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 |
|---|---|---|---|---|---|---|
Round $3.1415926$ to the nearest thousandth using the rounding rule, and determine the precision of the approximate number $3.0 \times 10^{6}$ up to which place. | 3.142 | 0.125 | 4,232.5 | 590.5 | 4,752.785714 | |
At the end of a professional bowling tournament, the top 5 bowlers have a playoff. First #5 bowls #4. The loser receives $5$th prize and the winner bowls #3 in another game. The loser of this game receives $4$th prize and the winner bowls #2. The loser of this game receives $3$rd prize and the winner bowls #1. The winn... | 16 | 0 | 6,896.1875 | -1 | 6,896.1875 | |
Given an isosceles triangle \( A B C \) where \( A B = A C \) and \( \angle A B C = 53^\circ \). Point \( K \) is such that \( C \) is the midpoint of \( A K \). Point \( M \) is chosen such that:
- \( B \) and \( M \) are on the same side of the line \( A C \);
- \( K M = A B \);
- the angle \( \angle M A K \) is the... | 44 | 0.0625 | 7,916.3125 | 8,133 | 7,901.866667 | |
A jar contains 97 marbles that are either red, green, or blue. Neil draws two marbles from the jar without replacement and notes that the probability that they would be the same color is $\frac{5}{12}$. After Neil puts his marbles back, Jerry draws two marbles from the jar with replacement. Compute the probability that... | \frac{41}{97} | Note that $\frac{5}{12}=\frac{40.97}{97 \cdot 96}$. Of all of the original ways we could've drawn marbles, we are adding 97 ways, namely drawing the same marble twice, all of which work. Thus, the answer is $$\frac{40 \cdot 97+97}{97 \cdot 96+97}=\frac{41}{97}$$ | 0.5625 | 6,058.5 | 4,511.777778 | 8,047.142857 |
Let $n > 2$ be an integer and let $\ell \in \{1, 2,\dots, n\}$. A collection $A_1,\dots,A_k$ of (not necessarily distinct) subsets of $\{1, 2,\dots, n\}$ is called $\ell$-large if $|A_i| \ge \ell$ for all $1 \le i \le k$. Find, in terms of $n$ and $\ell$, the largest real number $c$ such that the inequality
\[ \sum_{i... | \frac{\ell^2 - 2\ell + n}{n(n-1)} | To solve the problem, we need to find the largest real number \( c \) such that the inequality
\[
\sum_{i=1}^k \sum_{j=1}^k x_i x_j \frac{|A_i \cap A_j|^2}{|A_i| \cdot |A_j|} \ge c \left(\sum_{i=1}^k x_i\right)^2
\]
holds for all positive integers \( k \), all nonnegative real numbers \( x_1, x_2, \dots, x_k \), and ... | 0 | 8,192 | -1 | 8,192 |
In coordinate space, a particle starts at the point $(2,3,4)$ and ends at the point $(-1,-3,-3),$ along the line connecting the two points. Along the way, the particle intersects the unit sphere centered at the origin at two points. Then the distance between these two points can be expressed in the form $\frac{a}{\sq... | 59 | 0.9375 | 4,430.625 | 4,179.866667 | 8,192 | |
A certain unit is planning to build a rectangular room that is backed against a wall with a ground surface area of 12 square meters. Due to geographical constraints, the side length x of the room cannot exceed 5 meters. The cost of constructing the front face of the room is 400 yuan per square meter, the cost for the s... | 13000 | 0.6875 | 4,642.8125 | 3,725.363636 | 6,661.2 | |
A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have? | 9 | 1. **Define the Set $T$:**
Let $T$ be the set of all integral triples $(a, b, c)$ such that $a \ge b \ge c$, $b+c > a$ (triangle inequality), and $a, b, c < 5$.
2. **Enumerate Possible Triangles:**
We list all possible triangles that satisfy these conditions:
- $(4, 4, 4)$
- $(4, 4, 3)$
- $(4, 4, 2... | 0 | 7,845.1875 | -1 | 7,845.1875 |
Given $\sin\left(\theta - \frac{\pi}{6}\right) = \frac{1}{4}$ with $\theta \in \left( \frac{\pi}{6}, \frac{2\pi}{3}\right)$, calculate the value of $\cos\left(\frac{3\pi}{2} + \theta\right)$. | \frac{\sqrt{15} + \sqrt{3}}{8} | 0 | 5,212.6875 | -1 | 5,212.6875 | |
If $a,b,c$ are integers from the set of positive integers less than $7$ such that \begin{align*}
abc&\equiv 1\pmod 7,\\
5c&\equiv 2\pmod 7,\\
6b&\equiv 3+b\pmod 7,
\end{align*}then what is the remainder when $a+b+c$ is divided by $7$? | 4 | 1 | 1,926.8125 | 1,926.8125 | -1 | |
Given $\{a_{n}\}\left(n\in N*\right)$ is an arithmetic sequence with a common difference of $-2$, and $a_{6}$ is the geometric mean of $a_{2}$ and $a_{8}$. Let $S_{n}$ be the sum of the first $n$ terms of $\{a_{n}\}$. Find the value of $S_{10}$. | 90 | 1 | 2,430.5 | 2,430.5 | -1 | |
Given a $5\times 5$ chess board, how many ways can you place five distinct pawns on the board such that each column and each row contains exactly one pawn and no two pawns are positioned as if they were "attacking" each other in the manner of queens in chess? | 1200 | 0.3125 | 6,969.125 | 4,780.2 | 7,964.090909 | |
The Aeroflot cashier must deliver tickets to five groups of tourists. Three of these groups live in the hotels "Druzhba," "Russia," and "Minsk." The cashier will be given the address of the fourth group by the tourists from "Russia," and the address of the fifth group by the tourists from "Minsk." In how many ways can ... | 30 | 0 | 6,048.375 | -1 | 6,048.375 | |
For $n \geq 1$ , let $a_n$ be the number beginning with $n$ $9$ 's followed by $744$ ; eg., $a_4=9999744$ . Define $$ f(n)=\text{max}\{m\in \mathbb{N} \mid2^m ~ \text{divides} ~ a_n \} $$ , for $n\geq 1$ . Find $f(1)+f(2)+f(3)+ \cdots + f(10)$ . | 75 | 0 | 8,192 | -1 | 8,192 | |
If one can find a student with at least $k$ friends in any class which has $21$ students such that at least two of any three of these students are friends, what is the largest possible value of $k$ ? | 10 | 0.25 | 7,498.375 | 6,444.75 | 7,849.583333 | |
Find the sum of the distinct prime factors of $7^7 - 7^4$. | 31 | 0.9375 | 2,186.125 | 2,061 | 4,063 | |
A loaf of sausage is marked with thin transverse rings. If you cut along the red rings, you get 5 pieces; if along the yellow rings, you get 7 pieces; and if along the green rings, you get 11 pieces. How many pieces of sausage will you get if you cut along the rings of all three colors? | 21 | 0.125 | 7,112.0625 | 4,506.5 | 7,484.285714 | |
How many ways are there to place four points in the plane such that the set of pairwise distances between the points consists of exactly 2 elements? (Two configurations are the same if one can be obtained from the other via rotation and scaling.) | 6 | Let $A, B, C, D$ be the four points. There are 6 pairwise distances, so at least three of them must be equal. Case 1: There is no equilateral triangle. Then WLOG we have $A B=B C=C D=1$. - Subcase 1.1: $A D=1$ as well. Then $A C=B D \neq 1$, so $A B C D$ is a square. - Subcase 1.2: $A D \neq 1$. Then $A C=B D=A D$, so ... | 0 | 8,179.5 | -1 | 8,179.5 |
Triangle $GHI$ has sides of length 7, 24, and 25 units, and triangle $JKL$ has sides of length 9, 40, and 41 units. Both triangles have an altitude to the hypotenuse such that for $GHI$, the altitude splits the triangle into two triangles whose areas have a ratio of 2:3. For $JKL$, the altitude splits the triangle into... | \dfrac{7}{15} | 0.6875 | 5,550.1875 | 4,349.363636 | 8,192 | |
Given an ellipse $G$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{6}}{3}$, a right focus at $(2 \sqrt{2},0)$, and a line $l$ with a slope of $1$ intersecting the ellipse $G$ at points $A$ and $B$. An isosceles triangle is constructed with $AB$ as the base and $P(-3,2)$ ... | \frac{9}{2} | 0 | 8,192 | -1 | 8,192 | |
The keys of a safe with five locks are cloned and distributed among eight people such that any of five of eight people can open the safe. What is the least total number of keys? $ | 20 | 0.125 | 7,225.25 | 5,940.5 | 7,408.785714 | |
The number 119 has the following property:
- Division by 2 leaves a remainder of 1;
- Division by 3 leaves a remainder of 2;
- Division by 4 leaves a remainder of 3;
- Division by 5 leaves a remainder of 4;
- Division by 6 leaves a remainder of 5.
How many positive integers less than 2007 satisfy this property? | 32 | 0 | 3,647.3125 | -1 | 3,647.3125 | |
Lark has forgotten her locker combination. It is a sequence of three numbers, each in the range from 1 to 30, inclusive. She knows that the first number is odd, the second number is even, and the third number is a multiple of 3. How many combinations could possibly be Lark's? | 2250 | 1 | 1,575.8125 | 1,575.8125 | -1 | |
In the cells of a 9 × 9 square, there are non-negative numbers. The sum of the numbers in any two adjacent rows is at least 20, and the sum of the numbers in any two adjacent columns does not exceed 16. What can be the sum of the numbers in the entire table? | 80 | 0 | 8,192 | -1 | 8,192 | |
If $f(x) = 4-3x$ and $g(x) = x^2 +1$, find $f(g(\sqrt{2}))$. | -5 | 1 | 1,802 | 1,802 | -1 | |
Identify a six-digit number \( N \) composed of distinct digits such that the numbers \( 2N, 3N, 4N, 5N, \) and \( 6N \) are permutations of its digits. | 142857 | 0.5 | 6,378.125 | 4,564.25 | 8,192 | |
Points \( A \) and \( B \) are located on a straight highway running from west to east. Point \( B \) is 9 km east of \( A \). A car leaves point \( A \) heading east at a speed of 40 km/h. At the same time, a motorcycle leaves point \( B \) in the same direction with a constant acceleration of 32 km/h\(^2\). Determin... | 25 | 0.0625 | 6,926.875 | 2,956 | 7,191.6 | |
Any six points are taken inside or on a rectangle with dimensions $1 \times 2$. Let $b$ be the smallest possible value such that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$. | \frac{\sqrt{5}}{2} | 0 | 8,192 | -1 | 8,192 | |
If $2$ is a solution (root) of $x^3+hx+10=0$, then $h$ equals: | -9 | 1. **Substitute $x = 2$ into the equation**: Given that $2$ is a root of the polynomial $x^3 + hx + 10 = 0$, we substitute $2$ for $x$:
\[
2^3 + h \cdot 2 + 10 = 0
\]
2. **Simplify the equation**: Calculate $2^3$ and simplify the left-hand side:
\[
8 + 2h + 10 = 0
\]
\[
18 + 2h = 0
\]
3. **... | 1 | 2,096.75 | 2,096.75 | -1 |
Nikita usually leaves home at 8:00 AM, gets into Uncle Vanya's car, and his uncle drives him to school at a certain time. But on Friday, Nikita left home at 7:10 AM and ran in the opposite direction. Uncle Vanya waited for him and at 8:10 AM drove after him, caught up with Nikita, turned around, and took him to school,... | 13 | 0 | 7,554.4375 | -1 | 7,554.4375 | |
If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? | 39 | 1 | 1,024.1875 | 1,024.1875 | -1 | |
Given an arithmetic sequence $\{a_n\}$ where the sum of the first $n$ terms is $S_n = (a+1)n^2 + a$, if the sides of a certain triangle are $a_2$, $a_3$, and $a_4$, then the area of this triangle is ________. | \frac{15}{4} \sqrt{3} | 0 | 5,088.5625 | -1 | 5,088.5625 | |
For the Shanghai World Expo, 20 volunteers were recruited, with each volunteer assigned a unique number from 1 to 20. If four individuals are to be selected randomly from this group and divided into two teams according to their numbers, with the smaller numbers in one team and the larger numbers in another, what is the... | 21 | 0.1875 | 6,745.0625 | 6,270.333333 | 6,854.615385 | |
Each of two wheels contains numbers from 1 to 8. When the wheels are spun, a number is selected from each wheel. Find the probability that the sum of the two selected numbers is divisible by 4. | \frac{1}{4} | 0.9375 | 5,285.1875 | 5,091.4 | 8,192 | |
For a certain hyperbola
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\]where $a > b,$ the angle between the asymptotes is $60^\circ.$ Find $\frac{a}{b}.$ | \sqrt{3} | 1 | 2,977.0625 | 2,977.0625 | -1 | |
A projectile is launched, and its height (in meters) over time is represented by the equation $-20t^2 + 50t + 10$, where $t$ is the time in seconds after launch. Determine the maximum height reached by the projectile. | 41.25 | 0.875 | 4,131.25 | 3,551.142857 | 8,192 | |
During the November monthly exam at our school, approximately 1,000 science students participated, with mathematics scores distributed normally as $\xi \sim N(100, a^2), (a > 0)$, and a full score of 150. The statistics showed that about 60% of the students scored between 80 and 120 points. Therefore, approximately \_\... | 200 | 0.6875 | 4,657.375 | 4,048.818182 | 5,996.2 | |
Given the function $f(x)= \begin{cases} kx^{2}+2x-1, & x\in (0,1] \\ kx+1, & x\in (1,+\infty) \end{cases}$ has two distinct zeros $x_{1}$ and $x_{2}$, then the maximum value of $\dfrac {1}{x_{1}}+ \dfrac {1}{x_{2}}$ is ______. | \dfrac {9}{4} | 0.3125 | 6,938.125 | 5,830 | 7,441.818182 | |
A convoy of cars is moving on a highway at a speed of 80 km/h with a distance of 10 meters between the cars. Upon passing a speed limit sign, all cars reduce their speed to 60 km/h. What will be the distance between the cars in the convoy after passing the sign? | 7.5 | 0.0625 | 6,383.6875 | 6,593 | 6,369.733333 | |
Bob buys four burgers and three sodas for $\$5.00$, and Carol buys three burgers and four sodas for $\$5.40$. How many cents does a soda cost? | 94 | 0 | 7,604.1875 | -1 | 7,604.1875 | |
Through the vertices \(A\), \(C\), and \(D_1\) of a rectangular parallelepiped \(ABCD A_1 B_1 C_1 D_1\), a plane is drawn forming a dihedral angle of \(60^\circ\) with the base plane. The sides of the base are 4 cm and 3 cm. Find the volume of the parallelepiped. | \frac{144 \sqrt{3}}{5} | 0 | 5,577.1875 | -1 | 5,577.1875 | |
Determine the number of unordered triples of distinct points in the $4 \times 4 \times 4$ lattice grid $\{0,1,2,3\}^{3}$ that are collinear in $\mathbb{R}^{3}$ (i.e. there exists a line passing through the three points). | 376 | Define a main plane to be one of the $x y, y z, z x$ planes. Define a space diagonal to be a set of collinear points not parallel to a main plane. We classify the lines as follows: (a) Lines parallel to two axes (i.e. orthogonal to a main plane). Notice that given a plane of the form $v=k$, where $v \in\{x, y, z\}, k \... | 0 | 8,192 | -1 | 8,192 |
Paul needs to save 40 files onto flash drives, each with 2.0 MB space. 4 of the files take up 1.2 MB each, 16 of the files take up 0.9 MB each, and the rest take up 0.6 MB each. Determine the smallest number of flash drives needed to store all 40 files. | 20 | 0 | 8,192 | -1 | 8,192 | |
Consider the function $f: \mathbb{N}_{0} \rightarrow \mathbb{N}_{0}$, where $\mathbb{N}_{0}$ is the set of all non-negative integers, defined by the following conditions: (i) $f(0)=0$, (ii) $f(2n)=2f(n)$ and (iii) $f(2n+1)=n+2f(n)$ for all $n \geq 0$. (a) Determine the three sets $L:=\{n \mid f(n)<f(n+1)\}, E:=\{n \mid... | a_{k}=k2^{k-1}-2^{k}+1 | (a) Let $L_{1}:=\{2k: k>0\}, \quad E_{1}:=\{0\} \cup\{4k+1: k \geq 0\}, \quad \text { and } \quad G_{1}:=\{4k+3: k \geq 0\}$. We will show that $L_{1}=L, E_{1}=E$, and $G_{1}=G$. It suffices to verify that $L_{1} \subseteq E, E_{1} \subseteq E$, and $G_{1} \subseteq G$ because $L_{1}, E_{1}$, and $G_{1}$ are mutually d... | 0 | 8,192 | -1 | 8,192 |
A parallelogram has 2 sides of length 20 and 15. Given that its area is a positive integer, find the minimum possible area of the parallelogram. | 1 | The area of the parallelogram can be made arbitrarily small, so the smallest positive integer area is 1. | 0.75 | 5,804.5625 | 5,543.916667 | 6,586.5 |
Given the event "Randomly select a point P on the side CD of rectangle ABCD, such that the longest side of ΔAPB is AB", with a probability of 1/3, determine the ratio of AD to AB. | \frac{\sqrt{5}}{3} | 0 | 4,850.625 | -1 | 4,850.625 | |
A fair standard four-sided die is tossed four times. Given that the sum of the first three tosses equals the fourth toss, what is the probability that at least one "3" is rolled?
A) $\frac{1}{12}$
B) $\frac{1}{6}$
C) $\frac{3}{10}$
D) $\frac{1}{4}$
E) $\frac{1}{2}$ | \frac{3}{10} | 0 | 6,338.3125 | -1 | 6,338.3125 | |
Let $b(x)=x^{2}+x+1$. The polynomial $x^{2015}+x^{2014}+\cdots+x+1$ has a unique "base $b(x)$ " representation $x^{2015}+x^{2014}+\cdots+x+1=\sum_{k=0}^{N} a_{k}(x) b(x)^{k}$ where each "digit" $a_{k}(x)$ is either the zero polynomial or a nonzero polynomial of degree less than $\operatorname{deg} b=2$; and the "leadin... | -1006 | Comparing degrees easily gives $N=1007$. By ignoring terms of degree at most 2013, we see $a_{N}(x)\left(x^{2}+x+1\right)^{1007} \in x^{2015}+x^{2014}+O\left(x^{2013}\right)$. Write $a_{N}(x)=u x+v$, so $a_{N}(x)\left(x^{2}+x+1\right)^{1007} \in(u x+v)\left(x^{2014}+1007 x^{2013}+O\left(x^{2012}\right)\right) \subseteq... | 0 | 8,192 | -1 | 8,192 |
Let GCF(a, b) be the abbreviation for the greatest common factor of a and b, and let LCM(c, d) be the abbreviation for the least common multiple of c and d. What is GCF(LCM(8, 14), LCM(7, 12))? | 28 | 1 | 2,582.0625 | 2,582.0625 | -1 | |
Compute $\arcsin 1.$ Express your answer in radians. | \frac{\pi}{2} | 1 | 1,082.0625 | 1,082.0625 | -1 | |
The last three digits of \( 1978^n \) and \( 1978^m \) are the same. Find the positive integers \( m \) and \( n \) such that \( m+n \) is minimized (here \( n > m \geq 1 \)). | 106 | 0.125 | 7,794.3125 | 6,837.5 | 7,931 | |
Find all natural numbers \( n \) such that
\[
\sum_{\substack{d \mid n \\ 1 \leq d < n}} d^{2} = 5(n + 1)
\] | 16 | 0 | 8,192 | -1 | 8,192 | |
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? | \frac{1}{12} | 0.9375 | 4,308.0625 | 4,049.133333 | 8,192 | |
There are 100 people in a room with ages $1,2, \ldots, 100$. A pair of people is called cute if each of them is at least seven years older than half the age of the other person in the pair. At most how many pairwise disjoint cute pairs can be formed in this room? | 43 | For a cute pair $(a, b)$ we would have $$a \geq \frac{b}{2}+7, b \geq \frac{a}{2}+7$$ Solving the system, we get that $a$ and $b$ must both be at least 14. However 14 could only be paired with itself or a smaller number; therefore, only people with age 15 or above can be paired with someone of different age. Pairing co... | 0.0625 | 7,994.9375 | 5,039 | 8,192 |
Given a supermarket with 100 customers, the table shows the number of customers in each age group and payment method category. Determine the number of customers aged between 40 and 60 years old who do not use mobile payment. | \frac{2}{5} | 0 | 1,581.25 | -1 | 1,581.25 | |
$(1)$ Solve the system of inequalities: $\left\{\begin{array}{l}{2x-1＜5,①}\\{\frac{4-x}{2}≥1,②}\end{array}\right.$<br/>$(2)$ Simplify first, then find the value: $\frac{1}{a-1}÷(\frac{{a}^{2}+a}{{a}^{2}-1}+\frac{1}{a-1})$, where $a=(-2023)^0+(\frac{1}{2})^{-1}⋅$ | \frac{1}{4} | 1 | 3,426.0625 | 3,426.0625 | -1 | |
In the diagram, $\triangle ABC$ is isosceles with $AB = AC$ and $BC = 30 \mathrm{~cm}$. Square $EFGH$, which has a side length of $12 \mathrm{~cm}$, is inscribed in $\triangle ABC$, as shown. The area of $\triangle AEF$, in $\mathrm{cm}^{2}$, is | 48 | 0.125 | 7,236.875 | 7,026.5 | 7,266.928571 | |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, let $F_1$ and $F_2$ be the left and right foci, and let $P$ and $Q$ be two points on the right branch. If $\overrightarrow{PF_2} = 2\overrightarrow{F_2Q}$ and $\overrightarrow{F_1Q} \cdot \overrightarrow{PQ} = 0$, determine the eccentricity of this hyperbola. | \frac{\sqrt{17}}{3} | 0 | 7,381.875 | -1 | 7,381.875 | |
The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ for which $\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}$ is defined? | 2010 | To solve this problem, we need to understand the recursive definition of the tower function $T(n)$ and how logarithms interact with powers and products. We start by analyzing the expression for $B$ and then determine how many times we can apply the logarithm base 2 before the result is undefined (i.e., non-positive).
... | 0 | 7,682.875 | -1 | 7,682.875 |
The ratio of the areas of two squares is $\frac{192}{80}$. After rationalizing the denominator, the ratio of their side lengths can be expressed in the simplified form $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. What is the value of the sum $a+b+c$? | 22 | 1 | 1,737.6875 | 1,737.6875 | -1 | |
John rode his bicycle for 45 minutes at a speed of 20 mph, then walked for 120 minutes at a speed of 3 mph. Calculate the overall average speed in miles per hour. | 7.64 | 0.6875 | 2,339.375 | 2,025.636364 | 3,029.6 | |
Find all \( x \in [1,2) \) such that for any positive integer \( n \), the value of \( \left\lfloor 2^n x \right\rfloor \mod 4 \) is either 1 or 2. | 4/3 | 0 | 8,192 | -1 | 8,192 | |
Find $ \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} , $ where $ \# A $ is the cardinal of $ A . $ | 165 | 0 | 7,646 | -1 | 7,646 | |
Find the values of $ k$ such that the areas of the three parts bounded by the graph of $ y\equal{}\minus{}x^4\plus{}2x^2$ and the line $ y\equal{}k$ are all equal. | \frac{2}{3} | 0 | 8,192 | -1 | 8,192 | |
A cheetah takes strides of 2 meters each, while a fox takes strides of 1 meter each. The time it takes for the cheetah to run 2 strides is the same as the time it takes for the fox to run 3 strides. If the distance between the cheetah and the fox is 30 meters, how many meters must the cheetah run to catch up with the f... | 120 | 0.5625 | 5,855.1875 | 4,600.444444 | 7,468.428571 | |
On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained by adding the smallest non-zero digit of its decimal representation to the previous number. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th place in ... | 3001 | 0 | 8,192 | -1 | 8,192 | |
Complex numbers $z_1,$ $z_2,$ and $z_3$ are zeros of a polynomial $Q(z) = z^3 + pz + s,$ where $|z_1|^2 + |z_2|^2 + |z_3|^2 = 300$. The points corresponding to $z_1,$ $z_2,$ and $z_3$ in the complex plane are the vertices of a right triangle with the right angle at $z_3$. Find the square of the hypotenuse of this trian... | 450 | 0.5 | 6,685.5 | 5,496.5 | 7,874.5 | |
Find the value of the definite integral $\int_{0}^{1} ( \sqrt{1-(x-1)^{2}}-{x}^{2})dx$. | \frac{\pi}{4} - \frac{1}{3} | 0.875 | 3,087.875 | 3,061.357143 | 3,273.5 | |
There is a group of monkeys transporting peaches from location $A$ to location $B$. Every 3 minutes a monkey departs from $A$ towards $B$, and it takes 12 minutes for a monkey to complete the journey. A rabbit runs from $B$ to $A$. When the rabbit starts, a monkey has just arrived at $B$. On the way, the rabbit encount... | 300 | 0 | 7,979.9375 | -1 | 7,979.9375 | |
Find all permutations $a_1, a_2, \ldots, a_9$ of $1, 2, \ldots, 9$ such that \[ a_1+a_2+a_3+a_4=a_4+a_5+a_6+a_7= a_7+a_8+a_9+a_1 \]
and
\[ a_1^2+a_2^2+a_3^2+a_4^2=a_4^2+a_5^2+a_6^2+a_7^2= a_7^2+a_8^2+a_9^2+a_1^2 \] | (2, 9, 4, 5, 1, 6, 8, 3, 7) |
We are tasked with finding all permutations \( a_1, a_2, \ldots, a_9 \) of the numbers \( 1, 2, \ldots, 9 \) that satisfy the given conditions:
\[
a_1+a_2+a_3+a_4 = a_4+a_5+a_6+a_7 = a_7+a_8+a_9+a_1
\]
and
\[
a_1^2+a_2^2+a_3^2+a_4^2 = a_4^2+a_5^2+a_6^2+a_7^2 = a_7^2+a_8^2+a_9^2+a_1^2.
\]
### Step-by-step Solution:... | 0 | 8,192 | -1 | 8,192 |
If the function $f(x) = \frac{1}{2}(m-2)x^2 + (n-8)x + 1$ with $m \geq 0$ and $n \geq 0$ is monotonically decreasing in the interval $\left[\frac{1}{2}, 2\right]$, then the maximum value of $mn$ is __________. | 18 | 0.4375 | 6,848.1875 | 6,395.285714 | 7,200.444444 | |
The parabolas $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ intersect at four points $(x_1,y_1),$ $(x_2,y_2),$ $(x_3,y_3),$ and $(x_4,y_4).$ Find
\[x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.\] | 8 | 1 | 5,181.4375 | 5,181.4375 | -1 | |
Calculate $(3^5 \cdot 6^5)^2$. | 3570467226624 | 0.3125 | 7,533.1875 | 6,083.8 | 8,192 | |
Find all real numbers $x$ such that $$x^{2}+\left\lfloor\frac{x}{2}\right\rfloor+\left\lfloor\frac{x}{3}\right\rfloor=10$$ | -\sqrt{14} | Evidently $x^{2}$ must be an integer. Well, there aren't that many things to check, are there? Among positive $x, \sqrt{8}$ is too small and $\sqrt{9}$ is too big; among negative $x,-\sqrt{15}$ is too small and $-\sqrt{13}$ is too big. | 0.0625 | 8,176.0625 | 7,937 | 8,192 |
Let \( p, q, r \) be the roots of the polynomial \( x^3 - 8x^2 + 14x - 2 = 0 \). Define \( t = \sqrt{p} + \sqrt{q} + \sqrt{r} \). Find \( t^4 - 16t^2 - 12t \). | -8 | 0 | 8,173.625 | -1 | 8,173.625 | |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$? | \dfrac{1}{52} | 0.4375 | 7,695.6875 | 7,057.571429 | 8,192 | |
How many integers at minimum must be selected from the set $\{1, 2, \ldots, 20\}$ to ensure that this selection includes two integers \(a\) and \(b\) such that \(a - b = 2\)? | 11 | 0.5625 | 6,960.9375 | 6,019.555556 | 8,171.285714 | |
Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 10$ and
\[ab + ac + ad + bc + bd + cd = 20.\] Find the largest possible value of $d$. | \frac{5 + 5\sqrt{21}}{2} | 0 | 6,092.0625 | -1 | 6,092.0625 | |
A herd of elephants. Springs are bubbling at the bottom of the lake. A herd of 183 elephants could drink it dry in one day, and a herd of 37 elephants could do so in 5 days. How many days will it take for 1 elephant to drink the lake dry? | 365 | 0 | 7,577.875 | -1 | 7,577.875 | |
In triangle $ABC$, we have $\angle C = 90^\circ$, $AB = 13$, and $BC = 5$. What is $\tan A$? | \frac{5}{12} | 1 | 1,342.25 | 1,342.25 | -1 | |
Four different balls are to be placed into five boxes numbered 1, 2, 3, 4, and 5. (Write the result as a number)
(1) How many ways can there be with at least one ball in box number 1?
(2) How many ways are there to have exactly two empty boxes?
(3) How many ways are there to have exactly three empty boxes?
(4) ... | 375 | 0.6875 | 6,144.5625 | 5,452.454545 | 7,667.2 | |
Given the set $A=\{x\in \mathbb{R} | ax^2-3x+2=0, a\in \mathbb{R}\}$.
1. If $A$ is an empty set, find the range of values for $a$.
2. If $A$ contains only one element, find the value of $a$ and write down this element. | \frac{4}{3} | 1 | 2,960.0625 | 2,960.0625 | -1 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}| = 1$ and $\overrightarrow{a} \perp \overrightarrow{b}$, calculate the projection of the vector $2\overrightarrow{b} - \overrightarrow{a}$ in the direction of vector $\overrightarrow{a}$. | -1 | 0.9375 | 3,412.625 | 3,138.733333 | 7,521 | |
Consider all questions on this year's contest that ask for a single real-valued answer (excluding this one). Let \(M\) be the median of these answers. Estimate \(M\). | 18.5285921 | Looking back to the answers of previous problems in the round (or other rounds) can give you to a rough estimate. | 0 | 4,784.9375 | -1 | 4,784.9375 |
Let $\alpha$ and $\beta$ be acute angles, and $\cos \alpha = \frac{\sqrt{5}}{5}$, $\sin (\alpha + \beta) = \frac{3}{5}$. Find $\cos \beta$. | \frac{2\sqrt{5}}{25} | 0 | 5,613.4375 | -1 | 5,613.4375 | |
An isosceles, obtuse triangle has one angle with a degree measure that is 50$\%$ larger than the measure of a right angle. What is the measure, in degrees, of one of the two smallest angles in the triangle? Express your answer as a decimal to the nearest tenth. | 22.5^{\circ} | 1 | 1,459.375 | 1,459.375 | -1 | |
Given that the amount of cultural and tourism vouchers issued is $2.51 million yuan, express this amount in scientific notation. | 2.51 \times 10^{6} | 0 | 323.6875 | -1 | 323.6875 | |
Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book? | 250 | 1. **Calculate the total pages read in the first three days**: Isabella read an average of $36$ pages per day for the first three days. Therefore, the total number of pages she read in these three days is:
\[
3 \times 36 = 108 \text{ pages}
\]
2. **Calculate the total pages read in the next three days**: For ... | 1 | 1,384.1875 | 1,384.1875 | -1 |
If $\sqrt{5 + x} + \sqrt{20 - x} = 7$, what is the value of $(5 + x)(20 - x)$? | 144 | 1 | 1,770.25 | 1,770.25 | -1 | |
Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $50$. Determine how many values of $n$ make $q+r$ divisible by $13$.
A) 7000
B) 7200
C) 7400
D) 7600 | 7200 | 0 | 8,192 | -1 | 8,192 | |
A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $16$, $25$, and $36$, respectively. Furthermore, a circle centered at $Q$ inside $\triangle DEF$ cuts off... | 225 | 0.25 | 7,628 | 6,048.25 | 8,154.583333 | |
Four points in the order \( A, B, C, D \) lie on a circle with the extension of \( AB \) meeting the extension of \( DC \) at \( E \) and the extension of \( AD \) meeting the extension of \( BC \) at \( F \). Let \( EP \) and \( FQ \) be tangents to this circle with points of tangency \( P \) and \( Q \) respectively.... | 87 | 0.0625 | 7,957.3125 | 4,437 | 8,192 | |
Let \[P(x) = (3x^5 - 45x^4 + gx^3 + hx^2 + ix + j)(4x^3 - 60x^2 + kx + l),\] where $g, h, i, j, k, l$ are real numbers. Suppose that the set of all complex roots of $P(x)$ includes $\{1, 2, 3, 4, 5, 6\}$. Find $P(7)$. | 51840 | 0.0625 | 8,192 | 8,192 | 8,192 | |
A two-row triangle is created with a total of 15 pieces: nine unit rods and six connectors, as shown. What is the total number of pieces that would be used to create an eight-row triangle?
[asy]
draw((0,0)--(4,0)--(2,2sqrt(3))--(0,0)--cycle,linewidth(1));
draw((2,0)--(3,sqrt(3))--(1,sqrt(3))--(2,0)--cycle,linewidth(1)... | 153 | 0.625 | 6,361.25 | 5,681.2 | 7,494.666667 | |
Let \(ABCD\) be a square of side length 1. \(P\) and \(Q\) are two points on the plane such that \(Q\) is the circumcentre of \(\triangle BPC\) and \(D\) is the circumcentre of \(\triangle PQA\). Find the largest possible value of \(PQ^2\). Express the answer in the form \(a + \sqrt{b}\) or \(a - \sqrt{b}\), where \(a\... | 2 + \sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
The common ratio of the geometric sequence $a+\log_{2}3$, $a+\log_{4}3$, $a+\log_{8}3$ is __________. | \frac{1}{3} | 0.875 | 4,877.6875 | 4,404.214286 | 8,192 | |
What is the greatest integer less than or equal to $\frac{4^{50}+3^{50}}{4^{47}+3^{47}}$? | 64 | 0 | 7,779.125 | -1 | 7,779.125 | |
Let \( m \) be the smallest positive integer that is a multiple of \( 100 \) and has exactly \( 100 \) positive integral divisors, including 1 and itself. Find \( \frac{m}{100} \). | 4050 | 0 | 8,186.75 | -1 | 8,186.75 | |
Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15} \dotsm \frac{3n+6}{3n} \dotsm \frac{3003}{2997}.\] | 1001 | 0.375 | 3,898.1875 | 2,417.666667 | 4,786.5 | |
Find the domain of the function
\[f(x) = \sqrt{1 - \sqrt{2 - \sqrt{3 - x}}}.\] | [-1,2] | 1 | 2,790.4375 | 2,790.4375 | -1 |
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