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 |
|---|---|---|---|---|---|---|
What is the sum of the odd positive integers less than 50? | 625 | 1 | 2,054.5625 | 2,054.5625 | -1 | |
Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$ | -\ln 2 | 0.75 | 5,520.875 | 4,630.5 | 8,192 | |
Let \( S = \{1, 2, \cdots, 98\} \). Find the smallest positive integer \( n \) such that, in any subset of \( S \) with \( n \) elements, it is always possible to select 10 numbers, and no matter how these 10 numbers are evenly divided into two groups, there will always be one number in one group that is relatively pri... | 50 | 0 | 8,192 | -1 | 8,192 | |
Radii of five concentric circles $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$ form a geometric progression with common ratio $q$ in this order. What is the maximal value of $q$ for which it's possible to draw a broken line $A_0A_1A_2A_3A_4$ consisting of four equal segments such that $A_i$ lies on $\omega_... | \frac{1 + \sqrt{5}}{2} | 0 | 8,192 | -1 | 8,192 | |
Team A and Team B each have 7 players who compete in a predetermined order in a Go competition. Initially, Player 1 from each team competes. The loser is eliminated, and the winner competes next against the loser's team Player 2, and so on, until all players from one team are eliminated. The remaining team wins. How ma... | 3432 | 0 | 8,192 | -1 | 8,192 | |
Given the function $f\left(x\right)=|2x-3|+|x-2|$.<br/>$(1)$ Find the solution set $M$ of the inequality $f\left(x\right)\leqslant 3$;<br/>$(2)$ Under the condition of (1), let the smallest number in $M$ be $m$, and let positive numbers $a$ and $b$ satisfy $a+b=3m$, find the minimum value of $\frac{{{b^2}+5}}{a}+\frac{... | \frac{13}{2} | 0.875 | 5,945.5625 | 5,624.642857 | 8,192 | |
A circle with center \( Q \) and radius 2 rolls around the inside of a right triangle \( DEF \) with side lengths 9, 12, and 15, always remaining tangent to at least one side of the triangle. When \( Q \) first returns to its original position, through what distance has \( Q \) traveled? | 24 | 0 | 8,109.9375 | -1 | 8,109.9375 | |
Let \(f_{0}(x)=\frac{1}{c-x}\) and \(f_{n}(x)=f_{0}\left(f_{n-1}(x)\right)\), \(n=1,2,3, \ldots\). If \(f_{2000}(2000)=d\), find the value of \(d\). | 2000 | 0 | 8,192 | -1 | 8,192 | |
Grace has $\$4.80$ in U.S. coins. She has the same number of dimes and pennies. What is the greatest number of dimes she could have? | 43 | 0 | 5,994.875 | -1 | 5,994.875 | |
Four friends do yardwork for their neighbors over the weekend, earning $15, $20, $25, and $40, respectively. They decide to split their earnings equally among themselves. In total, how much will the friend who earned $40 give to the others? | 15 | 1. **Calculate the total earnings**: The four friends earn different amounts: $15, $20, $25, and $40. To find the total earnings, sum these amounts:
\[
15 + 20 + 25 + 40 = 100
\]
Therefore, the total earnings are $\$100$.
2. **Determine the equal share**: Since the friends decide to split their earnings eq... | 1 | 1,785.0625 | 1,785.0625 | -1 |
Choose any $2$ numbers from $-5$, $-3$, $-1$, $2$, and $4$. Let the maximum product obtained be denoted as $a$, and the minimum quotient obtained be denoted as $b$. Then the value of $\frac{a}{b}$ is ______. | -\frac{15}{4} | 0.1875 | 4,458.125 | 4,647.333333 | 4,414.461538 | |
Compute the sum of the series:
\[ 5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5)))))))))) \] | 305175780 | 0.375 | 6,268.75 | 4,798 | 7,151.2 | |
Let \( a, b, c \) be the roots of the cubic equation
\[
x^3 + 3x^2 + 5x + 7 = 0
\]
The cubic polynomial \( P \) satisfies the following conditions:
\[
\begin{array}{l}
P(a) = b + c, \quad P(b) = a + c, \\
P(c) = a + b, \quad P(a + b + c) = -16.
\end{array}
\]
Determine the value of \( P(0) \). | 11 | 0.875 | 3,623.6875 | 3,320.5 | 5,746 | |
How many positive integers $n$ satisfy the following condition:
$(130n)^{50} > n^{100} > 2^{200}$? | 125 | We are given the inequality $(130n)^{50} > n^{100} > 2^{200}$ and need to find the number of positive integers $n$ that satisfy this condition.
1. **First Inequality Analysis:**
\[
(130n)^{50} > n^{100}
\]
Simplifying, we get:
\[
130^{50} n^{50} > n^{100}
\]
Dividing both sides by $n^{50}$ (ass... | 1 | 3,391.75 | 3,391.75 | -1 |
Let $f$, $g$, and $h$ be polynomials such that $h(x) = f(x)\cdot g(x)$. If the constant term of $f(x)$ is $-4$ and the constant term of $h(x)$ is 3, what is $g(0)$? | -\frac{3}{4} | 1 | 1,119.1875 | 1,119.1875 | -1 | |
Compute the determinant of the following matrix:
\[
\begin{vmatrix} 3 & 1 & 0 \\ 8 & 5 & -2 \\ 3 & -1 & 6 \end{vmatrix}.
\] | 138 | 0 | 7,356.5625 | -1 | 7,356.5625 | |
Given the sequence ${a_n}$ that satisfies: $a_1=3$, $a_{n+1}=9\cdot 3a_{n} (n\geqslant 1)$, find $\lim\limits_{n\to∞}a_{n}=$ \_\_\_\_\_\_. | 27 | 0 | 2,150.375 | -1 | 2,150.375 | |
There is a garden with 3 rows and 2 columns of rectangular flower beds, each measuring 6 feet long and 2 feet wide. Between the flower beds, as well as around the garden, there is a 1-foot wide path. What is the total area \( S \) of the path in square feet? | 78 | 0 | 7,887.875 | -1 | 7,887.875 | |
As shown in the diagram, in the tetrahedron \(A B C D\), the face \(A B C\) intersects the face \(B C D\) at a dihedral angle of \(60^{\circ}\). The projection of vertex \(A\) onto the plane \(B C D\) is \(H\), which is the orthocenter of \(\triangle B C D\). \(G\) is the centroid of \(\triangle A B C\). Given that \(A... | \frac{4\sqrt{21}}{9} | 0 | 8,019.0625 | -1 | 8,019.0625 | |
Consider the operation $*$ defined by the following table:
\begin{tabular}{c|cccc} * & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ 2 & 2 & 4 & 1 & 3 \\ 3 & 3 & 1 & 4 & 2 \\ 4 & 4 & 3 & 2 & 1 \end{tabular}
For example, $3*2=1$. Then $(2*4)*(1*3)=$ | 4 | To solve the problem, we need to evaluate $(2*4)*(1*3)$ using the operation $*$ defined by the given table.
1. **Find $2*4$ and $1*3$ using the table:**
- From the table, $2*4 = 3$.
- From the table, $1*3 = 3$.
2. **Now find $(2*4)*(1*3)$ using the results from step 1:**
- We need to find $3*3$.
- From th... | 0.9375 | 3,195.375 | 2,862.266667 | 8,192 |
On a chessboard, a queen attacks every square it can reach by moving from its current square along a row, column, or diagonal without passing through a different square that is occupied by a chess piece. Find the number of ways in which three indistinguishable queens can be placed on an $8 \times 8$ chess board so that... | 864 | The configuration of three cells must come in a 45-45-90 triangle. There are two cases, both shown above: the triangle has legs parallel to the axes, or it has its hypotenuse parallel to an axis. The first case can be solved by noticing that each selection of four cells in the shape of a square corresponds to four such... | 0 | 7,943 | -1 | 7,943 |
A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top o... | 183 | 0 | 7,783.0625 | -1 | 7,783.0625 | |
A train arrives randomly some time between 2:00 and 4:00 PM, waits for 30 minutes, and then leaves. If Maria also arrives randomly between 2:00 and 4:00 PM, what is the probability that the train will be there when Maria arrives? | \frac{7}{32} | 0.5 | 6,929.8125 | 5,794.875 | 8,064.75 | |
The geometric series $a+ar+ar^2+\cdots$ has a sum of $12$, and the terms involving odd powers of $r$ have a sum of $5.$ What is $r$? | \frac{5}{7} | 1 | 2,363.625 | 2,363.625 | -1 | |
The expression $\cos 2x + \cos 6x + \cos 10x + \cos 14x$ can be written in the equivalent form
\[a \cos bx \cos cx \cos dx\] for some positive integers $a,$ $b,$ $c,$ and $d.$ Find $a + b + c + d.$ | 18 | 1 | 3,028.1875 | 3,028.1875 | -1 | |
1. How many four-digit numbers with no repeated digits can be formed using the digits 1, 2, 3, 4, 5, 6, 7, and the four-digit number must be even?
2. How many five-digit numbers with no repeated digits can be formed using the digits 0, 1, 2, 3, 4, 5, and the five-digit number must be divisible by 5? (Answer with number... | 216 | 0.625 | 4,649.6875 | 4,122.1 | 5,529 | |
Given the sets \( A=\{x \mid 5x - a \leqslant 0\} \) and \( B=\{x \mid 6x - b > 0\} \), where \( a, b \in \mathbf{N} \), and \( A \cap B \cap \mathbf{N} = \{2, 3, 4\} \), determine the number of integer pairs \( (a, b) \). | 30 | 0.25 | 6,518.0625 | 4,855 | 7,072.416667 | |
Let $a,$ $b,$ $c,$ be nonzero real numbers such that $a + b + c = 0.$ Find all possible values of
\[\frac{a^2 b^2}{(a^2 - bc)(b^2 - ac)} + \frac{a^2 c^2}{(a^2 - bc)(c^2 - ab)} + \frac{b^2 c^2}{(b^2 - ac)(c^2 - ab)}.\]Enter all possible values, separated by commas. | 1 | 0.75 | 5,396.625 | 4,464.833333 | 8,192 | |
If $x+y=10$ and $xy=12$, what is the value of $x^3-y^3$? | 176\sqrt{13} | 0.9375 | 5,123.8125 | 4,919.266667 | 8,192 | |
Evaluate $(2 + 1)(2^2 + 1^2)(2^4 + 1^4)$. | 255 | 1 | 2,543 | 2,543 | -1 | |
In a class of 35 students, everyone participates in physics and chemistry. Nine students received an A in chemistry and 15 received an A in physics, including five that received an A in both subjects. How many students did not receive an A in either of these two subjects? | 16 | 0.9375 | 1,128.5625 | 1,005.266667 | 2,978 | |
[asy] draw((0,0)--(0,3)--(4,0)--cycle,dot); draw((4,0)--(7,0)--(7,10)--cycle,dot); draw((0,3)--(7,10),dot); MP("C",(0,0),SW);MP("A",(0,3),NW);MP("B",(4,0),S);MP("E",(7,0),SE);MP("D",(7,10),NE); [/asy]
Triangle $ABC$ has a right angle at $C, AC=3$ and $BC=4$. Triangle $ABD$ has a right angle at $A$ and $AD=12$. Points $... | 128 | 0 | 5,848.375 | -1 | 5,848.375 | |
A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$? | 200 | 1. **Calculate the volume of the first box**:
The volume $V$ of a rectangular prism is calculated by the formula:
\[
V = \text{height} \times \text{width} \times \text{length}
\]
For the first box, the dimensions are given as height = $2$ cm, width = $3$ cm, and length = $5$ cm. Therefore, the volume o... | 0 | 1,566.125 | -1 | 1,566.125 |
Taylor is tiling his 12 feet by 16 feet living room floor. He plans to place 1 foot by 1 foot tiles along the edges to form a border, and then use 2 feet by 2 feet tiles to fill the remaining floor area, so find the total number of tiles he will use. | 87 | 0.375 | 879.4375 | 882.166667 | 877.8 | |
What is $\frac{0.\overline{72}}{0.\overline{27}}$? Express your answer as a common fraction in lowest terms. | \frac{8}{3} | 1 | 2,649.9375 | 2,649.9375 | -1 | |
[asy] draw(circle((4,1),1),black+linewidth(.75)); draw((0,0)--(8,0)--(8,6)--cycle,black+linewidth(.75)); MP("A",(0,0),SW);MP("B",(8,0),SE);MP("C",(8,6),NE);MP("P",(4,1),NW); MP("8",(4,0),S);MP("6",(8,3),E);MP("10",(4,3),NW); MP("->",(5,1),E); dot((4,1)); [/asy]The sides of $\triangle ABC$ have lengths $6,8,$ and $10$. ... | 12 | 0 | 7,573.8125 | -1 | 7,573.8125 | |
The simplest form of $1 - \frac{1}{1 + \frac{a}{1 - a}}$ is: | a | We start by simplifying the expression inside the fraction:
\[ 1 - \frac{1}{1 + \frac{a}{1 - a}}. \]
1. **Simplify the innermost fraction**:
\[ \frac{a}{1 - a}. \]
2. **Add this fraction to 1**:
\[ 1 + \frac{a}{1 - a} = \frac{1 - a + a}{1 - a} = \frac{1}{1 - a}. \]
3. **Take the reciprocal**:
\[ \frac{1}{\f... | 1 | 2,072.25 | 2,072.25 | -1 |
$A B$ is a diameter of circle $O . X$ is a point on $A B$ such that $A X=3 B X$. Distinct circles $\omega_{1}$ and $\omega_{2}$ are tangent to $O$ at $T_{1}$ and $T_{2}$ and to $A B$ at $X$. The lines $T_{1} X$ and $T_{2} X$ intersect $O$ again at $S_{1}$ and $S_{2}$. What is the ratio $\frac{T_{1} T_{2}}{S_{1} S_{2}}$... | \frac{3}{5} | Since the problem only deals with ratios, we can assume that the radius of $O$ is 1. As we have proven in Problem 5, points $S_{1}$ and $S_{2}$ are midpoints of arc $A B$. Since $A B$ is a diameter, $S_{1} S_{2}$ is also a diameter, and thus $S_{1} S_{2}=2$. Let $O_{1}, O_{2}$, and $P$ denote the center of circles $\om... | 0.1875 | 7,796.5 | 6,700.666667 | 8,049.384615 |
Given the ages of Daisy's four cousins are distinct single-digit positive integers, and the product of two of the ages is $24$ while the product of the other two ages is $35$, find the sum of the ages of Daisy's four cousins. | 23 | 0 | 7,747.5625 | -1 | 7,747.5625 | |
Rectangle $ABCD$ lies in a plane with $AB = CD = 3$ and $BC = DA = 8$. This rectangle is rotated $90^\circ$ clockwise around $D$, followed by another $90^\circ$ clockwise rotation around the new position of point $C$ after the first rotation. What is the length of the path traveled by point $A$?
A) $\frac{\pi(8 + \sqrt... | \frac{\pi(8 + \sqrt{73})}{2} | 0 | 6,901.6875 | -1 | 6,901.6875 | |
What is the maximum number of self-intersection points that a closed polyline with 7 segments can have? | 14 | 0.25 | 7,577.3125 | 5,841 | 8,156.083333 | |
For how many integers $x$ is the number $x^4 - 53x^2 + 150$ negative? | 12 | 0.9375 | 3,587.1875 | 3,600.4 | 3,389 | |
A classroom consists of a $5 \times 5$ array of desks, to be filled by anywhere from 0 to 25 students, inclusive. No student will sit at a desk unless either all other desks in its row or all others in its column are filled (or both). Considering only the set of desks that are occupied (and not which student sits at ea... | 962 | The set of empty desks must be of the form (non-full rows) $\times$ (non-full columns): each empty desk is in a non-full column and a non-full row, and the given condition implies that each desk in such a position is empty. So if there are fewer than 25 students, then both of these sets are nonempty; we have $2^{5}-1=3... | 0 | 8,192 | -1 | 8,192 |
A person flips a coin, where the probability of heads up and tails up is $\frac{1}{2}$ each. Construct a sequence $\left\{a_{n}\right\}$ such that
$$
a_{n}=\left\{
\begin{array}{ll}
1, & \text{if the } n \text{th flip is heads;} \\
-1, & \text{if the } n \text{th flip is tails.}
\end{array}
\right.
$$
Let $S_{n}=a_{1}+... | 13/128 | 0.25 | 7,143.5 | 4,421.5 | 8,050.833333 | |
A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths? | 195 | To solve this problem, we need to find all integer-coordinate points $(x, y)$ that a bug can reach on its path from $A = (-3, 2)$ to $B = (3, -2)$, given that the maximum allowable path length is 20 units. The bug moves only along lines parallel to the $x$-axis or $y$-axis.
1. **Calculate the shortest path from $A$ to... | 0 | 8,192 | -1 | 8,192 |
Trapezoid $EFGH$ has base $EF = 24$ units and base $GH = 36$ units. Diagonals $EG$ and $FH$ intersect at point $Y$. If the area of trapezoid $EFGH$ is $360$ square units, what is the area of triangle $FYH$? | 86.4 | 0 | 8,021.1875 | -1 | 8,021.1875 | |
Given an arithmetic-geometric sequence $\{ a_{n} \}$ that satisfies $a\_1 + a\_3 = 10$, $a\_2 + a\_4 = 5$, find the maximum value of the product $a\_1 a\_2 \ldots a\_n$. | 64 | 0 | 8,192 | -1 | 8,192 | |
If $r$ is the remainder when each of the numbers $1059$, $1417$, and $2312$ is divided by $d$, where $d$ is an integer greater than $1$, then $d-r$ equals | 15 | 1. We start with the given congruences:
\[
1059 \equiv r \pmod{d} \quad \text{(i)}
\]
\[
1417 \equiv r \pmod{d} \quad \text{(ii)}
\]
\[
2312 \equiv r \pmod{d} \quad \text{(iii)}
\]
2. Subtract congruence (i) from (ii) to eliminate $r$:
\[
1417 - 1059 \equiv r - r \pmod{d} \implies 358 ... | 1 | 2,397.875 | 2,397.875 | -1 |
In the triangle \(ABC\), it is known that \(AB=BC\) and \(\angle BAC=45^\circ\). The line \(MN\) intersects side \(AC\) at point \(M\), and side \(BC\) at point \(N\). Given that \(AM=2 \cdot MC\) and \(\angle NMC=60^\circ\), find the ratio of the area of triangle \(MNC\) to the area of quadrilateral \(ABNM\). | \frac{7 - 3\sqrt{3}}{11} | 0 | 7,648.375 | -1 | 7,648.375 | |
The numbers from 1 to 150, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is neither a perfect square nor a perfect cube? Express your answer as a common fraction. | \frac{9}{10} | 1 | 3,299.125 | 3,299.125 | -1 | |
Given $y=f(x)+x^2$ is an odd function, and $f(1)=1$, if $g(x)=f(x)+2$, then $g(-1)=$ . | -1 | 1 | 1,589.625 | 1,589.625 | -1 | |
The TV station is broadcasting 5 different advertisements, among which there are 3 different commercial advertisements and 2 different Olympic promotional advertisements. The last advertisement must be an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively... | 36 | 0.4375 | 6,888.0625 | 5,902.285714 | 7,654.777778 | |
A sphere is cut into three equal wedges. The circumference of the sphere is $18\pi$ inches. What is the volume of the intersection between one wedge and the top half of the sphere? Express your answer in terms of $\pi$. | 162\pi | 0.375 | 7,039.25 | 5,118 | 8,192 | |
Given that the terminal side of angle $\alpha$ passes through point $P(-4a, 3a) (a \neq 0)$, find the value of $\sin \alpha + \cos \alpha - \tan \alpha$. | \frac{19}{20} | 0 | 7,160.3125 | -1 | 7,160.3125 | |
On a two-lane highway where both lanes are single-directional, cars in both lanes travel at different constant speeds. The speed of cars in the left lane is 10 kilometers per hour higher than in the right lane. Cars follow a modified safety rule: the distance from the back of the car ahead to the front of the car in th... | 338 | 0.3125 | 6,722.5625 | 7,051.6 | 6,573 | |
At time $0$ , an ant is at $(1,0)$ and a spider is at $(-1,0)$ . The ant starts walking counterclockwise around the unit circle, and the spider starts creeping to the right along the $x$ -axis. It so happens that the ant's horizontal speed is always half the spider's. What will the shortest distance ever betwee... | \frac{\sqrt{14}}{4} | 0 | 7,921.5625 | -1 | 7,921.5625 | |
Find the sum of the values of $x$ such that $\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x$, where $x$ is measured in degrees and $100< x< 200.$ | 906 | Observe that $2\cos 4x\cos x = \cos 5x + \cos 3x$ by the sum-to-product formulas. Defining $a = \cos 3x$ and $b = \cos 5x$, we have $a^3 + b^3 = (a+b)^3 \rightarrow ab(a+b) = 0$. But $a+b = 2\cos 4x\cos x$, so we require $\cos x = 0$, $\cos 3x = 0$, $\cos 4x = 0$, or $\cos 5x = 0$.
Hence we see by careful analysis of ... | 0 | 8,118.8125 | -1 | 8,118.8125 |
Given $sn(α+ \frac {π}{6})= \frac {1}{3}$, and $\frac {π}{3} < α < \pi$, find $\sin ( \frac {π}{12}-α)$. | - \frac {4+ \sqrt {2}}{6} | 0 | 6,664.875 | -1 | 6,664.875 | |
The marble statue of Abraham Lincoln in the Lincoln Memorial has a height of 60 feet. A scale model of the statue has a height of 4 inches. How many feet of the statue does one inch of the model represent? | 15 | 1 | 1,847.1875 | 1,847.1875 | -1 | |
In triangle \( ABC \), the sides \( AC = 14 \) and \( AB = 6 \) are known. A circle with center \( O \), constructed on side \( AC \) as the diameter, intersects side \( BC \) at point \( K \). It turns out that \( \angle BAK = \angle ACB \). Find the area of triangle \( BOC \). | 21 | 0.1875 | 7,989.625 | 7,112.666667 | 8,192 | |
Given the functions $f(x)=x+e^{x-a}$ and $g(x)=\ln (x+2)-4e^{a-x}$, where $e$ is the base of the natural logarithm. If there exists a real number $x_{0}$ such that $f(x_{0})-g(x_{0})=3$, find the value of the real number $a$. | -\ln 2-1 | 0 | 7,620 | -1 | 7,620 | |
Diana is playing a card game against a computer. She starts with a deck consisting of a single card labeled 0.9. Each turn, Diana draws a random card from her deck, while the computer generates a card with a random real number drawn uniformly from the interval $[0,1]$. If the number on Diana's card is larger, she keeps... | 100 | By linearity of expectation, we can treat the number of turns each card contributes to the total independently. Let $f(x)$ be the expected number of turns a card of value $x$ contributes (we want $f(0.9)$). If we have a card of value $x$, we lose it after 1 turn with probability $1-x$. If we don't lose it after the fir... | 0 | 7,576.5 | -1 | 7,576.5 |
In the interval \([-6, 6]\), an element \(x_0\) is arbitrarily chosen. If the slope of the tangent line to the parabola \(y = x^2\) at \(x = x_0\) has an angle of inclination \(\alpha\), find the probability that \(\alpha \in \left[ \frac{\pi}{4}, \frac{3\pi}{4} \right]\). | \frac{11}{12} | 0.5625 | 6,298.625 | 5,908 | 6,800.857143 | |
Find the value of $10\cot(\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21).$ | 15 | On the coordinate plane, let $O=(0,0)$, $A_1=(3,0)$, $A_2=(3,1)$, $B_1=(21,7)$, $B_2=(20,10)$, $C_1=(260,130)$, $C_2=(250,150)$, $D_1=(5250,3150)$, $D_2=(5100,3400)$, and $H=(5100,0)$. We see that $\cot^{-1}(\angle A_2OA_1)=3$, $\cot^{-1}(\angle B_2OB_1)=7$, $\cot^{-1}(\angle C_2OC_1)=13$, and $\cot^{-1}(\angle D_2OD_1... | 0.875 | 4,139 | 3,560 | 8,192 |
Find the greatest negative value of the expression \( x - y \) for all pairs of numbers \((x, y)\) satisfying the equation
$$
(\sin x + \sin y)(\cos x - \cos y) = \frac{1}{2} + \sin(x - y) \cos(x + y)
$$ | -\frac{\pi}{6} | 0.5 | 7,375.4375 | 6,632.75 | 8,118.125 | |
At Archimedes Academy, there are three teachers, Mrs. Algebra, Mr. Calculus, and Ms. Statistics, teaching students preparing for the AMC 8 contest. Mrs. Algebra has 13 students, Mr. Calculus has 10 students, and Ms. Statistics has 12 students. If 3 students are enrolled in both Mrs. Algebra's and Ms. Statistics' classe... | 32 | 1 | 475.1875 | 475.1875 | -1 | |
Tatiana's teacher drew a $3 \times 3$ grid on the board, with zero in each cell. The students then took turns to pick a $2 \times 2$ square of four adjacent cells, and to add 1 to each of the numbers in the four cells. After a while, the grid looked like the diagram on the right (some of the numbers in the cells have b... | 16 | 0 | 7,924.25 | -1 | 7,924.25 | |
Given a sequence \( A = (a_1, a_2, \cdots, a_{10}) \) that satisfies the following four conditions:
1. \( a_1, a_2, \cdots, a_{10} \) is a permutation of \{1, 2, \cdots, 10\};
2. \( a_1 < a_2, a_3 < a_4, a_5 < a_6, a_7 < a_8, a_9 < a_{10} \);
3. \( a_2 > a_3, a_4 > a_5, a_6 > a_7, a_8 > a_9 \);
4. There does not exist ... | 42 | 0.125 | 7,613.375 | 4,926.5 | 7,997.214286 | |
Given a 50-term sequence $(b_1, b_2, \dots, b_{50})$, the Cesaro sum is 500. What is the Cesaro sum of the 51-term sequence $(2, b_1, b_2, \dots, b_{50})$? | 492 | 0 | 7,315.625 | -1 | 7,315.625 | |
Let's define a number as complex if it has at least two different prime divisors. Find the greatest natural number that cannot be represented as the sum of two complex numbers. | 23 | 0 | 8,138.6875 | -1 | 8,138.6875 | |
Nicholas is counting the sheep in a flock as they cross a road. The sheep begin to cross the road at 2:00 p.m. and cross at a constant rate of three sheep per minute. After counting 42 sheep, Nicholas falls asleep. He wakes up an hour and a half later, at which point exactly half of the total flock has crossed the road... | 624 | 0.9375 | 1,915.3125 | 1,804.933333 | 3,571 | |
Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has a diameter of 8 cm. In Crate A, the pipes are packed directly on top of each other in 25 rows of 8 pipes each across the width of the crate. In Crate B, pipes are packed in a staggered (hexagonal) pattern that res... | 200 - 96\sqrt{3} | 0 | 7,694.6875 | -1 | 7,694.6875 | |
The arithmetic mean of these six expressions is 30. What is the value of $y$? $$y + 10 \hspace{.5cm} 20 \hspace{.5cm} 3y \hspace{.5cm} 18 \hspace{.5cm} 3y + 6 \hspace{.5cm} 12$$ | \frac{114}{7} | 0.9375 | 3,025.1875 | 2,680.733333 | 8,192 | |
The side lengths of both triangles to the right are given in centimeters. What is the length of segment $AB$?
[asy]
pair A,B,C,D,E,F,G;
A=(0,0);
B=12*dir(0);
C=20*dir(120);
D=8+B;
E=D+6*dir(0);
F=D+10*dir(120);
draw(A--B--C--cycle);
draw(D--E--F--cycle);
label("A",F,N);
label("B",E+(1.4,0));
label("6",.5*(A+B),S);
... | 7 | 0 | 7,207.25 | -1 | 7,207.25 | |
In triangle $\triangle ABC$, it is known that $AB=2$, $AC=3$, $\angle A=60^{\circ}$, and $\overrightarrow{BM}=\frac{1}{3}\overrightarrow{BC}$, $\overrightarrow{AN}=\overrightarrow{NB}$. Find $\overrightarrow{AC}•\overrightarrow{NM}$. | \frac{7}{2} | 0.9375 | 4,334.9375 | 4,077.8 | 8,192 | |
Find the number of four-element subsets of $\{1,2,3,4,\dots, 20\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\{3,5,13,19\}$ and $\{6,10,20,18\}$ are two such subsets. | 210 | Let's say our four elements in our subset are $a,b,c,d$. We have two cases. Note that the order of the elements / the element letters themselves don't matter since they are all on equal grounds at the start.
$\textrm{Case } 1 \textrm{:}$ $a+b = 16$ and $c+d = 24$.
List out possibilities for $a+b$ $(\text{i.e. } 1+15, ... | 0 | 8,138.0625 | -1 | 8,138.0625 |
What is the smallest positive integer with exactly 12 positive integer divisors? | 72 | 0 | 4,569 | -1 | 4,569 | |
A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there? | 2400 | 1. **Identify the number of ways to reach the red arrows:**
- There is only one way to reach any of the red arrows from point A, as each red arrow has a unique path leading directly from A.
2. **Calculate the number of ways to reach the blue arrows from the red arrows:**
- From the first (top) red arrow, there a... | 0 | 5,207.875 | -1 | 5,207.875 |
A convex quadrilateral is determined by the points of intersection of the curves \( x^{4}+y^{4}=100 \) and \( x y=4 \); determine its area. | 4\sqrt{17} | By symmetry, the quadrilateral is a rectangle having \( x=y \) and \( x=-y \) as axes of symmetry. Let \( (a, b) \) with \( a>b>0 \) be one of the vertices. Then the desired area is \( (\sqrt{2}(a-b)) \cdot(\sqrt{2}(a+b))=2\left(a^{2}-b^{2}\right)=2 \sqrt{a^{4}-2 a^{2} b^{2}+b^{4}}=2 \sqrt{100-2 \cdot 4^{2}}=4 \sqrt{17... | 0 | 8,192 | -1 | 8,192 |
How many 10-digit numbers are there whose digits are all 1, 2, or 3 and in which adjacent digits differ by 1? | 64 | 0.3125 | 7,853.9375 | 7,110.2 | 8,192 | |
In $\triangle ABC$, $AB=7$, $BC=5$, $CA=6$, then $\overrightarrow{AB} \cdot \overrightarrow{BC} =$ \_\_\_\_\_\_. | -19 | 0.9375 | 4,634.75 | 4,397.6 | 8,192 | |
Given the system of equations \begin{align*}
xy &= 6 - 2x - 3y,\\
yz &= 6 - 4y - 2z,\\
xz &= 30 - 4x - 3z,
\end{align*}find the positive solution of $x$. | 3 | 0.8125 | 4,722.75 | 4,278.461538 | 6,648 | |
An investor has an open brokerage account with an investment company. In 2021, the investor received the following income from securities:
- Dividends from shares of the company PAO “Winning” amounted to 50,000 rubles.
- Coupon income from government bonds OFZ amounted to 40,000 rubles.
- Coupon income from corporate... | 11050 | 0 | 4,051.9375 | -1 | 4,051.9375 | |
Completely factor the expression: $$x^8-256$$ | (x^4+16)(x^2+4)(x+2)(x-2) | 0 | 6,370.4375 | -1 | 6,370.4375 | |
The graph of $y=x^6-10x^5+29x^4-4x^3+ax^2$ lies above the line $y=bx+c$ except at three values of $x$, where the graph and the line intersect. What is the largest of these values? | 4 |
1. **Identify the Polynomial and Intersection Points**:
The problem states that the graph of the polynomial $y = x^6 - 10x^5 + 29x^4 - 4x^3 + ax^2$ intersects the line $y = bx + c$ at three values of $x$. These intersections are the roots of the polynomial $x^6 - 10x^5 + 29x^4 - 4x^3 + ax^2 - bx - c = 0$.
2. **Mul... | 0.5625 | 6,239.125 | 4,720.222222 | 8,192 |
Let \( f \) be a function such that \( f(0) = 1 \), \( f'(0) = 2 \), and
\[ f''(t) = 4 f'(t) - 3 f(t) + 1 \]
for all \( t \). Compute the 4th derivative of \( f \), evaluated at 0. | 54 | 1 | 3,291.6875 | 3,291.6875 | -1 | |
In triangle \( ABC \), the angle bisector \( BL \) is drawn. Find the area of the triangle, given that \( AL=2 \), \( BL=3\sqrt{10} \), and \( CL=3 \). | \frac{15\sqrt{15}}{4} | 0 | 6,019.5625 | -1 | 6,019.5625 | |
Find the maximum value of $10^x - 100^x,$ over all real numbers $x.$ | \frac{1}{4} | 1 | 3,641.8125 | 3,641.8125 | -1 | |
Given a sequence of positive integers $a_1, a_2, a_3, \ldots, a_{100}$, where the number of terms equal to $i$ is $k_i$ ($i=1, 2, 3, \ldots$), let $b_j = k_1 + k_2 + \ldots + k_j$ ($j=1, 2, 3, \ldots$),
define $g(m) = b_1 + b_2 + \ldots + b_m - 100m$ ($m=1, 2, 3, \ldots$).
(I) Given $k_1 = 40, k_2 = 30, k_3 = 20, k... | -100 | 0 | 8,138.25 | -1 | 8,138.25 | |
Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded? | \frac{7}{16} |
#### Step 1: Understanding the Geometry
The regular octagon $ABCDEFGH$ is symmetric, and each side is equal. The center $O$ of the octagon is equidistant from all vertices. Since $X$ is the midpoint of $\overline{AB}$, the line segment $\overline{OX}$ bisects $\triangle ABO$ into two congruent right triangles, $\trian... | 0 | 7,800.75 | -1 | 7,800.75 |
Given that $\cos{\alpha}=-\frac{4}{5}$, where $\alpha$ is an angle in the third quadrant, find the value of $\sin{\left(\alpha-\frac{\pi}{4}\right)$. | \frac{\sqrt{2}}{10} | 0 | 5,165.5625 | -1 | 5,165.5625 | |
Let $M$ be the maximum possible value of $x_1x_2+x_2x_3+\cdots +x_5x_1$ where $x_1, x_2, \dots, x_5$ is a permutation of $(1,2,3,4,5)$ and let $N$ be the number of permutations for which this maximum is attained. Evaluate $M+N$. | 58 | 0 | 8,100.0625 | -1 | 8,100.0625 | |
Trapezoid $ABCD$ has parallel sides $\overline{AB}$ of length $33$ and $\overline {CD}$ of length $21$. The other two sides are of lengths $10$ and $14$. The angles $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$? | $25$ | 1. **Identify the right triangles and set up equations using the Pythagorean theorem:**
- Let $\overline{DE} \perp \overline{AB}$ and $\overline{FC} \perp \overline{AB}$.
- Denote $\overline{AE} = x$ and $\overline{DE} = h$. In right triangle $AED$, we have:
\[
x^2 + h^2 = 100 \quad \text{(since $AD = 1... | 0 | 5,018.25 | -1 | 5,018.25 |
Gwen, Eli, and Kat take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Kat will win? | \frac{1}{7} | 0.6875 | 5,821.875 | 4,744.545455 | 8,192 | |
Equilateral triangles $A B F$ and $B C G$ are constructed outside regular pentagon $A B C D E$. Compute $\angle F E G$. | 48^{\circ} | We have $\angle F E G=\angle A E G-\angle A E F$. Since $E G$ bisects $\angle A E D$, we get $\angle A E G=54^{\circ}$. Now, $\angle E A F=108^{\circ}+60^{\circ}=168^{\circ}$. Since triangle $E A F$ is isosceles, this means $\angle A E F=6^{\circ}$, so the answer is $54^{\circ}-6^{\circ}=48^{\circ}$. | 0 | 8,192 | -1 | 8,192 |
In $\triangle ABC$, where $C=60 ^{\circ}$, $AB= \sqrt {3}$, and the height from $AB$ is $\frac {4}{3}$, find the value of $AC+BC$. | \sqrt {11} | 0 | 6,253 | -1 | 6,253 | |
Find all integers $n$, not necessarily positive, for which there exist positive integers $a, b, c$ satisfying $a^{n}+b^{n}=c^{n}$. | \pm 1, \pm 2 | By Fermat's Last Theorem, we know $n<3$. Suppose $n \leq-3$. Then $a^{n}+b^{n}=c^{n} \Longrightarrow(b c)^{-n}+$ $(a c)^{-n}=(a b)^{-n}$, but since $-n \geq 3$, this is also impossible by Fermat's Last Theorem. As a result, $|n|<3$. Furthermore, $n \neq 0$, as $a^{0}+b^{0}=c^{0} \Longrightarrow 1+1=1$, which is false. ... | 0 | 6,656.75 | -1 | 6,656.75 |
Given the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + 1$, where $\overrightarrow{a} = (\sqrt{3}, 2\sin \frac{\omega x}{2})$ and $\overrightarrow{b} = (\sin \omega x, -\sin \frac{\omega x}{2})$, $\omega > 0$, and the smallest positive period of $f(x)$ is $\pi$.
(1) Find the value of $\omega$;
(2) F... | \frac{\pi}{12} | 0.375 | 7,030 | 5,979.833333 | 7,660.1 | |
A teacher has prepared three problems for the class to solve. In how many different ways can he present these problems to the students if there are 30 students in the class? | 24360 | 0.0625 | 4,757.25 | 3,229 | 4,859.133333 | |
In a school there are \( n \) students, each with a different student number. Each student number is a positive factor of \( 60^{60} \), and the H.C.F. of any two student numbers is not a student number in the school. Find the greatest possible value of \( n \). | 3721 | 0.125 | 7,700.375 | 4,590.5 | 8,144.642857 |
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