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 |
|---|---|---|---|---|---|---|
Given a set of data arranged in ascending order, which are -1, 0, 4, x, 7, 14, and the median is 5, find the variance of this set of data. | \frac{74}{3} | 0.8125 | 3,533.75 | 2,458.769231 | 8,192 | |
Find $2 \cdot 5^{-1} + 8 \cdot 11^{-1} \pmod{56}$.
Express your answer as an integer from $0$ to $55$, inclusive. | 50 | 1 | 3,376.6875 | 3,376.6875 | -1 | |
There are 8 participants in a performance, with a total of $m$ shows. In each show, 4 people perform on stage. It is required that any two of the 8 people perform together the same number of times. Find the minimum value of $m$. | 14 | 0.75 | 5,186.375 | 4,184.5 | 8,192 | |
Among all victims of zombie bites, 10% are prescribed the experimental drug Undetenin to treat them. Overall, 4% of the human population suffer an adverse reaction to Undetenin. Out of all the patients being treated with Undetenin, 2% suffer an adverse reaction to the drug. What is the probability that a patient allerg... | 0.05 | 0.75 | 4,945.5625 | 3,863.416667 | 8,192 | |
$ABC$ is triangle. $l_1$ - line passes through $A$ and parallel to $BC$ , $l_2$ - line passes through $C$ and parallel to $AB$ . Bisector of $\angle B$ intersect $l_1$ and $l_2$ at $X,Y$ . $XY=AC$ . What value can take $\angle A- \angle C$ ? | 60 | 0 | 8,192 | -1 | 8,192 | |
Calculate the value of the function $f(x)=3x^{6}-2x^{5}+x^{3}+1$ at $x=2$ using the Horner's method (also known as the Qin Jiushao algorithm) to determine the value of $v_{4}$. | 34 | 0.6875 | 3,602.5625 | 3,243.363636 | 4,392.8 | |
For how many of the given drawings can the six dots be labelled to represent the links between suspects? | 2 | Two of the five drawings can be labelled to represent the given data. | 0 | 4,400.875 | -1 | 4,400.875 |
Consider a unit square $ABCD$ whose bottom left vertex is at the origin. A circle $\omega$ with radius $\frac{1}{3}$ is inscribed such that it touches the square's bottom side at point $M$. If $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M$, where $A$ is at the top left corner of the square, find ... | \frac{1}{3} | 0 | 6,457.1875 | -1 | 6,457.1875 | |
A right triangle with legs $10$ and $8$ is similar to another right triangle with corresponding legs $x$ and $5$, respectively. What is the value of $x$? Express your answer as a decimal to the nearest hundredth. | 6.25 | 0.8125 | 4,222.3125 | 3,306.230769 | 8,192 | |
What is the inverse of $f(x)=4-5x$? | \frac{4-x}{5} | 0.5625 | 1,980.0625 | 1,790.666667 | 2,223.571429 | |
Let $\mathbf{v}_0$ be a vector. The vector $\mathbf{v}_0$ is projected onto $\begin{pmatrix} 3 \\ 1 \end{pmatrix},$ resulting in the vector $\mathbf{v}_1.$ The vector $\mathbf{v}_1$ is then projected onto $\begin{pmatrix} 1 \\ 1 \end{pmatrix},$ resulting in the vector $\mathbf{v}_2.$ Find the matrix that takes $\mat... | \begin{pmatrix} \frac{3}{5} & \frac{1}{5} \\ \frac{3}{5} & \frac{1}{5} \end{pmatrix} | 0.5625 | 6,839.8125 | 6,032.888889 | 7,877.285714 | |
Given vectors $\overrightarrow {a}, \overrightarrow {b}$ that satisfy $\overrightarrow {a}\cdot ( \overrightarrow {a}+ \overrightarrow {b})=5$, and $|\overrightarrow {a}|=2$, $|\overrightarrow {b}|=1$, find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$. | \frac{\pi}{3} | 0 | 1,529.0625 | -1 | 1,529.0625 | |
In the diagram, the area of triangle $ABC$ is 36 square units. What is the area of triangle $BCD$ if the length of segment $CD$ is 39 units?
[asy]
draw((0,0)--(39,0)--(10,18)--(0,0)); // Adjusted for new problem length
dot((0,0));
label("$A$",(0,0),SW);
label("9",(4.5,0),S); // New base length of ABC
dot((9,0));
label... | 156 | 0.9375 | 4,592.75 | 4,352.8 | 8,192 | |
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $E$: $x^2 + \frac{y^2}{b^2} = 1 (0 < b < 1)$, and the line $l$ passing through $F_1$ intersects $E$ at points $A$ and $B$. If the sequence $|AF_2|, |AB|, |BF_2|$ forms an arithmetic progression, then:
(1) Find $|AB|$;
(2) If the slope of line $l$ i... | \frac{\sqrt{2}}{2} | 0 | 7,367.5 | -1 | 7,367.5 | |
At 12 o'clock, the angle between the hour hand and the minute hand is 0 degrees. After that, at what time do the hour hand and the minute hand form a 90-degree angle for the 6th time? (12-hour format) | 3:00 | 0.0625 | 8,192 | 8,192 | 8,192 | |
A 24-hour digital clock shows the time in hours and minutes. How many times in one day will it display all four digits 2, 0, 1, and 9 in some order?
A) 6
B) 10
C) 12
D) 18
E) 24 | 10 | 0 | 7,056.75 | -1 | 7,056.75 | |
Let $A$ denote the set of all integers $n$ such that $1 \leq n \leq 10000$, and moreover the sum of the decimal digits of $n$ is 2. Find the sum of the squares of the elements of $A$. | 7294927 | From the given conditions, we want to calculate $$\sum_{i=0}^{3} \sum_{j=i}^{3}\left(10^{i}+10^{j}\right)^{2}$$ By observing the formula, we notice that each term is an exponent of $10.10^{6}$ shows up 7 times, $10^{5}$ shows up 2 times, $10^{4}$ shows up 9 times, $10^{3}$ shows up 4 times, $10^{2}$ shows up 9 times, 1... | 0.1875 | 7,676.5625 | 6,066 | 8,048.230769 |
The sum of two numbers is \( t \) and the positive difference between the squares of these two numbers is 208. What is the larger of the two numbers? | 53 | 0 | 6,267.4375 | -1 | 6,267.4375 | |
In the sequence $\{a\_n\}$, $a\_1=1$, $a\_{n+1}=3a\_n (n∈N^{})$，then $a\_3=$ _______ , $S\_5=$ _______ . | 121 | 1 | 1,496.75 | 1,496.75 | -1 | |
How many three-digit numbers are there in which the hundreds digit is greater than both the tens digit and the units digit? | 285 | 0.6875 | 5,163.9375 | 4,292.727273 | 7,080.6 | |
Let $ABCD$ be an inscribed trapezoid such that the sides $[AB]$ and $[CD]$ are parallel. If $m(\widehat{AOD})=60^\circ$ and the altitude of the trapezoid is $10$ , what is the area of the trapezoid? | 100\sqrt{3} | 0.125 | 7,894.8125 | 5,974.5 | 8,169.142857 | |
Let \( ABCD \) and \( WXYZ \) be two squares that share the same center such that \( WX \parallel AB \) and \( WX < AB \). Lines \( CX \) and \( AB \) intersect at \( P \), and lines \( CZ \) and \( AD \) intersect at \( Q \). If points \( P, W \), and \( Q \) are collinear, compute the ratio \( AB / WX \). | \sqrt{2} + 1 | 0 | 8,151.125 | -1 | 8,151.125 | |
Suppose $w$ is a complex number such that $w^3 = 64 - 48i$. Find $|w|$. | 2\sqrt[3]{10} | 1 | 2,145.9375 | 2,145.9375 | -1 | |
Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $\frac{1}{6}$, independent of the outcome o... | \frac{36}{91} | 1. **Identify the Event Sequence**:
- Alice, Bob, and Carol take turns tossing a die in the order: Alice, Bob, Carol, Alice, Bob, Carol, and so on.
- We need to find the probability that Carol is the first to toss a six.
2. **Calculate the Probability of Carol Winning in the First Cycle**:
- Alice does not ge... | 0 | 6,060.75 | -1 | 6,060.75 |
Given the hyperbola $C:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with its right vertex at point $A$, a circle $A$ is created with center at $A$ and radius $b$. Circle $A$ intersects with one of the asymptotes of hyperbola $C$ at points $M$ and $N$. If $\angle MAN = 60^{\circ}$, find the eccentricity of ... | \frac{2\sqrt{3}}{3} | 0 | 6,063.4375 | -1 | 6,063.4375 | |
Petya and Vasya participated in a bicycle race. All participants started at the same time and finished with different times. Petya finished immediately after Vasya and ended up in tenth place. How many people participated in the race if Vasya was fifteenth from the end? | 23 | 0.8125 | 3,705.875 | 3,234.384615 | 5,749 | |
If $a$ is a constant such that $9x^2 + 24x + a$ is the square of a binomial, then what is $a$? | 16 | 1 | 1,504 | 1,504 | -1 | |
Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area of $ABCD.$ Find $\sqrt2 \cdot K.$ | 270 | 1. **Identify the Trapezoid and Points**: Let $ABCD$ be an isosceles trapezoid with $AD = BC$ and $AB < CD$. Define $P$, $Q$, and $R$ as the feet of the perpendiculars from $A$ to $BC$, $CD$, and $BD$, respectively.
2. **Use of Simson Line**: Since $ABCD$ is an isosceles trapezoid, it is also a cyclic quadrilateral. T... | 0 | 7,949.8125 | -1 | 7,949.8125 |
Let $S$ be the set of all 3-tuples $(a, b, c)$ of positive integers such that $a + b + c = 2013$ . Find $$ \sum_{(a,b,c)\in S} abc. $$ | \binom{2015}{5} | 0 | 7,834 | -1 | 7,834 | |
$ABCD$ is a trapezium such that $\angle ADC=\angle BCD=60^{\circ}$ and $AB=BC=AD=\frac{1}{2}CD$. If this trapezium is divided into $P$ equal portions $(P>1)$ and each portion is similar to trapezium $ABCD$ itself, find the minimum value of $P$.
The sum of tens and unit digits of $(P+1)^{2001}$ is $Q$. Find the value o... | 34 | 0 | 8,191.75 | -1 | 8,191.75 | |
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumulate... | 831 | After the first game, there are $10$ games we care about-- those involving $A$ or $B$. There are $3$ cases of these $10$ games: $A$ wins more than $B$, $B$ wins more than $A$, or $A$ and $B$ win the same number of games. Also, there are $2^{10} = 1024$ total outcomes. By symmetry, the first and second cases are equally... | 0.375 | 7,231.875 | 6,509 | 7,665.6 |
The notation $[x]$ stands for the greatest integer that is less than or equal to $x$. Calculate $[-1.2]$. | -2 | 0.9375 | 2,245.0625 | 2,051.333333 | 5,151 | |
In a class of 50 students, it is decided to use systematic sampling to select 10 students. The 50 students are randomly assigned numbers from 1 to 50 and divided into groups, with the first group being 1-5, the second group 6-10, ..., and the tenth group 45-50. If a student with the number 12 is selected from the third... | 37 | 0.4375 | 6,415.0625 | 5,152 | 7,397.444444 | |
Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$ . She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always mu... | 2022 | 0.4375 | 7,489.125 | 6,585.428571 | 8,192 | |
Given a quadratic function $f(x)$ with a second-degree coefficient $a$, and the inequality $f(x) > -2x$ has the solution set $(1,3)$:
(1) If the function $y = f(x) + 6a$ has exactly one zero, find the explicit form of $f(x)$.
(2) Let $h(a)$ be the maximum value of $f(x)$, find the minimum value of $h(a)$. | -2 | 0.5 | 6,407.1875 | 5,078.625 | 7,735.75 | |
Each of the nine letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "TEST"? Express your answer as a common fraction. | \frac{2}{3} | 0.4375 | 5,084.0625 | 4,010.142857 | 5,919.333333 | |
For how many integers $x$ does a triangle with side lengths $10, 24$ and $x$ have all its angles acute? | 4 | To determine the number of integers $x$ for which a triangle with sides $10, 24,$ and $x$ has all acute angles, we need to consider both the triangle inequality and the condition for all angles to be acute.
#### Step 1: Apply the Triangle Inequality
The triangle inequality states that the sum of the lengths of any two... | 0.8125 | 6,227.8125 | 5,774.538462 | 8,192 |
In a right triangle \( ABC \) with \( AC = 16 \) and \( BC = 12 \), a circle with center at \( B \) and radius \( BC \) is drawn. A tangent to this circle is constructed parallel to the hypotenuse \( AB \) (the tangent and the triangle lie on opposite sides of the hypotenuse). The leg \( BC \) is extended to intersect ... | 15 | 0.5 | 5,423 | 5,125.125 | 5,720.875 | |
Determine how many ordered pairs $(a, b)$, where $a$ is a positive real number and $b$ is an integer between $1$ and $210$, inclusive, satisfy the equation $(\log_b a)^{2023} = \log_b(a^{2023})$. | 630 | 0 | 6,898.0625 | -1 | 6,898.0625 | |
Given that odd prime numbers \( x, y, z \) satisfy
\[ x \mid (y^5 + 1), \quad y \mid (z^5 + 1), \quad z \mid (x^5 + 1). \]
Find the minimum value of the product \( xyz \). | 2013 | 0 | 8,159.5 | -1 | 8,159.5 | |
Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even? | \frac{5}{9} |
To solve this problem, we need to determine the probability that the product of two randomly drawn chips, one from each box, is even. Each box contains chips numbered $1$, $2$, and $3$.
#### Step 1: Determine Total Outcomes
Each box has $3$ chips, and a chip is drawn from each box. Therefore, the total number of out... | 0.75 | 4,806.0625 | 3,924.333333 | 7,451.25 |
A circle with its center on the line \( y = b \) intersects the parabola \( y = \frac{3}{4} x^{2} \) at least at three points; one of these points is the origin, and two of the remaining points lie on the line \( y = \frac{3}{4} x + b \). Find all values of \( b \) for which the described configuration is possible. | \frac{25}{12} | 0.1875 | 7,856.6875 | 6,403.666667 | 8,192 | |
What is the sum of the squares of the lengths of the medians of a triangle whose side lengths are $13, 13,$ and $10$? | 432 | 0 | 6,186.0625 | -1 | 6,186.0625 | |
A sequence of figures is formed using tiles. Each tile is an equilateral triangle with side length 7 cm. The first figure consists of 1 tile. Each figure after the first is formed by adding 1 tile to the previous figure. How many tiles are used to form the figure in the sequence with perimeter 91 cm? | 11 | The first figure consists of one tile with perimeter $3 \times 7 \mathrm{~cm} = 21 \mathrm{~cm}$. Each time an additional tile is added, the perimeter of the figure increases by 7 cm (one side length of a tile), because one side length of the previous figure is 'covered up' and two new side lengths of a tile are added ... | 0.375 | 5,217.25 | 4,472 | 5,664.4 |
Let $a_1$ , $a_2$ , $\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\gcd(a_{k-1},a_k)$ for all $k\geq 2$ . Compute the smallest possible value of $a_1+a_2+\cdots+a_{10}$ . | 440 | 0.0625 | 8,168.6875 | 7,819 | 8,192 | |
Calculate $\frac{3 \cdot 5! + 15\cdot 4!}{6!}$ | 1 | 1 | 2,491.1875 | 2,491.1875 | -1 | |
Given the function $f(x) = |2x+1| + |3x-2|$, and the solution set of the inequality $f(x) \leq 5$ is $\left\{x \mid -\frac{4a}{5} \leq x \leq \frac{3a}{5}\right\}$, where $a, b \in \mathbb{R}$.
1. Find the values of $a$ and $b$;
2. For any real number $x$, the inequality $|x-a| + |x+b| \geq m^2 - 3m$ holds, find the ma... | \frac{3 + \sqrt{21}}{2} | 0 | 8,192 | -1 | 8,192 | |
Let $g(x) = dx^3 + ex^2 + fx + g$, where $d$, $e$, $f$, and $g$ are integers. Suppose that $g(1) = 0$, $70 < g(5) < 80$, $120 < g(6) < 130$, $10000m < g(50) < 10000(m+1)$ for some integer $m$. What is $m$? | 12 | 0 | 8,078.875 | -1 | 8,078.875 | |
Consider a trapezoid field with base lengths of 120 meters and 180 meters and non-parallel sides each measuring 130 meters. The angles adjacent to the longer base are $60^\circ$. At harvest, the crops at any point in the field are brought to the nearest point on the field's perimeter. Determine the fraction of the crop... | \frac{1}{2} | 0 | 8,162.25 | -1 | 8,162.25 | |
A set of numbers $\{-3, 1, 5, 8, 10, 14\}$ needs to be rearranged with new rules:
1. The largest isn't in the last position, but it is in one of the last four places.
2. The smallest isn’t in the first position, but it is in one of the first four places.
3. The median isn't in the middle positions.
What is the produc... | -24 | 0 | 8,002.125 | -1 | 8,002.125 | |
On square $ABCD$, points $E,F,G$, and $H$ lie on sides $\overline{AB},\overline{BC},\overline{CD},$ and $\overline{DA},$ respectively, so that $\overline{EG} \perp \overline{FH}$ and $EG=FH = 34$. Segments $\overline{EG}$ and $\overline{FH}$ intersect at a point $P$, and the areas of the quadrilaterals $AEPH, BFPE, CGP... | 850 | Continue in the same way as solution 1 to get that $POK$ has area $3a$, and $OK = \frac{d}{10}$. You can then find $PK$ has length $\frac 32$.
Then, if we drop a perpendicular from $H$ to $BC$ at $L$, We get $\triangle HLF \sim \triangle OPK$.
Thus, $LF = \frac{15\cdot 34}{d}$, and we know $HL = d$, and $HF = 34$. Th... | 0 | 8,135.3125 | -1 | 8,135.3125 |
What is the area of the region defined by the equation $x^2+y^2 - 7 = 2y-8x+1$? | 25\pi | 0.9375 | 1,782.5625 | 1,736 | 2,481 | |
Given the sequence \( S_{1} = 1, S_{2} = 1 - 2, S_{3} = 1 - 2 + 3, S_{4} = 1 - 2 + 3 - 4, S_{5} = 1 - 2 + 3 - 4 + 5, \cdots \), find the value of \( S_{1} + S_{2} + S_{3} + \cdots + S_{299} \). | 150 | 0.6875 | 6,315.5625 | 5,462.636364 | 8,192 | |
John borrows $2000$ from Mary, who charges an interest rate of $6\%$ per month (which compounds monthly). What is the least integer number of months after which John will owe more than triple what he borrowed? | 19 | 0.625 | 5,615.125 | 4,069 | 8,192 | |
The number of games won by five baseball teams are displayed on a chart. The team names are missing, and we have several clues to help identify them:
1. The Sharks won fewer games than the Raptors.
2. The Royals won more games than the Dragons, but fewer games than the Knights.
3. The Dragons won more than 30 games.
... | 50 | 0.3125 | 824.875 | 880.2 | 799.727273 | |
If $$\sin\alpha= \frac {4}{7} \sqrt {3}$$ and $$\cos(\alpha+\beta)=- \frac {11}{14}$$, and $\alpha$, $\beta$ are acute angles, then $\beta= \_\_\_\_\_\_$. | \frac {\pi}{3} | 0.1875 | 6,299.875 | 4,855.333333 | 6,633.230769 | |
A teacher tells the class,
"Think of a number, add 1 to it, and double the result. Give the answer to your partner. Partner, subtract 1 from the number you are given and double the result to get your answer."
Ben thinks of $6$, and gives his answer to Sue. What should Sue's answer be? | 26 | 1. **Ben's Calculation:**
- Ben thinks of a number, which is $6$.
- He adds $1$ to this number: $6 + 1 = 7$.
- He then doubles the result: $7 \times 2 = 14$.
- Ben gives the number $14$ to Sue.
2. **Sue's Calculation:**
- Sue receives the number $14$.
- She subtracts $1$ from this number: $14 - 1 = 1... | 1 | 617.6875 | 617.6875 | -1 |
An inscribed circle is drawn inside isosceles trapezoid \(ABCD\) with \(AB = CD\). Let \(M\) be the point where the circle touches side \(CD\), \(K\) be the intersection point of the circle with segment \(AM\), and \(L\) be the intersection point of the circle with segment \(BM\). Calculate the value of \(\frac{AM}{AK}... | 10 | 0 | 8,137.75 | -1 | 8,137.75 | |
A particular triangle has sides of length 14 cm, 8 cm and 9 cm. In centimeters, what is the perimeter of the triangle? | 31 | 1 | 1,232 | 1,232 | -1 | |
A bookstore has a sale on days of the month that are multiples of 5 (such as June 5, June 10...). A shoe store has a sale every 6 days. The shoe store has a sale on July 3. How many times in July do the two stores have sales on the same date? | 1 | 0.8125 | 4,644.6875 | 4,206.615385 | 6,543 | |
Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a + b + c = 8.$ Find the maximum value of
\[\sqrt{3a + 2} + \sqrt{3b + 2} + \sqrt{3c + 2}.\] | 3\sqrt{10} | 0.75 | 5,573.1875 | 4,700.25 | 8,192 | |
A circle with a radius of 2 units has its center at $(0, 0)$. A circle with a radius of 7 units has its center at $(15, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction. | \frac{10}{3} | 0.875 | 5,070.0625 | 4,624.071429 | 8,192 | |
A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\ell > 0$ such that there exists a square of side $\ell$ contained in a cube with edge $1$ . | \frac{\sqrt{6}}{2} | 0 | 8,192 | -1 | 8,192 | |
Given an arithmetic sequence $\{a_n\}$, if $d < 0$, $T_n$ is the sum of the first $n$ terms of $\{a_n\}$, and $T_3=15$, also $a_1+1, a_2+3, a_3+9$ form a geometric sequence with common ratio $q$, calculate the value of $q$. | \frac{1}{2} | 0.875 | 4,095.75 | 3,555.785714 | 7,875.5 | |
Determine the largest value the expression $$ \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} $$ may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-negative real numbers, and add up to $ 1. $ Find also the specific values of this numbers that make the above sum achieve the asked maximum. | 3/4 | 0 | 8,192 | -1 | 8,192 | |
A television's price was discounted $40\%$ and then slashed another $40\%$ during a liquidation sale. By what percent has the original price been reduced? | 64\% | 1 | 1,928 | 1,928 | -1 | |
Given an ellipse $C$: $\frac{{x}^{2}}{3}+{y}^{2}=1$ with left focus and right focus as $F_{1}$ and $F_{2}$ respectively, the line $y=x+m$ intersects $C$ at points $A$ and $B$. Determine the value of $m$ such that the area of $\triangle F_{1}AB$ is twice the area of $\triangle F_{2}AB$. | -\frac{\sqrt{2}}{3} | 0 | 7,371.875 | -1 | 7,371.875 | |
Compute the number of intersection points of the graphs of
\[(x - \lfloor x \rfloor)^2 + y^2 = x - \lfloor x \rfloor\]and $y = \frac{1}{5} x.$ | 11 | 0.0625 | 7,874.3125 | 6,293 | 7,979.733333 | |
A sequence $a_1,$ $a_2,$ $a_3,$ $\dots,$ is defined recursively by $a_1 = 1,$ $a_2 = 1,$ and for $k \ge 3,$
\[a_k = \frac{1}{3} a_{k - 1} + \frac{1}{4} a_{k - 2}.\]Evaluate $a_1 + a_2 + a_3 + \dotsb.$ | 4 | 0.5 | 7,060.125 | 5,928.25 | 8,192 | |
For a natural number \( x \), five statements are made:
$$
3x > 91
$$
$$
\begin{aligned}
& x < 120 \\
& 4x > 37 \\
& 2x \geq 21 \\
& x > 7
\end{aligned}
$$
It is known that only three of these statements are true, and two are false. Determine \( x \). | 10 | 0.625 | 6,655.8125 | 6,003.5 | 7,743 | |
Given a hyperbola with eccentricity $e$ and an ellipse with eccentricity $\frac{\sqrt{2}}{2}$ share the same foci $F_{1}$ and $F_{2}$. If $P$ is a common point of the two curves and $\angle F_{1}PF_{2}=60^{\circ}$, then $e=$ ______. | \frac{\sqrt{6}}{2} | 0 | 6,284 | -1 | 6,284 | |
What is the smallest positive four-digit number divisible by 9 which has two even and two odd digits? | 1089 | 0.0625 | 7,229.75 | 7,873 | 7,186.866667 | |
For how many integers $n$ between 1 and 2005, inclusive, is $2 \cdot 6 \cdot 10 \cdots(4 n-2)$ divisible by $n!$? | 2005 | Note that $$\begin{aligned} 2 \cdot 6 \cdot 10 \cdots(4 n-2) & =2^{n} \cdot 1 \cdot 3 \cdot 5 \cdots(2 n-1) \\ & =2^{n} \cdot \frac{1 \cdot 2 \cdot 3 \cdots 2 n}{2 \cdot 4 \cdot 6 \cdots 2 n} \\ & =\frac{1 \cdot 2 \cdot 3 \cdots 2 n}{1 \cdot 2 \cdot 3 \cdots n} \end{aligned}$$ that is, it is just $(2 n)!/ n$ !. Therefo... | 0.1875 | 8,059.875 | 7,487.333333 | 8,192 |
What is the largest positive integer with only even digits that is less than $10,000$ and is a multiple of $9$? | 8820 | 0.125 | 8,155.1875 | 7,897.5 | 8,192 | |
In triangle $ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and it is given that $a\sin C= \sqrt{3}c\cos A$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $a= \sqrt{13}$ and $c=3$, find the area of triangle $ABC$. | 3\sqrt{3} | 1 | 4,336.4375 | 4,336.4375 | -1 | |
A random binary string of length 1000 is chosen. Let \(L\) be the expected length of its longest (contiguous) palindromic substring. Estimate \(L\). | 23.120 | The probability that there exists a palindromic substring of length \(2n+1\) is approximately \(2^{-n} \cdot 1000\). Thus, we can expect to often see a length 21 palindrome, and sometimes longer ones. This leads to a guess a bit above 21. \(L\) was approximated with \(10^{7}\) simulations (the answer is given with a st... | 0 | 8,192 | -1 | 8,192 |
The diagonals of parallelogram $ABCD$ intersect at $E$ . The bisectors of $\angle DAE$ and $\angle EBC$ intersect at $F$. Assume $ECFD$ is a parellelogram . Determine the ratio $AB:AD$. | {\sqrt{3}} |
Let's denote the lengths \( AB = x \) and \( AD = y \). We are tasked with finding the ratio \( \frac{x}{y} \).
Since \( ABCD \) is a parallelogram, its diagonals \( AC \) and \( BD \) bisect each other at \( E \). Therefore, \( AE = EC \) and \( BE = ED \).
Given that \( ECFD \) is a parallelogram, \( EC \parallel ... | 0 | 7,983.375 | -1 | 7,983.375 |
It is given that \( k \) is a positive integer not exceeding 99. There are no natural numbers \( x \) and \( y \) such that \( x^{2} - k y^{2} = 8 \). Find the difference between the maximum and minimum possible values of \( k \). | 96 | 0.0625 | 8,052.3125 | 5,957 | 8,192 | |
Calculate the lengths of the arcs of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\left(t^{2}-2\right) \sin t+2 t \cos t \\
y=\left(2-t^{2}\right) \cos t+2 t \sin t
\end{array}\right. \\
& 0 \leq t \leq 2 \pi
\end{aligned}
$$ | \frac{8\pi^3}{3} | 0.875 | 3,036.6875 | 3,012.785714 | 3,204 | |
Find the area of the triangle formed by the axis of the parabola $y^{2}=8x$ and the two asymptotes of the hyperbola $(C)$: $\frac{x^{2}}{8}-\frac{y^{2}}{4}=1$. | 2\sqrt{2} | 0.0625 | 7,850.375 | 6,545 | 7,937.4 | |
In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have? | 7 | Let $c$ be the number of correct answers Olivia had, and $w$ be the number of incorrect answers. Since there are 10 problems in total, we have:
\[ c + w = 10 \]
For each correct answer, Olivia gains 5 points, and for each incorrect answer, she loses 2 points. Therefore, her total score can be expressed as:
\[ 5c - 2w ... | 1 | 2,269.625 | 2,269.625 | -1 |
In a convex pentagon \(ABCDE\), \(AB = BC\), \(CD = DE\), \(\angle ABC = 100^\circ\), \(\angle CDE = 80^\circ\), and \(BD^2 = \frac{100}{\sin 100^\circ}\). Find the area of the pentagon. | 50 | 0 | 8,192 | -1 | 8,192 | |
Find the number of ways to arrange the numbers 1 through 7 in a circle such that the numbers are increasing along each arc from 1. | 32 | First, we can fix the position of the 1. Then, by the condition that the numbers are increasing along each arc from 1, we know that the 2 must be adjacent to the 1; so we have two options for its placement. Similarly, we have two options for placing each of $3,4,5,6$ in that order. Finally, the 7 must go in the remaini... | 0 | 8,192 | -1 | 8,192 |
Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge. | \frac{9\sqrt{3}}{4} | 0 | 8,192 | -1 | 8,192 | |
Consider an $n$ -by- $n$ board of unit squares for some odd positive integer $n$ . We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ ... | \[
\left(\frac{n+1}{2}\right)^2
\] | We claim the answer is $(\frac{n+1}{2})^2$ .
First, consider a checkerboard tiling of the board with 4 colors: R, G, B, Y. Number each column from $1$ to $n$ from left to right and each row from $1$ to $n$ from top to bottom. We color a tile R if its row and column are odd, a tile G is its row is even but its column is... | 0 | 8,014 | -1 | 8,014 |
An electronic clock always displays the date as an eight-digit number. For example, January 1, 2011, is displayed as 20110101. What is the last day of 2011 that can be evenly divided by 101? The date is displayed as $\overline{2011 \mathrm{ABCD}}$. What is $\overline{\mathrm{ABCD}}$? | 1221 | 0.1875 | 7,747.875 | 5,823.333333 | 8,192 | |
Add $704_{12} + 159_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary. | 861_{12} | 1 | 2,976 | 2,976 | -1 | |
Given the function $f(x)=e^{x}$, for real numbers $m$, $n$, $p$, it is known that $f(m+n)=f(m)+f(n)$ and $f(m+n+p)=f(m)+f(n)+f(p)$. Determine the maximum value of $p$. | 2\ln2-\ln3 | 0 | 7,606.0625 | -1 | 7,606.0625 | |
Given $\tan \theta = 5,$ find
\[\frac{1 - \cos \theta}{\sin \theta} - \frac{\sin \theta}{1 + \cos \theta}.\] | 0 | 0.9375 | 4,045.0625 | 3,768.6 | 8,192 | |
Let $z$ be a complex number satisfying $12|z|^2=2|z+2|^2+|z^2+1|^2+31.$ What is the value of $z+\frac 6z?$ | -2 | 1. **Rewrite the given equation using $w \overline{w} = |w|^2$:**
\[
12|z|^2 = 2|z+2|^2 + |z^2+1|^2 + 31
\]
Expressing in terms of $z$ and $\overline{z}$:
\[
12z\overline{z} = 2(z+2)(\overline{z}+2) + (z^2+1)(\overline{z}^2+1) + 31
\]
Expanding the terms:
\[
12z\overline{z} = 2z\overline{z... | 0.3125 | 7,357.5 | 5,521.6 | 8,192 |
Given a positive number \( r \), let the set \( T = \left\{(x, y) \mid x, y \in \mathbb{R}, \text{ and } x^{2} + (y-7)^{2} \leq r^{2} \right\} \). This set \( T \) is a subset of the set \( S = \{(x, y) \mid x, y \in \mathbb{R}, \text{ and for any } \theta \in \mathbb{R}, \ \cos 2\theta + x \cos \theta + y \geq 0\} \).... | 4\sqrt{2} | 0.125 | 8,098.75 | 7,853.5 | 8,133.785714 | |
The minimum value of the function \( f(x)=(x+1)(x+2)(x+3)(x+4)+35 \) is: | 34 | 0.8125 | 5,715.75 | 5,144.307692 | 8,192 | |
In trapezoid \(PQRS\), the lengths of the bases \(PQ\) and \(RS\) are 10 and 23 respectively. The legs of the trapezoid are extended beyond \(P\) and \(Q\) to meet at point \(T\). What is the ratio of the area of triangle \(TPQ\) to the area of trapezoid \(PQRS\)? Express your answer as a common fraction. | \frac{100}{429} | 0.6875 | 5,254.3125 | 3,919 | 8,192 | |
In the diagram, \(PQ\) is a diameter of a larger circle, point \(R\) is on \(PQ\), and smaller semi-circles with diameters \(PR\) and \(QR\) are drawn. If \(PR = 6\) and \(QR = 4\), what is the ratio of the area of the shaded region to the area of the unshaded region? | 2: 3 | 0 | 5,371.625 | -1 | 5,371.625 | |
Evaluate the expression $3 + 2\sqrt{3} + \frac{1}{3 + 2\sqrt{3}} + \frac{1}{2\sqrt{3} - 3}$. | 3 + \frac{10\sqrt{3}}{3} | 0 | 4,316.5625 | -1 | 4,316.5625 | |
Rhombus $ABCD$ has side length $2$ and $\angle B = 120^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? | \frac{2\sqrt{3}}{3} | 1. **Identify the region $R$:**
Region $R$ consists of all points inside rhombus $ABCD$ that are closer to vertex $B$ than to any other vertex $A$, $C$, or $D$.
2. **Use of perpendicular bisectors:**
Let $\ell_{BC}$, $\ell_{BD}$, and $\ell_{BA}$ be the perpendicular bisectors of sides $BC$, $BD$, and $BA$ resp... | 0 | 8,140.9375 | -1 | 8,140.9375 |
The numbers 1, 2, ..., 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ..., 3k+1th, ... numbers in the list are erased. The process is repeated on the remaining list (i.e., erase the 1st, 4th, 7th, ... 3k+1th numbers in the new list). This continues until no numbers are left. What is the last number ... | 1598 | 0 | 8,192 | -1 | 8,192 | |
The absolute value of -9 is ; the reciprocal of -3 is . | -\frac{1}{3} | 0.75 | 278.375 | 275.916667 | 285.75 | |
The radian measure of 300° is $$\frac {5π}{3}$$ | \frac{5\pi}{3} | 1 | 2,178.375 | 2,178.375 | -1 | |
The quantity $\sqrt{45} - 2\sqrt{5} + \frac{\sqrt{360}}{\sqrt{2}}$ can be expressed as $\sqrt{N}$, where $N$ is an integer. Find $N$. | 245 | 1 | 1,632.375 | 1,632.375 | -1 |
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