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 |
|---|---|---|---|---|---|---|
At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for $5. This week they are on sale at 5 boxes for $4. The percent decrease in the price per box during the sale was closest to | 35\% | 1. **Calculate the original price per box:**
Last week, the boxes were sold at 4 boxes for $5. Therefore, the price per box last week was:
\[
\frac{5}{4} = 1.25 \text{ dollars per box}
\]
2. **Calculate the new price per box:**
This week, the boxes are on sale at 5 boxes for $4. Therefore, the price per... | 0 | 4,107.4375 | -1 | 4,107.4375 |
Given a moving circle $E$ that passes through the point $F(1,0)$, and is tangent to the line $l: x=-1$.
(Ⅰ) Find the equation of the trajectory $G$ for the center $E$ of the moving circle.
(Ⅱ) Given the point $A(3,0)$, if a line with a slope of $1$ intersects with the line segment $OA$ (not passing through the origin $... | \frac{32 \sqrt{3}}{9} | 0 | 7,038.875 | -1 | 7,038.875 | |
Polly has three circles cut from three pieces of colored card. She originally places them on top of each other as shown. In this configuration, the area of the visible black region is seven times the area of the white circle.
Polly moves the circles to a new position, as shown, with each pair of circles touching each ... | 7:6 | 0 | 7,865.6875 | -1 | 7,865.6875 | |
In \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively. Given that \\(a=2\\), \\(c=3\\), and \\(\cos B= \dfrac {1}{4}\\),
\\((1)\\) find the value of \\(b\\);
\\((2)\\) find the value of \\(\sin C\\). | \dfrac {3 \sqrt {6}}{8} | 0 | 3,015.5 | -1 | 3,015.5 | |
Evaluate $\log_3\frac{1}{3}$. | -1 | 1 | 1,924.3125 | 1,924.3125 | -1 | |
Calculate the integrals:
1) $\int_{0}^{1} x e^{-x} \, dx$
2) $\int_{1}^{2} x \log_{2} x \, dx$
3) $\int_{1}^{e} \ln^{2} x \, dx$ | e - 2 | 0.875 | 3,874.375 | 3,600.071429 | 5,794.5 | |
In the complex plane, the points corresponding to the complex number $1+ \sqrt {3}i$ and $- \sqrt {3}+i$ are $A$ and $B$, respectively, with $O$ as the coordinate origin. Calculate the measure of $\angle AOB$. | \dfrac {\pi}{2} | 0.125 | 2,592.5 | 3,242.5 | 2,499.642857 | |
In trapezoid \(ABCD\) with bases \(AB\) and \(CD\), it holds that \(|AD| = |CD|\), \(|AB| = 2|CD|\), \(|BC| = 24 \text{ cm}\), and \(|AC| = 10 \text{ cm}\).
Calculate the area of trapezoid \(ABCD\). | 180 | 0.5 | 6,483.6875 | 5,220.75 | 7,746.625 | |
Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle? | \angle B=80^{\circ},\angle C=40^{\circ} |
Given a triangle \( ABC \) with the angle \( \angle BAC = 60^\circ \), we need to determine the other angles \(\angle B\) and \(\angle C\) given that \( AP \) bisects \( \angle BAC \) and \( BQ \) bisects \( \angle ABC \), where \( P \) is on \( BC \) and \( Q \) is on \( AC \), and the condition \( AB + BP = AQ + QB ... | 0 | 8,192 | -1 | 8,192 |
Given non-negative real numbers $a_{1}, a_{2}, \cdots, a_{2008}$ whose sum equals 1, determine the maximum value of $a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{2007} a_{2008} + a_{2008} a_{1}$. | \frac{1}{4} | 0.125 | 8,082.8125 | 7,606 | 8,150.928571 | |
Let $\mathcal{H}$ be a regular hexagon with side length one. Peter picks a point $P$ uniformly and at random within $\mathcal{H}$, then draws the largest circle with center $P$ that is contained in $\mathcal{H}$. What is this probability that the radius of this circle is less than $\frac{1}{2}$? | \frac{2 \sqrt{3}-1}{3} | We first cut the regular hexagon $\mathcal{H}$ by segments connecting its center to each vertex into six different equilateral triangles with side lengths 1. Therefore, each point inside $\mathcal{H}$ is contained in some equilateral triangle. We first see that for each point inside an equilateral triangle, the radius ... | 0 | 7,748.75 | -1 | 7,748.75 |
A cubical cake with edge length $2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $M$ is the midpoint of a top edge. The piece whose top is triangle $B$ contains $c$ cubic inches of cake and $s$ square inches of icing. What is $c+s$? | \frac{32}{5} |
1. **Understanding the Problem:**
- We have a cube with edge length $2$ inches, which is iced on the sides and the top.
- The cube is cut into three pieces, and we focus on the piece with a triangular top view labeled as triangle $B$.
- We need to find the sum of the volume $c$ of this piece and the area $s$ ... | 0 | 7,819.5 | -1 | 7,819.5 |
Let $n, k \geq 3$ be integers, and let $S$ be a circle. Let $n$ blue points and $k$ red points be chosen uniformly and independently at random on the circle $S$. Denote by $F$ the intersection of the convex hull of the red points and the convex hull of the blue points. Let $m$ be the number of vertices of the convex po... | \frac{2 k n}{n+k-1}-2 \frac{k!n!}{(k+n-1)! | We prove that $$E(m)=\frac{2 k n}{n+k-1}-2 \frac{k!n!}{(k+n-1)!}$$ Let $A_{1}, \ldots, A_{n}$ be blue points. Fix $i \in\{1, \ldots, n\}$. Enumerate our $n+k$ points starting from a blue point $A_{i}$ counterclockwise as $A_{i}, X_{1, i}, X_{2, i}, \ldots, X_{(n+k-1), i}$. Denote the minimal index $j$ for which the poi... | 0 | 8,146.5625 | -1 | 8,146.5625 |
Choose one of the following conditions from (1) $a\sin \left(B+C\right)+c\sin C-b\sin B=2a\sin C\sin B$, (2) $\frac{cosB}{cosC}+\frac{b}{c-\sqrt{2}a}=0$, (3) $2a^{2}=(a^{2}+b^{2}-c^{2})(1+\tan C)$, and fill in the blank in the question below, and answer the corresponding questions.
Given $\triangle ABC$ with sides $a... | \frac{3\sqrt{2}}{5} | 0 | 7,086.5 | -1 | 7,086.5 | |
In triangle \(ABC\), angle \(B\) is \(120^\circ\), and \(AB = 2BC\). The perpendicular bisector of side \(AB\) intersects \(AC\) at point \(D\). Find the ratio \(AD:DC\). | 3/2 | 0 | 6,925.875 | -1 | 6,925.875 | |
Let \( a \) be a nonzero real number. In the Cartesian coordinate plane \( xOy \), the focal distance of the conic section \( x^2 + a y^2 + a^2 = 0 \) is 4. Find the value of \( a \). | \frac{1 - \sqrt{17}}{2} | 0 | 6,610.875 | -1 | 6,610.875 | |
Some positive integers are initially written on a board, where each $2$ of them are different.
Each time we can do the following moves:
(1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$ (2) If there are 2 numbers (written in the board) in the form $n, n+... | -3 | 0 | 8,192 | -1 | 8,192 | |
In the quadrilateral \( ABCD \), angle \( B \) is \( 150^{\circ} \), angle \( C \) is a right angle, and the sides \( AB \) and \( CD \) are equal.
Find the angle between side \( BC \) and the line passing through the midpoints of sides \( BC \) and \( AD \). | 60 | 0.625 | 6,391.6875 | 5,651.9 | 7,624.666667 | |
Squares $ABCD$ and $EFGH$ have a common center and $\overline{AB} || \overline{EF}$. The area of $ABCD$ is 2016, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the differenc... | 840 | Letting $AI=a$ and $IB=b$, we have \[IJ^{2}=a^{2}+b^{2} \geq 1008\] by AM-GM inequality. Also, since $EFGH||ABCD$, the angles that each square cuts another are equal, so all the triangles are formed by a vertex of a larger square and $2$ adjacent vertices of a smaller square are similar. Therefore, the areas form a geo... | 0 | 8,192 | -1 | 8,192 |
What is the result when $\frac{2}{3}$ is subtracted from $2\frac{1}{4}$? | 1\frac{7}{12} | 0.6875 | 4,191.0625 | 3,976.363636 | 4,663.4 | |
Given the hyperbola $\frac {x^{2}}{4}-y^{2}$=1, a line $l$ with a slope angle of $\frac {π}{4}$ passes through the right focus $F\_2$ and intersects the right branch of the hyperbola at points $M$ and $N$. The midpoint of the line segment $MN$ is $P$. Determine the vertical coordinate of point $P$. | \frac{\sqrt {5}}{3} | 0 | 4,212.375 | -1 | 4,212.375 | |
Given that $\sin \alpha - \cos \alpha = \frac{1}{5}$, and $0 \leqslant \alpha \leqslant \pi$, find the value of $\sin (2\alpha - \frac{\pi}{4})$ = $\_\_\_\_\_\_\_\_$. | \frac{31\sqrt{2}}{50} | 0 | 5,663.8125 | -1 | 5,663.8125 | |
In the Cartesian coordinate system $xOy$, the ellipse $C: \frac{x^2}{2} + \frac{y^2}{3} = 1$ has a focus on the positive y-axis denoted as $F$. A line $l$ passing through $F$ with a slope angle of $\frac{3\pi}{4}$ intersects $C$ at points $M$ and $N$. The quadrilateral $OMPN$ is a parallelogram.
(1) Determine the posit... | \frac{4}{5}\sqrt{6} | 0 | 7,207.875 | -1 | 7,207.875 | |
In $\vartriangle ABC, AB=AC=14 \sqrt2 , D$ is the midpoint of $CA$ and $E$ is the midpoint of $BD$ . Suppose $\vartriangle CDE$ is similar to $\vartriangle ABC$ . Find the length of $BD$ . | 14 | 0.875 | 4,925.875 | 4,459.285714 | 8,192 | |
What is the largest four-digit negative integer congruent to $2 \pmod{17}$? | -1001 | 0.3125 | 6,364.375 | 4,260.4 | 7,320.727273 | |
Add $123.4567$ to $98.764$ and round your answer to the nearest hundredth. Then, subtract $0.02$ from the rounded result. | 222.20 | 1 | 356.375 | 356.375 | -1 | |
Given the expansion of $(1+\frac{a}{x}){{(2x-\frac{1}{x})}^{5}}$, find the constant term. | 80 | 0 | 6,089.25 | -1 | 6,089.25 | |
In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive $x$-axis as the polar axis, the parametric equation of curve $C_1$ is
$$
\begin{cases}
x=2+ \sqrt {3}\cos \theta \\
y= \sqrt {3}\sin \theta
\end{cases}
(\theta \text{ is the parameter}),
$$
and the polar equation of curve $C_2$ is... | 2 \sqrt {2} | 0 | 4,975.3125 | -1 | 4,975.3125 | |
Let \( f(x) = \left\{ \begin{array}{cc} 1 & 1 \leqslant x \leqslant 2 \\ x-1 & 2 < x \leqslant 3 \end{array} \right. \). For any \( a \,(a \in \mathbb{R}) \), define \( v(a) = \max \{ f(x) - a x \mid x \in [1,3] \} - \min \{ f(x) - a x \mid x \in [1,3] \} \). Draw the graph of \( v(a) \) and find the minimum value of \... | \frac{1}{2} | 0.4375 | 6,970.375 | 5,543.142857 | 8,080.444444 | |
Determine the common rational root \( k \) of the following polynomial equations which is not integral:
\[45x^4 + ax^3 + bx^2 + cx + 8 = 0\]
\[8x^5 + dx^4 + ex^3 + fx^2 + gx + 45 = 0\]
This root \( k \) is assumed to be a negative non-integer. | -\frac{1}{3} | 0 | 8,192 | -1 | 8,192 | |
Given $\overrightarrow{a}=(2\sin x,1)$ and $\overrightarrow{b}=(2\cos (x-\frac{\pi }{3}),\sqrt{3})$, let $f(x)=\overrightarrow{a}\bullet \overrightarrow{b}-2\sqrt{3}$.
(I) Find the smallest positive period and the zeros of $f(x)$;
(II) Find the maximum and minimum values of $f(x)$ on the interval $[\frac{\pi }{24},\f... | -\sqrt{2} | 0.75 | 4,725.125 | 4,392.833333 | 5,722 | |
Calculate the product: $500 \times 2019 \times 0.02019 \times 5.$ | 0.25 \times 2019^2 | 0 | 6,825.125 | -1 | 6,825.125 | |
A set of positive numbers has the triangle property if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\{4, 5, 6, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible... | 253 | Out of all ten-element subsets with distinct elements that do not possess the triangle property, we want to find the one with the smallest maximum element. Call this subset $\mathcal{S}$. Without loss of generality, consider any $a, b, c \,\in \mathcal{S}$ with $a < b < c$. $\,\mathcal{S}$ does not possess the triangle... | 0.5 | 6,199.125 | 5,132.875 | 7,265.375 |
On a trip from the United States to Canada, Isabella took $d$ U.S. dollars. At the border she exchanged them all, receiving $10$ Canadian dollars for every $7$ U.S. dollars. After spending $60$ Canadian dollars, she had $d$ Canadian dollars left. What is the sum of the digits of $d$? | 5 | 1. **Understanding the exchange rate and the amount exchanged**:
Isabella exchanges her U.S. dollars to Canadian dollars at a rate where $7$ U.S. dollars yield $10$ Canadian dollars. This means for every $7$ U.S. dollars, she receives $10$ Canadian dollars.
2. **Calculating the total Canadian dollars received**:
... | 1 | 1,400.625 | 1,400.625 | -1 |
At Wednesday's basketball game, the Cayley Comets scored 90 points. At Friday's game, they scored $80\%$ as many points as they scored on Wednesday. How many points did they score on Friday? | 72 | On Friday, the Cayley Comets scored $80\%$ of 90 points. This is equal to $\frac{80}{100} \times 90 = \frac{8}{10} \times 90 = 8 \times 9 = 72$ points. Alternatively, since $80\%$ is equivalent to 0.8, then $80\%$ of 90 is equal to $0.8 \times 90 = 72$. | 1 | 1,119 | 1,119 | -1 |
The Grunters play the Screamers 4 times. The Grunters are the much better team, and are $75\%$ likely to win any given game. What is the probability that the Grunters will win all 4 games? Express your answer as a common fraction. | \frac{81}{256} | 1 | 2,068.3125 | 2,068.3125 | -1 | |
There are $20n$ members in the Trumpington marching band, and when they line up in rows of 26, there are 4 band members left over. If $n$ is an integer and there are fewer than 1000 band members, what is the maximum number of people that could be in the Trumpington marching band? | 940 | 1 | 3,319.5625 | 3,319.5625 | -1 | |
The Antarctican language has an alphabet of just 16 letters. Interestingly, every word in the language has exactly 3 letters, and it is known that no word's first letter equals any word's last letter (for instance, if the alphabet were $\{a, b\}$ then $a a b$ and aaa could not both be words in the language because $a$ ... | 1024 | 1024 Every letter can be the first letter of a word, or the last letter of a word, or possibly neither, but not both. If there are $a$ different first letters and $b$ different last letters, then we can form $a \cdot 16 \cdot b$ different words (and the desired conditions will be met). Given the constraints $0 \leq a, ... | 0.0625 | 7,609 | 6,310 | 7,695.6 |
Given that points $P$ and $Q$ are moving points on the curve $y=xe^{-2x}$ and the line $y=x+2$ respectively, find the minimum distance between points $P$ and $Q$. | \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Determine the appropriate value of $h$ so that the following equation in base $h$ is accurate:
$$\begin{array}{c@{}c@{}c@{}c@{}c@{}c}
&&8&6&7&4_h \\
&+&4&3&2&9_h \\
\cline{2-6}
&1&3&0&0&3_h.
\end{array}$$ | 10 | 0.875 | 4,768.125 | 4,279 | 8,192 | |
Let $x,$ $y,$ $z$ be real numbers, all greater than 3, so that
\[\frac{(x + 2)^2}{y + z - 2} + \frac{(y + 4)^2}{z + x - 4} + \frac{(z + 6)^2}{x + y - 6} = 36.\]Enter the ordered triple $(x,y,z).$ | (10,8,6) | 0.8125 | 5,035.6875 | 4,307.307692 | 8,192 | |
At the beginning of a trip, the mileage odometer read $56,200$ miles. The driver filled the gas tank with $6$ gallons of gasoline. During the trip, the driver filled his tank again with $12$ gallons of gasoline when the odometer read $56,560$. At the end of the trip, the driver filled his tank again with $20$ gallons o... | 26.9 | 1. **Identify the total distance traveled**:
The odometer readings at the start and end of the trip are given as $56,200$ miles and $57,060$ miles, respectively. Therefore, the total distance traveled during the trip is:
\[
57,060 - 56,200 = 860 \text{ miles}
\]
2. **Calculate the total gasoline used**:
... | 0.5 | 6,319.6875 | 5,044.875 | 7,594.5 |
The Fibonacci sequence is the sequence 1, 1, 2, 3, 5, $\ldots$ where the first and second terms are 1 and each term after that is the sum of the previous two terms. What is the remainder when the $100^{\mathrm{th}}$ term of the sequence is divided by 8? | 3 | 0.75 | 5,268.625 | 4,468.5 | 7,669 | |
Let \( ABC \) be a triangle with \( AB = 5 \), \( AC = 4 \), \( BC = 6 \). The angle bisector of \( \angle C \) intersects side \( AB \) at \( X \). Points \( M \) and \( N \) are drawn on sides \( BC \) and \( AC \), respectively, such that \( \overline{XM} \parallel \overline{AC} \) and \( \overline{XN} \parallel \ov... | \frac{3 \sqrt{14}}{5} | 0 | 6,722 | -1 | 6,722 | |
Jar C initially contains 6 red buttons and 12 green buttons. Michelle removes the same number of red buttons as green buttons from Jar C and places them into an empty Jar D. After the removal, Jar C is left with $\frac{3}{4}$ of its initial button count. If Michelle were to randomly choose a button from Jar C and a but... | \frac{5}{14} | 0 | 6,456.75 | -1 | 6,456.75 | |
Given that $\{a_n\}$ is a geometric sequence with a common ratio of $q$, and $a_m$, $a_{m+2}$, $a_{m+1}$ form an arithmetic sequence.
(Ⅰ) Find the value of $q$;
(Ⅱ) Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Determine whether $S_m$, $S_{m+2}$, $S_{m+1}$ form an arithmetic sequence and exp... | -\frac{1}{2} | 0.0625 | 7,054.4375 | 8,192 | 6,978.6 | |
Given $\sqrt{20} \approx 4.472, \sqrt{2} \approx 1.414$, find $-\sqrt{0.2} \approx$____. | -0.4472 | 0.25 | 512.0625 | 578.5 | 489.916667 | |
Given a moving circle that passes through the fixed point $F(1,0)$ and is tangent to the fixed line $l$: $x=-1$.
(1) Find the equation of the trajectory $C$ of the circle's center;
(2) The midpoint of the chord $AB$ formed by the intersection of line $l$ and $C$ is $(2,1}$. $O$ is the coordinate origin. Find the value ... | \sqrt{35} | 0.125 | 7,745.5625 | 6,806 | 7,879.785714 | |
Calculate: $|\sqrt{8}-2|+(\pi -2023)^{0}+(-\frac{1}{2})^{-2}-2\cos 60^{\circ}$. | 2\sqrt{2}+2 | 0.125 | 2,315.9375 | 2,320.5 | 2,315.285714 | |
What is the largest value of $x$ that satisfies the equation $\sqrt{2x}=4x$? Express your answer in simplest fractional form. | \frac18 | 1 | 1,985.125 | 1,985.125 | -1 | |
Five guys are eating hamburgers. Each one puts a top half and a bottom half of a hamburger bun on the grill. When the buns are toasted, each guy randomly takes two pieces of bread off of the grill. What is the probability that each guy gets a top half and a bottom half? | 8/63 | 0.8125 | 5,880.625 | 5,445.384615 | 7,766.666667 | |
What is the sum of the fractions of the form $\frac{2}{n(n+2)}$, where $n$ takes on odd positive integers from 1 to 2011? Express your answer as a decimal to the nearest thousandth. | 0.999 | 0 | 8,192 | -1 | 8,192 | |
Calculate the definite integral:
$$
\int_{\frac{\pi}{2}}^{2 \operatorname{arctg} 2} \frac{d x}{\sin ^{2} x(1-\cos x)}
$$ | 55/96 | 0.125 | 7,608.875 | 5,415 | 7,922.285714 | |
Max picks two different cards without replacement from a standard 52-card deck. What is the probability that the cards are of different suits? | \frac{13}{17} | 1 | 3,180.1875 | 3,180.1875 | -1 | |
Simplify $\frac{8xy^2}{6x^2y}$ with $x=2$ and $y=3.$ | 2 | 1 | 2,426.3125 | 2,426.3125 | -1 | |
If $a, b, c$, and $d$ are pairwise distinct positive integers that satisfy \operatorname{lcm}(a, b, c, d)<1000$ and $a+b=c+d$, compute the largest possible value of $a+b$. | 581 | Let $a^{\prime}=\frac{\operatorname{lcm}(a, b, c, d)}{a}$. Define $b^{\prime}, c^{\prime}$, and $d^{\prime}$ similarly. We have that $a^{\prime}, b^{\prime}, c^{\prime}$, and $d^{\prime}$ are pairwise distinct positive integers that satisfy $$\frac{1}{a^{\prime}}+\frac{1}{b^{\prime}}=\frac{1}{c^{\prime}}+\frac{1}{d^{\p... | 0 | 8,192 | -1 | 8,192 |
In convex quadrilateral $ABCD$ , $\angle ADC = 90^\circ + \angle BAC$ . Given that $AB = BC = 17$ , and $CD = 16$ , what is the maximum possible area of the quadrilateral?
*Proposed by Thomas Lam* | 529/2 | 0.125 | 7,969.3125 | 6,729 | 8,146.5 | |
Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than $30$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$. | 0 | 1 | 2,798.5625 | 2,798.5625 | -1 | |
The ratio of boys to girls in Mr. Brown's math class is $2:3$. If there are $30$ students in the class, how many more girls than boys are in the class? | 6 | 1. **Understanding the Ratio**: The problem states that the ratio of boys to girls in Mr. Brown's math class is $2:3$. This means for every 2 boys, there are 3 girls.
2. **Setting Up Variables**: Let the number of boys be $2x$ and the number of girls be $3x$. Here, $x$ is a common multiplier that will help us find the... | 1 | 1,170.875 | 1,170.875 | -1 |
Given $\sin (\frac{\pi }{3}-\theta )=\frac{3}{4}$, find $\cos (\frac{\pi }{3}+2\theta )$. | \frac{1}{8} | 0.875 | 5,056.1875 | 4,608.214286 | 8,192 | |
First $a$ is chosen at random from the set $\{1,2,3,\cdots,99,100\}$, and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is | \frac{3}{16} | 1. **Identify the Cyclic Nature of Units Digits for Powers of 3 and 7**:
- The units digit of powers of $3$ cycles through $1, 3, 9, 7$. This can be verified by calculating $3^1, 3^2, 3^3, 3^4$, etc., and observing the units digits.
- Similarly, the units digit of powers of $7$ cycles through $7, 9, 3, 1$. This i... | 0.9375 | 4,121 | 4,027.133333 | 5,529 |
In the Cartesian coordinate system $xoy$, the parametric equation of line $l$ is $\begin{cases} x=1- \frac { \sqrt {3}}{2}t \\ y= \frac {1}{2}t\end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the equation o... | 2 \sqrt {3} | 0 | 5,645.4375 | -1 | 5,645.4375 | |
A cat is going up a stairwell with ten stairs. The cat can jump either two or three stairs at each step, or walk the last step if necessary. How many different ways can the cat go from the bottom to the top? | 12 | 0 | 7,588.4375 | -1 | 7,588.4375 | |
Given that $x \in (0, \frac{1}{2})$, find the minimum value of $\frac{2}{x} + \frac{9}{1-2x}$. | 25 | 0.9375 | 5,163.1875 | 4,961.266667 | 8,192 | |
Given the coordinates of the three vertices of $\triangle ABC$ are $A(0,1)$, $B(1,0)$, $C(0,-2)$, and $O$ is the origin, if a moving point $M$ satisfies $|\overrightarrow{CM}|=1$, calculate the maximum value of $|\overrightarrow{OA}+ \overrightarrow{OB}+ \overrightarrow{OM}|$. | \sqrt{2}+1 | 0 | 5,635.1875 | -1 | 5,635.1875 | |
Let \[\begin{aligned} a &= \sqrt{2}+\sqrt{3}+\sqrt{6}, \\ b &= -\sqrt{2}+\sqrt{3}+\sqrt{6}, \\ c&= \sqrt{2}-\sqrt{3}+\sqrt{6}, \\ d&=-\sqrt{2}-\sqrt{3}+\sqrt{6}. \end{aligned}\]Evaluate $\left(\frac1a + \frac1b + \frac1c + \frac1d\right)^2.$ | \frac{96}{529} | 0.125 | 7,775.8125 | 5,342.5 | 8,123.428571 | |
A circle is inscribed in a square, and within this circle, a smaller square is inscribed such that one of its sides coincides with a side of the larger square and two vertices lie on the circle. Calculate the percentage of the area of the larger square that is covered by the smaller square. | 50\% | 0 | 7,722.6875 | -1 | 7,722.6875 | |
Arrange 3 volunteer teachers to 4 schools, with at most 2 people per school. How many different distribution plans are there? (Answer with a number) | 60 | 0.625 | 6,504.5625 | 5,492.1 | 8,192 | |
Let $g(x)$ be a function piecewise defined as \[g(x) = \left\{
\begin{array}{cl}
-x & x\le 0, \\
2x-41 & x>0.
\end{array}
\right.\] If $a$ is negative, find $a$ so that $g(g(g(10.5)))=g(g(g(a)))$. | a=-30.5 | 0.6875 | 5,392.1875 | 4,647.545455 | 7,030.4 | |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | \sqrt{55} | 0.875 | 3,633.1875 | 2,981.928571 | 8,192 | |
Vasya has 9 different books by Arkady and Boris Strugatsky, each containing a single work by the authors. Vasya wants to arrange these books on a shelf in such a way that:
(a) The novels "Beetle in the Anthill" and "Waves Extinguish the Wind" are next to each other (in any order).
(b) The stories "Restlessness" and "A ... | 4 \cdot 7! | 0 | 5,074.625 | -1 | 5,074.625 | |
A spinner has eight congruent sections, each labeled with numbers 1 to 8. Jane and her brother each spin this spinner once. Jane wins if the non-negative difference of their numbers is less than three; otherwise, her brother wins. Determine the probability of Jane winning. Express your answer as a common fraction. | \frac{17}{32} | 0.4375 | 7,271.625 | 6,088.285714 | 8,192 | |
Consider all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$f(f(x)+2 x+20)=15$$ Call an integer $n$ good if $f(n)$ can take any integer value. In other words, if we fix $n$, for any integer $m$, there exists a function $f$ such that $f(n)=m$. Find the sum of all good integers $x$. | -35 | For almost all integers $x, f(x) \neq-x-20$. If $f(x)=-x-20$, then $$f(-x-20+2 x+20)=15 \Longrightarrow-x-20=15 \Longrightarrow x=-35$$ Now it suffices to prove that the $f(-35)$ can take any value. $f(-35)=15$ in the function $f(x) \equiv 15$. Otherwise, set $f(-35)=c$, and $f(x)=15$ for all other $x$. It is easy to c... | 0 | 8,110.625 | -1 | 8,110.625 |
In the diagram, $PQ$ and $RS$ are diameters of a circle with radius 6. $PQ$ and $RS$ intersect perpendicularly at the center $O$. The line segments $PR$ and $QS$ subtend central angles of 60° and 120° respectively at $O$. What is the area of the shaded region formed by $\triangle POR$, $\triangle SOQ$, sector $POS$, an... | 36 + 18\pi | 0 | 7,370.5 | -1 | 7,370.5 | |
Four people are sitting at four sides of a table, and they are dividing a 32-card Hungarian deck equally among themselves. If one selected player does not receive any aces, what is the probability that the player sitting opposite them also has no aces among their 8 cards? | 130/759 | 0.1875 | 7,890 | 7,206.333333 | 8,047.769231 | |
The polynomial $f(x)=x^3-4\sqrt{3}x^2+13x-2\sqrt{3}$ has three real roots, $a$ , $b$ , and $c$ . Find $$ \max\{a+b-c,a-b+c,-a+b+c\}. $$ | 2\sqrt{3} + 2\sqrt{2} | 0.75 | 6,311.25 | 5,773.166667 | 7,925.5 | |
Mr. J left his entire estate to his wife, his daughter, his son, and the cook. His daughter and son got half the estate, sharing in the ratio of $4$ to $3$. His wife got twice as much as the son. If the cook received a bequest of $\textdollar{500}$, then the entire estate was: | 7000 | Let's denote the total estate by $E$. According to the problem, the daughter and son together received half of the estate, and they share this half in the ratio of $4:3$. Let's denote the shares of the daughter and son as $4x$ and $3x$ respectively. Therefore, we have:
\[ 4x + 3x = \frac{1}{2}E \]
\[ 7x = \frac{1}{2}E ... | 0.9375 | 2,017 | 1,605.333333 | 8,192 |
Robinson had 200,000 strands of hair when he ended up on the deserted island. At that time, his hair strands were $5 \mathrm{~cm}$ long. The hair strands grew $0.5 \mathrm{~mm}$ per day, but Robinson did not cut his hair because he did not have the proper tools, and he lost 50 strands of hair each day without replaceme... | 1950 | 0.5625 | 7,234.125 | 6,650.333333 | 7,984.714286 | |
It costs 5 cents to copy 3 pages. How many pages can you copy for $\$20$? | 1200 | 1 | 2,218.4375 | 2,218.4375 | -1 | |
If $x+\frac1x=-5$, what is $x^3+\frac1{x^3}$? | -110 | 1 | 3,332.875 | 3,332.875 | -1 | |
A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in the given diagram). In each square of the eleventh ro... | 640 | 0.3125 | 7,215.5625 | 6,359.2 | 7,604.818182 | |
Given that point $P$ is a moving point on circle $C_{1}$: $\left(x-1\right)^{2}+y^{2}=1$, point $Q$ is a moving point on circle $C_{2}$: $\left(x-4\right)^{2}+\left(y-1\right)^{2}=4$, and point $R$ moves on the line $l: x-y+1=0$, find the minimum value of $|PR|+|QR|$. | \sqrt{26}-3 | 0.25 | 7,925.5 | 7,616.25 | 8,028.583333 | |
Let \( P(x) = x^{4} + a x^{3} + b x^{2} + c x + d \), where \( a, b, c, \) and \( d \) are real coefficients. Given that
\[ P(1) = 7, \quad P(2) = 52, \quad P(3) = 97, \]
find the value of \(\frac{P(9) + P(-5)}{4}\). | 1202 | 0.3125 | 7,145.9375 | 5,205.4 | 8,028 | |
Find all solutions to $(m^2+n)(m + n^2)= (m - n)^3$ , where m and n are non-zero integers.
Do it | \[
\{(-1,-1), (8,-10), (9,-6), (9,-21)\}
\] | Expanding both sides, \[m^3+mn+m^2n^2+n^3=m^3-3m^2n+3mn^2-n^3\] Note that $m^3$ can be canceled and as $n \neq 0$ , $n$ can be factored out.
Writing this as a quadratic equation in $n$ : \[2n^2+(m^2-3m)n+(3m^2+m)=0\] .
The discriminant $b^2-4ac$ equals \[(m^2-3m)^2-8(3m^2+m)\] \[=m^4-6m^3-15m^2-8m\] , which we want to... | 0 | 8,192 | -1 | 8,192 |
On the 5 by 5 square grid below, each dot is 1 cm from its nearest horizontal and vertical neighbors. What is the product of the value of the area of square $ABCD$ (in cm$^2$) and the value of the perimeter of square $ABCD$ (in cm)? Express your answer in simplest radical form.
[asy]unitsize(1cm);
defaultpen(linewidth... | 40\sqrt{10} | 1 | 3,582.25 | 3,582.25 | -1 | |
If $f(x)=3x+2$ for all real $x$, then the statement:
"$|f(x)+4|<a$ whenever $|x+2|<b$ and $a>0$ and $b>0$"
is true when | $b \le a/3$ | 1. **Substitute $f(x)$ into the inequality**: Given $f(x) = 3x + 2$, we substitute this into the inequality $|f(x) + 4| < a$:
\[
|3x + 2 + 4| = |3x + 6|.
\]
Simplifying further, we have:
\[
|3x + 6| = |3(x + 2)| = 3|x + 2|.
\]
2. **Relate $3|x + 2|$ to $a$**: From the above, we know:
\[
3|x ... | 0 | 3,769.6875 | -1 | 3,769.6875 |
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy = 1$ and both branches of the hyperbola $xy = -1$. (A set $S$ in the plane is called \emph{convex} if for any two points in $S$ the line segment connecting them is contained in $S$.) | 4 | The minimum is 4, achieved by the square with vertices $(\pm 1, \pm 1)$.
\textbf{First solution:}
To prove that 4 is a lower bound, let $S$ be a convex set of the desired form. Choose $A,B,C,D \in S$ lying on the branches of the two hyperbolas, with $A$ in the upper right quadrant, $B$ in the upper left, $C$ in the lo... | 0.0625 | 8,122.0625 | 7,073 | 8,192 |
Compute
\[\cos^2 0^\circ + \cos^2 1^\circ + \cos^2 2^\circ + \dots + \cos^2 90^\circ.\] | \frac{91}{2} | 0.375 | 7,561.9375 | 6,511.833333 | 8,192 | |
Simplify $3\cdot\frac{11}{4}\cdot \frac{16}{-55}$. | -\frac{12}{5} | 1 | 3,255.0625 | 3,255.0625 | -1 | |
A bag contains 2 red balls, 3 white balls, and 4 yellow balls. If 4 balls are randomly selected from the bag, what is the probability that the selection includes balls of all three colors? | 4/7 | 0.5 | 7,026.5625 | 5,861.125 | 8,192 | |
If
\[\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 3 \quad \text{and} \quad \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 0,\]find $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}.$ | 9 | 1 | 3,360.5 | 3,360.5 | -1 | |
Given the function $f(x)=x^2+x+b\ (b\in\mathbb{R})$ with a value range of $[0,+\infty)$, the solution to the equation $f(x) < c$ is $m+8$. Determine the value of $c$. | 16 | 0.3125 | 6,903.8125 | 5,908.2 | 7,356.363636 | |
If the vector $\overrightarrow{a} = (x, y-1)$ is collinear with the vector $\overrightarrow{b} = (3, -2)$, then the minimum value of $z = \log_{2}(4^x + 8^y)$ is ______. | \frac{5}{2} | 1 | 2,924.625 | 2,924.625 | -1 | |
Consider a circle of radius \( R \) centered at the origin on the Cartesian plane. Specify at least one value of \( R \) for which there are exactly 32 integer points (a point is called an integer point if both its x-coordinate and y-coordinate are integers) lying on this circle. | \sqrt{1105} | 0 | 8,089.9375 | -1 | 8,089.9375 | |
If the orthocenter of \( \triangle OAB \) is exactly the focus of the parabola \( y^2 = 4x \), where \( O \) is the origin and points \( A \) and \( B \) lie on the parabola, then the area of \( \triangle OAB \) is equal to ____. | 10\sqrt{5} | 0.8125 | 5,787.5625 | 5,232.692308 | 8,192 | |
In triangle \(ABC\), angle \(A\) is the largest angle. Points \(M\) and \(N\) are symmetric to vertex \(A\) with respect to the angle bisectors of angles \(B\) and \(C\) respectively. Find \(\angle A\) if \(\angle MAN = 50^\circ\). | 80 | 0.25 | 7,804.5 | 6,657.75 | 8,186.75 | |
Consider the set $\{8, -7, 2, -4, 20\}$. Find the smallest sum that can be achieved by adding three different numbers from this set. | -9 | 0.9375 | 3,147.4375 | 2,813.533333 | 8,156 | |
Let the function $f(x)=2\tan \frac{x}{4}\cdot \cos^2 \frac{x}{4}-2\cos^2\left(\frac{x}{4}+\frac{\pi }{12}\right)+1$.
(Ⅰ) Find the smallest positive period and the domain of $f(x)$;
(Ⅱ) Find the intervals of monotonicity and the extremum of $f(x)$ in the interval $[-\pi,0]$; | -\frac{\sqrt{3}}{2} | 0 | 6,432.1875 | -1 | 6,432.1875 | |
Determine the sum of all real numbers $x$ that are not in the domain of the function $$g(x) = \frac{1}{2+\frac{1}{3+\frac{1}{x}}}.$$ | -\frac{13}{21} | 1 | 3,775.0625 | 3,775.0625 | -1 | |
If $A = 3009 \div 3$, $B = A \div 3$, and $Y = A - B$, then what is the value of $Y$? | 669 | 0 | 313.1875 | -1 | 313.1875 |
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