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 |
|---|---|---|---|---|---|---|
How many of the integers between 30 and 50, inclusive, are not possible total scores if a multiple choice test has 10 questions, each correct answer is worth 5 points, each unanswered question is worth 1 point, and each incorrect answer is worth 0 points? | 6 | If 10 of 10 questions are answered correctly, the total score is $10 \times 5=50$ points. If 9 of 10 questions are answered correctly, the score is either $9 \times 5=45$ or $46$ points. If 8 of 10 questions are answered correctly, the score is $40,41,42$ points. If 7 of 10 questions are answered correctly, the score i... | 0.1875 | 7,758.3125 | 8,097.333333 | 7,680.076923 |
In $\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, and points $E$ and $F$ lie on $\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$. | 150 | 1. **Identify the type of triangle $PQR$**:
Given $PR=15$, $QR=20$, and $PQ=25$, we check if $\triangle PQR$ is a right triangle using the Pythagorean theorem:
\[
PR^2 + QR^2 = 15^2 + 20^2 = 225 + 400 = 625 = 25^2 = PQ^2.
\]
Since $PR^2 + QR^2 = PQ^2$, $\triangle PQR$ is a right triangle with $PQ$ as the... | 0 | 7,938.3125 | -1 | 7,938.3125 |
A soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $3$ substitutions, where any one of the $11$ players in the game is replaced by one of the substitutes. No player removed from the game... | 122 | 0.1875 | 6,851.4375 | 6,917.666667 | 6,836.153846 | |
From the diagram below, find the positive difference in the $x$-coordinates when lines $l$ and $m$ reach $y = 15.$ [asy]
import cse5; import olympiad;
size(120);
add(grid(8,8));
draw((0,0)--(8,0),linewidth(1.2));
draw((0,0)--(0,8),linewidth(1.2));
label("$x$",(8,0),E);
label("$y$",(0,8),N);
draw((0,5)--(3,0),Arrows);
d... | 39.5 | 0.0625 | 5,061.0625 | 8,192 | 4,852.333333 | |
In Idaho, 472 people were asked what they call soft drinks. The results of the survey are shown in the pie chart. The central angle of the ``Pop'' sector of the graph is $251^\circ$, to the nearest whole degree. How many of the people surveyed chose ``Pop''? Express your answer as a whole number.
[asy]import graph;
si... | 329 | 0.125 | 495.3125 | 481 | 497.357143 | |
Angle $EAB$ is a right angle, and $BE=9$ units. What is the number of square units in the sum of the areas of the two squares $ABCD$ and $AEFG$?
[asy]
draw((0,0)--(1,1)--(0,2)--(-1,1)--cycle);
draw((0,2)--(2,4)--(0,6)--(-2,4)--cycle);
draw((1,1)--(2,4));
draw((-1,1)--(-2,4));
label("A", (0,2), S);
label("B", (1,1), S... | 81 | 1 | 3,337.5 | 3,337.5 | -1 | |
Given point \( A(2,0) \), point \( B \) lies on the curve \( y = \sqrt{1 - x^2} \), and the triangle \( \triangle ABC \) is an isosceles right triangle with \( A \) as the right angle vertex. Determine the maximum value of \( |OC| \). | 2\sqrt{2} + 1 | 0.125 | 7,408.75 | 6,223 | 7,578.142857 | |
Rectangle $PQRS$ is inscribed in triangle $XYZ$, with side $PS$ of the rectangle on side $XZ$. Triangle $XYZ$ has an altitude from $Y$ to side $XZ$ of 8 inches, and the side $XZ$ measures 15 inches. The length of segment $PQ$ is one-third the length of segment $PS$. Calculate the area of rectangle $PQRS$, and express y... | \frac{4800}{169} | 0.375 | 6,824.5625 | 5,293.333333 | 7,743.3 | |
What is 0.3 more than 29.8? | 30.1 | 0.8125 | 219.625 | 216.384615 | 233.666667 | |
Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$?
| 589 | 0.25 | 7,460.625 | 6,768.25 | 7,691.416667 | |
Given a function f(n) defined on the set of positive integers, where f(1) = 2: For even n, f(n) = f(n-1) + 2; For odd n > 1, f(n) = f(n-2) + 2. Calculate the value of f(2017). | 2018 | 1 | 5,253.4375 | 5,253.4375 | -1 | |
Given 3 zeros and 2 ones are randomly arranged in a row, calculate the probability that the 2 ones are not adjacent. | \frac{3}{5} | 0.8125 | 4,980.25 | 4,239.076923 | 8,192 | |
In triangle $ABC$, $AB=20$ and $AC=11$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive ... | 51 | 0.75 | 4,866.625 | 4,037.5 | 7,354 | |
Consider two solid spherical balls, one centered at $\left( 0, 0, \frac{21}{2} \right),$ with radius 6, and the other centered at $(0,0,1)$ with radius $\frac{9}{2}.$ How many points $(x,y,z)$ with only integer coefficients are there in the intersection of the balls? | 13 | 0.6875 | 6,228.3125 | 5,335.727273 | 8,192 | |
A cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\circ$ angle with the plane of the first cut. The intersection of these two planes has e... | 216 | The volume of the wedge is half the volume of a cylinder with height $12$ and radius $6$. (Imagine taking another identical wedge and sticking it to the existing one). Thus, $V=\dfrac{6^2\cdot 12\pi}{2}=216\pi$, so $n=\boxed{216}$.
Part 1:
Part 2:
- AMBRIGGS | 0 | 8,192 | -1 | 8,192 |
As shown in the figure below, a circular park consists of an outer-ring path for walkers (white) and a ring-shaped flower garden (gray) surrounding a central circular fountain (black). The walking path is six feet wide in all places, the garden ring is eight feet wide in all places, and the fountain has a diameter of 1... | 38 | 1 | 1,505.6875 | 1,505.6875 | -1 | |
A strip of size $1 \times 10$ is divided into unit squares. The numbers $1, 2, \ldots, 10$ are written in these squares. First, the number 1 is written in one of the squares, then the number 2 is written in one of the neighboring squares, then the number 3 is written in one of the squares neighboring those already occu... | 512 | 0.0625 | 8,165.625 | 7,770 | 8,192 | |
Four identical regular tetrahedrons are thrown simultaneously on a table. Calculate the probability that the product of the four numbers on the faces touching the table is divisible by 4. | \frac{13}{16} | 0.125 | 7,924.8125 | 6,054.5 | 8,192 | |
In isosceles triangle $\triangle ABC$, $CA=CB=6$, $\angle ACB=120^{\circ}$, and point $M$ satisfies $\overrightarrow{BM}=2 \overrightarrow{MA}$. Determine the value of $\overrightarrow{CM} \cdot \overrightarrow{CB}$. | 12 | 0 | 3,793.625 | -1 | 3,793.625 | |
Let $b_n$ be the integer obtained by writing the integers from $5$ to $n+4$ from left to right. For example, $b_2 = 567$, and $b_{10} = 567891011121314$. Compute the remainder when $b_{25}$ is divided by $55$ (which is the product of $5$ and $11$ for the application of the Chinese Remainder Theorem). | 39 | 0.125 | 7,401 | 5,856.5 | 7,621.642857 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\frac {2a+b}{c}= \frac {\cos (A+C)}{\cos C}$.
(I) Find the magnitude of angle $C$,
(II) If $c=2$, find the maximum area of $\triangle ABC$. | \frac { \sqrt {3}}{3} | 0 | 5,397.9375 | -1 | 5,397.9375 | |
Let \( a_{n} = \frac{1}{3} + \frac{1}{12} + \frac{1}{30} + \frac{1}{60} + \cdots + \frac{2}{n(n-1)(n-2)} + \frac{2}{(n+1) n(n-1)} \), find \( \lim_{n \rightarrow \infty} a_{n} \). | \frac{1}{2} | 0.1875 | 7,781.1875 | 6,001 | 8,192 | |
$\triangle ABC$ has side lengths $AB=20$ , $BC=15$ , and $CA=7$ . Let the altitudes of $\triangle ABC$ be $AD$ , $BE$ , and $CF$ . What is the distance between the orthocenter (intersection of the altitudes) of $\triangle ABC$ and the incenter of $\triangle DEF$ ? | 15 | 0.125 | 8,088.875 | 7,367 | 8,192 | |
How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit? | 296 | 0.125 | 7,735.0625 | 5,987.5 | 7,984.714286 | |
You walk for 90 minutes at a rate of 3 mph, then rest for 15 minutes, and then cycle for 45 minutes at a rate of 20 kph. Calculate the total distance traveled in 2 hours and 30 minutes. | 13.82 | 0.3125 | 1,970.625 | 1,774 | 2,060 | |
Given \( P = 3659893456789325678 \) and \( 342973489379256 \), the product \( P \) is calculated. The number of digits in \( P \) is: | 34 | 0.4375 | 7,195.875 | 6,765.857143 | 7,530.333333 | |
A certain school has $7$ members in its student art department (4 males and 3 females). Two members are to be selected to participate in the school's art performance event.
$(1)$ Find the probability that only one female member is selected.
$(2)$ Given that a male member, let's call him A, is selected, find the pro... | \frac{1}{6} | 0.4375 | 4,297.25 | 2,845.571429 | 5,426.333333 | |
Let \( P \) be an arbitrary point on the graph of the function \( y = x + \frac{2}{x} \) (where \( x > 0 \)). From point \( P \), perpendiculars are drawn to the line \( y = x \) and the \( y \)-axis, with the foots of these perpendiculars being points \( A \) and \( B \) respectively. Determine the value of \( \overri... | -1 | 0.9375 | 3,093.875 | 2,754 | 8,192 | |
Let $\alpha$ and $\beta$ be real numbers. Find the minimum value of
\[(3 \cos \alpha + 4 \sin \beta - 7)^2 + (3 \sin \alpha + 4 \cos \beta - 12)^2.\] | 48 | 0 | 8,192 | -1 | 8,192 | |
Find the sum of the values of $x$ which satisfy $x^2 +1992x = 1993$. | -1992 | 1 | 2,293.75 | 2,293.75 | -1 | |
Suppose point $P$ is on the curve represented by the equation $\sqrt{(x-5)^2+y^2} - \sqrt{(x+5)^2+y^2} = 6$, and $P$ is also on the line $y=4$. Determine the x-coordinate of point $P$. | -3\sqrt{2} | 0.625 | 6,121.625 | 5,509.1 | 7,142.5 | |
The length of the escalator is 200 steps. When Petya walks down the escalator, he counts 50 steps. How many steps will he count if he runs twice as fast? | 80 | 0.75 | 5,530.375 | 4,781.75 | 7,776.25 | |
The sum of all of the digits of the integers from 1 to 2008 is: | 28054 | 0.1875 | 7,859.125 | 7,666.333333 | 7,903.615385 | |
A lattice point in the $x,y$-plane is a point both of whose coordinates are integers (not necessarily positive). How many lattice points lie on the graph of the equation $x^2-y^2=47$? | 4 | 1 | 4,424.9375 | 4,424.9375 | -1 | |
The numbers $2^{0}, 2^{1}, \cdots, 2^{15}, 2^{16}=65536$ are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one from the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when the... | 131069 | If we reverse the order of the numbers in the final subtraction we perform, then the final number will be negated. Thus, the possible final numbers come in pairs with opposite signs. Therefore, the largest possible number is the negative of the smallest possible number. To get the smallest possible number, clearly we c... | 0 | 8,154.1875 | -1 | 8,154.1875 |
Given Laila had a 10-hour work day and attended three different meetings during this period, the first meeting took 50 minutes, the second meeting twice as long as the first, and the third meeting half as long as the second meeting, calculate the percentage of her work day that was spent attending meetings. | 33.33\% | 0.5625 | 582.4375 | 593.222222 | 568.571429 | |
Let $a$ and $b$ be positive whole numbers such that $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$ .
Find the fraction $\frac{a}{b}$ for which the sum $a+b$ is as small as possible.
Justify your answer | 3/7 | 0.5625 | 7,539.25 | 7,031.555556 | 8,192 | |
Given the taxi fare of $3.50 for the first 0.5 mile, and an additional charge of $0.30 for each 0.1 mile thereafter, and a $3 tip, calculate the total number of miles that can be ridden with a $15 budget. | 3.333 | 0.25 | 649.375 | 695.5 | 634 | |
What is the least positive integer which when divided by 5 gives a remainder of 4, when divided by 6 gives a remainder of 5, when divided by 7 gives a remainder of 6, when divided by 8 gives a remainder of 7, when divided by 9 gives a remainder of 8, and when divided by 10 gives a remainder of 9? | 2519 | 1 | 2,061.5 | 2,061.5 | -1 | |
The sum of the first $n$ terms in the infinite geometric sequence $\left\{1,\frac{1}{3},\frac{1}{9},\frac{1}{27},\dots \right\}$ is $\frac{121}{81}$. Find $n$. | 5 | 1 | 2,228.0625 | 2,228.0625 | -1 | |
Two distinct numbers $a$ and $b$ are chosen randomly from the set $\{2, 2^2, 2^3, ..., 2^{25}\}$. What is the probability that $\log_a b$ is an integer? | \frac{31}{300} | 1. **Define the problem in terms of logarithms and powers of 2:**
Let $a = 2^x$ and $b = 2^y$ where $x$ and $y$ are integers such that $1 \leq x, y \leq 25$. We are given that $\log_a b$ is an integer, say $z$. Therefore, we have:
\[
\log_a b = z \implies a^z = b \implies (2^x)^z = 2^y \implies 2^{xz} = 2^y \i... | 0 | 7,916.25 | -1 | 7,916.25 |
Given that point $P$ moves on the circle $x^{2}+(y-2)^{2}=1$, and point $Q$ moves on the ellipse $\frac{x^{2}}{9}+y^{2}=1$, find the maximum value of the distance $PQ$. | \frac{3\sqrt{6}}{2} + 1 | 0 | 8,192 | -1 | 8,192 | |
The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping, he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at hi... | 6:00 | 1. **Identify the rate of gain of the car clock**:
The car clock shows 12:35 when the accurate watch shows 12:30. This means in 30 minutes of real time, the car clock gains an extra 5 minutes.
2. **Determine the rate of the car clock relative to real time**:
In 30 minutes of real time, the car clock shows 35 ... | 0.875 | 4,254.125 | 4,123.142857 | 5,171 |
Determine the maximal size of a set of positive integers with the following properties:
1. The integers consist of digits from the set {1,2,3,4,5,6}.
2. No digit occurs more than once in the same integer.
3. The digits in each integer are in increasing order.
4. Any two integers have at least one digit in common (poss... | 32 | 0.0625 | 8,139.6875 | 7,355 | 8,192 | |
Consider a $3 \times 3$ block of squares as the center area in an array of unit squares. The first ring around this center block contains unit squares that directly touch the block. If the pattern continues as before, how many unit squares are in the $10^{th}$ ring? | 88 | 0.8125 | 4,490.0625 | 4,026.076923 | 6,500.666667 | |
Compute the number of sequences of integers $(a_{1}, \ldots, a_{200})$ such that the following conditions hold. - $0 \leq a_{1}<a_{2}<\cdots<a_{200} \leq 202$. - There exists a positive integer $N$ with the following property: for every index $i \in\{1, \ldots, 200\}$ there exists an index $j \in\{1, \ldots, 200\}$ suc... | 20503 | Let $m:=203$ be an integer not divisible by 3. We'll show the answer for general such $m$ is $m\left\lceil\frac{m-1}{2}\right\rceil$. Let $x, y, z$ be the three excluded residues. Then $N$ works if and only if $\{x, y, z\} \equiv\{N-x, N-y, N-z\} (\bmod m)$. Since $x, y, z(\bmod m)$ has opposite orientation as $N-x, N-... | 0 | 8,021.25 | -1 | 8,021.25 |
Determine the smallest integral value of $n$ such that the quadratic equation
\[3x(nx+3)-2x^2-9=0\]
has no real roots.
A) -2
B) -1
C) 0
D) 1 | -1 | 0 | 6,118.6875 | -1 | 6,118.6875 | |
How many integers satisfy the inequality $(x+3)^{2}\leq1$? | 3 | 1 | 1,378.1875 | 1,378.1875 | -1 | |
In $\triangle ABC$, if $AB=2$, $AC=\sqrt{2}BC$, find the maximum value of $S_{\triangle ABC}$. | 2\sqrt{2} | 0.6875 | 6,918.5 | 6,339.636364 | 8,192 | |
How many words are there in a language that are 10 letters long and begin with a vowel, given that the language uses only the letters A, B, C, D, and E, where A and E are vowels, and B, C, and D are consonants, and a word does not include the same letter twice in a row or two vowels in a row? | 199776 | Using the given rules, the words that are 1 letter long are A, B, C, D, E. Using the given rules, the words that are 2 letters long are AB, AC, AD, BA, BC, BD, BE, CA, CB, CD, CE, DA, DB, DC, DE, EB, EC, ED. Let $v_{n}$ be the number of words that are $n$ letters long and that begin with a vowel. Note that $v_{1}=2$ an... | 0 | 7,992.8125 | -1 | 7,992.8125 |
The root(s) of $\frac {15}{x^2 - 4} - \frac {2}{x - 2} = 1$ is (are): | -3 \text{ and } 5 | 1. **Identify the common denominator and simplify the equation:**
Given the equation:
\[
\frac{15}{x^2 - 4} - \frac{2}{x - 2} = 1
\]
We recognize that $x^2 - 4 = (x-2)(x+2)$. Thus, the common denominator for the fractions is $x^2 - 4$. Rewriting the second term with this denominator:
\[
\frac{2}{x-... | 0 | 2,650.25 | -1 | 2,650.25 |
Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations $$
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
$$has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$. Find $\rho^2.$ | \frac{4}{3} | 0 | 8,192 | -1 | 8,192 | |
On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take? | 80 | 1. **Define the speeds and times:**
Let $v$ be the speed on the coastal road. Then the speed on the highway is $3v$ since he drives three times as fast on the highway.
2. **Calculate the time spent on the coastal road:**
Anh spent 30 minutes driving on the coastal road. Since time is distance divided by speed, w... | 1 | 1,934.25 | 1,934.25 | -1 |
A school plans to purchase volleyball and basketball in one go. The price of each basketball is $30$ yuan more expensive than a volleyball. It costs a total of $340$ yuan to buy $2$ volleyballs and $3$ basketballs.<br/>$(1)$ Find the price of each volleyball and basketball.<br/>$(2)$ If the school purchases a total of ... | 3660 | 0.875 | 2,882.4375 | 2,813.714286 | 3,363.5 | |
Given a sequence $\{a_n\}$ that satisfies $a_{n+1}^2=a_na_{n+2}$, and $a_1= \frac{1}{3}$, $a_4= \frac{1}{81}$.
(1) Find the general formula for the sequence $\{a_n\}$.
(2) Let $f(x)=\log_3x$, $b_n=f(a_1)+f(a_2)+\ldots+f(a_n)$, $T_n= \frac{1}{b_1}+ \frac{1}{b_2}+\ldots+ \frac{1}{b_n}$, find $T_{2017}$. | \frac{-2017}{1009} | 0 | 5,595.1875 | -1 | 5,595.1875 | |
Jia and Yi are playing a guessing game with the following rules: It is known that there are five cards, each with the numbers $1-\left( \frac{1}{2} \right)^n$ ($n\in \mathbf{N}^*, 1\leqslant n\leqslant 5$) written on them. Now, Jia and Yi each randomly draw one card and then try to guess who has the larger number based... | \frac{7}{8} | 0.4375 | 7,822 | 7,346.285714 | 8,192 | |
Let $g(x)$ be a function defined for all positive real numbers such that $g(x) > 0$ for all $x > 0$ and
\[g(x + y) = \sqrt{g(xy) + 3}\] for all $x > y > 0.$ Determine $g(2023).$ | \frac{1 + \sqrt{13}}{2} | 0 | 6,971.0625 | -1 | 6,971.0625 | |
How many 9-digit numbers that are divisible by 5 can be formed by permuting the digits of the number 377353752? | 1120 | 0.25 | 5,520.125 | 3,288.25 | 6,264.083333 | |
Calculate the sum of $0.\overline{6}$ and $0.\overline{7}$ as a common fraction. | \frac{13}{9} | 1 | 1,999.875 | 1,999.875 | -1 | |
A monthly cell phone plan costs $30 per month, plus $10 cents per text message, plus $15 cents for each minute used over 25 hours, and an additional $5 for every gigabyte of data used over 15GB. In February, Emily sent 150 text messages, talked for 26 hours, and used 16GB of data. How much did she have to pay?
**A)** $... | 59.00 | 0.3125 | 696.625 | 642.4 | 721.272727 | |
How many ways can 6 people stand in a row such that persons A and B are not next to each other and neither is at either end of the row? | 96 | 0 | 7,635.375 | -1 | 7,635.375 | |
Let $f(x) = |x-1| + |x+1|$, where $x \in \mathbb{R}$.
(Ⅰ) Solve the inequality $f(x) \leq 4$;
(Ⅱ) If there exists a non-zero real number $b$ such that the inequality $f(x) \geq \frac{|2b+1| + |1-b|}{|b|}$ holds, find the maximum value of $x$ when $x$ is a negative number. | -1.5 | 0 | 6,627 | -1 | 6,627 | |
A rabbit and a hedgehog participated in a running race on a 550 m long circular track, both starting and finishing at the same point. The rabbit ran clockwise at a speed of 10 m/s and the hedgehog ran anticlockwise at a speed of 1 m/s. When they met, the rabbit continued as before, but the hedgehog turned around and ra... | 545 | 0 | 6,916.25 | -1 | 6,916.25 | |
Given an ellipse $$\frac {x^{2}}{a^{2}}$$ + $$\frac {y^{2}}{b^{2}}$$ = 1 with its right focus F, a line passing through the origin O intersects the ellipse C at points A and B. If |AF| = 2, |BF| = 4, and the eccentricity of the ellipse C is $$\frac {\sqrt {7}}{3}$$, calculate the area of △AFB. | 2\sqrt{3} | 0.25 | 7,891.5 | 6,990 | 8,192 | |
Say that an integer $A$ is yummy if there exist several consecutive non-negative integers, including $A$, that add up to 2023. What is the smallest yummy integer? | 1011 | 0 | 7,281.625 | -1 | 7,281.625 | |
In triangle $PQR,$ $PQ = 25$ and $QR = 10.$ Find the largest possible value of $\tan B.$ | \frac{2}{\sqrt{21}} | 0 | 8,192 | -1 | 8,192 | |
Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$? | 4 | 1. **Define the sets and probabilities**: Let $A$ be the set of all configurations where the balls are distributed as $3{-}5{-}4{-}4{-}4$ among the bins, and let $B$ be the set of all configurations where the balls are distributed as $4{-}4{-}4{-}4{-}4$. Define $p = \frac{|A|}{N}$ and $q = \frac{|B|}{N}$, where $N$ is ... | 0 | 5,345.625 | -1 | 5,345.625 |
Given that 216 sprinters enter a 100-meter dash competition, and the track has 6 lanes, determine the minimum number of races needed to find the champion sprinter. | 43 | 0.5 | 6,369.3125 | 4,680.25 | 8,058.375 | |
What is the smallest positive integer with eight positive odd integer divisors and sixteen positive even integer divisors? | 60 | 0 | 6,284.625 | -1 | 6,284.625 | |
What is the value of $\frac{2013^3-2 \cdot 2013^2 \cdot 2014+3 \cdot 2013 \cdot 2014^2-2014^3+1}{2013 \cdot 2014}$? | 2013 | 0.875 | 4,169.5625 | 4,096.857143 | 4,678.5 | |
Find the remainder when $6x^4-14x^3-4x^2+2x-26$ is divided by $2x - 6.$ | 52 | 1 | 3,210.125 | 3,210.125 | -1 | |
Calculate: $|\sqrt{3}-2|+(\pi -\sqrt{10})^{0}-\sqrt{12}$. | 3-3\sqrt{3} | 0.875 | 653.0625 | 638.642857 | 754 | |
If a positive integer \( N \) can be expressed as \( \lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor \) for some real number \( x \), then we say that \( N \) is "visible"; otherwise, we say that \( N \) is "invisible". For example, 8 is visible since \( 8 = \lfloor 1.5 \rfloor + \lfloor 2(1.5) \rfloor + \l... | 6034 | 0.1875 | 7,775.375 | 7,569 | 7,823 | |
Given that $689\Box\Box\Box20312 \approx 69$ billion (rounded), find the number of ways to fill in the three-digit number. | 500 | 0 | 8,192 | -1 | 8,192 | |
What is the sum of the first 10 odd positive integers? | 100 | 1 | 1,400.5625 | 1,400.5625 | -1 | |
$ABCD$ is a rectangle (see the accompanying diagram) with $P$ any point on $\overline{AB}$. $\overline{PS} \perp \overline{BD}$ and $\overline{PR} \perp \overline{AC}$. $\overline{AF} \perp \overline{BD}$ and $\overline{PQ} \perp \overline{AF}$. Then $PR + PS$ is equal to: | $AF$ | 1. **Identify Parallel Lines**: Since $\overline{PQ}$ and $\overline{BD}$ are both perpendicular to $\overline{AF}$, it follows that $\overline{PQ} \parallel \overline{BD}$.
2. **Angle Relationships**: Because $\overline{PQ} \parallel \overline{BD}$ and $\overline{AB}$ is a transversal, $\angle APQ = \angle ABD$. Sinc... | 0 | 7,261.625 | -1 | 7,261.625 |
Given $sinα+cosα=-\frac{{\sqrt{10}}}{5}, α∈(-\frac{π}{2},\frac{π}{2})$.
$(1)$ Find the value of $\tan \alpha$;
$(2)$ Find the value of $2\sin ^{2}\alpha +\sin \alpha \cos \alpha -1$. | \frac{1}{2} | 0.625 | 6,182.3125 | 4,976.5 | 8,192 | |
A lattice point is a point whose coordinates are integers. How many lattice points are on the boundary or inside the region bounded by \( y = |x| \) and \( y = -x^2 + 8 \)? | 33 | 0 | 7,415.3125 | -1 | 7,415.3125 | |
Given Lucy starts with an initial term of 8 in her sequence, where each subsequent term is generated by either doubling the previous term and subtracting 2 if a coin lands on heads, or halving the previous term and subtracting 2 if a coin lands on tails, determine the probability that the fourth term in Lucy's sequence... | \frac{3}{4} | 0.1875 | 7,339.5625 | 5,266 | 7,818.076923 | |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $c=2$ and $C=\frac{\pi}{3}$.
1. If the area of $\triangle ABC$ is $\sqrt{3}$, find $a$ and $b$.
2. If $\sin B = 2\sin A$, find the area of $\triangle ABC$. | \frac{4\sqrt{3}}{3} | 0 | 5,527.125 | -1 | 5,527.125 | |
The set of vectors $\mathbf{v}$ such that
\[\mathbf{v} \cdot \mathbf{v} = \mathbf{v} \cdot \begin{pmatrix} 10 \\ -40 \\ 8 \end{pmatrix}\]forms a solid in space. Find the volume of this solid. | 12348 \pi | 1 | 2,882.4375 | 2,882.4375 | -1 | |
A square piece of paper has a side length of 1. It is folded such that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{BC}$ intersects edge $\overline{AB}$ at point $E$. Given $C'D = \frac{1}{4}$, find the perimeter of triangle $\bigtriangleup AEC'$.
**A)** $\frac{25}{12}$
**B)** $\frac{33}{12... | \frac{10}{3} | 0 | 8,192 | -1 | 8,192 | |
In a novel that consists of 530 pages, each page number is printed once, starting from page 1 up to page 530. How many more 3's are printed than 7's throughout the book? | 100 | 0.0625 | 7,068.4375 | 5,238 | 7,190.466667 | |
Let $\alpha \neq 1$ be a complex number such that the distance from $\alpha^2$ to 1 is twice the distance from $\alpha$ to 1, while the distance from $\alpha^4$ to 1 is four times the distance from $\alpha$ to 1. Enter all possible values of $\alpha,$ separated by commas. | i \sqrt{3}, -i \sqrt{3} | 0 | 6,560.4375 | -1 | 6,560.4375 | |
A rectangle of size $1000 \times 1979$ is divided into cells. Into how many parts will it be divided if one diagonal is drawn in it? | 2978 | 0.625 | 6,112 | 5,132.6 | 7,744.333333 | |
Given the function $f(x)=\cos ( \sqrt {3}x+\phi)- \sqrt {3}\sin ( \sqrt {3}x+\phi)$, find the smallest positive value of $\phi$ such that $f(x)$ is an even function. | \frac{2\pi}{3} | 0.4375 | 6,888.1875 | 5,716.571429 | 7,799.444444 | |
In $\triangle ABC$, $BC=a$, $AC=b$, and $a$, $b$ are the roots of the equation $x^{2}-2 \sqrt{3}x+2=0$, $2\cos (A+B)=1$
$(1)$ Find the degree of angle $C$.
$(2)$ Find the length of $AB$. | \sqrt{10} | 1 | 3,757.75 | 3,757.75 | -1 | |
Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days f... | 84 | 1 | 2,198.4375 | 2,198.4375 | -1 | |
In triangle $ABC,$ the side lengths $a,$ $b,$ and $c$ satisfy $a + 2b + 2c = a^2$ and $a + 2b - 2c = -3.$ Find the largest angle of the triangle, in degrees.
Note: $a$ is the side length opposite $\angle A,$ etc. | 120^\circ | 0.5 | 6,523.1875 | 5,372.375 | 7,674 | |
Four fair eight-sided dice (with faces showing 1 to 8) are rolled. What is the probability that the sum of the numbers on the top faces equals 32? | \frac{1}{4096} | 0.9375 | 3,895.0625 | 3,608.6 | 8,192 | |
In triangle $DEF,$ $\cot D \cot F = \frac{1}{3}$ and $\cot E \cot F = \frac{1}{8}.$ Find $\tan F.$ | 12 + \sqrt{136} | 0 | 4,814.875 | -1 | 4,814.875 | |
For a science project, Sammy observed a chipmunk and squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide... | 48 | 1. **Define Variables:**
Let $x$ be the number of holes the chipmunk dug. Each hole contains 3 acorns, so the total number of acorns the chipmunk hid is $3x$.
Let $y$ be the number of holes the squirrel dug. Each hole contains 4 acorns, so the total number of acorns the squirrel hid is $4y$.
2. **Set Up the Equa... | 1 | 1,225.625 | 1,225.625 | -1 |
Gretchen has ten socks, two of each color: red, blue, green, yellow, and purple. She randomly draws six socks. What is the probability that she ends up with exactly two pairs, each of a different color? | \frac{4}{7} | 0.5625 | 6,450.25 | 5,912.222222 | 7,142 | |
Joe is studying a bacteria population. There are 20 bacteria present at 3:00 p.m. and the population doubles every 3 minutes. Assuming none of the bacteria die, how many bacteria are present at 3:15 p.m. the same day? | 640 | 1 | 1,695.9375 | 1,695.9375 | -1 | |
The graph of $xy = 4$ is a hyperbola. Find the distance between the foci of this hyperbola. | 4\sqrt{2} | 0.0625 | 4,437.5625 | 5,908 | 4,339.533333 | |
The side lengths of a triangle are 14 cm, 48 cm and 50 cm. How many square centimeters are in the area of the triangle? | 336 | 1 | 2,165.0625 | 2,165.0625 | -1 | |
Given sin(x + $\frac{π}{4}$) = $\frac{1}{3}$, find the value of sin4x - 2cos3xsinx = ___. | -\frac{7}{9} | 0.8125 | 6,742.5625 | 6,408.076923 | 8,192 | |
Given that $\alpha \in \left(\frac{\pi}{2}, \pi\right)$ and $\sin \alpha = \frac{4}{5}$, calculate the value of $\sin 2\alpha$. | -\frac{24}{25} | 1 | 2,031.25 | 2,031.25 | -1 | |
Five people are at a party. Each pair of them are friends, enemies, or frenemies (which is equivalent to being both friends and enemies). It is known that given any three people $A, B, C$ : - If $A$ and $B$ are friends and $B$ and $C$ are friends, then $A$ and $C$ are friends; - If $A$ and $B$ are enemies and $B$ and $... | 17 | If $A$ and $B$ are frenemies, then regardless of whether another person $C$ is friends or enemies with $A$, $C$ will have to be frenemies with $B$ and vice versa. Therefore, if there is one pair of frenemies then all of them are frenemies with each other, and there is only one possibility. If there are no frenemies, th... | 0 | 8,192 | -1 | 8,192 |
In triangle $ABC$, the measure of $\angle A$ is $86$ degrees. The measure of $\angle B$ is $22$ degrees more than three times the measure of $\angle C$. What is the measure, in degrees, of $\angle C$? | 18 \text{ degrees} | 1 | 1,193.625 | 1,193.625 | -1 |
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