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 |
|---|---|---|---|---|---|---|
To popularize knowledge of fire safety, a certain school organized a competition on related knowledge. The competition is divided into two rounds, and each participant must participate in both rounds. If a participant wins in both rounds, they are considered to have won the competition. It is known that in the first ro... | \frac{223}{300} | 0.8125 | 5,234 | 4,792.846154 | 7,145.666667 | |
What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock? | 60 | 1. **Understanding the Clock's Degree System**: A clock face is a full circle, which contains 360 degrees. Since there are 12 hours marked on the clock, the angle between each hour is calculated by dividing the total degrees by the number of hours:
\[
\frac{360^\circ}{12} = 30^\circ
\]
This means each hour ... | 1 | 2,150.4375 | 2,150.4375 | -1 |
Given an arithmetic sequence $\{a_n\}$, its sum of the first $n$ terms is $S_n$. It is known that $a_2=2$, $S_5=15$, and $b_n=\frac{1}{a_{n+1}^2-1}$. Find the sum of the first 10 terms of the sequence $\{b_n\}$. | \frac {175}{264} | 0.5625 | 7,085.375 | 6,224.666667 | 8,192 | |
Let $A_{1} A_{2} \ldots A_{19}$ be a regular nonadecagon. Lines $A_{1} A_{5}$ and $A_{3} A_{4}$ meet at $X$. Compute $\angle A_{7} X A_{5}$. | \frac{1170^{\circ}}{19} | Inscribing the nondecagon in a circle, note that $$\angle A_{3} X A_{5}=\frac{1}{2}(\widehat{A_{1} A_{3}}-\widehat{A_{4} A_{5}})=\frac{1}{2} \widehat{A_{5} A_{3} A_{4}}=\angle A_{5} A_{3} X$$ Thus $A_{5} X=A_{5} A_{3}=A_{5} A_{7}$, so $$\begin{aligned} \angle A_{7} X A_{5} & =90^{\circ}-\frac{1}{2} \angle X A_{5} A_{7}... | 0 | 8,158.5 | -1 | 8,158.5 |
Consider five-dimensional Cartesian space $\mathbb{R}^{5}=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \mid x_{i} \in \mathbb{R}\right\}$ and consider the hyperplanes with the following equations: - $x_{i}=x_{j}$ for every $1 \leq i<j \leq 5$; - $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=-1$ - $x_{1}+x_{2}+x_{3}+x_{4}+x_{... | 480 | Note that given a set of plane equations $P_{i}\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)=0$, for $i=1,2, \ldots, n$, each region that the planes separate the space into correspond to a $n$-tuple of -1 and 1 , representing the sign of $P_{1}, P_{2}, \ldots P_{n}$ for all points in that region. Therefore, the first ... | 0.125 | 8,036.875 | 8,186.5 | 8,015.5 |
Calculate:<br/>$(1)2(\sqrt{3}-\sqrt{5})+3(\sqrt{3}+\sqrt{5})$;<br/>$(2)-{1}^{2}-|1-\sqrt{3}|+\sqrt[3]{8}-(-3)×\sqrt{9}$. | 11 - \sqrt{3} | 0.5625 | 2,011.25 | 1,752 | 2,344.571429 | |
In $\triangle ABC$, the lengths of the sides opposite to angles A, B, and C are a, b, and c respectively. Given that a = 3, cosC = $- \frac{1}{15}$, and 5sin(B + C) = 3sin(A + C).
(1) Find the length of side c.
(2) Find the value of sin(B - $\frac{\pi}{3}$). | \frac{2\sqrt{14} - 5\sqrt{3}}{18} | 0 | 8,192 | -1 | 8,192 | |
Given $\cos \left(\alpha- \frac {\beta}{2}\right)=- \frac {1}{9}$ and $\sin \left( \frac {\alpha}{2}-\beta\right)= \frac {2}{3}$, with $0 < \beta < \frac {\pi}{2} < \alpha < \pi$, find $\sin \frac {\alpha+\beta}{2}=$ ______. | \frac {22}{27} | 0.625 | 5,923.1875 | 4,690.8 | 7,977.166667 | |
The sum of the digits in the product of $\overline{A A A A A A A A A} \times \overline{B B B B B B B B B}$. | 81 | 0 | 8,192 | -1 | 8,192 | |
Compute $\arctan ( \tan 65^\circ - 2 \tan 40^\circ )$. (Express your answer in degrees as an angle between $0^\circ$ and $180^\circ$.) | 25^\circ | 0.8125 | 6,637.4375 | 6,278.692308 | 8,192 | |
(1) Given the hyperbola $C$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, its right vertex is $A$, and a circle $A$ with center $A$ and radius $b$ intersects one of the asymptotes of the hyperbola $C$ at points $M$ and $N$. If $\angle MAN = 60^{\circ}$, then the eccentricity of $C$ is ______.
(2... | \dfrac{4}{5} | 0 | 8,192 | -1 | 8,192 | |
A bicycle trip is 30 km long. Ari rides at an average speed of 20 km/h. Bri rides at an average speed of 15 km/h. If Ari and Bri begin at the same time, how many minutes after Ari finishes the trip will Bri finish? | 30 | Riding at 15 km/h, Bri finishes the 30 km in $\frac{30 \text{ km}}{15 \text{ km/h}} = 2 \text{ h}$. Riding at 20 km/h, Ari finishes the 30 km in $\frac{30 \text{ km}}{20 \text{ km/h}} = 1.5 \text{ h}$. Therefore, Bri finishes 0.5 h after Ari, which is 30 minutes. | 1 | 1,572.8125 | 1,572.8125 | -1 |
Find $x+y+z$ when $$ a_1x+a_2y+a_3z= a $$ $$ b_1x+b_2y+b_3z=b $$ $$ c_1x+c_2y+c_3z=c $$ Given that $$ a_1\left(b_2c_3-b_3c_2\right)-a_2\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c_2-b_2c_1\right)=9 $$ $$ a\left(b_2c_3-b_3c_2\right)-a_2\left(bc_3-b_3c\right)+a_3\left(bc_2-b_2c\right)=17 $$ $$ a_1\left(bc... | 16/9 | 0.6875 | 5,108.9375 | 4,417.181818 | 6,630.8 | |
For how many ordered pairs of positive integers $(a, b)$ such that $a \le 50$ is it true that $x^2 - ax + b$ has integer roots? | 625 | 0.125 | 8,147.0625 | 7,832.5 | 8,192 | |
The perimeter of a square with side length $x$ units is equal to the circumference of a circle with radius 2 units. What is the value of $x$? Express your answer as a decimal to the nearest hundredth. | 3.14 | 1 | 2,376.4375 | 2,376.4375 | -1 | |
A $7 \times 7$ board is either empty or contains an invisible $2 \times 2$ ship placed "by the cells." You are allowed to place detectors in some cells of the board and then activate them all at once. An activated detector signals if its cell is occupied by the ship. What is the minimum number of detectors needed to gu... | 16 | 0 | 8,135.1875 | -1 | 8,135.1875 | |
Travel along the alley clockwise.
In 1 hour of walking, the pedestrian walked 6 kilometers and did not reach point $B$ (a whole $2 \pi - 6$ km!), so the third option is clearly longer than the first and can be excluded.
In the first case, when moving along the alley, they would need to cover a distance of 6 km, and i... | 0.21 | 0 | 8,047.75 | -1 | 8,047.75 | |
Two ferries cross a river with constant speeds, turning at the shores without losing time. They start simultaneously from opposite shores and meet for the first time 700 feet from one shore. They continue to the shores, return, and meet for the second time 400 feet from the opposite shore. Determine the width of the ri... | 1400 | 0.125 | 6,779.4375 | 6,150.5 | 6,869.285714 | |
A non-increasing sequence of 100 non-negative reals has the sum of the first two terms at most 100 and the sum of the remaining terms at most 100. What is the largest possible value for the sum of the squares of the terms? | 10000 | 0.0625 | 8,118.0625 | 7,009 | 8,192 | |
Let \( A, B, C \) be points on the same plane with \( \angle ACB = 120^\circ \). There is a sequence of circles \( \omega_0, \omega_1, \omega_2, \ldots \) on the same plane (with corresponding radii \( r_0, r_1, r_2, \ldots \) where \( r_0 > r_1 > r_2 > \cdots \)) such that each circle is tangent to both segments \( CA... | \frac{3}{2} + \sqrt{3} | 0.0625 | 7,147.875 | 6,872 | 7,166.266667 | |
Place each of the digits 6, 7, 8 and 9 in exactly one square to make the smallest possible product. What is this product? [asy]draw((0,.5)--(10,.5),linewidth(1));
draw((4,1)--(6,1)--(6,3)--(4,3)--(4,1),linewidth(1));
draw((7,1)--(9,1)--(9,3)--(7,3)--(7,1),linewidth(1));
draw((7,4)--(9,4)--(9,6)--(7,6)--(7,4),linewidth(... | 5372 | 0.1875 | 7,118.4375 | 6,216.333333 | 7,326.615385 | |
A six place number is formed by repeating a three place number; for example, $256256$ or $678678$, etc. Any number of this form is always exactly divisible by: | 1001 | 1. **Express the number in a general form**:
Let the three-digit number be represented as $\overline{abc}$, where $a$, $b$, and $c$ are its digits. The six-digit number formed by repeating $\overline{abc}$ is $\overline{abcabc}$.
2. **Convert the six-digit number into a mathematical expression**:
The number ... | 0.75 | 3,883.6875 | 3,049.083333 | 6,387.5 |
Compute
\[\sin^2 4^\circ + \sin^2 8^\circ + \sin^2 12^\circ + \dots + \sin^2 176^\circ.\] | \frac{45}{2} | 0.5 | 7,829.5625 | 7,467.125 | 8,192 | |
The least common multiple of $x$ and $y$ is $18$, and the least common multiple of $y$ and $z$ is $20$. Determine the least possible value of the least common multiple of $x$ and $z$. | 90 | 0 | 6,423.125 | -1 | 6,423.125 | |
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$ . (Note: $n$ is written in the usual base ten notation.) | 9999 | 0 | 8,192 | -1 | 8,192 | |
Given the line $y=-x+1$ and the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, they intersect at points $A$ and $B$. $OA \perp OB$, where $O$ is the origin. If the eccentricity of the ellipse $e \in [\frac{1}{2}, \frac{\sqrt{3}}{2}]$, find the maximum value of $a$. | \frac{\sqrt{10}}{2} | 0 | 6,908.4375 | -1 | 6,908.4375 | |
Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8)=8$ and $\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x))=3$? | 10 | 1. **Define the function and its application**: Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8) = 8$ and $\clubsuit(123) = 1 + 2 + 3 = 6$.
2. **Determine possible values of $y = \clubsuit(x)$**: Since $x$ is a two-digit number, the maximum value of $x$ is 99. The... | 0.9375 | 5,304.6875 | 5,112.2 | 8,192 |
On the board we write a series of $n$ numbers, where $n \geq 40$ , and each one of them is equal to either $1$ or $-1$ , such that the following conditions both hold:
(i) The sum of every $40$ consecutive numbers is equal to $0$ .
(ii) The sum of every $42$ consecutive numbers is not equal to $0$ .
We den... | 20 | 0 | 8,192 | -1 | 8,192 | |
Alice wants to write down a list of prime numbers less than 100, using each of the digits 1, 2, 3, 4, and 5 once and no other digits. Which prime number must be in her list? | 41 | 0.25 | 6,787.375 | 5,576 | 7,191.166667 | |
In a division problem, the dividend is 12, and the divisor is a natural number less than 12. What is the sum of all possible different remainders? | 15 | 0.6875 | 4,464.125 | 4,394.636364 | 4,617 | |
At the "Economics and Law" congress, a "Best of the Best" tournament was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. During one match, participants had to ask each other questions within a limited time and record correct answers. Each participant played with each other... | 105 | 0 | 8,192 | -1 | 8,192 | |
Given four one-inch squares are placed with their bases on a line. The second square from the left is lifted out and rotated 30 degrees before reinserting it such that it just touches the adjacent square on its right. Determine the distance in inches from point B, the highest point of the rotated square, to the line on... | \frac{2 + \sqrt{3}}{4} | 0 | 8,192 | -1 | 8,192 | |
Let $P(x) = b_0 + b_1x + \dots + b_nx^n$ be a polynomial with integer coefficients, and $0 \le b_i < 5$ for all $0 \le i \le n$.
Given that $P(\sqrt{5}) = 40 + 31\sqrt{5}$, compute $P(3)$. | 381 | 0.5625 | 6,235.9375 | 4,976.666667 | 7,855 | |
The length of a rectangle is three times its width. The perimeter is 160 cm. What is the number of square centimeters in the area of the rectangle? | 1200 | 1 | 1,365.6875 | 1,365.6875 | -1 | |
Given a convex quadrilateral \(ABCD\) with \(\angle C = 57^{\circ}\), \(\sin \angle A + \sin \angle B = \sqrt{2}\), and \(\cos \angle A + \cos \angle B = 2 - \sqrt{2}\), find the measure of angle \(D\) in degrees. | 168 | 0.5625 | 6,984.875 | 6,046 | 8,192 | |
In $\triangle ABC$, side $a = \sqrt{3}$, side $b = \sqrt{3}$, and side $c > 3$. Let $x$ be the largest number such that the magnitude, in degrees, of the angle opposite side $c$ exceeds $x$. Then $x$ equals: | 120^{\circ} | 1. **Identify the given information**: In $\triangle ABC$, we have $a = \sqrt{3}$, $b = \sqrt{3}$, and $c > 3$. We need to find the largest $x$ such that the angle opposite to side $c$, denoted as $\angle C$, exceeds $x$ degrees.
2. **Apply the Law of Cosines**: The Law of Cosines states that for any triangle with sid... | 0.8125 | 4,651.25 | 3,834.153846 | 8,192 |
From 50 products, 10 are selected for inspection. The total number of items is \_\_\_\_\_\_\_, and the sample size is \_\_\_\_\_\_. | 10 | 0.3125 | 232.875 | 255.4 | 222.636364 | |
The measures of the three interior angles of a triangle are $50^\circ$, $55^\circ$ and $x^\circ$. What is the degree measure of the largest interior angle of this triangle? | 75^\circ | 1 | 1,142.125 | 1,142.125 | -1 | |
There is a unique two-digit positive integer $t$ for which the last two digits of $13 \cdot t$ are $26$. | 62 | 0 | 8,192 | -1 | 8,192 | |
There are two red, two black, two white, and a positive but unknown number of blue socks in a drawer. It is empirically determined that if two socks are taken from the drawer without replacement, the probability they are of the same color is $\frac{1}{5}$. How many blue socks are there in the drawer? | 4 | Let the number of blue socks be $x>0$. Then the probability of drawing a red sock from the drawer is $\frac{2}{6+x}$ and the probability of drawing a second red sock from the drawer is $\frac{1}{6+x-1}=\frac{1}{5+x}$, so the probability of drawing two red socks from the drawer without replacement is $\frac{2}{(6+x)(5+x... | 1 | 2,940 | 2,940 | -1 |
Compute $\arccos (\sin 3)$. All functions are in radians. | 3 - \frac{\pi}{2} | 0.5625 | 6,707.8125 | 5,553.444444 | 8,192 | |
Square \(ABCD\) has sides of length 14. A circle is drawn through \(A\) and \(D\) so that it is tangent to \(BC\). What is the radius of the circle? | 8.75 | 0 | 3,961.8125 | -1 | 3,961.8125 | |
Four brothers have together forty-eight Kwanzas. If the first brother's money were increased by three Kwanzas, if the second brother's money were decreased by three Kwanzas, if the third brother's money were triplicated and if the last brother's money were reduced by a third, then all brothers would have the same quant... | 6, 12, 3, 27 |
Let \( x_1, x_2, x_3, \) and \( x_4 \) be the amounts of money that the first, second, third, and fourth brothers have, respectively. According to the problem, we have the following equation describing their total amount of money:
\[
x_1 + x_2 + x_3 + x_4 = 48
\]
We are also given conditions on how these amounts are... | 0 | 6,816.4375 | -1 | 6,816.4375 |
For natural numbers \\(m\\) greater than or equal to \\(2\\) and their powers of \\(n\\), the following decomposition formula is given:
\\(2^{2}=1+3\\) \\(3^{2}=1+3+5\\) \\(4^{2}=1+3+5+7\\) \\(…\\)
\\(2^{3}=3+5\\) \\(3^{3}=7+9+11\\) \\(…\\)
\\(2^{4}=7+9\\) \\(…\\)
Following this pattern, the third nu... | 125 | 0.125 | 7,960 | 6,336 | 8,192 | |
Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one vertex on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$? | \sqrt{95} | 1. **Assigning Coordinates**: Assume without loss of generality that vertex $A$ is on the $x$-axis, vertex $B$ is on the $y$-axis, and vertex $C$ is on the $z$-axis. Let the coordinates of $A$, $B$, and $C$ be $(a,0,0)$, $(0,b,0)$, and $(0,0,c)$ respectively.
2. **Using Triangle Side Lengths**: Given the side lengths ... | 0.9375 | 5,230.1875 | 5,032.733333 | 8,192 |
Let $S$ be the set of $10$-tuples of non-negative integers that have sum $2019$. For any tuple in $S$, if one of the numbers in the tuple is $\geq 9$, then we can subtract $9$ from it, and add $1$ to the remaining numbers in the tuple. Call thus one operation. If for $A,B\in S$ we can get from $A$ to $B$ in finitely ma... | 10^8 |
### Part 1:
We need to find the smallest integer \( k \) such that if the minimum number in \( A, B \in S \) are both \(\geq k\), then \( A \rightarrow B \) implies \( B \rightarrow A \).
We claim that the smallest integer \( k \) is \( 8 \).
**Proof:**
1. **\( k \leq 7 \) does not satisfy the condition:**
Con... | 0 | 8,149.4375 | -1 | 8,149.4375 |
Find all possible positive integers represented in decimal as $13 x y 45 z$, which are divisible by 792, where $x, y, z$ are unknown digits. | 1380456 | 0.5625 | 6,837.8125 | 5,784.555556 | 8,192 | |
A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a $2 / 3$ chance of catching each individual error still in the article. After 3 days,... | \frac{416}{729} | Consider the error that was introduced on day 1. The probability that the Moderator misses this error on all three checks is $1 / 3^{3}$, so the probability that this error gets removed is $1-\frac{1}{3^{3}}$. Similarly, the probability that the moderator misses the other two errors are $1-\frac{1}{3^{2}}$ and $1-\frac... | 0 | 7,867.75 | -1 | 7,867.75 |
Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([3, 5]\). Find \(\underbrace{f(f(\ldots f}_{2017}\left(\frac{7+\sqrt{15}}{2}\right)) \ldots)\). Round the answer to hundredths if necessary. | 1.56 | 0.0625 | 8,182.8125 | 8,045 | 8,192 | |
Consider a cube PQRSTUVW with a side length s. Let M and N be the midpoints of edges PU and RW, and let K be the midpoint of QT. Find the ratio of the area of triangle MNK to the area of one of the faces of the cube. | \frac{1}{4} | 0.25 | 7,060.5625 | 6,626.75 | 7,205.166667 | |
The shape of a bridge arch is a parabola. It is known that the width of the parabolic arch is 8 meters, and the area of the parabolic arch is 160 square meters. Then, the height of the parabolic arch is | 30 | 1 | 3,274.75 | 3,274.75 | -1 | |
What is the largest power of 2 by which the number \(10^{10} - 2^{10}\) is divisible? | 13 | 0.1875 | 6,978.4375 | 6,063 | 7,189.692308 | |
How many subsets $S$ of the set $\{1,2, \ldots, 10\}$ satisfy the property that, for all $i \in[1,9]$, either $i$ or $i+1$ (or both) is in $S$? | 144 | We do casework on the number of $i$ 's not in $S$. Notice that these $i$ 's that are not in $S$ cannot be consecutive, otherwise there exists an index $i$ such that both $i$ and $i+1$ are both not in $S$. Hence if there are $k i$ 's not in $S$, we want to arrange $k$ black balls and $10-k$ white balls such that no two ... | 0.25 | 7,628 | 6,331 | 8,060.333333 |
Given the lines $l_{1}$: $x+ay-a+2=0$ and $l_{2}$: $2ax+(a+3)y+a-5=0$.
$(1)$ When $a=1$, find the coordinates of the intersection point of lines $l_{1}$ and $l_{2}$.
$(2)$ If $l_{1}$ is parallel to $l_{2}$, find the value of $a$. | a = \frac{3}{2} | 0.3125 | 4,047.875 | 4,110.8 | 4,019.272727 | |
Given that $-1 - 4\sqrt{2}$ is a root of the equation \[x^3 + ax^2 + bx + 31 = 0\]and that $a$ and $b$ are rational numbers, compute $a.$ | 1 | 1 | 2,805.9375 | 2,805.9375 | -1 | |
There are 3 rods with several golden disks of different sizes placed on them. Initially, 5 disks are arranged on the leftmost rod (A) in descending order of size. According to the rule that only one disk can be moved at a time and a larger disk can never be placed on top of a smaller one, the goal is to move all 5 disk... | 31 | 0.8125 | 3,593.9375 | 2,532.846154 | 8,192 | |
Let $\mathbf{a}$ and $\mathbf{b}$ be two vectors such that
\[\|\mathbf{a} + \mathbf{b}\| = \|\mathbf{b}\|.\]Find the angle between the vectors $\mathbf{a} + 2 \mathbf{b}$ and $\mathbf{a},$ in degrees | 90^\circ | 0.9375 | 5,182.375 | 4,981.733333 | 8,192 | |
The digits $1,2,3,4,5,6$ are randomly chosen (without replacement) to form the three-digit numbers $M=\overline{A B C}$ and $N=\overline{D E F}$. For example, we could have $M=413$ and $N=256$. Find the expected value of $M \cdot N$. | 143745 | By linearity of expectation and symmetry, $$\mathbb{E}[M N]=\mathbb{E}[(100 A+10 B+C)(100 D+10 E+F)]=111^{2} \cdot \mathbb{E}[A D]$$ Since $$\mathbb{E}[A D]=\frac{(1+2+3+4+5+6)^{2}-\left(1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}\right)}{6 \cdot 5}=\frac{350}{30}$$ our answer is $111 \cdot 35 \cdot 37=111 \cdot 1295=143745$. | 0 | 8,192 | -1 | 8,192 |
If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 30 | 0 | 8,192 | -1 | 8,192 | |
1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996= | 0 | 1. **Group the terms**: We group the terms in sets of four as follows:
\[
(1-2-3+4) + (5-6-7+8) + (9-10-11+12) + \cdots + (1993-1994-1995+1996)
\]
2. **Calculate the sum of each group**:
- For the first group:
\[
1 - 2 - 3 + 4 = (1 + 4) - (2 + 3) = 5 - 5 = 0
\]
- For the second group:
... | 0.4375 | 6,598.9375 | 4,730.714286 | 8,052 |
We call a positive integer $t$ good if there is a sequence $a_{0}, a_{1}, \ldots$ of positive integers satisfying $a_{0}=15, a_{1}=t$, and $a_{n-1} a_{n+1}=\left(a_{n}-1\right)\left(a_{n}+1\right)$ for all positive integers $n$. Find the sum of all good numbers. | 296 | By the condition of the problem statement, we have $a_{n}^{2}-a_{n-1} a_{n+1}=1=a_{n-1}^{2}-a_{n-2} a_{n}$. This is equivalent to $\frac{a_{n-2}+a_{n}}{a_{n-1}}=\frac{a_{n-1}+a_{n+1}}{a_{n}}$. Let $k=\frac{a_{0}+a_{2}}{a_{1}}$. Then we have $\frac{a_{n-1}+a_{n+1}}{a_{n}}=\frac{a_{n-2}+a_{n}}{a_{n-1}}=\frac{a_{n-3}+a_{n... | 0 | 8,181.5 | -1 | 8,181.5 |
In a square, 20 points were marked and connected by non-intersecting segments with each other and with the vertices of the square, dividing the square into triangles. How many triangles were formed? | 42 | 0.875 | 4,864.8125 | 4,389.5 | 8,192 | |
In an isosceles triangle \(ABC\), the base \(AC\) is equal to \(x\), and the lateral side is equal to 12. On the ray \(AC\), point \(D\) is marked such that \(AD = 24\). From point \(D\), a perpendicular \(DE\) is dropped to the line \(AB\). Find \(x\) given that \(BE = 6\). | 18 | 0.0625 | 8,110 | 8,192 | 8,104.533333 | |
In a certain city, vehicle license plates are numbered consecutively from "10000" to "99999". How many license plates out of these 90,000 have the digit 9 appearing at least once and where the sum of the digits is a multiple of 9? | 4168 | 0.375 | 7,585.3125 | 6,584 | 8,186.1 | |
Eastbound traffic flows at 80 miles per hour and westbound traffic flows at 60 miles per hour. An eastbound driver observes 30 westbound vehicles in a 10-minute period. Calculate the number of westbound vehicles in a 150-mile section of the highway. | 193 | 0.0625 | 2,126.25 | 7,757 | 1,750.866667 | |
The ratio of the sums of the first \( n \) terms of two arithmetic sequences is \(\frac{9n+2}{n+7}\). Find the ratio of their 5th terms. | \frac{83}{16} | 0.625 | 6,173 | 5,041.5 | 8,058.833333 | |
For each ordered pair of real numbers $(x,y)$ satisfying \[\log_2(2x+y) = \log_4(x^2+xy+7y^2)\]there is a real number $K$ such that \[\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).\]Find the product of all possible values of $K$. | 189 | Using the logarithmic property $\log_{a^n}b^n = \log_{a}b$, we note that \[(2x+y)^2 = x^2+xy+7y^2\] That gives \[x^2+xy-2y^2=0\] upon simplification and division by $3$. Factoring $x^2+xy-2y^2=0$ by Simon's Favorite Factoring Trick gives \[(x+2y)(x-y)=0\] Then, \[x=y \text{ or }x=-2y\] From the second equation, \[9x^2+... | 0.8125 | 4,433.0625 | 3,962.461538 | 6,472.333333 |
When the height of a cylinder is doubled and its radius is increased by $200\%$, the cylinder's volume is multiplied by a factor of $X$. What is the value of $X$? | 18 | 1 | 1,147.5625 | 1,147.5625 | -1 | |
Given the sequence \(\left\{a_{n}\right\}\) with the general term
\[ a_{n} = n^{4} + 6n^{3} + 11n^{2} + 6n, \]
find the sum of the first 12 terms \( S_{12} \). | 104832 | 0.625 | 6,455.0625 | 5,412.9 | 8,192 | |
An urn contains $k$ balls labeled with $k$, for all $k = 1, 2, \ldots, 2016$. What is the minimum number of balls we must draw, without replacement and without looking at the balls, to ensure that we have 12 balls with the same number? | 22122 | 0.3125 | 4,601.1875 | 5,171.2 | 4,342.090909 | |
Given that $\operatorname{tg} \theta$ and $\operatorname{ctg} \theta$ are the real roots of the equation $2x^{2} - 2kx = 3 - k^{2}$, and $\alpha < \theta < \frac{5 \pi}{4}$, find the value of $\cos \theta - \sin \theta$. | -\sqrt{\frac{5 - 2\sqrt{5}}{5}} | 0 | 8,192 | -1 | 8,192 | |
A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a $512$ by $1$ strip of double thickness.... | 593 | We can keep track of the position of the square labeled 942 in each step. We use an $(x,y)$ coordinate system, so originally the 942 square is in the position $(942,1)$. In general, suppose that we've folded the strip into an array $r=2^k$ squares wide and $c=1024/r=2^{10-k}$ squares tall (so we've made $10-k$ folds). ... | 0 | 8,192 | -1 | 8,192 |
What is the integer formed by the rightmost two digits of the integer equal to \(4^{127} + 5^{129} + 7^{131}\)? | 52 | We start by looking for patterns in the rightmost two digits of powers of 4, powers of 5, and powers of 7. The first few powers of 5 are \(5^{1} = 5\), \(5^{2} = 25\), \(5^{3} = 125\), \(5^{4} = 625\), \(5^{5} = 3125\). It appears that, starting with \(5^{2}\), the rightmost two digits of powers of 5 are always 25. To ... | 1 | 4,518.5625 | 4,518.5625 | -1 |
If $p, q,$ and $r$ are three non-zero integers such that $p + q + r = 26$ and\[\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1,\] compute $pqr$.
| 576 | 0.5625 | 6,526.6875 | 5,231.444444 | 8,192 | |
A square with a side length of 100 cm is drawn on a board. Alexei intersected it with two lines parallel to one pair of its sides. After that, Danil intersected the square with two lines parallel to the other pair of sides. As a result, the square was divided into 9 rectangles, with the dimensions of the central rectan... | 2400 | 0.3125 | 7,244.1875 | 5,159 | 8,192 | |
Determine the product of the solutions of the equation $-21 = -x^2 + 4x$. | -21 | 1 | 1,788.4375 | 1,788.4375 | -1 | |
How many of the numbers in Grace's sequence, starting from 43 and each number being 4 less than the previous one, are positive? | 11 | We write out the numbers in the sequence until we obtain a negative number: $43,39,35,31,27,23,19,15,11,7,3,-1$. Since each number is 4 less than the number before it, then once a negative number is reached, every following number will be negative. Thus, Grace writes 11 positive numbers in the sequence. | 1 | 2,655.4375 | 2,655.4375 | -1 |
In triangle $ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\overline{BE}=\overline{ED}$. Point $F$ is on $AB$. Then, if $\overline{BF}=5$, $\overline{BA}$ equals: | 15 | 0.875 | 4,428.75 | 4,467.357143 | 4,158.5 | |
For how many integers $n$ with $1 \le n \le 2023$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]equal to zero, where $n$ needs to be an even multiple of $5$? | 202 | 0 | 8,096.625 | -1 | 8,096.625 | |
Find the area of rhombus $EFGH$ given that the radii of the circles circumscribed around triangles $EFG$ and $EGH$ are $15$ and $30$, respectively. | 60 | 0 | 7,820.5 | -1 | 7,820.5 | |
There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer "Yes" or "No" to the question: "Are there more knights on the island than liars?". They answered in turn such that everyone else could hear. Knights told the truth... | 1009 | 0 | 8,192 | -1 | 8,192 | |
Alice places a coin, heads up, on a table then turns off the light and leaves the room. Bill enters the room with 2 coins and flips them onto the table and leaves. Carl enters the room, in the dark, and removes a coin at random. Alice reenters the room, turns on the light and notices that both coins are heads. What is ... | 3/5 | 0.0625 | 7,974.3125 | 6,381 | 8,080.533333 | |
$908 \times 501 - [731 \times 1389 - (547 \times 236 + 842 \times 731 - 495 \times 361)] =$ | 5448 | 0.5 | 5,056.5 | 5,820.375 | 4,292.625 | |
There are 8 sprinters in the Olympic 100-meter finals. Three of the sprinters are Americans. The gold medal goes to first place, silver to second, and bronze to third. In how many ways can the medals be awarded if at most one American gets a medal? | 240 | 0.6875 | 5,041.1875 | 3,937.727273 | 7,468.8 | |
Let $m$ and $n$ satisfy $mn = 6$ and $m+n = 7$. Additionally, suppose $m^2 - n^2 = 13$. Find the value of $|m-n|$. | \frac{13}{7} | 0.5 | 7,322.375 | 6,857.875 | 7,786.875 | |
Define a sequence of convex polygons \( P_n \) as follows. \( P_0 \) is an equilateral triangle with side length 1. \( P_{n+1} \) is obtained from \( P_n \) by cutting off the corners one-third of the way along each side (for example, \( P_1 \) is a regular hexagon with side length \(\frac{1}{3}\)). Find \( \lim_{n \to... | \frac{\sqrt{3}}{7} | 0 | 8,192 | -1 | 8,192 | |
A regular decagon is formed by connecting three sequentially adjacent vertices of the decagon. Find the probability that all three sides of the triangle are also sides of the decagon. | \frac{1}{12} | 0.4375 | 6,428.75 | 5,030.142857 | 7,516.555556 | |
Given that the diagonals of a rhombus are always perpendicular bisectors of each other, what is the area of a rhombus with side length $\sqrt{113}$ units and diagonals that differ by 10 units? | 72 | 0 | 3,352.5 | -1 | 3,352.5 | |
The function $y=f(x)$ is an even function with the smallest positive period of 4, and when $x \in [-2, 0]$, $f(x) = 2x + 1$. If there exist $x_1, x_2, \ldots, x_n$ satisfying $0 \leq x_1 < x_2 < \ldots < x_n$, and $|f(x_1) - f(x_2)| + |f(x_2) - f(x_3)| + \ldots + |f(x_{n-1}) - f(x_n)| = 2016$, then the minimum value of... | 1513 | 0.5 | 7,414.0625 | 6,636.125 | 8,192 | |
Let \(x\) and \(y\) be positive real numbers such that
\[
\frac{1}{x + 1} + \frac{1}{y + 1} = \frac{1}{2}.
\]
Find the minimum value of \(x + 3y.\) | 4 + 4 \sqrt{3} | 0.625 | 7,464.125 | 7,027.4 | 8,192 | |
Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.
Hint
\[\color{red}\boxed{\boxed{\color{blue}\textbf{Use Vieta's Formulae!}}}\] | 420 | Because the coefficient of $x^2$ in both $p(x)$ and $q(x)$ is 0, the remaining root of $p(x)$ is $-(r+s)$, and the remaining root of $q(x)$ is $-(r+s+1)$. The coefficients of $x$ in $p(x)$ and $q(x)$ are both equal to $a$, and equating the two coefficients gives \[rs-(r+s)^2 = (r+4)(s-3)-(r+s+1)^2\]from which $s = \tfr... | 0.125 | 8,013 | 7,555.5 | 8,078.357143 |
(1) If the terminal side of angle $\theta$ passes through $P(-4t, 3t)$ ($t>0$), find the value of $2\sin\theta + \cos\theta$.
(2) Given that a point $P$ on the terminal side of angle $\alpha$ has coordinates $(x, -\sqrt{3})$ ($x\neq 0$), and $\cos\alpha = \frac{\sqrt{2}}{4}x$, find $\sin\alpha$ and $\tan\alpha$. | \frac{2}{5} | 0 | 5,351.875 | -1 | 5,351.875 | |
How many five-digit numbers are there that are divisible by 5 and do not contain repeating digits? | 5712 | 0.4375 | 6,119.875 | 4,997 | 6,993.222222 | |
Two distinct natural numbers end with 8 zeros and have exactly 90 divisors. Find their sum. | 700000000 | 0.25 | 7,865.125 | 6,884.5 | 8,192 | |
When each edge of a cube is increased by $50\%$, by what percent is the surface area of the cube increased? | 125\% | 1 | 1,694.1875 | 1,694.1875 | -1 | |
How many distinct four-digit numbers are divisible by 3 and have 23 as their last two digits? | 30 | 0.75 | 5,257.125 | 4,278.833333 | 8,192 | |
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer stan... | \frac{3}{16} | 1. **Calculate Rachel's running details:**
- Rachel completes a lap every 90 seconds.
- In 10 minutes (600 seconds), Rachel completes $\frac{600}{90} = 6\frac{2}{3}$ laps. This means she completes 6 full laps and is $\frac{2}{3}$ of a lap into her seventh lap.
- $\frac{2}{3}$ of a lap corresponds to $\frac{2}{... | 0.0625 | 7,828.3125 | 6,390 | 7,924.2 |
Zan has created this iterative rule for generating sequences of whole numbers:
1) If a number is 25 or less, double the number.
2) If a number is greater than 25, subtract 12 from it.
Let $F$ be the first number in a sequence generated by the rule above. $F$ is a "sweet number" if 16 is not a term in the sequence th... | 16 | 0 | 8,099.75 | -1 | 8,099.75 | |
What is the smallest possible sum of two consecutive integers whose product is greater than 420? | 43 | 0.6875 | 5,780.4375 | 4,746.090909 | 8,056 | |
Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.
a) How much time will Petya need to defi... | \frac{29}{120} | 0 | 5,828.25 | -1 | 5,828.25 |
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