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 |
|---|---|---|---|---|---|---|
Supposed that $x$ and $y$ are nonzero real numbers such that $\frac{3x+y}{x-3y}=-2$. What is the value of $\frac{x+3y}{3x-y}$? | 2 | 1. Start with the given equation:
\[
\frac{3x+y}{x-3y} = -2
\]
2. Cross-multiply to eliminate the fraction:
\[
3x + y = -2(x - 3y)
\]
3. Distribute the -2 on the right-hand side:
\[
3x + y = -2x + 6y
\]
4. Rearrange the equation to isolate terms involving \(x\) and \(y\) on opposite sides:... | 1 | 2,777.5 | 2,777.5 | -1 |
One and one-half of what number is 30? | 20 | 1 | 1,484.5 | 1,484.5 | -1 | |
A four-digit palindrome is defined as any four-digit natural number that has the same digit in the units place as in the thousands place, and the same digit in the tens place as in the hundreds place. How many pairs of four-digit palindromes exist whose difference is 3674? | 35 | 0.125 | 7,758.125 | 4,721 | 8,192 | |
Given that the equation of line $l_{1}$ is $y=x$, and the equation of line $l_{2}$ is $y=kx-k+1$, find the value of $k$ for which the area of triangle $OAB$ is $2$. | \frac{1}{5} | 0 | 7,566.125 | -1 | 7,566.125 | |
Let $a,b,c$ be the roots of $x^3-9x^2+11x-1=0$, and let $s=\sqrt{a}+\sqrt{b}+\sqrt{c}$. Find $s^4-18s^2-8s$. | -37 | 0.875 | 4,257.0625 | 3,694.928571 | 8,192 | |
Find the largest integer $x$ such that the number
$$
4^{27} + 4^{1000} + 4^{x}
$$
is a perfect square. | 1972 | 0.3125 | 7,802.4375 | 6,945.4 | 8,192 | |
For all positive integers $n$ greater than 2, the greatest common divisor of $n^5 - 5n^3 + 4n$ is. | 120 | 0.6875 | 6,101.625 | 5,385.545455 | 7,677 | |
Given that (1+ex)<sup>2019</sup>=a<sub>0</sub>+a<sub>1</sub>x+a<sub>2</sub>x<sup>2</sup>+……+a<sub>2019</sub>x<sup>2019</sup>, find the value of:
- $$\frac {a_{1}}{e}$$+ $$\frac {a_{2}}{e^{2}}$$\- $$\frac {a_{3}}{e^{3}}$$+ $$\frac {a_{4}}{e^{4}}$$\-……- $$\frac {a_{2019}}{e^{2019}}$$ | -1 | 0 | 7,226 | -1 | 7,226 | |
Given a sequence $\{a_n\}$, let $S_n$ denote the sum of its first $n$ terms. Define $T_n = \frac{S_1 + S_2 + \dots + S_n}{n}$ as the "ideal number" of the sequence $a_1, a_2, \dots, a_n$. If the "ideal number" of the sequence $a_1, a_2, \dots, a_{502}$ is $2012$, calculate the "ideal number" of the sequence $2, a_1, a_... | 2010 | 0.75 | 5,494.625 | 4,595.5 | 8,192 | |
A sequence \( b_1, b_2, b_3, \dots \) is defined recursively by \( b_1 = 2, b_2 = 2, \) and for \( k \ge 3, \)
\[
b_k = \frac{1}{2} b_{k - 1} + \frac{1}{3} b_{k - 2}.
\]
Evaluate \( b_1 + b_2 + b_3 + \dotsb. \) | 18 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Compute
\[
\sin^2 0^\circ + \sin^2 10^\circ + \sin^2 20^\circ + \dots + \sin^2 180^\circ.
\] | 10 | 0 | 7,496.3125 | -1 | 7,496.3125 | |
Every pair of communities in a county are linked directly by one mode of transportation; bus, train, or airplane. All three methods of transportation are used in the county with no community being serviced by all three modes and no three communities being linked pairwise by the same mode. Determine the largest number o... | 4 |
Let us consider a set of communities, denoted as vertices in a graph, where each edge between a pair of communities is labeled with one of the following modes of transportation: bus, train, or airplane. The problem imposes the following conditions:
1. All three modes of transportation (bus, train, and airplane) are u... | 0 | 8,125.6875 | -1 | 8,125.6875 |
In triangle ABC, BR = RC, CS = 3SA, and (AT)/(TB) = p/q. If the area of △RST is twice the area of △TBR, determine the value of p/q. | \frac{7}{3} | 0.625 | 4,896.6875 | 4,482.8 | 5,586.5 | |
Given point P(a, -1) (a∈R), draw the tangent line to the parabola C: $y=x^2$ at point P, and let the tangent points be A($x_1$, $y_1$) and B($x_2$, $y_2$) (where $x_1<x_2$).
(Ⅰ) Find the values of $x_1$ and $x_2$ (expressed in terms of a);
(Ⅱ) If a circle E with center at point P is tangent to line AB, find the minim... | 3\pi | 0.625 | 6,236.125 | 5,559.5 | 7,363.833333 | |
Solve for $x$: $5 - x = 8$. | -3 | 1 | 1,602.5 | 1,602.5 | -1 | |
Find the product of the divisors of $50$. | 125,\!000 | 0 | 2,356.75 | -1 | 2,356.75 | |
The integers that can be expressed as a sum of three distinct numbers chosen from the set $\{4,7,10,13, \ldots,46\}$. | 37 | 0 | 7,854.1875 | -1 | 7,854.1875 | |
A ship travels from Port A to Port B against the current at a speed of 24 km/h. After arriving at Port B, it returns to Port A with the current. It is known that the journey with the current takes 5 hours less than the journey against the current. The speed of the current is 3 km/h. Find the distance between Port A and... | 350 | 0 | 2,512.5625 | -1 | 2,512.5625 | |
Let $S_1 = \{(x, y)|\log_{10}(1 + x^2 + y^2) \le 1 + \log_{10}(x+y)\}$ and $S_2 = \{(x, y)|\log_{10}(2 + x^2 + y^2) \le 2 + \log_{10}(x+y)\}$. What is the ratio of the area of $S_2$ to the area of $S_1$? | 102 | 1. **Analyzing the constraint for $S_1$:**
- Given: $\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)$
- Since $\log_{10}(ab) = \log_{10}(a) + \log_{10}(b)$, we rewrite the inequality:
\[
\log_{10}(1+x^2+y^2) \le \log_{10}(10) + \log_{10}(x+y)
\]
- This simplifies to:
\[
\log_{10}(1+x^2+y^2) \... | 1 | 3,646.6875 | 3,646.6875 | -1 |
Sasha wrote down numbers from one to one hundred, and Misha erased some of them. Among the remaining numbers, 20 have the digit one in their recording, 19 have the digit two in their recording, and 30 numbers have neither the digit one nor the digit two. How many numbers did Misha erase? | 33 | 0 | 6,097.0625 | -1 | 6,097.0625 | |
The sequence $(x_n)$ is defined by $x_1 = 150$ and $x_k = x_{k - 1}^2 - x_{k - 1}$ for all $k \ge 2.$ Compute
\[\frac{1}{x_1 + 1} + \frac{1}{x_2 + 1} + \frac{1}{x_3 + 1} + \dots.\] | \frac{1}{150} | 0 | 7,876.75 | -1 | 7,876.75 | |
The numbers \( 2^{2021} \) and \( 5^{2021} \) are written out one after the other. How many digits are written in total? | 2022 | 0.6875 | 6,012.6875 | 5,022.090909 | 8,192 | |
What is $\frac{2}{5}$ divided by 3? | \frac{2}{15} | 1 | 969.1875 | 969.1875 | -1 | |
A and B are playing a series of Go games, with the first to win 3 games declared the winner. Assuming in a single game, the probability of A winning is 0.6 and the probability of B winning is 0.4, with the results of each game being independent. It is known that in the first two games, A and B each won one game.
(1) ... | 2.48 | 0 | 8,076.6875 | -1 | 8,076.6875 | |
The length of a rectangle is three times its width. Given that its perimeter and area are both numerically equal to $k>0$, find $k$. | \frac{64}{3} | Let $a$ be the width of the rectangle. Then the length of the rectangle is $3 a$, so the perimeter is $2(a+3 a)=8 a$, and the area is $3 a^{2}$. Since the length is numerically equal to the width, we know that $$8 a=3 a^{2}=k$$ Because $k>0$, the rectangle is non-degenerate. It follows that $8=3 a$, so $a=\frac{8}{3}$.... | 1 | 1,772.3125 | 1,772.3125 | -1 |
In a right triangle ABC, with right angle at A, side AB measures 5 units and side BC measures 13 units. Find $\sin C$. | \frac{12}{13} | 0 | 2,417.3125 | -1 | 2,417.3125 | |
Find \(\sin \alpha\) if \(\cos \alpha = \operatorname{tg} \beta\), \(\cos \beta = \operatorname{tg} \gamma\), \(\cos \gamma = \operatorname{tg} \alpha\) \((0 < \alpha < \frac{\pi}{2}, 0 < \beta < \frac{\pi}{2}, 0 < \gamma < \frac{\pi}{2})\). | \frac{\sqrt{2}}{2} | 0 | 7,076.1875 | -1 | 7,076.1875 | |
Given that 600 athletes are numbered from 001 to 600 and divided into three color groups (red: 001 to 311, white: 312 to 496, and yellow: 497 to 600), calculate the probability of randomly drawing an athlete wearing white clothing. | \frac{8}{25} | 0 | 474.875 | -1 | 474.875 | |
Brothers Lyosha and Sasha decided to get from home to the skate park. They left at the same time, but Lyosha walked with the skateboard in his hands, while Sasha rode the skateboard. It is known that Sasha rides the skateboard 3 times faster than Lyosha walks with the skateboard. After some time, they simultaneously c... | 1100 | 0.125 | 6,823.4375 | 6,450 | 6,876.785714 | |
Given data: $2$, $5$, $7$, $9$, $11$, $8$, $7$, $8$, $10$, the $80$th percentile is ______. | 10 | 0.375 | 5,924.1875 | 5,647.833333 | 6,090 | |
The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer (1,000 meters)? | 11^{\text{th}} | 1. **Identify the sequence**: The problem describes a geometric sequence where the first term \(a_1 = 1\) meter and each subsequent term doubles the previous term. This can be expressed as:
\[
a_n = 2^{n-1}
\]
where \(n\) is the jump number.
2. **Determine the condition**: We need to find the smallest \(n\... | 0 | 2,648.375 | -1 | 2,648.375 |
Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips? | 5 | 1. **Identify the chips in each bag:**
- Bag A contains odd numbers: 1, 3, 5.
- Bag B contains even numbers: 2, 4, 6.
2. **Understand the nature of sums:**
- The sum of an odd number (from Bag A) and an even number (from Bag B) is always odd.
3. **Calculate the possible sums:**
- The smallest sum is obtai... | 0.8125 | 4,111.1875 | 3,169.461538 | 8,192 |
28 apples weigh 3 kilograms. If they are evenly divided into 7 portions, each portion accounts for $\boxed{\frac{1}{7}}$ of all the apples, and each portion weighs $\boxed{\frac{3}{7}}$ kilograms. | \frac{3}{7} | 0.6875 | 395.25 | 374.909091 | 440 | |
What is the area of a quadrilateral with vertices at $(0,0)$, $(4,3)$, $(7,0)$, and $(4,4)$? | 3.5 | 0 | 7,236.9375 | -1 | 7,236.9375 | |
Given that
\begin{align*}x_{1}&=211,\\ x_{2}&=375,\\ x_{3}&=420,\\ x_{4}&=523,\ \text{and}\\ x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\ \text{when}\ n\geq5, \end{align*}
find the value of $x_{531}+x_{753}+x_{975}$. | 898 | Calculate the first few terms:
\[211,375,420,523,267,-211,-375,-420,-523,\dots\]
At this point it is pretty clear that the sequence is periodic with period 10 (one may prove it quite easily like in solution 1) so our answer is obviously $211+420+267=\boxed{898}$
~Dhillonr25
~ pi_is_3.14 | 0.8125 | 4,130.625 | 4,131 | 4,129 |
Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$ | 550 | 0.125 | 8,066.875 | 7,191 | 8,192 | |
In the equation below, $A$ and $B$ are consecutive positive integers, and $A$, $B$, and $A+B$ represent number bases: \[132_A+43_B=69_{A+B}.\]What is $A+B$? | 13 | 1. **Convert the given equation to base 10:**
The equation given is $132_A + 43_B = 69_{A+B}$. We need to express each number in base 10.
- For $132_A$, it represents $1 \cdot A^2 + 3 \cdot A + 2$.
- For $43_B$, it represents $4 \cdot B + 3$.
- For $69_{A+B}$, it represents $6 \cdot (A+B) + 9$.
2. **Se... | 1 | 2,592.625 | 2,592.625 | -1 |
Find the integer values of $m$, $n$, and $p$ such that the roots of the equation $4x(2x - 5) = -4$ can be expressed in the forms $\frac{m+\sqrt{n}}{p}$ and $\frac{m-\sqrt{n}}{p}$, and find $m+n+p$. | 26 | 1 | 2,122.75 | 2,122.75 | -1 | |
In triangle $ABC$, $AB = BC$, and $\overline{BD}$ is an altitude. Point $E$ is on the extension of $\overline{AC}$ such that $BE =
10$. The values of $\tan \angle CBE$, $\tan \angle DBE$, and $\tan \angle ABE$ form a geometric progression, and the values of $\cot \angle DBE$, $\cot \angle CBE$, $\cot \angle DBC$ form ... | \frac{50}{3} | 0.0625 | 8,177.4375 | 7,959 | 8,192 | |
Two circular poles, with diameters of 8 inches and 24 inches, touch each other at a single point. A wire is wrapped around them such that it goes around the entire configuration including a straight section tangential to both poles. Find the length of the shortest wire that sufficiently encloses both poles.
A) $16\sqrt... | 16\sqrt{3} + 32\pi | 0 | 8,192 | -1 | 8,192 | |
What is the least positive integer that satisfies the following conditions?
a) When divided by 2, the remainder is 1.
b) When divided by 3, the remainder is 2.
c) When divided by 4, the remainder is 3.
d) When divided by 5, the remainder is 4. | 59 | 1 | 3,162.25 | 3,162.25 | -1 | |
What is the largest integer that must divide the product of any $5$ consecutive integers? | 60 | 0 | 6,768.5625 | -1 | 6,768.5625 | |
Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet? | 50 | 1. **Calculate the total surface area of the cube**:
The cube has 6 faces, and each face is a square with an edge length of 10 feet. The area of one face is:
\[
10 \times 10 = 100 \text{ square feet}
\]
Therefore, the total surface area of the cube is:
\[
6 \times 100 = 600 \text{ square feet}
... | 0.9375 | 3,223.375 | 2,892.133333 | 8,192 |
Students in the class of Peter practice the addition and multiplication of integer numbers.The teacher writes the numbers from $1$ to $9$ on nine cards, one for each number, and places them in an ballot box. Pedro draws three cards, and must calculate the sum and the product of the three corresponding numbers. Ana and... | 2, 6, 7 |
Let's analyze the information provided about Pedro and Ana to solve the problem and find out which numbers Julian removed.
### Pedro's Drawn Numbers
We are told that Pedro picks three consecutive numbers whose product is 5 times their sum. Let \( a \), \( a+1 \), and \( a+2 \) be the consecutive numbers drawn by Ped... | 0.1875 | 6,937.125 | 4,762 | 7,439.076923 |
What is the difference between the largest and smallest numbers in the list $0.023,0.302,0.203,0.320,0.032$? | 0.297 | We write the list in increasing order: $0.023,0.032,0.203,0.302,0.320$. The difference between the largest and smallest of these numbers is $0.320-0.023=0.297$. | 1 | 1,316.375 | 1,316.375 | -1 |
A set containing three real numbers can be represented as $\{a, \frac{b}{a}, 1\}$, and also as $\{a^2, a+b, 0\}$. Find the value of $a+b$. | -1 | 0.625 | 5,712.625 | 4,405 | 7,892 | |
Let $M$ be the number of positive integers that are less than or equal to $2048$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$. | 24 | 0 | 8,192 | -1 | 8,192 | |
A sphere is tangent to all the edges of the pyramid \( SABC \), specifically the lateral edges \( SA, SB, \) and \( SC \) at the points \( A', B' \), and \( C' \) respectively. Find the volume of the pyramid \( SA'B'C' \), given that \( AB = BC = SB = 5 \) and \( AC = 4 \). | \frac{2 \sqrt{59}}{15} | 0 | 8,192 | -1 | 8,192 | |
There are 5 students taking a test, and each student's score ($a, b, c, d, e$) is an integer between 0 and 100 inclusive. It is known that $a \leq b \leq c \leq d \leq e$. If the average score of the 5 students is $p$, then the median score $c$ is at least $\qquad$ . | 40 | 0 | 8,075.5 | -1 | 8,075.5 | |
A box of 25 chocolate candies costs $\$6$. How many dollars does it cost to buy 600 chocolate candies? | 144 | 1 | 2,182 | 2,182 | -1 | |
Given \\(a > 0\\), \\(b > 0\\), and \\(a+4b={{(ab)}^{\\frac{3}{2}}}\\).
\\((\\)I\\()\\) Find the minimum value of \\(a^{2}+16b^{2}\\);
\\((\\)II\\()\\) Determine whether there exist \\(a\\) and \\(b\\) such that \\(a+3b=6\\), and explain the reason. | 32 | 0.1875 | 7,911.75 | 6,697.333333 | 8,192 | |
Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\vec{a}-\vec{b}|=\sqrt{3}|\vec{a}+\vec{b}|$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac{2\pi}{3} | 0.25 | 2,100.9375 | 2,121.5 | 2,094.083333 | |
Today is 17.02.2008. Natasha noticed that in this date, the sum of the first four digits is equal to the sum of the last four digits. When will this coincidence happen for the last time this year? | 25.12.2008 | 0 | 7,669.6875 | -1 | 7,669.6875 | |
Real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2000} \left| x_{k} - x_{k+1} \right| = 2001 \). Let \( y_{k} = \frac{1}{k} \left( x_{1} + x_{2} + \cdots + x_{k} \right) \) for \( k = 1, 2, \cdots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} | y_{k} - y_{k+1} | \). (2001 Sha... | 2000 | 0 | 8,192 | -1 | 8,192 | |
Find the remainder when $r^{13} + 1$ is divided by $r - 1$. | 2 | 0.875 | 3,778.6875 | 3,148.214286 | 8,192 | |
Find the set of values for parameter \(a\) for which the sum of the cubes of the roots of the equation \(x^{2} + ax + a + 1 = 0\) is equal to 1. | -1 | 0.0625 | 6,512 | 5,983 | 6,547.266667 | |
Given two arithmetic sequences \\(\{a_n\}\) and \\(\{b_n\}\) with the sum of the first \\(n\) terms denoted as \\(S_n\) and \\(T_n\) respectively. If \\( \dfrac {S_n}{T_n}= \dfrac {2n}{3n+1}\), then \\( \dfrac {a_2}{b_3+b_7}+ \dfrac {a_8}{b_4+b_6}=\) ______. | \dfrac {9}{14} | 0.625 | 4,185.1875 | 3,685.2 | 5,018.5 | |
9 judges each award 20 competitors a rank from 1 to 20. The competitor's score is the sum of the ranks from the 9 judges, and the winner is the competitor with the lowest score. For each competitor, the difference between the highest and lowest ranking (from different judges) is at most 3. What is the highest score the... | 24 | 0 | 8,192 | -1 | 8,192 | |
The horizontal and vertical distances between adjacent points equal 1 unit. What is the area of triangle $ABC$? | \frac{1}{2} | To solve this problem, we need to first understand the positions of points $A$, $B$, $C$, and $D$ on the grid. However, the problem statement does not provide specific coordinates for these points, and the solution provided seems to assume a specific configuration without describing it. Let's assume a configuration bas... | 0.5 | 6,074.75 | 5,544.625 | 6,604.875 |
How many ways are there to place 31 knights in the cells of an $8 \times 8$ unit grid so that no two attack one another? | 68 | Consider coloring the squares of the chessboard so that 32 are black and 32 are white, and no two squares of the same color share a side. Then a knight in a square of one color only attacks squares of the opposite color. Any arrangement of knights in which all 31 are placed on the same color therefore works: there are ... | 0 | 7,695.625 | -1 | 7,695.625 |
Let $x_{0}$ be a zero of the function $f(x)=\sin \pi x$, and suppose it satisfies $|x_{0}| + f\left(x_{0}+ \frac{1}{2}\right) < 11$. Calculate the number of such zeros. | 21 | 0.625 | 5,878.3125 | 5,131.7 | 7,122.666667 | |
The cells of a $100 \times 100$ table are colored white. In one move, it is allowed to select some $99$ cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring?
*S. Berlov* | 100 | 0 | 8,192 | -1 | 8,192 | |
Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V-E+F=2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P+10T+V$?
| 250 | 0.5625 | 6,975.875 | 6,030 | 8,192 | |
An omino is a 1-by-1 square or a 1-by-2 horizontal rectangle. An omino tiling of a region of the plane is a way of covering it (and only it) by ominoes. How many omino tilings are there of a 2-by-10 horizontal rectangle? | 7921 | There are exactly as many omino tilings of a 1-by-$n$ rectangle as there are domino tilings of a 2-by-$n$ rectangle. Since the rows don't interact at all, the number of omino tilings of an $m$-by-$n$ rectangle is the number of omino tilings of a 1-by-$n$ rectangle raised to the $m$ th power, $F_{n}^{m}$. The answer is ... | 0 | 7,551.9375 | -1 | 7,551.9375 |
The hypotenuse of a right triangle whose legs are consecutive even numbers is 50 units. What is the sum of the lengths of the two legs? | 70 | 0.0625 | 8,135.5 | 7,288 | 8,192 | |
Math City has eight streets, all of which are straight. No street is parallel to another street. One police officer is stationed at each intersection. What is the greatest number of police officers needed? | 28 | 1 | 1,269.5 | 1,269.5 | -1 | |
How many positive integers less than $555$ are either a perfect cube or a perfect square? | 29 | 0.875 | 4,167.125 | 3,974.928571 | 5,512.5 | |
Among all triangles $ABC,$ find the maximum value of $\cos A + \cos B \cos C.$ | \frac{1}{\sqrt{2}} | 0 | 7,867.75 | -1 | 7,867.75 | |
Let $S = \{5^k | k \in \mathbb{Z}, 0 \le k \le 2004 \}$. Given that $5^{2004} = 5443 \cdots 0625$ has $1401$ digits, how many elements of $S$ begin with the digit $1$?
| 604 | 0.3125 | 7,848.625 | 7,398.6 | 8,053.181818 | |
The diagram shows a quadrilateral \(PQRS\) made from two similar right-angled triangles, \(PQR\) and \(PRS\). The length of \(PQ\) is 3, the length of \(QR\) is 4, and \(\angle PRQ = \angle PSR\).
What is the perimeter of \(PQRS\)? | 22 | 0.5625 | 5,592.1875 | 4,559.444444 | 6,920 | |
Let $m$ be a positive integer, and let $a_0, a_1,\ldots,a_m$ be a sequence of reals such that $a_0 = 37, a_1 = 72, a_m = 0,$ and $a_{k+1} = a_{k-1} - \frac 3{a_k}$ for $k = 1,2,\ldots, m-1.$ Find $m.$ | 889 | For $0 < k < m$, we have
$a_{k}a_{k+1} = a_{k-1}a_{k} - 3$.
Thus the product $a_{k}a_{k+1}$ is a monovariant: it decreases by 3 each time $k$ increases by 1. For $k = 0$ we have $a_{k}a_{k+1} = 37\cdot 72$, so when $k = \frac{37 \cdot 72}{3} = 888$, $a_{k}a_{k+1}$ will be zero for the first time, which implies that $m... | 0.625 | 6,061 | 4,901.3 | 7,993.833333 |
Given the function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, determine the value of $f(-\frac{{5π}}{{12}})$. | \frac{\sqrt{3}}{2} | 0 | 7,581.875 | -1 | 7,581.875 | |
Let \( a \leq b < c \) be the side lengths of a right triangle. Find the maximum constant \( M \) such that \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq \frac{M}{a+b+c} \). | 5 + 3 \sqrt{2} | 0.125 | 8,189.25 | 8,170 | 8,192 | |
In triangle \( \triangle ABC \), \( BC=a \), \( AC=b \), \( AB=c \), and \( \angle C = 90^{\circ} \). \( CD \) and \( BE \) are two medians of \( \triangle ABC \), and \( CD \perp BE \). Express the ratio \( a:b:c \) in simplest form. | 1 : \sqrt{2} : \sqrt{3} | 0.875 | 3,302.1875 | 3,184.285714 | 4,127.5 | |
The numbers $1,2, \ldots, 20$ are put into a hat. Claire draws two numbers from the hat uniformly at random, $a<b$, and then puts them back into the hat. Then, William draws two numbers from the hat uniformly at random, $c<d$. Let $N$ denote the number of integers $n$ that satisfy exactly one of $a \leq n \leq b$ and $... | \frac{181}{361} | The number of integers that satisfy exactly one of the two inequalities is equal to the number of integers that satisfy the first one, plus the number of integers that satisfy the second one, minus twice the number of integers that satisfy both. Parity-wise, this is just the number of integers that satisfy the first on... | 0.25 | 7,139.9375 | 5,700.5 | 7,619.75 |
The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names? | 4 |
To find the median length of the names represented in the bar graph, we need to determine the position of the median in a sorted list of the name lengths. Since there are 19 names, the median will be the length of the 10th name when the names are arranged in increasing order of their lengths.
1. **Count the Total Num... | 0.3125 | 4,223.0625 | 4,245 | 4,213.090909 |
Let points $A = (0,0)$, $B = (2,4)$, $C = (6,6)$, and $D = (8,0)$. Quadrilateral $ABCD$ is cut into two pieces by a line passing through $A$ and intersecting $\overline{CD}$ such that the area above the line is twice the area below the line. This line intersects $\overline{CD}$ at a point $\left(\frac{p}{q}, \frac{r}{s... | 28 | 0.4375 | 6,065.625 | 5,017.571429 | 6,880.777778 | |
Consider integers c and d where c consists of 1986 nines, and d consists of 1986 sixes. What is the sum of the digits of the resulting number in base 10 when these numbers are added? | 9931 | 0 | 5,162.25 | -1 | 5,162.25 | |
Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased $-2, -1, 0$ he would replace them with $0$ .) If $M$ is the maximum possible final value remaining on the board, and if m is the mini... | 2014 | 0 | 8,192 | -1 | 8,192 | |
Given an arithmetic sequence $\{a_n\}$ with the first term $a_1=11$ and common difference $d=2$, and $a_n=2009$, find $n$. | 1000 | 1 | 1,632.5625 | 1,632.5625 | -1 | |
Calculate the degree of ionization using the formula:
$$
\alpha=\sqrt{ } K_{\mathrm{HCN}} \mathrm{C}
$$
Given values:
$$
\alpha_{\text {ion }}=\sqrt{ }\left(7,2 \cdot 10^{-10}\right) / 0,1=\sqrt{ } 7,2 \cdot 10^{-9}=8,5 \cdot 10^{-5}, \text{ or } 8,5 \cdot 10^{-5} \cdot 10^{2}=0,0085\%
$$
Alternatively, if the co... | 0.0085 | 0.5 | 5,311.5 | 4,724.375 | 5,898.625 | |
Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the $8$ people to be seated at the table unde... | 520 | We see that for every 2 adjacent spots on the table, there is exactly one way for an ambassador and his or her partner to sit. There are 2 cases:
There is an ambassador at the 12th chair and his or her partner at the 1st chair
There is no pair that has their chairs numbered as 12 and 1
For the first case, there are $\... | 0 | 8,171.125 | -1 | 8,171.125 |
The arithmetic mean (average) of four numbers is $85$. If the largest of these numbers is $97$, then the mean of the remaining three numbers is | 81.0 | 1. Let the four numbers be $a, b, c,$ and $97$. Given that the arithmetic mean of these numbers is $85$, we can set up the equation:
\[
\frac{a+b+c+97}{4} = 85
\]
2. Multiply both sides of the equation by $4$ to eliminate the fraction:
\[
a+b+c+97 = 340
\]
3. To find the sum of the three numbers $a,... | 0 | 1,027.8125 | -1 | 1,027.8125 |
Suppose $x$ is an integer that satisfies the following congruences:
\begin{align*}
2+x &\equiv 3^2 \pmod{2^4}, \\
3+x &\equiv 2^3 \pmod{3^4}, \\
4+x &\equiv 3^3 \pmod{2^3}.
\end{align*}
What is the remainder when $x$ is divided by $24$? | 23 | 0.9375 | 4,339.25 | 4,082.4 | 8,192 | |
Let \(\alpha\) and \(\beta\) be angles such that
\[
\frac{\cos^2 \alpha}{\cos \beta} + \frac{\sin^2 \alpha}{\sin \beta} = 2,
\]
Find the sum of all possible values of
\[
\frac{\sin^2 \beta}{\sin \alpha} + \frac{\cos^2 \beta}{\cos \alpha}.
\] | \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Given an arithmetic sequence $\{a_n\}$ with the common difference $d$ being an integer, and $a_k=k^2+2$, $a_{2k}=(k+2)^2$, where $k$ is a constant and $k\in \mathbb{N}^*$
$(1)$ Find $k$ and $a_n$
$(2)$ Let $a_1 > 1$, the sum of the first $n$ terms of $\{a_n\}$ is $S_n$, the first term of the geometric sequence $\{b... | \frac{\sqrt{13}-1}{2} | 0 | 7,918.8125 | -1 | 7,918.8125 | |
Divide the natural numbers from 1 to 30 into two groups such that the product $A$ of all numbers in the first group is divisible by the product $B$ of all numbers in the second group. What is the minimum value of $\frac{A}{B}$? | 1077205 | 0.0625 | 8,024.1875 | 5,507 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\cos \frac{A}{2}= \frac{2 \sqrt{5}}{5}, \overrightarrow{AB} \cdot \overrightarrow{AC}=15$.
$(1)$ Find the area of $\triangle ABC$;
$(2)$ If $\tan B=2$, find the value of $a$. | 2 \sqrt{5} | 0.625 | 5,825.3125 | 5,022.4 | 7,163.5 | |
Evaluate the following product of sequences: $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{2187} \cdot \frac{6561}{1}$. | 81 | 0.25 | 6,229.3125 | 4,861 | 6,685.416667 | |
Given the function $f(x)=4\cos (ωx- \frac {π}{6})\sin (π-ωx)-\sin (2ωx- \frac {π}{2})$, where $ω > 0$.
(1) Find the range of the function $f(x)$.
(2) If $y=f(x)$ is an increasing function in the interval $[- \frac {3π}{2}, \frac {π}{2}]$, find the maximum value of $ω$. | \frac{1}{6} | 0.375 | 7,607.25 | 6,632.666667 | 8,192 | |
Let $n$ be a positive integer. Determine, in terms of $n$, the largest integer $m$ with the following property: There exist real numbers $x_1,\dots,x_{2n}$ with $-1 < x_1 < x_2 < \cdots < x_{2n} < 1$ such that the sum of the lengths of the $n$ intervals \[ [x_1^{2k-1}, x_2^{2k-1}], [x_3^{2k-1},x_4^{2k-1}], \dots, [x_{2... | n | The largest such $m$ is $n$. To show that $m \geq n$, we take \[ x_j = \cos \frac{(2n+1-j)\pi}{2n+1} \qquad (j=1,\dots,2n). \] It is apparent that $-1 < x_1 < \cdots < x_{2n} < 1$. The sum of the lengths of the intervals can be interpreted as \begin{align*} & -\sum_{j=1}^{2n} ((-1)^{2n+1-j} x_j)^{2k-1} \\ &= -\sum_{j=1... | 0 | 8,192 | -1 | 8,192 |
Ken is the best sugar cube retailer in the nation. Trevor, who loves sugar, is coming over to make an order. Ken knows Trevor cannot afford more than 127 sugar cubes, but might ask for any number of cubes less than or equal to that. Ken prepares seven cups of cubes, with which he can satisfy any order Trevor might make... | 64 | The only way to fill seven cups to satisfy the above condition is to use a binary scheme, so the cups must contain $1,2,4,8,16,32$, and 64 cubes of sugar. | 0.9375 | 2,847.9375 | 2,491.666667 | 8,192 |
Calculate $[(6^{6} \div 6^{5})^3 \cdot 8^3] \div 4^3$. | 1728 | 0.9375 | 2,875.75 | 2,852.266667 | 3,228 | |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $(E)$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$, $M$ and $N$ are the endpoints of its minor axis, and the perimeter of the quadrilateral $MF\_1NF\_2$ is $4$, let line $(l)$ pass through $F\_1$ intersecting $(E)$ at points $A$... | \frac{2}{3} | 0 | 8,192 | -1 | 8,192 | |
In triangle \\(ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively, and it is given that \\(A < B < C\\) and \\(C = 2A\\).
\\((1)\\) If \\(c = \sqrt{3}a\\), find the measure of angle \\(A\\).
\\((2)\\) If \\(a\\), \\(b\\), and \\(c\\) are three consecutive... | \dfrac{15\sqrt{7}}{4} | 0 | 6,094.25 | -1 | 6,094.25 | |
A solid cube of side length $2$ is removed from each corner of a larger solid cube of side length $4$. Find the number of edges of the remaining solid. | 36 | 0.5 | 7,465.3125 | 7,087.5 | 7,843.125 | |
Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction. | \frac{1}{12} | 1 | 4,195.9375 | 4,195.9375 | -1 | |
Consider the function $g(x)$ represented by the line segments in the graph below. The graph consists of four line segments as follows: connecting points (-3, -4) to (-1, 0), (-1, 0) to (0, -1), (0, -1) to (2, 3), and (2, 3) to (3, 2). Find the sum of the $x$-coordinates where $g(x) = x + 2$. | -1.5 | 0 | 7,202.4375 | -1 | 7,202.4375 | |
Given $cos({\frac{π}{4}+α})=\frac{{\sqrt{2}}}{3}$, then $\frac{{sin2α}}{{1-sinα+cosα}}=$____. | \frac{1}{3} | 0.875 | 4,868.6875 | 4,393.928571 | 8,192 | |
Given that connecting all the vertices of a polygon from a point on one of the edges results in 2022 triangles, determine the number of sides of this polygon. | 2023 | 0.625 | 5,548.5 | 4,804.2 | 6,789 |
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