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 |
|---|---|---|---|---|---|---|
In triangle $ABC$, angles $A$ and $B$ measure $60$ degrees and $45$ degrees, respectively. The bisector of angle $A$ intersects $\overline{BC}$ at $T$, and $AT=24$. The area of triangle $ABC$ can be written in the form $a+b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square ... | 291 | After chasing angles, $\angle ATC=75^{\circ}$ and $\angle TCA=75^{\circ}$, meaning $\triangle TAC$ is an isosceles triangle and $AC=24$.
Using law of sines on $\triangle ABC$, we can create the following equation:
$\frac{24}{\sin(\angle ABC)}$ $=$ $\frac{BC}{\sin(\angle BAC)}$
$\angle ABC=45^{\circ}$ and $\angle BAC=6... | 0.6875 | 7,201.9375 | 6,751.909091 | 8,192 |
A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\... | \dfrac{11}{850} | 1 | 3,453.1875 | 3,453.1875 | -1 | |
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.
For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$ $7 \t... | 83 | We start by understanding the function used by the machine. If $N$ is the input, the output $O$ is given by:
- If $N$ is odd, $O = 3N + 1$.
- If $N$ is even, $O = \frac{N}{2}$.
We need to find the inverse of this function to trace back from $O = 1$ to the original input $N$ after six steps. The inverse function can be... | 0.1875 | 7,876.4375 | 7,146.666667 | 8,044.846154 |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, perpendicular lines to the $x$-axis are drawn through points $F\_1$ and $F\_2$ intersecting the ellipse at four points to form a square, determine the eccentricity $e$ of the ellipse. | \frac{\sqrt{5} - 1}{2} | 0 | 4,041.5 | -1 | 4,041.5 | |
Let the number $9999\cdots 99$ be denoted by $N$ with $94$ nines. Then find the sum of the digits in the product $N\times 4444\cdots 44$. | 846 | 0.5 | 7,339.5625 | 6,487.125 | 8,192 | |
If two distinct members of the set $\{ 2, 4, 10, 12, 15, 20, 50 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 100? Express your answer as a common fraction. | \frac{1}{3} | 0.0625 | 8,134.25 | 7,268 | 8,192 | |
A ferry boat begins shuttling tourists to an island every hour starting at 10 AM, with its last trip starting at 4 PM. On the 10 AM trip, there were 100 tourists on the ferry, and on each successive trip, the number of tourists decreased by 2 from the previous trip. Calculate the total number of tourists transported to... | 658 | 0.6875 | 1,824.6875 | 1,964.818182 | 1,516.4 | |
Given $A=\{-3,-2,-1,0,1,2,3\}$, $a$ and $b$ are elements of $A$. How many cases are there where $|a| \lt |b|$? | 18 | 0.4375 | 7,063.3125 | 5,612.142857 | 8,192 | |
A circle on a plane divides the plane into 2 parts. A circle and a line can divide the plane into a maximum of 4 parts. A circle and 2 lines can divide the plane into a maximum of 8 parts. A circle and 5 lines can divide the plane into a maximum of how many parts? | 26 | 0 | 8,010.4375 | -1 | 8,010.4375 | |
A regular polygon has interior angles of 160 degrees, and each side is 4 units long. How many sides does the polygon have, and what is its perimeter? | 72 | 1 | 1,271.625 | 1,271.625 | -1 | |
On the Cartesian plane in which each unit is one foot, a dog is tied to a post on the point $(4,3)$ by a $10$ foot rope. What is the greatest distance the dog can be from the origin? | 15 | 0.875 | 4,074.625 | 3,486.428571 | 8,192 | |
Compute the exact value of the expression
\[|\pi - |\pi - 7||.\]Write your answer using only integers and $\pi,$ without any absolute value signs. | 7 - 2 \pi | 1 | 1,589 | 1,589 | -1 | |
I'm going to dinner at a large restaurant which my friend recommended, unaware that I am vegan and have both gluten and dairy allergies. Initially, there are 6 dishes that are vegan, which constitutes one-sixth of the entire menu. Unfortunately, 4 of those vegan dishes contain either gluten or dairy. How many dishes on... | \frac{1}{18} | 0 | 468.3125 | -1 | 468.3125 | |
In triangle $ABC$ , angle $A$ is twice angle $B$ , angle $C$ is obtuse , and the three side lengths $a, b, c$ are integers. Determine, with proof, the minimum possible perimeter . | \(\boxed{77}\) | Solution 1
[asy] import olympiad; pair A, B, C, D, extensionAC; real angleABC; path braceBC; A = (0, 0); B = (2, 0); D = (1, .5); angleABC = atan(.5); //y = 4x/3 and x+2y = 2 (sides AC and BC, respectively) intersect here: C = (6/11, 8/11); braceBC = brace(C, B, .1); label("$\mathsf{A}$", A, W); label("$\mathsf{B... | 0 | 8,192 | -1 | 8,192 |
If $f^{-1}(g(x))=x^4-4$ and $g$ has an inverse, find $g^{-1}(f(15))$. | \sqrt[4]{19} | 0.75 | 5,660.375 | 4,816.5 | 8,192 | |
Each square in a $3 \times 10$ grid is colored black or white. Let $N$ be the number of ways this can be done in such a way that no five squares in an 'X' configuration (as shown by the black squares below) are all white or all black. Determine $\sqrt{N}$. | 25636 | Note that we may label half of the cells in our board the number 0 and the other half 1, in such a way that squares labeled 0 are adjacent only to squares labeled 1 and vice versa. In other words, we make this labeling in a 'checkerboard' pattern. Since cells in an 'X' formation are all labeled with the same number, th... | 0 | 8,162.625 | -1 | 8,162.625 |
Matilda has a summer job delivering newspapers. She earns \$6.00 an hour plus \$0.25 per newspaper delivered. Matilda delivers 30 newspapers per hour. How much money will she earn during a 3-hour shift? | \$40.50 | During a 3-hour shift, Matilda will deliver \( 3 \times 30=90 \) newspapers. Therefore, she earns a total of \( 3 \times \$6.00+90 \times \$0.25=\$18.00+\$22.50=\$40.50 \) during her 3-hour shift. | 0.625 | 416.0625 | 420.6 | 408.5 |
If a die is rolled, event \( A = \{1, 2, 3\} \) consists of rolling one of the faces 1, 2, or 3. Similarly, event \( B = \{1, 2, 4\} \) consists of rolling one of the faces 1, 2, or 4.
The die is rolled 10 times. It is known that event \( A \) occurred exactly 6 times.
a) Find the probability that under this conditio... | \frac{16}{3} | 0.0625 | 6,458.9375 | 3,710 | 6,642.2 | |
Let \( p \) and \( q \) be positive integers such that \( \frac{5}{8} < \frac{p}{q} < \frac{7}{8} \). What is the smallest value of \( p \) so that \( p + q = 2005 \)? | 772 | 0.8125 | 5,847.125 | 5,306 | 8,192 | |
Sasha has $\$3.20$ in U.S. coins. She has the same number of quarters and nickels. What is the greatest number of quarters she could have? | 10 | 0.9375 | 5,250.6875 | 5,054.6 | 8,192 | |
Vasya has three cans of paint of different colors. In how many different ways can he paint a fence of 10 boards such that any two adjacent boards are of different colors, and all three colors are used? | 1530 | 0.125 | 7,763.6875 | 5,930.5 | 8,025.571429 | |
A and B plays a game on a pyramid whose base is a $2016$ -gon. In each turn, a player colors a side (which was not colored before) of the pyramid using one of the $k$ colors such that none of the sides with a common vertex have the same color. If A starts the game, find the minimal value of $k$ for which $B$ can... | 2016 | 0 | 7,771.8125 | -1 | 7,771.8125 | |
The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers? | 4 | 1. **Define the variables**: Let the two nonzero real numbers be $x$ and $y$.
2. **Translate the given condition into an equation**: We are given that the sum of $x$ and $y$ is $4$ times their product. This can be written as:
\[
x + y = 4xy
\]
3. **Manipulate the equation to find the sum of the reciprocals**... | 1 | 2,282.5 | 2,282.5 | -1 |
Given two lines $$l_{1}: \sqrt {3}x+y-1=0$$ and $$l_{2}: ax+y=1$$, and $l_{1}$ is perpendicular to $l_{2}$, then the slope angle of $l_{1}$ is \_\_\_\_\_\_, and the distance from the origin to $l_{2}$ is \_\_\_\_\_\_. | \frac { \sqrt {3}}{2} | 0 | 2,855.375 | -1 | 2,855.375 | |
What is the sum of the digits of the square of the number 22222? | 46 | 0.5625 | 6,606 | 5,372.444444 | 8,192 | |
How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits? | 90 | 0.9375 | 3,148.125 | 3,091.866667 | 3,992 | |
The increasing sequence of positive integers $b_1, b_2, b_3, \ldots$ follows the rule:
\[ b_{n + 2} = b_{n + 1} + b_n \]
for all $n \geq 1$. If $b_5 = 55$, find $b_6$. | 84 | 0 | 7,989.875 | -1 | 7,989.875 | |
Given the function $f(x) = x^2 - 2\cos{\theta}x + 1$, where $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{2}\right]$.
(1) When $\theta = \frac{\pi}{3}$, find the maximum and minimum values of $f(x)$.
(2) If $f(x)$ is a monotonous function on $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{2}\right]$ and $\theta \in [0, 2\pi)... | \frac{16 + 4\sqrt{11}}{5} | 0 | 7,689.0625 | -1 | 7,689.0625 | |
You have $5$ red shirts, $5$ green shirts, $6$ pairs of pants, $8$ green hats and $8$ red hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats? | 480 | 0.6875 | 5,304.6875 | 3,992.272727 | 8,192 | |
Given an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, its left and right foci are $F_1$ and $F_2$ respectively. Point $P(1, \frac{\sqrt{2}}{2})$ is on the ellipse, and $|PF_1| + |PF_2| = 2\sqrt{2}$.
$(1)$ Find the standard equation of ellipse $C$;
$(2)$ A line $l$ passing through $F_2$ intersects ... | \frac{\sqrt{2}}{2} | 0 | 8,192 | -1 | 8,192 | |
Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\frac{1}{2}$-mile stretch in a school zone where his average speed is $20... | 3 \frac{3}{4} | To determine which route is quicker and by how many minutes, we need to calculate the time taken for each route and then compare them.
#### Route A:
- **Distance:** $6$ miles
- **Speed:** $30$ miles per hour
Using the formula for time, $t = \frac{d}{s}$, where $d$ is distance and $s$ is speed:
\[ t_A = \frac{6 \text{... | 0 | 3,107.1875 | -1 | 3,107.1875 |
At Clover View Junior High, one half of the students go home on the school bus. One fourth go home by automobile. One tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home? | \frac{3}{20} | To find the fractional part of the students who walk home, we need to subtract the fractions of students who use other means of transportation from the whole.
1. **Identify the fractions for each mode of transportation:**
- School bus: $\frac{1}{2}$
- Automobile: $\frac{1}{4}$
- Bicycle: $\frac{1}{10}$
2. **... | 1 | 1,716.5 | 1,716.5 | -1 |
$ABC$ is a triangle with $AB = 33$ , $AC = 21$ and $BC = m$ , an integer. There are points $D$ , $E$ on the sides $AB$ , $AC$ respectively such that $AD = DE = EC = n$ , an integer. Find $m$ .
| 30 | 0.125 | 8,041.25 | 6,986 | 8,192 | |
Given that point $P$ moves on the ellipse $\frac{x^{2}}{4}+y^{2}=1$, find the minimum distance from point $P$ to line $l$: $x+y-2\sqrt{5}=0$. | \frac{\sqrt{10}}{2} | 0 | 6,569.875 | -1 | 6,569.875 | |
Calculate: (1) $(\sqrt[3]{1.5} \times \sqrt[6]{12})^{2} + 8{1}^{0.75} - (-\frac{1}{4})^{-2} - 5 \times 0.12{5}^{0}$; (2) $\lg 25 + \lg 2 \cdot \lg 50 + (\lg 2)^{2} - e^{3\ln 2}$. | -6 | 0.75 | 6,548.9375 | 6,001.25 | 8,192 | |
A cube with an edge length of 1 and its circumscribed sphere intersect with a plane to form a cross section that is a circle and an inscribed equilateral triangle. What is the distance from the center of the sphere to the plane of the cross section? | $\frac{\sqrt{3}}{6}$ | 0 | 7,353.5625 | -1 | 7,353.5625 | |
In a gymnastics competition, the position where Qiqi stands is the 6th from the front, the 12th from the back, the 15th from the left, and the 11th from the right. If the number of people in each row is the same and the number of people in each column is also the same, how many people are there in total participating i... | 425 | 0.5625 | 2,378.3125 | 1,295.444444 | 3,770.571429 | |
Given that $a \in \{0,1,2\}$ and $b \in \{-1,1,3,5\}$, the probability that the function $f(x) = ax^2 - 2bx$ is an increasing function in the interval $(1, +\infty)$ is $(\quad\quad)$. | \frac{1}{3} | 0.25 | 5,874.3125 | 5,382 | 6,038.416667 | |
Consider two fictional states: Alpha and Beta. Alpha issues license plates with a format of two letters followed by four numbers, and then ending with one letter (LLNNNNL). Beta issues plates with three letters followed by three numbers and lastly one letter (LLLNNNL). Assume all 10 digits and 26 letters are equally li... | 281216000 | 0.625 | 5,926 | 5,763.1 | 6,197.5 | |
Given a trapezoid \(ABCD\) with bases \(AB\) and \(CD\), and angles \(\angle C = 30^\circ\) and \(\angle D = 80^\circ\). Find \(\angle ACB\), given that \(DB\) is the bisector of \(\angle D\). | 10 | 0 | 8,192 | -1 | 8,192 | |
Given an angle \(\theta\), consider the polynomial \(P(x)=\sin(\theta)x^{2}+\cos(\theta)x+\tan(\theta)x+1\). Given that \(P\) only has one real root, find all possible values of \(\sin(\theta)\). | 0, \frac{\sqrt{5}-1}{2} | Note that if \(\sin(\theta)=0\), then the polynomial has 1 root. Now assume this is not the case then the polynomial is a quadratic in \(x\). Factor the polynomial as \((\tan(\theta)x+1)(x+\sec(\theta))\). Then the condition is equivalent to \(\sec(\theta)=\frac{1}{\tan(\theta)}\), which is equivalent to \(\sin(\theta)... | 0 | 6,879.6875 | -1 | 6,879.6875 |
Six people stand in a row, with exactly two people between A and B. Calculate the number of different ways for them to stand. | 144 | 0.375 | 6,604.75 | 5,414.5 | 7,318.9 | |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 0.3125 | 6,710.1875 | 4,961.8 | 7,504.909091 | |
Each of the four digits of the integer 2024 is even. How many integers between 1000 and 9999, inclusive, have the property that all four of their digits are even? | 500 | The integers between 1000 and 9999, inclusive, are all four-digit positive integers of the form $abcd$. We want each of $a, b, c$, and $d$ to be even. There are 4 choices for $a$, namely $2, 4, 6, 8$. ($a$ cannot equal 0.) There are 5 choices for each of $b, c$ and $d$, namely $0, 2, 4, 6, 8$. The choice of each digit ... | 1 | 1,758.625 | 1,758.625 | -1 |
In the center of a circular field, there is a geologist's cabin. From it extend 6 straight roads, dividing the field into 6 equal sectors. Two geologists start a journey from their cabin at a speed of 4 km/h each on a randomly chosen road. Determine the probability that the distance between them will be at least 6 km a... | 0.5 | 0.0625 | 7,271.125 | 4,946 | 7,426.133333 | |
In the given figure, hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$? | $12$ | 1. **Determine the length of $\overline{FE}$ and $\overline{BC}$:**
Since $FEHG$ is a square with area $32$, we have:
\[
\overline{FE}^2 = 32 \implies \overline{FE} = \sqrt{32} = 4\sqrt{2}.
\]
Given that $\overline{FE} = \overline{BC}$, it follows that:
\[
\overline{BC} = 4\sqrt{2}.
\]
2. **Det... | 0 | 7,981.375 | -1 | 7,981.375 |
Find the remainder when $x^4 + 1$ is divided by $x^2 - 3x + 5.$ | -3x - 19 | 0.8125 | 5,509.125 | 4,890 | 8,192 | |
A cube has side length 1. Find the product of the lengths of the diagonals of this cube (a diagonal is a line between two vertices that is not an edge). | 576 | There are 12 diagonals that go along a face and 4 that go through the center of the cube, so the answer is $\sqrt{2}^{12} \cdot \sqrt{3}^{4}=576$. | 0.75 | 4,249.625 | 4,252.25 | 4,241.75 |
Simplify:<br/>$(1)(-\frac{1}{2}+\frac{2}{3}-\frac{1}{4})÷(-\frac{1}{24})$;<br/>$(2)3\frac{1}{2}×(-\frac{5}{7})-(-\frac{5}{7})×2\frac{1}{2}-\frac{5}{7}×(-\frac{1}{2})$. | -\frac{5}{14} | 0.75 | 1,669.5625 | 1,530.333333 | 2,087.25 | |
Kacey is handing out candy for Halloween. She has only $15$ candies left when a ghost, a goblin, and a vampire arrive at her door. She wants to give each trick-or-treater at least one candy, but she does not want to give any two the same number of candies. How many ways can she distribute all $15$ identical candies... | 72 | 0.375 | 7,496.3125 | 6,568.833333 | 8,052.8 | |
Quadrilateral $ABCD$ is inscribed in a circle, $M$ is the point of intersection of its diagonals, $O_1$ and $O_2$ are the centers of the inscribed circles of triangles $ABM$ and $CMD$ respectively, $K$ is the midpoint of the arc $AD$ that does not contain points $B$ and $C$, $\angle O_1 K O_2 = 60^{\circ}$, $K O_1 = 10... | 10 | 0.5625 | 7,330.5625 | 6,692.888889 | 8,150.428571 | |
Given the line $l: y = kx + m$ intersects the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ at points $A$ and $P$, and intersects the $x$-axis and $y$-axis at points $N$ and $M$ respectively, with $PM = MN$. Point $Q$ is the reflection of point $P$ across the $x$-axis, and the extension of line $QM$ in... | \frac{1}{2} | 0.0625 | 8,034.25 | 5,668 | 8,192 | |
Use the method of random simulation to estimate the probability that it will rain on exactly two of the three days. Using a calculator, generate random integer values between 0 and 9, where 1, 2, 3, and 4 represent raining days and 5, 6, 7, 8, 9, and 0 signify non-raining days. Then, group every three random numbers to... | 0.25 | 0.0625 | 4,443.9375 | 2,862 | 4,549.4 | |
The shaded region shown consists of 11 unit squares and rests along the $x$-axis and the $y$-axis. The shaded region is rotated about the $x$-axis to form a solid. In cubic units, what is the volume of the resulting solid? Express your answer in simplest form in terms of $\pi$.
[asy]
defaultpen(linewidth(0.7));
fill((... | 37\pi | 0.625 | 4,685.1875 | 3,268.5 | 7,046.333333 | |
A box contains a collection of triangular and square tiles. There are $25$ tiles in the box, containing $84$ edges total. How many square tiles are there in the box? | 9 | 1. **Define Variables:**
Let $a$ be the number of triangular tiles and $b$ be the number of square tiles in the box.
2. **Set Up Equations:**
- Each triangle has 3 edges, and each square has 4 edges.
- The total number of tiles is given by the equation:
\[
a + b = 25
\]
- The total number of... | 1 | 1,441.4375 | 1,441.4375 | -1 |
A large rectangle measures 15 units by 20 units. One-quarter of this rectangle is shaded. If half of this quarter rectangle is shaded, what fraction of the large rectangle is shaded?
A) $\frac{1}{24}$
B) $\frac{1}{12}$
C) $\frac{1}{10}$
D) $\frac{1}{8}$
E) $\frac{1}{6}$ | \frac{1}{8} | 0.8125 | 804.625 | 506.615385 | 2,096 | |
On the Island of Misfortune, there are knights, who always tell the truth, and liars, who always lie. One day, $n$ islanders gathered in a room.
The first person said: "Exactly 1 percent of the people present in this room are liars."
The second person said: "Exactly 2 percent of the people present in this room are li... | 100 | 0.0625 | 8,155.0625 | 7,601 | 8,192 | |
The three-digit integer $63\underline{\hphantom{0}}$ is a multiple of 3. What is the greatest possible difference between two of the possibilities for the units digit? | 9 | 1 | 1,769.875 | 1,769.875 | -1 | |
What is the value of \[\frac{x^1\cdot x^2\cdot x^3\cdots x^{15}}{x^2\cdot x^4 \cdot x^6 \cdots x^{20}}\]if $x=2$? | 1024 | 0.875 | 3,707.8125 | 3,067.214286 | 8,192 | |
I have five different pairs of socks. Every day for five days, I pick two socks at random without replacement to wear for the day. Find the probability that I wear matching socks on both the third day and the fifth day. | \frac{1}{63} | I get a matching pair on the third day with probability $\frac{1}{9}$ because there is a $\frac{1}{9}$ probability of the second sock matching the first. Given that I already removed a matching pair of the third day, I get a matching pair on the fifth day with probability $\frac{1}{7}$. We multiply these probabilities ... | 0.125 | 8,143.5625 | 8,047 | 8,157.357143 |
Snow White has a row of 101 plaster dwarfs in her garden, arranged by weight from heaviest to lightest, with the weight difference between each pair of adjacent dwarfs being the same. Once, Snow White weighed the dwarfs and discovered that the first, heaviest dwarf weighs exactly $5 \mathrm{~kg}$. Snow White was most s... | 2.5 | 0.75 | 4,525.125 | 4,082.166667 | 5,854 | |
Given $|\vec{a}|=1$, $|\vec{b}|= \sqrt{2}$, and $\vec{a} \perp (\vec{a} - \vec{b})$, find the angle between the vectors $\vec{a}$ and $\vec{b}$. | \frac{\pi}{4} | 0.5 | 2,267.75 | 2,196.25 | 2,339.25 | |
A box contains 5 black ties, 7 gold ties, and 8 pink ties. What is the probability that Stephen randomly chooses a pink tie? | \frac{2}{5} | There are \(5+7+8=20\) ties in the box, 8 of which are pink. When Stephen removes a tie at random, the probability of choosing a pink tie is \(\frac{8}{20}\) which is equivalent to \(\frac{2}{5}\). | 1 | 493.1875 | 493.1875 | -1 |
How many rectangles can be formed where each of the four vertices are points on a 4x4 grid with points spaced evenly along the grid lines? | 36 | 0.1875 | 5,991.375 | 5,684.333333 | 6,062.230769 | |
Given that the approximate ratio of the three cases drawn, DD, Dd, dd, is 1:2:1, calculate the probability of drawing dd when two students who have drawn cards are selected, and one card is drawn from each of these two students. | \frac{1}{4} | 0.0625 | 5,667.5 | 7,130 | 5,570 | |
Given $\sin \alpha + \cos \alpha = - \frac{\sqrt{5}}{2}$, and $\frac{5\pi}{4} < \alpha < \frac{3\pi}{2}$, calculate the value of $\cos \alpha - \sin \alpha$. | \frac{\sqrt{3}}{2} | 0 | 6,809.8125 | -1 | 6,809.8125 | |
A company plans to advertise on TV stations A and B for a total time of no more than 300 minutes in 2011, with an advertising budget of no more than 90,000 yuan. The advertising rates for TV stations A and B are 500 yuan/minute and 200 yuan/minute, respectively. Assuming that for every minute of advertising done by TV ... | 70 | 0.5 | 5,198.1875 | 5,012.75 | 5,383.625 | |
Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible? | 5 | 0.375 | 7,961.625 | 7,577.666667 | 8,192 | |
What is the value of $1^{3}+2^{3}+3^{3}+4^{3}$? | 10^{2} | Expanding and simplifying, $1^{3}+2^{3}+3^{3}+4^{3}=1 \times 1 \times 1+2 \times 2 \times 2+3 \times 3 \times 3+4 \times 4 \times 4=1+8+27+64=100$. Since $100=10^{2}$, then $1^{3}+2^{3}+3^{3}+4^{3}=10^{2}$. | 0 | 605.125 | -1 | 605.125 |
Find the smallest natural number for which there exist that many natural numbers such that the sum of the squares of their squares is equal to $ 1998. $ *Gheorghe Iurea* | 15 | 0 | 7,811.875 | -1 | 7,811.875 | |
A $9\times 1$ board is completely covered by $m\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $9\times 1$ board in which all three colors are used at leas... | 838 | 0 | 8,053.625 | -1 | 8,053.625 | |
Given the sequence $\{a_{n}\}$ satisfies $a_{1}=1$, $({{a}\_{n+1}}-{{a}\_{n}}={{(-1)}^{n+1}}\dfrac{1}{n(n+2)})$, find the sum of the first 40 terms of the sequence $\{(-1)^{n}a_{n}\}$. | \frac{20}{41} | 0 | 8,192 | -1 | 8,192 | |
On a grid, there are three locations $A, B, C$, with each small square having a side length of 100 meters. If roads are to be built along the grid lines to connect the three locations, what is the minimum total length of the roads in meters? | 1000 | 0 | 7,333.5625 | -1 | 7,333.5625 | |
Find the largest possible value of $k$ for which $3^{13}$ is expressible as the sum of $k$ consecutive positive integers. | 1458 | 0.0625 | 8,184.8125 | 8,077 | 8,192 | |
The integers \(1,2,3,4,5,6,7,8,9,10\) are written on a blackboard. Each day, a teacher chooses one of the integers uniformly at random and decreases it by 1. Let \(X\) be the expected value of the number of days which elapse before there are no longer positive integers on the board. Estimate \(X\). An estimate of \(E\)... | 120.75280458176904 | Answer: 120.75280458176904 | 0 | 8,192 | -1 | 8,192 |
Let $(b_1, b_2, b_3, \ldots, b_{10})$ be a permutation of $(1, 2, 3, \ldots, 10)$ such that $b_1 > b_2 > b_3 > b_4 > b_5$ and $b_5 < b_6 < b_7 < b_8 < b_9 < b_{10}$. An example of such a permutation is $(5, 4, 3, 2, 1, 6, 7, 8, 9, 10)$. Find the number of such permutations. | 126 | 0.5625 | 6,586.5 | 5,337.777778 | 8,192 | |
The points $B(1, 1)$, $I(2, 4)$ and $G(5, 1)$ are plotted in the standard rectangular coordinate system to form triangle $BIG$. Triangle $BIG$ is translated five units to the left and two units upward to triangle $B'I'G'$, in such a way that $B'$ is the image of $B$, $I'$ is the image of $I$, and $G'$ is the image of $... | (-2,3) | 0.9375 | 2,141 | 1,737.6 | 8,192 | |
A teacher wrote a sequence of consecutive odd numbers starting from 1 on the blackboard: $1, 3, 5, 7, 9, 11, \cdots$ After writing, the teacher erased two numbers, dividing the sequence into three segments. If the sums of the first two segments are 961 and 1001 respectively, what is the sum of the two erased odd number... | 154 | 0 | 8,192 | -1 | 8,192 | |
Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him 1 Joule of energy to hop one step north or one step south, and 1 Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with 100 Joules of energy, and hops till he falls asleep with 0 energy.... | 10201 | It is easy to see that the coordinates of the frog's final position must have the same parity. Suppose that the frog went to sleep at $(x, y)$. Then, we have that $-100 \leq y \leq 100$ and $|x| \leq 100-|y|$, so $x$ can take on the values $-100+|y|,-98+|y|, \ldots, 100-|y|$. There are $101-|y|$ such values, so the tot... | 0 | 7,508.375 | -1 | 7,508.375 |
Given that the math test scores, X, follow a normal distribution with a mean of 100 and a standard deviation of δ, and the number of students scoring between 80 and 120 points accounted for about 3/4 of the total 1600 students, calculate the number of students who scored no less than 80 points in this final exam. | 1400 | 0.625 | 6,321.625 | 5,315.2 | 7,999 | |
Find the number of solutions in positive integers $(k ; a_{1}, a_{2}, \ldots, a_{k} ; b_{1}, b_{2}, \ldots, b_{k})$ to the equation $$a_{1}(b_{1})+a_{2}(b_{1}+b_{2})+\cdots+a_{k}(b_{1}+b_{2}+\cdots+b_{k})=7$$ | 15 | Let $k, a_{1}, \ldots, a_{k}, b_{1}, \ldots, b_{k}$ be a solution. Then $b_{1}, b_{1}+b_{2}, \ldots, b_{1}+\cdots+b_{k}$ is just some increasing sequence of positive integers. Considering the $a_{i}$ as multiplicities, the $a_{i}$ 's and $b_{i}$ 's uniquely determine a partition of 7. Likewise, we can determine $a_{i}$... | 0 | 8,074 | -1 | 8,074 |
A square has sides of length 3 units. A second square is formed having sides that are $120\%$ longer than the sides of the first square. This process is continued sequentially to create a total of five squares. What will be the percent increase in the perimeter from the first square to the fifth square? Express your an... | 107.4\% | 0.5625 | 5,678.8125 | 3,724.111111 | 8,192 | |
Given that circle $\odot M$ passes through the point $(1,0)$ and is tangent to the line $x=-1$, $S$ is a moving point on the trajectory of the center $M$ of the circle, and $T$ is a moving point on the line $x+y+4=0$. Find the minimum value of $|ST|$. | \frac{3\sqrt{2}}{2} | 0 | 6,853.6875 | -1 | 6,853.6875 | |
Let $S$ be the sum of all the real coefficients of the expansion of $(1+i x)^{2009}$. What is $\log _{2}(S)$ ? | 1004 | The sum of all the coefficients is $(1+i)^{2009}$, and the sum of the real coefficients is the real part of this, which is $\frac{1}{2}((1+i)^{2009}+(1-i)^{2009})=2^{1004}$. Thus $\log _{2}(S)=1004$. | 0.6875 | 5,718.125 | 4,593.636364 | 8,192 |
Let $\mathbf{P}$ be the matrix for projecting onto the vector $\begin{pmatrix} -3 \\ -2 \end{pmatrix}.$ Find $\mathbf{P}^{-1}.$
If the inverse does not exist, then enter the zero matrix. | \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} | 0.9375 | 1,836.625 | 1,812.466667 | 2,199 | |
A certain machine has a display showing an integer $x$, and two buttons, $\mathrm{A}$ and $\mathrm{B}$. When button $\mathrm{A}$ is pressed, the number $x$ on the display is replaced by $2x + 1$. When button $\mathrm{B}$ is pressed, the number $x$ on the display is replaced by $3x - 1$. What is the largest two-digit nu... | 95 | 0.125 | 8,086.375 | 7,347 | 8,192 | |
A cube has edge length 4 m. One end of a rope of length 5 m is anchored to the centre of the top face of the cube. What is the integer formed by the rightmost two digits of the integer closest to 100 times the area of the surface of the cube that can be reached by the other end of the rope? | 81 | The top face of the cube is a square, which we label $ABCD$, and we call its centre $O$. Since the cube has edge length 4, then the side length of square $ABCD$ is 4. This means that $O$ is a perpendicular distance of 2 from each of the sides of square $ABCD$, and thus is a distance of $\sqrt{2^{2}+2^{2}}=\sqrt{8}$ fro... | 0 | 7,988.9375 | -1 | 7,988.9375 |
Simplify the following expressions:
(1) $\sin^{4}{\alpha} + \tan^{2}{\alpha} \cdot \cos^{4}{\alpha} + \cos^{2}{\alpha}$
(2) $\frac{\cos{(180^{\circ} + \alpha)} \cdot \sin{(\alpha + 360^{\circ})}}{\sin{(-\alpha - 180^{\circ})} \cdot \cos{(-180^{\circ} - \alpha)}}$ | -1 | 0 | 4,506 | -1 | 4,506 | |
Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is ... | 12 | 1. **Understanding the Problem:**
We are given a line of balls consisting of one red ball and $N$ green balls, where $N$ is a multiple of 5. We need to find the probability $P(N)$ that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball.
2. **Initial Observations:**
- For $N=5$, all gr... | 0.1875 | 7,685.25 | 5,489.333333 | 8,192 |
The ratio $AC:CB$ is $3:4$, in $\triangle ABC$. The external angle bisector of $\angle C$ intersects the extension of $BA$ at $P$, where $A$ is between $P$ and $B$. Find the ratio $PA:AB$. | 3:1 | 0.8125 | 4,555.5625 | 4,112.538462 | 6,475.333333 | |
John has 12 marbles of different colors, including one red, one green, and one blue marble. In how many ways can he choose 4 marbles, if exactly one of the chosen marbles is red, green, or blue? | 252 | 1 | 3,055.3125 | 3,055.3125 | -1 | |
The function $f(x)=\frac{1}{3}x^{3}-ax^{2}+4$, and $x=2$ is a local minimum point of the function $f(x)$.
(1) Find the value of the real number $a$;
(2) Find the maximum and minimum values of $f(x)$ on the interval $[-1,3]$. | \frac{8}{3} | 0.9375 | 3,631.6875 | 3,327.666667 | 8,192 | |
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. The area of the triangle is S, and it is given that 2$\sqrt {3}$S - $\overrightarrow {AB}$•$\overrightarrow {AC}$ = 0, and c = 2.
(I) Find the measure of angle A.
(II) If a² + b² - c² = $\frac {6}{5}$ab, find the value of b. | \frac{3 + 4\sqrt{3}}{4} | 0 | 5,052.125 | -1 | 5,052.125 | |
Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{12}$ be the 12 zeroes of the polynomial $z^{12} - 2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $iz_j$. Find the maximum possible value of the real part of
\[\sum_{j = 1}^{12} w_j.\] | 16 + 16 \sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
In a row, there are 99 people - knights and liars (knights always tell the truth, and liars always lie). Each of them said one of two phrases: "To my left, there are twice as many knights as liars" or "To my left, there are as many knights as liars." In reality, there were more knights than liars, and more than 50 peop... | 49 | 0 | 8,192 | -1 | 8,192 | |
An octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 108 centimeters. What is the number of square centimeters in the area of the octagon? | 567 | 0.0625 | 7,983.4375 | 4,855 | 8,192 | |
Given 12 points in a diagram, calculate the number of groups of 3 points that can be formed to create a triangle. | 200 | 0 | 1,290 | -1 | 1,290 | |
Let $n$ be an odd integer with exactly 12 positive divisors. Find the number of positive divisors of $27n^3$. | 256 | 0 | 8,192 | -1 | 8,192 | |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} & x=3\cos \alpha \\ & y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of line $l$ is $\r... | \dfrac {18 \sqrt {2}}{5} | 0 | 6,298.6875 | -1 | 6,298.6875 | |
$18 \cdot 92$ A square $ABCD$ has side lengths of 1. Points $E$ and $F$ lie on sides $AB$ and $AD$ respectively, such that $AE = AF$. If the quadrilateral $CDFE$ has the maximum area, what is the maximum area? | $\frac{5}{8}$ | 0 | 4,984.3125 | -1 | 4,984.3125 |
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