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 |
|---|---|---|---|---|---|---|
Triangle $\triangle P Q R$, with $P Q=P R=5$ and $Q R=6$, is inscribed in circle $\omega$. Compute the radius of the circle with center on $\overline{Q R}$ which is tangent to both $\omega$ and $\overline{P Q}$. | \frac{20}{9} | Solution 1: Denote the second circle by $\gamma$. Let $T$ and $r$ be the center and radius of $\gamma$, respectively, and let $X$ and $H$ be the tangency points of $\gamma$ with $\omega$ and $\overline{P Q}$, respectively. Let $O$ be the center of $\omega$, and let $M$ be the midpoint of $\overline{Q R}$. Note that $Q ... | 0.75 | 5,513.5625 | 5,031 | 6,961.25 |
Consider numbers of the form \(10n + 1\), where \(n\) is a positive integer. We shall call such a number 'grime' if it cannot be expressed as the product of two smaller numbers, possibly equal, both of which are of the form \(10k + 1\), where \(k\) is a positive integer. How many 'grime numbers' are there in the sequen... | 87 | 0 | 8,192 | -1 | 8,192 | |
In the plane Cartesian coordinate system \(xOy\), the set of points
$$
\begin{aligned}
K= & \{(x, y) \mid(|x|+|3 y|-6) \cdot \\
& (|3 x|+|y|-6) \leqslant 0\}
\end{aligned}
$$
corresponds to an area in the plane with the measurement of ______. | 24 | 0.0625 | 7,763.9375 | 7,857 | 7,757.733333 | |
The distances from point \( P \), which lies inside an equilateral triangle, to its vertices are 3, 4, and 5. Find the area of the triangle. | 9 + \frac{25\sqrt{3}}{4} | 0 | 7,780.0625 | -1 | 7,780.0625 | |
Two teachers and 4 students need to be divided into 2 groups, each consisting of 1 teacher and 2 students. Calculate the number of different arrangements. | 12 | 0.1875 | 5,958.25 | 7,297.666667 | 5,649.153846 | |
I have a drawer with 4 shirts, 5 pairs of shorts, and 6 pairs of socks in it. If I reach in and randomly remove three articles of clothing, what is the probability that I get one shirt, one pair of shorts, and one pair of socks? (Treat pairs of socks as one article of clothing.) | \frac{24}{91} | 1 | 2,224.625 | 2,224.625 | -1 | |
In the expansion of $(2x +3y)^{20}$ , find the number of coefficients divisible by $144$ .
*Proposed by Hannah Shen* | 15 | 0 | 8,192 | -1 | 8,192 | |
A student correctly added the two two-digit numbers on the left of the board and got the answer 137. What answer will she obtain if she adds the two four-digit numbers on the right of the board? | 13837 | 0.1875 | 4,881.875 | 6,309 | 4,552.538462 | |
Let \(ABCD\) be a convex quadrilateral such that \(AB + BC = 2021\) and \(AD = CD\). We are also given that \(\angle ABC = \angle CDA = 90^\circ\). Determine the length of the diagonal \(BD\). | \frac{2021}{\sqrt{2}} | 0 | 7,777.875 | -1 | 7,777.875 | |
Hou Yi shot three arrows at each of three targets. On the first target, he scored 29 points, and on the second target, he scored 43 points. How many points did he score on the third target? | 36 | 0.125 | 458.375 | 476 | 455.857143 | |
Zhang Hua, Li Liang, and Wang Min each sent out x, y, z New Year's cards, respectively. If it is known that the least common multiple of x, y, z is 60, the greatest common divisor of x and y is 4, and the greatest common divisor of y and z is 3, then how many New Year's cards did Zhang Hua send out? | 20 | 0.125 | 7,569.375 | 3,626.5 | 8,132.642857 | |
What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, \dots, 30\}$? (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.) | \frac{2}{3} | We define an outcome as $\left( a_1 ,\cdots, a_5 \right)$ with $1 \leq a_1 < a_2 < a_3 < a_4 < a_5 \leq 30$.
We denote by $\Omega$ the sample space. Hence, $| \Omega | = \binom{30}{5}$.
#### Case 1: There is only 1 pair of consecutive integers.
- **Case 1.1**: $\left( a_1 , a_2 \right)$ is the single pair of consecuti... | 0.9375 | 5,749.6875 | 5,586.866667 | 8,192 |
Four positive integers $A$, $B$, $C$ and $D$ have a sum of 36. If $A+2 = B-2 = C \times 2 = D \div 2$, what is the value of the product $A \times B \times C \times D$? | 3840 | 1 | 2,220.3125 | 2,220.3125 | -1 | |
If $f(x)=\dfrac{5x+1}{x-1}$, find the value of $f(7)$. | 6 | 1 | 1,902.125 | 1,902.125 | -1 | |
For an arithmetic sequence $a_1,$ $a_2,$ $a_3,$ $\dots,$ let
\[S_n = a_1 + a_2 + a_3 + \dots + a_n,\]and let
\[T_n = S_1 + S_2 + S_3 + \dots + S_n.\]If you are told the value of $S_{2019},$ then you can uniquely determine the value of $T_n$ for some integer $n.$ What is this integer $n$? | 3028 | 0.875 | 5,905.375 | 5,578.714286 | 8,192 | |
Let $S_n$ be the sum of the first $n$ terms of a geometric sequence $\{a_n\}$. Given that $a_3 = 8a_6$, find the value of $\frac{S_4}{S_2}$. | \frac{5}{4} | 1 | 4,405.375 | 4,405.375 | -1 | |
If triangle $ABC$ has sides of length $AB = 6,$ $AC = 5,$ and $BC = 4,$ then calculate
\[\frac{\cos \frac{A - B}{2}}{\sin \frac{C}{2}} - \frac{\sin \frac{A - B}{2}}{\cos \frac{C}{2}}.\] | \frac{5}{3} | 0.875 | 4,487.375 | 3,958.142857 | 8,192 | |
For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\sin x = \sin (n^2 x)$ on the interval $[0, 2\pi]$. What is $\sum_{n=2}^{100} G(n)$? | 676797 | 0.1875 | 7,819.1875 | 7,312.666667 | 7,936.076923 | |
Let $N$ be the number of ways in which the letters in "HMMTHMMTHMMTHMMTHMMTHMMT" ("HMMT" repeated six times) can be rearranged so that each letter is adjacent to another copy of the same letter. For example, "MMMMMMTTTTTTHHHHHHHHHHHH" satisfies this property, but "HMMMMMTTTTTTHHHHHHHHHHHM" does not. Estimate $N$. An es... | 78556 | We first count the number of arrangements for which each block of consecutive identical letters has even size. Pair up the letters into 3 pairs of $H, 6$ pairs of $M$, and 3 pairs of $T$, then rearrange the pairs. There are $\frac{12!}{6!3!3!}=18480$ ways to do this. In the original problem, we may estimate the number ... | 0 | 8,083.1875 | -1 | 8,083.1875 |
Let f be a function that satisfies the following conditions: $(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$ , then $f(z) = v + z$ , for some number $z$ between $x$ and $y$ . $(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not sm... | 1988 | 0.5625 | 6,240.5625 | 4,824.555556 | 8,061.142857 | |
Compute the limit of the function:
$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{e^{\sin 2 x}-e^{\tan 2 x}}{\ln \left(\frac{2 x}{\pi}\right)}$$ | -2\pi | 0.875 | 5,114.25 | 4,674.571429 | 8,192 | |
Person A, Person B, Person C, and Person D share 2013 candies. Person A gets 10 more candies than twice the amount Person B has, 18 more candies than three times the amount Person C has, and 55 less candies than five times the amount Person D has. How many candies does Person A get? | 990 | 0.75 | 2,001.125 | 2,021.25 | 1,940.75 | |
Alice and Carol each have a rectangular sheet of paper. Alice has a sheet of paper measuring 10 inches by 12 inches and rolls it into a tube by taping the two 10-inch sides together. Carol rolls her sheet, which measures 8 inches by 15 inches, by taping the two 15-inch sides together. Calculate $\pi$ times the positive... | 150 | 0.5625 | 1,932.8125 | 2,016.222222 | 1,825.571429 | |
Given $(x^2+1)(x-2)^8 = a + a_1(x-1) + a_2(x-1)^2 + \ldots + a_{10}(x-1)^{10}$, find the value of $a_1 + a_2 + \ldots + a_{10}$. | -2 | 0.25 | 7,206.5625 | 5,505 | 7,773.75 | |
Find all the solutions to
\[\sqrt{(2 + \sqrt{3})^x} + \sqrt{(2 - \sqrt{3})^x} = 4.\]Enter all the solutions, separated by commas. | 2,-2 | 0 | 5,071.625 | -1 | 5,071.625 | |
In triangle $ABC$, $AB = 3$, $BC = 4$, $AC = 5$, and $BD$ is the angle bisector from vertex $B$. If $BD = k \sqrt{2}$, then find $k$. | \frac{12}{7} | 1 | 3,424.5625 | 3,424.5625 | -1 | |
Find the matrix $\mathbf{M},$ with real entries, such that
\[\mathbf{M}^3 - 4 \mathbf{M}^2 + 5 \mathbf{M} = \begin{pmatrix} 10 & 20 \\ 5 & 10 \end{pmatrix}.\] | \begin{pmatrix} 2 & 4 \\ 1 & 2 \end{pmatrix} | 0.3125 | 7,157.5 | 4,881.6 | 8,192 | |
In the equation $\frac{1}{m} + \frac{1}{n} = \frac{1}{4}$, where $m$ and $n$ are positive integers, determine the sum of all possible values for $n$. | 51 | 0.9375 | 2,300.5 | 2,356.066667 | 1,467 | |
Find the value of \( \cos (\angle OBC + \angle OCB) \) in triangle \( \triangle ABC \), where angle \( \angle A \) is an obtuse angle, \( O \) is the orthocenter, and \( AO = BC \). | -\frac{\sqrt{2}}{2} | 0 | 7,794.1875 | -1 | 7,794.1875 | |
If $x + 2y = 30$, what is the value of $\frac{x}{5} + \frac{2y}{3} + \frac{2y}{5} + \frac{x}{3}$? | 16 | Since $x + 2y = 30$, then $\frac{x}{5} + \frac{2y}{3} + \frac{2y}{5} + \frac{x}{3} = \frac{x}{5} + \frac{2y}{5} + \frac{x}{3} + \frac{2y}{3} = \frac{1}{5}x + \frac{1}{5}(2y) + \frac{1}{3}x + \frac{1}{3}(2y) = \frac{1}{5}(x + 2y) + \frac{1}{3}(x + 2y) = \frac{1}{5}(30) + \frac{1}{3}(30) = 6 + 10 = 16$ | 1 | 2,481.5 | 2,481.5 | -1 |
The concept of negative numbers first appeared in the ancient Chinese mathematical book "Nine Chapters on the Mathematical Art." If income of $5$ yuan is denoted as $+5$ yuan, then expenses of $5$ yuan are denoted as $-5$ yuan. | -5 | 0.8125 | 1,917.8125 | 1,966.769231 | 1,705.666667 | |
What is the smallest positive integer with exactly 12 positive integer divisors? | 150 | 0 | 3,626.0625 | -1 | 3,626.0625 | |
Find the least upper bound for the set of values \((x_1 x_2 + 2x_2 x_3 + x_3 x_4) / (x_1^2 + x_2^2 + x_3^2 + x_4^2)\), where \(x_i\) are real numbers, not all zero. | \frac{\sqrt{2}+1}{2} | 0 | 7,539.5625 | -1 | 7,539.5625 | |
For any real number a and positive integer k, define
$\binom{a}{k} = \frac{a(a-1)(a-2)\cdots(a-(k-1))}{k(k-1)(k-2)\cdots(2)(1)}$
What is
$\binom{-\frac{1}{2}}{100} \div \binom{\frac{1}{2}}{100}$? | -199 | 1. **Define the binomial coefficient for non-integer upper index**:
\[
{a \choose k} = \frac{a(a-1)(a-2)\cdots(a-(k-1))}{k(k-1)(k-2)\cdots(2)(1)}
\]
This definition extends the binomial coefficient to cases where $a$ is not necessarily a non-negative integer.
2. **Simplify the expression**:
We need to e... | 0.6875 | 6,775.3125 | 6,131.363636 | 8,192 |
Find the greatest common divisor of $8!$ and $(6!)^2.$ | 2880 | 0 | 3,765.9375 | -1 | 3,765.9375 | |
On a table, there are 2004 boxes, each containing one ball. It is known that some of the balls are white, and their number is even. You are allowed to point to any two boxes and ask if there is at least one white ball in them. What is the minimum number of questions needed to guarantee the identification of a box that... | 2003 | 0 | 8,192 | -1 | 8,192 | |
The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is: | -12 | 1. **Understanding the percentage changes**:
- A 25% increase in population means the population becomes 125% of its original, which can be expressed as a multiplication by $\frac{5}{4}$.
- A 25% decrease means the population becomes 75% of its previous amount, which can be expressed as a multiplication by $\fra... | 0.1875 | 5,388.875 | 6,653.666667 | 5,097 |
Given an ellipse E: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ ($$a > b > 0$$) passing through point Q ($$\frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}$$), the product of the slopes of the lines connecting the moving point P on the ellipse to the two endpoints of the minor axis is $$-\frac{1}{2}$$.
1. Find the equation of the... | \frac{4}{3} | 0.5625 | 6,965.25 | 6,181.555556 | 7,972.857143 | |
Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing around a Christmas tree. Afterwards, each was asked if the girl to her right was in a blue dress. It turned out that only those who stood between two girls in dresses of the same color answered correctly. How many girls could have answered affirmati... | 17 | 0 | 8,192 | -1 | 8,192 | |
Find all odd positive integers $n>1$ such that there is a permutation $a_{1}, a_{2}, \ldots, a_{n}$ of the numbers $1,2, \ldots, n$, where $n$ divides one of the numbers $a_{k}^{2}-a_{k+1}-1$ and $a_{k}^{2}-a_{k+1}+1$ for each $k, 1 \leq k \leq n$ (we assume $a_{n+1}=a_{1}$ ). | n=3 | Since $\{a_{1}, a_{2}, \ldots, a_{n}\}=\{1,2, \ldots, n\}$ we conclude that $a_{i}-a_{j}$ : $n$ only if $i=j$. From the problem conditions it follows that $$a_{k+1}=a_{k}^{2}+\varepsilon_{k}-n b_{k}$$ where $b_{k} \in \mathbb{Z}$ and $\varepsilon_{k}= \pm 1$. We have $a_{k+1}-a_{l+1}=\left(a_{k}-a_{l}\right)\left(a_{k}... | 0.125 | 8,149.0625 | 7,848.5 | 8,192 |
In triangle $\triangle ABC$, $sin2C=\sqrt{3}sinC$.
$(1)$ Find the value of $\angle C$;
$(2)$ If $b=6$ and the perimeter of $\triangle ABC$ is $6\sqrt{3}+6$, find the area of $\triangle ABC$. | 6\sqrt{3} | 0.5625 | 5,988.25 | 4,736.222222 | 7,598 | |
Find the number of six-digit palindromes. | 900 | 1 | 2,261.5 | 2,261.5 | -1 | |
What is the height of Jack's house, in feet, if the house casts a shadow 56 feet long at the same time a 21-foot tree casts a shadow that is 24 feet long? Express your answer to the nearest whole number. | 49 | 1 | 1,652.3125 | 1,652.3125 | -1 | |
Given real numbers $x$ and $y$ that satisfy the system of inequalities $\begin{cases} x - 2y - 2 \leqslant 0 \\ x + y - 2 \leqslant 0 \\ 2x - y + 2 \geqslant 0 \end{cases}$, if the minimum value of the objective function $z = ax + by + 5 (a > 0, b > 0)$ is $2$, determine the minimum value of $\frac{2}{a} + \frac{3}{b}$... | \frac{10 + 4\sqrt{6}}{3} | 0 | 7,812.25 | -1 | 7,812.25 | |
Suppose a regular tetrahedron \( P-ABCD \) has all edges equal in length. Using \(ABCD\) as one face, construct a cube \(ABCD-EFGH\) on the other side of the regular tetrahedron. Determine the cosine of the angle between the skew lines \( PA \) and \( CF \). | \frac{2 + \sqrt{2}}{4} | 0 | 7,457.125 | -1 | 7,457.125 | |
The decimal expansion of $8/11$ is a repeating decimal. What is the least number of digits in a repeating block of 8/11? | 2 | 1 | 1,572.0625 | 1,572.0625 | -1 | |
How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? | 42 | 0.9375 | 3,921.3125 | 3,636.6 | 8,192 | |
The number $24!$ has many positive integer divisors. What is the probability that a divisor randomly chosen from these is odd? | \frac{1}{23} | 0.6875 | 5,419 | 4,456.454545 | 7,536.6 | |
In the expansion of $(1-x)^{2}(2-x)^{8}$, find the coefficient of $x^{8}$. | 145 | 0.5 | 7,175.6875 | 6,159.375 | 8,192 | |
Express $0.6\overline{03}$ as a common fraction. | \frac{104}{165} | 0 | 5,983.5 | -1 | 5,983.5 | |
Given the function $f(x)=\sin (2x+\varphi)$, if the graph is shifted to the left by $\dfrac {\pi}{6}$ units and the resulting graph is symmetric about the $y$-axis, determine the possible value of $\varphi$. | \dfrac {\pi}{6} | 0.3125 | 6,706.5625 | 5,590.6 | 7,213.818182 | |
How many positive integers less than $1000$ are either a perfect cube or a perfect square? | 37 | 0.875 | 3,548.375 | 2,885 | 8,192 | |
A line passing through any two vertices of a cube has a total of 28 lines. Calculate the number of pairs of skew lines among them. | 174 | 0 | 8,192 | -1 | 8,192 | |
When the value of $y$ is doubled and then this increased value is divided by 5, the result is 10. What is the value of $y$? | 25 | 1 | 1,582.375 | 1,582.375 | -1 | |
Given that $O$ is any point in space, and $A$, $B$, $C$, $D$ are four points such that no three of them are collinear, but they are coplanar, and $\overrightarrow{OA}=2x\cdot \overrightarrow{BO}+3y\cdot \overrightarrow{CO}+4z\cdot \overrightarrow{DO}$, find the value of $2x+3y+4z$. | -1 | 0.25 | 6,398.625 | 3,987.5 | 7,202.333333 | |
Determine the largest prime number less than 5000 of the form \( a^n - 1 \), where \( a \) and \( n \) are positive integers, and \( n \) is greater than 1. | 127 | 0 | 8,192 | -1 | 8,192 | |
For any integer $n>1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is:
$\text{(A) } 0\quad\qquad \text{(B) } 1\quad\\ \text{(C) } \frac{n}{2} \text{ for n even, } \frac{n+1}{2} \text{ for n odd}\quad\\ \text{(D) } n-1\quad \text{(E) } n$
| 0 | 3,907.8125 | -1 | 3,907.8125 | ||
A factory assigns five newly recruited employees, including A and B, to three different workshops. Each workshop must be assigned at least one employee, and A and B must be assigned to the same workshop. The number of different ways to assign the employees is \_\_\_\_\_\_. | 36 | 0 | 8,192 | -1 | 8,192 | |
Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together? | 120 | 1. **Understanding the problem**: We are given that Brenda received $36$ votes, which represents $\frac{3}{10}$ of the total votes in the election.
2. **Calculating the total number of votes**:
- Since $36$ votes is $\frac{3}{10}$ of the total votes, we can find the total number of votes by setting up the equation:... | 0.1875 | 5,521.75 | 4,369 | 5,787.769231 |
To open the safe, you need to enter a code — a number consisting of seven digits: twos and threes. The safe will open if there are more twos than threes, and the code is divisible by both 3 and 4. Create a code that opens the safe. | 2222232 | 0.25 | 1,650.5 | 2,182 | 1,473.333333 | |
Let $\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f \colon \mathbb{R}^+ \to \mathbb{R}^+$ such that, for all $x,y \in \mathbb{R}^+$,
$$f(xy+f(x))=xf(y)+2.$$ | f(x) = x + 1 |
Let \( f: \mathbb{R}^+ \to \mathbb{R}^+ \) be a function satisfying the functional equation for all \( x, y \in \mathbb{R}^+ \):
\[
f(xy + f(x)) = x f(y) + 2.
\]
To find \( f \), consider substituting specific values for \( x \) and \( y \) to gain insights into the function’s form.
### Step 1: Functional Equation ... | 0 | 7,301 | -1 | 7,301 |
In the diagram, three lines meet at the points \( A, B \), and \( C \). If \( \angle ABC = 50^\circ \) and \( \angle ACB = 30^\circ \), the value of \( x \) is: | 80 | 0.4375 | 4,293.125 | 4,645.428571 | 4,019.111111 | |
Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana? | \frac{5}{3} | 1. **Define Variables:**
Let $m$ represent the cost of one muffin and $b$ represent the cost of one banana.
2. **Set Up Equations:**
According to the problem, Susie's total cost for $4$ muffins and $3$ bananas is:
\[
4m + 3b
\]
Calvin spends twice as much as Susie for $2$ muffins and $16$ bananas, so... | 1 | 1,583.3125 | 1,583.3125 | -1 |
What is the median of the numbers in the list $19^{20}, \frac{20}{19}, 20^{19}, 2019, 20 \times 19$? | 2019 | Since $\frac{20}{19}$ is larger than 1 and smaller than 2, and $20 \times 19 = 380$, then $\frac{20}{19} < 20 \times 19 < 2019$. We note that $19^{20} > 10^{20} > 10000$ and $20^{19} > 10^{19} > 10000$. This means that both $19^{20}$ and $20^{19}$ are greater than 2019. In other words, of the five numbers $19^{20}, \fr... | 0.25 | 6,749.875 | 3,864.25 | 7,711.75 |
A sequence \( b_1, b_2, b_3, \ldots \) is defined recursively by \( b_1 = 2 \), \( b_2 = 3 \), and for \( k \geq 3 \),
\[ b_k = \frac{1}{2} b_{k-1} + \frac{1}{3} b_{k-2}. \]
Evaluate \( b_1 + b_2 + b_3 + \dotsb. \) | 24 | 0.3125 | 7,736.1875 | 6,733.4 | 8,192 | |
Calculate the sum:
\[
\sum_{n=1}^\infty \frac{n^3 + n^2 + n - 1}{(n+3)!}
\] | \frac{2}{3} | 0 | 8,173.9375 | -1 | 8,173.9375 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $(\overrightarrow{a}+\overrightarrow{b}) \perp \overrightarrow{a}$, and $(2\overrightarrow{a}+\overrightarrow{b}) \perp \overrightarrow{b}$, calculate the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{3\pi}{4} | 0.5 | 2,396.8125 | 2,406.75 | 2,386.875 | |
George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, how many minutes after starting did they pass each other the second time? | 4\frac{1}{2} | 1. **Understanding the Problem:**
George and Henry start from opposite ends of the pool and meet in the center after $1.5$ minutes. This means each of them has traveled half the length of the pool in $1.5$ minutes.
2. **Analyzing the Speeds:**
Since they meet in the center, they have traveled half the pool's len... | 0 | 6,573.875 | -1 | 6,573.875 |
Through the end of a chord that divides the circle in the ratio 3:5, a tangent is drawn. Find the acute angle between the chord and the tangent. | 67.5 | 0.6875 | 5,251.9375 | 4,243.818182 | 7,469.8 | |
If the sines of the internal angles of $\triangle ABC$ form an arithmetic sequence, what is the minimum value of $\cos C$? | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
Suppose that $b$ is a positive integer greater than or equal to $2.$ When $197$ is converted to base $b$, the resulting representation has $4$ digits. What is the number of possible values for $b$? | 2 | 0.8125 | 6,578.75 | 6,206.461538 | 8,192 | |
Given that $-4\leq x\leq-2$ and $2\leq y\leq4$, what is the largest possible value of $\frac{x+y}{x}$? | \frac{1}{2} | 0.8125 | 6,765.75 | 6,436.615385 | 8,192 | |
For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 60 | 0 | 5,551.1875 | -1 | 5,551.1875 | |
The FISS World Cup is a very popular football event among high school students worldwide. China successfully obtained the hosting rights for the International Middle School Sports Federation (FISS) World Cup in 2024, 2026, and 2028. After actively bidding by Dalian City and official recommendation by the Ministry of Ed... | 10 | 0 | 5,085.0625 | -1 | 5,085.0625 | |
In the triangle $ABC$ it is known that $\angle A = 75^o, \angle C = 45^o$ . On the ray $BC$ beyond the point $C$ the point $T$ is taken so that $BC = CT$ . Let $M$ be the midpoint of the segment $AT$ . Find the measure of the $\angle BMC$ .
(Anton Trygub) | 45 | 0.625 | 7,031.8125 | 6,792.6 | 7,430.5 | |
Given three numbers $1$, $3$, $4$, find the value of x such that the set $\{1, 3, 4, x\}$ forms a proportion. | 12 | 0.5 | 2,662.6875 | 495.125 | 4,830.25 | |
Bob has a seven-digit phone number and a five-digit postal code. The sum of the digits in his phone number and the sum of the digits in his postal code are the same. Bob's phone number is 346-2789. What is the largest possible value for Bob's postal code, given that no two digits in the postal code are the same? | 98765 | 0 | 8,192 | -1 | 8,192 | |
On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling ... | 375 | 0.1875 | 7,835.25 | 6,289.333333 | 8,192 | |
Given $a$, $b$, $c > 0$ and $$a(a+b+c)+bc=4-2 \sqrt {3}$$, calculate the minimum value of $2a+b+c$. | 2\sqrt{3}-2 | 0.4375 | 7,296.9375 | 6,146.142857 | 8,192 | |
We write the equation on the board:
$$
(x-1)(x-2) \ldots (x-2016) = (x-1)(x-2) \ldots (x-2016) .
$$
We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimal number of factors that need to be erased to achieve this? | 2016 | 0 | 8,192 | -1 | 8,192 | |
What is the sum of the seven smallest distinct positive integer multiples of 9? | 252 | 1 | 2,372.5625 | 2,372.5625 | -1 | |
The center of one sphere is on the surface of another sphere with an equal radius. How does the volume of the intersection of the two spheres compare to the volume of one of the spheres? | \frac{5}{16} | 0.5625 | 6,622.3125 | 5,401.444444 | 8,192 | |
Among five numbers, if we take the average of any four numbers and add the remaining number, the sums will be 74, 80, 98, 116, and 128, respectively. By how much is the smallest number less than the largest number among these five numbers? | 72 | 0.6875 | 4,837.25 | 3,719.727273 | 7,295.8 | |
Let three non-identical complex numbers \( z_1, z_2, z_3 \) satisfy the equation \( 4z_1^2 + 5z_2^2 + 5z_3^2 = 4z_1z_2 + 6z_2z_3 + 4z_3z_1 \). Denote the lengths of the sides of the triangle in the complex plane, with vertices at \( z_1, z_2, z_3 \), from smallest to largest as \( a, b, c \). Find the ratio \( a : b : ... | 2:\sqrt{5}:\sqrt{5} | 0 | 8,165.375 | -1 | 8,165.375 | |
The product \( \left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right) \) is equal to what? | \frac{2}{5} | Simplifying, \( \left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right)=\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\left(\frac{4}{5}\right) \). We can simplify further by dividing equal numerators and denominators to obtain a final value of \( \frac{2}{5} \). | 1 | 2,093.625 | 2,093.625 | -1 |
The sum of two positive integers $a$ and $b$ is 1001. What is the largest possible value of $\gcd(a,b)$? | 143 | 1 | 3,473.25 | 3,473.25 | -1 | |
In $\triangle ABC$, $AB= 400$, $BC=480$, and $AC=560$. An interior point $P$ is identified, and segments are drawn through $P$ parallel to the sides of the triangle. These three segments are of equal length $d$. Determine $d$. | 218\frac{2}{9} | 0 | 8,060.1875 | -1 | 8,060.1875 | |
Complex numbers \( p, q, r \) form an equilateral triangle with a side length of 24 in the complex plane. If \( |p + q + r| = 48 \), find \( |pq + pr + qr| \). | 768 | 0 | 8,192 | -1 | 8,192 | |
Find the area of a trapezoid with bases 4 and 7 and side lengths 4 and 5.
| 22 | 0.9375 | 4,043.3125 | 4,185.8 | 1,906 | |
Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | \frac{5}{7} | 0.0625 | 7,604.9375 | 6,415 | 7,684.266667 | |
Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and a die with 8 faces numbered 1 to 8 is rolled. Determine the probability that the product of the numbers on the tile and the die will be a square. | \frac{7}{48} | 0.125 | 6,548.9375 | 5,912.5 | 6,639.857143 | |
Given that $α \in (0, \frac{π}{3})$ satisfies the equation $\sqrt{6} \sin α + \sqrt{2} \cos α = \sqrt{3}$, find the values of:
1. $\cos (α + \frac{π}{6})$
2. $\cos (2α + \frac{π}{12})$ | \frac{\sqrt{30} + \sqrt{2}}{8} | 0 | 5,972.0625 | -1 | 5,972.0625 | |
How many whole numbers are between $\sqrt[3]{50}$ and $\sqrt{200}$? | 11 | 0.9375 | 4,778.625 | 4,927.733333 | 2,542 | |
Let \(a, b, c\) be the side lengths of a right triangle, with \(a \leqslant b < c\). Determine the maximum constant \(k\) such that the inequality \(a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c\) holds for all right triangles, and specify when equality occurs. | 2 + 3\sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Given that $α \in \left( \frac{π}{2}, π \right)$ and $3\cos 2α = \sin \left( \frac{π}{4} - α \right)$, find the value of $\sin 2α$. | -\frac{17}{18} | 0.375 | 7,264.4375 | 6,268.333333 | 7,862.1 | |
How many sequences of 0s and 1s are there of length 10 such that there are no three 0s or 1s consecutively anywhere in the sequence? | 178 | We can have blocks of either 1 or 20s and 1s, and these blocks must be alternating between 0s and 1s. The number of ways of arranging blocks to form a sequence of length $n$ is the same as the number of omino tilings of a $1-b y-n$ rectangle, and we may start each sequence with a 0 or a 1, making $2 F_{n}$ or, in this ... | 0 | 8,192 | -1 | 8,192 |
The graphs of a function $f(x)=3x+b$ and its inverse function $f^{-1}(x)$ intersect at the point $(-3,a)$. Given that $b$ and $a$ are both integers, what is the value of $a$? | -3 | 1 | 1,786.75 | 1,786.75 | -1 | |
A car travels the 120 miles from $A$ to $B$ at 60 miles per hour, and then returns to $A$ on the same road. If the average rate of the round trip is 45 miles per hour, what is the rate, in miles per hour, of the car traveling back from $B$ to $A$? | 36 | 1 | 1,915.0625 | 1,915.0625 | -1 | |
Determine the length of side $PQ$ in the right-angled triangle $PQR$, where $PR = 15$ units and $\angle PQR = 45^\circ$. | 15 | 0 | 4,877.875 | -1 | 4,877.875 | |
Below is the graph of $y = a \sin (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$ Find $a.$
[asy]import TrigMacros;
size(400);
real f(real x)
{
return 2*sin(3*x + pi) + 1;
}
draw(graph(f,-3*pi,3*pi,n=700,join=operator ..),red);
trig_axes(-3*pi,3*pi,-4,4,pi/2,1);
layer();
rm_trig_labels(-5,5, 2);... | 2 | 1 | 2,636.875 | 2,636.875 | -1 |
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