r_{j}$, where $i$ and $j$ are integers for which $1 \leq i<j \leq 7$. Determine the value of $K^{2}$. | 117649 | We first note that $x^{7}-7=\left(x-r_{1}\right)\left(x-r_{2}\right) \cdots\left(x-r_{7}\right)$, which implies, replacing $x$ by $-x$ and taking the negative of the equation, that $\left(x+r_{1}\right)\left(x+r_{2}\right) \cdots\left(x+r_{7}\right)=x^{7}+7$. Also note that the product of the $r_{i}$ is just the consta... | 0.125 | 7,933.25 | 8,119 | 7,906.714286 |
Is there an integer $x$ such that $x \equiv 1 \ (\text{mod} \ 6)$, $x \equiv 9 \ (\text{mod} \ 14)$, and $x \equiv 7 \ (\text{mod} \ 15)$? | 37 | 1 | 2,446 | 2,446 | -1 | |
In the coordinate plane, let $A = (-8, 0)$ , $B = (8, 0)$ , and $C = (t, 6)$ . What is the maximum value of $\sin m\angle CAB \cdot \sin m\angle CBA$ , over all real numbers $t$ ? | 3/8 | 0.9375 | 5,896.4375 | 5,743.4 | 8,192 | |
The letters of the alphabet are each assigned a random integer value, and $H=10$. The value of a word comes from the sum of its letters' values. If $MATH$ is 35 points, $TEAM$ is 42 points and $MEET$ is 38 points, what is the value of $A$? | 21 | 0.875 | 2,505.9375 | 1,865.571429 | 6,988.5 | |
Calculate the definite integral:
$$
\int_{0}^{\sqrt{2} / 2} \frac{x^{4} \cdot d x}{\sqrt{\left(1-x^{2}\right)^{3}}}
$$ | \frac{5}{4} - \frac{3\pi}{8} | 0 | 6,740.9375 | -1 | 6,740.9375 | |
Let $N$ denote the number of permutations of the $15$-character string $AAAABBBBBCCCCCC$ such that
None of the first four letters is an $A$.
None of the next five letters is a $B$.
None of the last six letters is a $C$.
Find the remainder when $N$ is divided by $1000$.
| 320 | 0.0625 | 7,821.25 | 4,650 | 8,032.666667 | |
Given that in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $\left(a+b\right)\left(\sin A-\sin B\right)=\left(c-b\right)\sin C$ with $a=2$, find the maximum area of triangle $\triangle ABC$. | \sqrt{3} | 0.5 | 7,254.3125 | 6,589.125 | 7,919.5 | |
Given \( a_{n} = 50 + n^{2} \) for \( n = 1, 2, \cdots \), find the maximum value of the greatest common divisor \( d_{n} = \gcd(a_{n}, a_{n+1}) \). | 201 | 0.8125 | 5,583.375 | 4,981.384615 | 8,192 | |
In the diagram, points $U$, $V$, $W$, $X$, $Y$, and $Z$ lie on a straight line with $UV=VW=WX=XY=YZ=5$. Semicircles with diameters $UZ$, $UV$, $VW$, $WX$, $XY$, and $YZ$ create the shape shown. What is the area of the shaded region?
[asy]
size(5cm); defaultpen(fontsize(9));
pair one = (1, 0);
pair u = (0, 0); pair v ... | \frac{325}{4}\pi | 0 | 7,581.6875 | -1 | 7,581.6875 | |
Given vectors $\overrightarrow{a} = (1, 2)$, $\overrightarrow{b} = (x, 1)$,
1. If $\langle \overrightarrow{a}, \overrightarrow{b} \rangle$ forms an acute angle, find the range of $x$.
2. Find the value of $x$ when $(\overrightarrow{a}+2\overrightarrow{b}) \perp (2\overrightarrow{a}-\overrightarrow{b})$. | \frac{7}{2} | 0.875 | 3,365.625 | 3,335 | 3,580 | |
In the plane quadrilateral $\mathrm{ABCD}$, given $\mathrm{AB}=1, \mathrm{BC}=4, \mathrm{CD}=2, \mathrm{DA}=3$, find the value of $\overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{BD}}$. | 10 | 0.25 | 7,393.5 | 7,087 | 7,495.666667 | |
A book that is to be recorded onto compact discs takes $412$ minutes to read aloud. Each disc can hold up to $56$ minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain? | 51.5 | 1. **Calculate the number of discs needed if each disc is fully utilized:**
Given that each disc can hold up to $56$ minutes and the total reading time is $412$ minutes, the number of discs required if each disc is fully utilized is calculated by dividing the total minutes by the minutes each disc can hold:
\[
... | 0.625 | 5,794.75 | 5,529.2 | 6,237.333333 |
Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit perfect square!”
Claire asks, “If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?”
... | 25 | 0.0625 | 7,160.3125 | 4,142 | 7,361.533333 | |
Let $\angle ABC = 24^\circ$ and $\angle ABD = 20^\circ$. What is the smallest possible degree measure for $\angle CBD$? | 4 | 1. **Identify the Relationship Between Angles**: Given $\angle ABC = 24^\circ$ and $\angle ABD = 20^\circ$, we know that these two angles share the common ray $AB$.
2. **Expression for $\angle CBD$**: Since $\angle ABC$ is the full angle formed at point $B$ between rays $AB$ and $BC$, and $\angle ABD$ is the angle fo... | 1 | 2,864.625 | 2,864.625 | -1 |
Calculate the value of $15 \times 30 + 45 \times 15$. | 1125 | 0.9375 | 258 | 254.2 | 315 | |
Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge... | \frac{13}{16} | To solve this problem, we will calculate the probability that Frieda reaches a corner square within four hops, starting from the center of a $3 \times 3$ grid. We will use a state-based approach to model Frieda's possible positions and transitions.
#### Definitions:
- **State**: A position on the grid.
- **Transition*... | 0 | 8,025.9375 | -1 | 8,025.9375 |
The left and right foci of the hyperbola $E$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0, b > 0$) are $F_1$ and $F_2$, respectively. Point $M$ is a point on the asymptote of hyperbola $E$, and $MF_1 \perpendicular MF_2$. If $\sin \angle MF_1F_2 = \dfrac{1}{3}$, then the eccentricity of this hyperbola is ______. | \dfrac{9}{7} | 0.8125 | 5,500.6875 | 4,879.615385 | 8,192 | |
A point $P$ lies in the same plane as a given square of side $2$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $B, C$ and $D$, respectively, be $v, w$ and $t$. What is the greatest distance that $P$ can be from $A$ if $v^2 + w^2 = t^2$?
A) $\sqrt{8}$
... | \sqrt{10} | 0 | 8,192 | -1 | 8,192 | |
A circle touches the extensions of two sides \( AB \) and \( AD \) of a square \( ABCD \) with a side length of 4 cm. From point \( C \), two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is \( 60^{\circ} \). | 4 (\sqrt{2} + 1) | 0.0625 | 6,302.5 | 6,542 | 6,286.533333 | |
Given that \\(F_1\\) and \\(F_2\\) are the left and right foci of the hyperbola \\( \frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a > 0,b > 0)\\), if there exists a point \\(P\\) on the left branch of the hyperbola that is symmetric to point \\(F_2\\) with respect to the line \\(y= \frac {bx}{a}\\), then the eccentricit... | \sqrt {5} | 0 | 6,861.25 | -1 | 6,861.25 | |
Xiao Ming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, Xiao Ming took the subway first and then transferred to the bus, taking a total of 40 minutes to reach school, with the transfer process taking 6 minutes. How many minutes did Xiao Ming take the bus that day? | 10 | 0 | 507.0625 | -1 | 507.0625 | |
After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored $15$ points. None of the other $7$ team members scored more than $2$ points. What was the total number of points scored by the other $... | 11 | Let $x$ be the total number of points scored by the team. According to the problem, the points can be broken down as follows:
- Alexa scored $\frac{1}{4}x$ points.
- Brittany scored $\frac{2}{7}x$ points.
- Chelsea scored $15$ points.
- The other $7$ team members scored $y$ points in total.
The total points scored by ... | 1 | 2,817.0625 | 2,817.0625 | -1 |
Given that $x_{0}$ is a zero of the function $f(x)=2a\sqrt{x}+b-{e}^{\frac{x}{2}}$, and $x_{0}\in [\frac{1}{4}$,$e]$, find the minimum value of $a^{2}+b^{2}$. | \frac{{e}^{\frac{3}{4}}}{4} | 0 | 6,609.875 | -1 | 6,609.875 | |
There is a unique two-digit positive integer $u$ for which the last two digits of $15\cdot u$ are $45$, and $u$ leaves a remainder of $7$ when divided by $17$. | 43 | 0 | 8,192 | -1 | 8,192 | |
Let $R$ be a rectangle. How many circles in the plane of $R$ have a diameter both of whose endpoints are vertices of $R$? | 5 | 1. **Identify the vertices of the rectangle**: Let the vertices of rectangle $R$ be labeled as $A$, $B$, $C$, and $D$. Assume $ABCD$ is a rectangle with $AB$ parallel to $CD$ and $AD$ parallel to $BC$.
2. **Count the pairs of vertices**: There are $\binom{4}{2} = 6$ ways to choose 2 vertices from 4 vertices. These pai... | 0.375 | 6,587.625 | 6,032.5 | 6,920.7 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.