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 |
|---|---|---|---|---|---|---|
The square
$\begin{tabular}{|c|c|c|} \hline 50 & \textit{b} & \textit{c} \\ \hline \textit{d} & \textit{e} & \textit{f} \\ \hline \textit{g} & \textit{h} & 2 \\ \hline \end{tabular}$
is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries ... | 35 | 1. **Identify the products of rows, columns, and diagonals:** Given the multiplicative magic square, we know that the product of the numbers in each row, column, and diagonal must be the same. Let's denote this common product by $P$. Thus, we have:
- $50 \cdot b \cdot c = P$
- $d \cdot e \cdot f = P$
- $g \cdo... | 0.25 | 6,784.4375 | 4,183.25 | 7,651.5 |
The Yellers are coached by Coach Loud. The Yellers have 15 players, but three of them, Max, Rex, and Tex, refuse to play together in any combination. How many starting lineups (of 5 players) can Coach Loud make, if the starting lineup can't contain any two of Max, Rex, and Tex together? | 2277 | 0.625 | 5,912.75 | 4,545.2 | 8,192 | |
Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface ... | 749 | 1. **Identify the side lengths of the cubes**: The volumes of the cubes are given as $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units. The side lengths of these cubes are the cube roots of these volumes, which are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ units respectively.
2. **Determine the surface area of eac... | 0 | 7,798.3125 | -1 | 7,798.3125 |
For how many positive integers $x$ is $\log_{10}(x-40) + \log_{10}(60-x) < 2$? | 18 | 1. **Identify the domain of the function**:
The expression $\log_{10}(x-40) + \log_{10}(60-x)$ is defined only when both $x-40$ and $60-x$ are positive. This implies:
\[
x-40 > 0 \quad \text{and} \quad 60-x > 0
\]
Simplifying these inequalities, we get:
\[
x > 40 \quad \text{and} \quad x < 60
\... | 1 | 3,923.3125 | 3,923.3125 | -1 |
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League? | 91 | 1. **Calculate the cost of one T-shirt**:
Given that each T-shirt costs $5 more than a pair of socks, and a pair of socks costs $4, the cost of one T-shirt is:
\[
4 + 5 = 9 \text{ dollars}
\]
2. **Calculate the total cost for one member**:
Each member needs 2 pairs of socks and 2 T-shirts. The cost ... | 1 | 1,251.3125 | 1,251.3125 | -1 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 0} \frac{e^{4 x}-1}{\sin \left(\pi\left(\frac{x}{2}+1\right)\right)}$$ | -\frac{8}{\pi} | 0.5625 | 5,703.5625 | 3,768.111111 | 8,192 | |
If the graph of the line $y = ax + b$ passes through the points $(4,5)$ and $(8,17)$, what is $a - b$? | 10 | 1 | 1,326.75 | 1,326.75 | -1 | |
A box of chocolates in the shape of a cuboid was full of chocolates arranged in rows and columns. Míša ate some of them, and the remaining chocolates were rearranged to fill three entire rows completely, except for one space. Míša ate the remaining chocolates from another incomplete row. Then he rearranged the remainin... | 25 | 0.0625 | 7,518.5 | 2,898 | 7,826.533333 | |
Find the range of $f(x) = \sin^4 x - \sin x \cos x +\cos^4 x.$ | \left[ 0, \frac{9}{8} \right] | 0.8125 | 6,924.875 | 6,632.461538 | 8,192 | |
Calculate the difference $(2001 + 2002 + 2003 + \cdots + 2100) - (51 + 53 + 55 + \cdots + 149)$. | 200050 | 0.4375 | 7,431.75 | 6,454.285714 | 8,192 | |
To be continuous at $x = -1$, the value of $\frac{x^3 + 1}{x^2 - 1}$ is taken to be: | -\frac{3}{2} | 1. **Identify the point of discontinuity**: We start by noting that the function $\frac{x^3 + 1}{x^2 - 1}$ is undefined at $x = -1$ and $x = 1$ because the denominator becomes zero at these points. To find the value that makes the function continuous at $x = -1$, we need to simplify the expression and evaluate the limi... | 1 | 1,923.4375 | 1,923.4375 | -1 |
Evaluate the following expression: $$ 0 - 1 -2 + 3 - 4 + 5 + 6 + 7 - 8 + ... + 2000 $$ The terms with minus signs are exactly the powers of two.
| 1996906 | 0.8125 | 5,414.875 | 4,774 | 8,192 | |
Pick a random digit in the decimal expansion of $\frac{1}{99999}$. What is the probability that it is 0? | \frac{4}{5} | The decimal expansion of $\frac{1}{99999}$ is $0.\overline{00001}$. Therefore, the probability that a random digit is 0 is $\frac{4}{5}$. | 0.75 | 4,336.6875 | 4,304 | 4,434.75 |
Given that $a$, $b$, and $c$ are the sides opposite the angles $A$, $B$, and $C$ in $\triangle ABC$ respectively, and the equation $\sqrt{3}b\sin A - a\cos B - 2a = 0$ holds, then the measure of $\angle B$ is ______. | \frac{2\pi}{3} | 0.125 | 3,591.9375 | 4,567.5 | 3,452.571429 | |
Reading material: After studying square roots, Kang Kang found that some expressions containing square roots can be written as the square of another expression, such as $3+2\sqrt{2}=({1+\sqrt{2}})^2$. With his good thinking skills, Kang Kang made the following exploration: Let $a+b\sqrt{2}=({m+n\sqrt{2}})^2$ (where $a$... | 1+\sqrt{5} | 0.9375 | 3,916.75 | 3,868.333333 | 4,643 | |
The projection of $\begin{pmatrix} -8 \\ b \end{pmatrix}$ onto $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ is
\[-\frac{13}{5} \begin{pmatrix} 2 \\ 1 \end{pmatrix}.\]Find $b.$ | 3 | 0.9375 | 2,906.875 | 2,554.533333 | 8,192 | |
Let $(a_1, a_2, a_3, \ldots, a_{15})$ be a permutation of $(1, 2, 3, \ldots, 15)$ for which
$a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7 \mathrm{\ and \ } a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13} < a_{14} < a_{15}.$
Find the number of such permutations. | 3003 | 0.375 | 5,069.375 | 3,868.166667 | 5,790.1 | |
11 people were standing in line under the rain, each holding an umbrella. They stood so close together that the umbrellas touched each other. Once the rain stopped, people closed their umbrellas and maintained a distance of 50 cm between each other. By how many times did the length of the queue decrease? Assume people ... | 2.2 | 0.0625 | 5,653.6875 | 2,698 | 5,850.733333 | |
Let $ABC$ be a triangle with $AB=5$, $BC=6$, and $AC=7$. Let its orthocenter be $H$ and the feet of the altitudes from $A, B, C$ to the opposite sides be $D, E, F$ respectively. Let the line $DF$ intersect the circumcircle of $AHF$ again at $X$. Find the length of $EX$. | \frac{190}{49} | Since $\angle AFH=\angle AEH=90^{\circ}$, $E$ is on the circumcircle of $AHF$. So $\angle XEH=\angle HFD=\angle HBD$, which implies that $XE \parallel BD$. Hence $\frac{EX}{BD}=\frac{EY}{YB}$. Let $DF$ and $BE$ intersect at $Y$. Note that $\angle EDY=180^{\circ}-\angle BDF-\angle CDE=180^{\circ}-2 \angle A$, and $\angl... | 0 | 8,192 | -1 | 8,192 |
Points $A$, $B$, $C$ and $D$ are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square? [asy]
pair a=(0,1),b=(1,0),c=(0,-1),d=(-1,0);
draw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle);
draw(a--b--c--d--cycle);
label("$A$", a, N);
label("$B$", b, E);
label("$C$"... | 30 | 1 | 3,592.125 | 3,592.125 | -1 | |
The coefficient of \\(x^4\\) in the expansion of \\((1+x+x^2)(1-x)^{10}\\). | 135 | 0.875 | 5,966.5625 | 5,648.642857 | 8,192 | |
Given that a, b, and c are the sides opposite to angles A, B, and C respectively in triangle ABC, and c = 2, sinC(cosB - $\sqrt{3}$sinB) = sinA.
(1) Find the measure of angle C;
(2) If cosA = $\frac{2\sqrt{2}}{3}$, find the length of side b. | \frac{4\sqrt{2} - 2\sqrt{3}}{3} | 0 | 6,228.8125 | -1 | 6,228.8125 | |
What is the measure, in degrees, of one interior angle of a regular hexagon? | 120 | 1 | 1,436.5625 | 1,436.5625 | -1 | |
If $2^{3x} = 7$, evaluate $8^{x+1}$. | 56 | 1 | 2,003.25 | 2,003.25 | -1 | |
Find the sum of all positive integers such that their expression in base $7$ digits is the reverse of their expression in base $16$ digits. Express your answer in base $10$. | 58 | 0.3125 | 7,937.125 | 7,376.4 | 8,192 | |
When a student multiplied the number $66$ by the repeating decimal, \(1.\overline{ab}\), where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit number $ab?$ | 75 | 1. **Understanding the Problem:**
We are given a repeating decimal $1.\overline{ab}$, where $a$ and $b$ are digits, and the student mistakenly multiplied $66$ by $1.ab$ instead of $1.\overline{ab}$. The error in the calculation resulted in an answer that was $0.5$ less than the correct answer. We need to find the tw... | 1 | 2,432.8125 | 2,432.8125 | -1 |
Every morning when Tim wakes up, he groggily searches around his sock drawer and picks two socks randomly. If he has 10 gray-bottomed socks and 8 white-bottomed socks in his drawer, what is the probability that he picks a matching pair? | \frac{73}{153} | 1 | 2,567.125 | 2,567.125 | -1 | |
Evaluate the expression $a^2\cdot a^5$ if $a= 3$. | 2187 | 1 | 1,294.3125 | 1,294.3125 | -1 | |
Construct a five-digit number without repeated digits using 0, 1, 2, 3, and 4, with the condition that even and odd digits must be adjacent to each other. Find the total number of such five-digit numbers. | 20 | 0.0625 | 7,304.0625 | 4,427 | 7,495.866667 | |
Ten numbers have an average (mean) of 87. Two of those numbers are 51 and 99. What is the average of the other eight numbers? | 90 | Since 10 numbers have an average of 87, their sum is $10 \times 87 = 870$. When the numbers 51 and 99 are removed, the sum of the remaining 8 numbers is $870 - 51 - 99$ or 720. The average of these 8 numbers is $\frac{720}{8} = 90$. | 1 | 1,873.375 | 1,873.375 | -1 |
Let $a, b, c, x$ be reals with $(a+b)(b+c)(c+a) \neq 0$ that satisfy $$\frac{a^{2}}{a+b}=\frac{a^{2}}{a+c}+20, \quad \frac{b^{2}}{b+c}=\frac{b^{2}}{b+a}+14, \quad \text { and } \quad \frac{c^{2}}{c+a}=\frac{c^{2}}{c+b}+x$$ Compute $x$. | -34 | Note that $$\begin{aligned} \frac{a^{2}}{a+b}+\frac{b^{2}}{b+c}+\frac{c^{2}}{c+a}-\frac{a^{2}}{c+a}-\frac{b^{2}}{a+b}-\frac{c^{2}}{b+c} & =\frac{a^{2}-b^{2}}{a+b}+\frac{b^{2}-c^{2}}{b+c}+\frac{c^{2}-a^{2}}{c+a} \\ & =(a-b)+(b-c)+(c-a) \\ & =0 \end{aligned}$$ Thus, when we sum up all the given equations, we get that $20... | 0 | 8,192 | -1 | 8,192 |
Given an arithmetic-geometric sequence {$a_n$} with the first term as $\frac{4}{3}$ and a common ratio of $- \frac{1}{3}$. The sum of its first n terms is represented by $S_n$. If $A ≤ S_{n} - \frac{1}{S_{n}} ≤ B$ holds true for any n∈N*, find the minimum value of B - A. | \frac{59}{72} | 0.5625 | 7,779.5625 | 7,673.333333 | 7,916.142857 | |
A telephone number has the form \text{ABC-DEF-GHIJ}, where each letter represents
a different digit. The digits in each part of the number are in decreasing
order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore,
$D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive o... | 8 | 1. **Identify the constraints**: The telephone number is in the form $\text{ABC-DEF-GHIJ}$ where each segment has digits in decreasing order. Additionally, $D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd digits; and $A + B + C = 9$.
2. **Analyze the consecutive even and odd di... | 0.6875 | 5,837 | 4,766.545455 | 8,192 |
Using the digits 1, 2, and 3 to form four-digit numbers, where each digit must appear, and identical digits cannot be adjacent, how many such four-digit numbers are there? | 18 | 0.5 | 6,410.5 | 4,974.75 | 7,846.25 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. It is given that $(2c-a)\cos B = b\cos A$.
1. Find angle $B$.
2. If $b=6$ and $c=2a$, find the area of $\triangle ABC$. | 6\sqrt{3} | 0.9375 | 4,126.3125 | 3,855.266667 | 8,192 | |
A motorcyclist left point A for point B, and at the same time, a pedestrian left point B for point A. When they met, the motorcyclist took the pedestrian on his motorcycle to point A and then immediately went back to point B. As a result, the pedestrian reached point A 4 times faster than if he had walked the entire di... | 2.75 | 0 | 7,241.6875 | -1 | 7,241.6875 | |
Given $\cos (\frac{\pi}{4}-\frac{\theta}{2})=\frac{2}{3}$, find the value of $\sin \theta$. | -\frac{1}{9} | 0.9375 | 4,260.875 | 3,998.8 | 8,192 | |
Suppose in a right triangle where angle \( Q \) is at the origin and \( \cos Q = 0.5 \). If the length of \( PQ \) is \( 10 \), what is \( QR \)? | 20 | 0.25 | 3,786.6875 | 5,247.25 | 3,299.833333 | |
Consider a wardrobe that consists of $6$ red shirts, $7$ green shirts, $8$ blue shirts, $9$ pairs of pants, $10$ green hats, $10$ red hats, and $10$ blue hats. Additionally, you have $5$ ties in each color: green, red, and blue. Every item is distinct. How many outfits can you make consisting of one shirt, one pair of ... | 18900 | 0.0625 | 7,859.4375 | 5,011 | 8,049.333333 | |
In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random, looks at t... | \frac{5}{6} | If Bob predicts that he will draw two balls of the same color, then there are two possible plays: he draws from the same box, or he draws from a different box. If he draws from the same box, then in the $\frac{2}{3}$ chance that he originally picked box 1, he will always win, and in the $\frac{1}{3}$ chance that he pic... | 0 | 7,869.5625 | -1 | 7,869.5625 |
Given that a water tower stands 60 meters high and contains 150,000 liters of water, and a model of the tower holds 0.15 liters, determine the height of Liam's model tower. | 0.6 | 0.125 | 586.6875 | 580.5 | 587.571429 | |
A positive integer $N$ with three digits in its base ten representation is chosen at random, with each three digit number having an equal chance of being chosen. The probability that $\log_2 N$ is an integer is | \frac{1}{300} | 1. **Identify the Condition for $\log_2 N$ to be an Integer:**
To have $\log_2 N$ as an integer, $N$ must be a power of $2$, i.e., $N = 2^k$ for some integer $k$.
2. **Determine the Range of $k$:**
Since $N$ is a three-digit number, $100 \leq N \leq 999$. We need to find $k$ such that $100 \leq 2^k \leq 999$.
3... | 1 | 2,762 | 2,762 | -1 |
Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $P(x) = 2x^3-2ax^2+(a^2-81)x-c$ for some positive integers $a$ and $c$. Can you tell me the values of $a$ and $c$?"
After some calculations, Jon says, "There is more than one such polynomial."
Steve sa... | 440 | Since each of the roots is positive, the local maximum of the function must occur at a positive value of $x$. Taking $\frac{d}{dx}$ of the polynomial yields $6x^2-4ax+a^2-81$, which is equal to $0$ at the local maximum. Since this is a quadratic in $a$, we can find an expression for $a$ in terms of $x$. The quadratic f... | 0.375 | 7,303.875 | 6,118.333333 | 8,015.2 |
Three men, Alpha, Beta, and Gamma, working together, do a job in 6 hours less time than Alpha alone, in 1 hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together, to do the job. Then $h$ equals: | \frac{4}{3} | 1. **Define Variables:**
Let $A$, $B$, and $C$ be the number of hours needed by Alpha, Beta, and Gamma, respectively, to complete the job alone. Their respective rates of work are $\frac{1}{A}$, $\frac{1}{B}$, and $\frac{1}{C}$ jobs per hour.
2. **Set Up Equations:**
- When Alpha, Beta, and Gamma work together, ... | 0.9375 | 4,404.0625 | 4,151.533333 | 8,192 |
For every integer $k$ with $k > 0$, let $R(k)$ be the probability that
\[
\left[\frac{n}{k}\right] + \left[\frac{200 - n}{k}\right] = \left[\frac{200}{k}\right]
\]
for an integer $n$ randomly chosen from the interval $1 \leq n \leq 199$. What is the minimum possible value of $R(k)$ over the integers $k$ in the interval... | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
The vertices of an equilateral triangle lie on the hyperbola $xy = 4$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle? | 1728 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Given \\(\alpha \in (0^{\circ}, 90^{\circ})\\) and \\(\sin (75^{\circ} + 2\alpha) = -\frac{3}{5}\\), calculate \\(\sin (15^{\circ} + \alpha) \cdot \sin (75^{\circ} - \alpha)\\). | \frac{\sqrt{2}}{20} | 0 | 7,685.625 | -1 | 7,685.625 | |
On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k \ge 0$ lines. How many different values of $k$ are possible? | 5 | To determine the number of different values of $k$, the number of lines simultaneously tangent to both circles, we analyze the possible configurations of the two circles:
1. **Concentric Circles**: If the circles are concentric (i.e., they have the same center), no tangent lines can be drawn that are tangent to both c... | 0.9375 | 3,259.875 | 2,931.066667 | 8,192 |
Two students are preparing to register for the independent admission tests of Zhejiang University, Fudan University, and Shanghai Jiao Tong University, with the requirement that each person can choose at most two schools. Calculate the number of different registration results. | 36 | 0.75 | 4,822.5625 | 4,051.75 | 7,135 | |
Let $ABC$ be a triangle with $\angle BAC=60^\circ$ . Consider a point $P$ inside the triangle having $PA=1$ , $PB=2$ and $PC=3$ . Find the maximum possible area of the triangle $ABC$ . | \frac{3\sqrt{3}}{2} | 0 | 8,192 | -1 | 8,192 | |
It is given that \( a = 103 \times 97 \times 10009 \). Find \( a \). | 99999919 | 0.625 | 6,057.375 | 4,776.6 | 8,192 | |
Convert the base 2 number \(1011111010_2\) to its base 4 representation. | 23322_4 | 0.6875 | 5,347.75 | 4,054.909091 | 8,192 | |
The English alphabet, which has 26 letters, is randomly permuted. Let \(p_{1}\) be the probability that \(\mathrm{AB}, \mathrm{CD}\), and \(\mathrm{EF}\) all appear as contiguous substrings. Let \(p_{2}\) be the probability that \(\mathrm{ABC}\) and \(\mathrm{DEF}\) both appear as contiguous substrings. Compute \(\frac... | 23 | There are 23! ways to arrange the alphabet such that AB, CD, and EF all appear as contiguous substrings: treat each of these pairs of letters as a single merged symbol, which leaves 23 symbols to permute. Similarly, there are 22! ways to arrange the alphabet such that ABC and DEF both appear as contiguous substrings. T... | 0.3125 | 5,402.4375 | 4,381.2 | 5,866.636364 |
Let $f(x)$ be a function defined on $\mathbb{R}$ with a period of $2$, and for any real number $x$, it always holds that $f(x)-f(-x)=0$. When $x \in [0,1]$, $f(x)=-\sqrt{1-x^{2}}$. Determine the number of zeros of the function $g(x)=f(x)-e^{x}+1$ in the interval $[-2018,2018]$. | 2018 | 0 | 8,192 | -1 | 8,192 | |
The legs \( AC \) and \( CB \) of the right triangle \( ABC \) are 15 and 8, respectively. A circular arc with radius \( CB \) is drawn from center \( C \), cutting off a part \( BD \) from the hypotenuse. Find \( BD \). | \frac{128}{17} | 0.5625 | 7,648.25 | 7,225.333333 | 8,192 | |
Four spheres of radius 1 are placed so that each touches the other three. What is the radius of the smallest sphere that contains all four spheres? | \sqrt{\frac{3}{2}} + 1 | 0 | 5,437.0625 | -1 | 5,437.0625 | |
Let $z$ be a complex number with $|z| = \sqrt{2}.$ Find the maximum value of
\[|(z - 1)^2 (z + 1)|.\] | 4 \sqrt{2} | 0.5 | 7,019.25 | 5,846.5 | 8,192 | |
Let $C_1$ and $C_2$ be circles defined by $$
(x-10)^2+y^2=36
$$and $$
(x+15)^2+y^2=81,
$$respectively. What is the length of the shortest line segment $\overline{PQ}$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$? | 20 | 0.375 | 7,561 | 6,509.333333 | 8,192 | |
For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + 2x + 15\] has three distinct roots, which are also roots of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 75x + c.\] Determine the value of $f(-1)$. | -2773 | 0 | 4,063.75 | -1 | 4,063.75 | |
A driver left point A and headed towards point D, which are 100 km apart. The road from A to D passes through points B and C. At point B, the navigator showed that there were 30 minutes left to drive, and the driver immediately reduced their speed by 10 km/h. At point C, the navigator indicated that there were 20 km le... | 100 | 0.25 | 7,410.4375 | 5,065.75 | 8,192 | |
Two dice are thrown one after the other, and the numbers obtained are denoted as $a$ and $b$.
(Ⅰ) Find the probability that $a^2 + b^2 = 25$;
(Ⅱ) Given that the lengths of three line segments are $a$, $b$, and $5$, find the probability that these three line segments can form an isosceles triangle. | \dfrac{7}{18} | 0.1875 | 7,449.3125 | 7,100 | 7,529.923077 | |
The equation $x^2-4x+7=19$ has two solutions, $a$ and $b$, with $a\geq b$. What is the value of $2a+b$? | 10 | 1 | 2,738.375 | 2,738.375 | -1 | |
Find \( b \) if \( b \) is the remainder when \( 1998^{10} \) is divided by \( 10^{4} \). | 1024 | 0.75 | 6,225.875 | 5,570.5 | 8,192 | |
A circular table has 60 chairs around it. There are $N$ people seated at this table in such a way that the next person seated must sit next to someone. What is the smallest possible value for $N$? | 20 | To find the smallest possible value for $N$, we need to ensure that any new person seated must sit next to someone already seated. We will analyze the seating arrangement to determine the minimum $N$ that satisfies this condition.
1. **Understanding the seating arrangement**: The table has 60 chairs arranged in a circ... | 0.6875 | 6,504.5 | 5,737.454545 | 8,192 |
The curvature of a polyhedron is $2 \pi$ minus the sum of the face angles (the internal angles of the faces of the polyhedron). For example, a cube's face angle is $2 \pi - 3 \times \frac{\pi}{2} = \frac{\pi}{2}$, and its total curvature is $\frac{\pi}{2} \times 8 = 4 \pi$. What is the total curvature of a polyhedron w... | 4\pi | 0.375 | 7,175.3125 | 5,822 | 7,987.3 | |
The longest seminar session and the closing event lasted a total of $4$ hours and $45$ minutes plus $135$ minutes, plus $500$ seconds. Convert this duration to minutes and determine the total number of minutes. | 428 | 0 | 5,723.0625 | -1 | 5,723.0625 | |
Given the task of selecting 10 individuals to participate in a quality education seminar from 7 different schools, with the condition that at least one person must be chosen from each school, determine the total number of possible allocation schemes. | 84 | 0.0625 | 7,258.9375 | 2,111 | 7,602.133333 | |
In the rectangular coordinate system xOy, a polar coordinate system is established with the origin O of the rectangular coordinate system as the pole and the positive semi-axis of the x-axis as the polar axis. The parametric equations of the line l are given by $$\begin{cases} x= \frac {1}{2}+ \frac {1}{2}t \\ y= \frac... | \frac{8}{3} | 1 | 4,437.75 | 4,437.75 | -1 | |
Given the function $f(x)=\left\{\begin{array}{l}{x^2}-2ax+8, x\leq 1\\ x+\frac{4}{x}+2a, x>1\end{array}\right.$, if the minimum value of $f(x)$ is $f(1)$, find the value of the real number $a$. | \frac{5}{4} | 0.125 | 8,192 | 8,192 | 8,192 | |
Which type of conic section is described by the equation \[(x+5)^2 = (4y-3)^2 - 140?\]Enter "C" for circle, "P" for parabola, "E" for ellipse, "H" for hyperbola, and "N" for none of the above. | (\text{H}) | 0 | 3,271.4375 | -1 | 3,271.4375 | |
A line through the points $(5, -12)$ and $(k, 23)$ is parallel to the line $4x + 6y = 12$. What is the value of $k$? | -47.5 | 0.0625 | 2,474.25 | 1,958 | 2,508.666667 | |
In a regular octagon, there are two types of diagonals - one that connects alternate vertices (shorter) and another that skips two vertices between ends (longer). What is the ratio of the shorter length to the longer length? Express your answer as a common fraction in simplest form. | \frac{\sqrt{2}}{2} | 0 | 7,980.625 | -1 | 7,980.625 | |
Teresa the bunny has a fair 8-sided die. Seven of its sides have fixed labels $1,2, \ldots, 7$, and the label on the eighth side can be changed and begins as 1. She rolls it several times, until each of $1,2, \ldots, 7$ appears at least once. After each roll, if $k$ is the smallest positive integer that she has not rol... | 104 | Let $n=7$ and $p=\frac{1}{4}$. Let $q_{k}$ be the probability that $n$ is the last number rolled, if $k$ numbers less than $n$ have already been rolled. We want $q_{0}$ and we know $q_{n-1}=1$. We have the relation $$q_{k}=(1-p) \frac{k}{n-1} q_{k}+\left[1-(1-p) \frac{k+1}{n-1}\right] q_{k+1}$$ This rearranges to $$\le... | 0 | 8,192 | -1 | 8,192 |
Pascal's Triangle starting with row 1 has the sum of elements in row $n$ given by $2^{n-1}$. What is the sum of the interior numbers of the ninth row, considering interior numbers are all except the first and last numbers in the row? | 254 | 0.625 | 4,675.5625 | 2,565.7 | 8,192 | |
$f : \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $m+f(m+f(n+f(m))) = n + f(m)$ for every integers $m,n$. Given that $f(6) = 6$, determine $f(2012)$. | -2000 | 0.625 | 5,966.1875 | 4,640.8 | 8,175.166667 | |
How many ways are there to put 5 balls in 3 boxes if the balls are not distinguishable but the boxes are? | 21 | 0.9375 | 3,050.0625 | 2,707.266667 | 8,192 | |
Solve for $x$ in the equation $(-1)(2)(x)(4)=24$. | -3 | Since $(-1)(2)(x)(4)=24$, then $-8x=24$ or $x=\frac{24}{-8}=-3$. | 1 | 1,400.5 | 1,400.5 | -1 |
A person orders 4 pairs of black socks and some pairs of blue socks. The price of each pair of black socks is twice the price of each pair of blue socks. However, the colors were reversed on the order form, causing his expenditure to increase by 50%. What is the original ratio of the number of pairs of black socks to t... | 1: 4 | 0.3125 | 3,366.1875 | 3,321.4 | 3,386.545455 | |
Given that $| \overrightarrow{a}|=1$, $| \overrightarrow{b}|= \sqrt {2}$, and $\overrightarrow{a} \perp ( \overrightarrow{a}- \overrightarrow{b})$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac {\pi}{4} | 0.5 | 2,153.5625 | 1,950 | 2,357.125 | |
Calculate the sum of the squares of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 3 = 0,\] given that all roots are real and nonnegative. | 46 | 0 | 6,226.3125 | -1 | 6,226.3125 | |
Let \( x, y, z, u, v \in \mathbf{R}_{+} \). Determine the maximum value of \( f = \frac{xy + yz + zu + uv}{2x^2 + y^2 + 2z^2 + u^2 + 2v^2} \). | 1/2 | 0.0625 | 7,951.25 | 6,678 | 8,036.133333 | |
Jo adds up all the positive integers from 1 to 50. Kate does a similar thing with the first 50 positive integers; however, she first rounds every integer to its nearest multiple of 10 (rounding 5s up) and then adds the 50 values. What is the positive difference between Jo's sum and Kate's sum? | 25 | 0.4375 | 6,617.5625 | 5,098.285714 | 7,799.222222 | |
The Lucas sequence is the sequence 1, 3, 4, 7, 11, $\ldots$ where the first term is 1, the second term is 3 and each term after that is the sum of the previous two terms. What is the remainder when the $100^{\mathrm{th}}$ term of the sequence is divided by 8? | 7 | 0.625 | 5,854.875 | 4,592.8 | 7,958.333333 | |
Let \( f(n) \) be the number of 0's in the decimal representation of the positive integer \( n \). For example, \( f(10001123) = 3 \) and \( f(1234567) = 0 \). Find the value of
\[ f(1) + f(2) + f(3) + \ldots + f(99999) \] | 38889 | 0 | 8,166.25 | -1 | 8,166.25 | |
In hexagon $ABCDEF$, $AC$ and $CE$ are two diagonals. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $\frac{AM}{AC}=\frac{CN}{CE}=r$. Given that points $B$, $M$, and $N$ are collinear, find $r$. | \frac{\sqrt{3}}{3} | 0 | 7,392.6875 | -1 | 7,392.6875 | |
Determine the value of $\frac{3b^{-1} - \frac{b^{-1}}{3}}{b^2}$ when $b = \tfrac{1}{3}$. | 72 | 1 | 2,143.8125 | 2,143.8125 | -1 | |
Terrell usually lifts two 20-pound weights 12 times. If he uses two 15-pound weights instead, how many times must Terrell lift them in order to lift the same total weight? | 16 | 1 | 1,191.625 | 1,191.625 | -1 | |
Cube $ABCDEFGH,$ labeled as shown below, has edge length $2$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. Find the volume of the smaller of the two solids. | \frac{1}{6} | 0 | 8,192 | -1 | 8,192 | |
Determine all positive integers $n$ for which the equation $$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$ has an integer as a solution. | n=1 | If $n$ is even, $x^{n}+(2+x)^{n}+(2-x)^{n}>0$, so $n$ is odd. For $n=1$, the equation reduces to $x+(2+x)+(2-x)=0$, which has the unique solution $x=-4$. For $n>1$, notice that $x$ is even, because $x, 2-x$, and $2+x$ have all the same parity. Let $x=2 y$, so the equation reduces to $$y^{n}+(1+y)^{n}+(1-y)^{n}=0$$ Look... | 0.25 | 8,150.6875 | 8,026.75 | 8,192 |
Cindy leaves school at the same time every day. If she cycles at \(20 \ \text{km/h}\), she arrives home at 4:30 in the afternoon. If she cycles at \(10 \ \text{km/h}\), she arrives home at 5:15 in the afternoon. Determine the speed, in \(\text{km/h}\), at which she must cycle to arrive home at 5:00 in the afternoon. | 12 | 0.4375 | 5,019.4375 | 3,490.428571 | 6,208.666667 | |
The curve parameterized by $(x,y) = (2t + 4, 4t - 5)$ is a line, where $t$ is a real number. Find the equation of the line. Enter the equation in the form "$y = mx + b$". | y = 2x - 13 | 1 | 1,604.6875 | 1,604.6875 | -1 | |
Given that $-6 \leq x \leq -3$ and $1 \leq y \leq 5$, what is the largest possible value of $\frac{x+y}{x}$? | \frac{1}{6} | 0 | 6,708.6875 | -1 | 6,708.6875 | |
On the side \( BC \) of an equilateral triangle \( ABC \), points \( K \) and \( L \) are marked such that \( BK = KL = LC \). On the side \( AC \), point \( M \) is marked such that \( AM = \frac{1}{3} AC \). Find the sum of the angles \( \angle AKM \) and \( \angle ALM \). | 30 | 0.625 | 7,234 | 6,659.2 | 8,192 | |
Suppose the domain of function $y=f(x)$ is $D$. If for any $x_{1}, x_{2} \in D$, when $x_{1} + x_{2} = 2a$, it always holds that $f(x_{1}) + f(x_{2}) = 2b$, then the point $(a,b)$ is called the center of symmetry of the graph of the function $y=f(x)$. Investigate a center of symmetry for the function $f(x) = 2x + 3\cos... | -4035 | 0.6875 | 6,399.9375 | 6,237.636364 | 6,757 | |
Suppose $p(x)$ is a function such that $p(x) + (x^5+3x^3+9x) = (7x^3+24x^2+25x+1)$. Express $p(x)$ as a polynomial with the degrees of the terms in decreasing order. | -x^5+4x^3+24x^2+16x+1 | 0.875 | 2,274.9375 | 2,341.5 | 1,809 | |
For dessert, Melinda eats a spherical scoop of ice cream with diameter 2 inches. She prefers to eat her ice cream in cube-like shapes, however. She has a special machine which, given a sphere placed in space, cuts it through the planes $x=n, y=n$, and $z=n$ for every integer $n$ (not necessarily positive). Melinda cent... | 7+\frac{13 \pi}{3} | Note that if we consider the division of \mathbb{R}^{3}$ into unit cubes by the given planes, we only need to compute the sum of the probabilities that the ice cream scoop intersects each cube. There are three types of cubes that can be intersected: - The cube $0 \leq x, y, z \leq 1$ in which the center lies, as well a... | 0 | 8,192 | -1 | 8,192 |
Find the maximum value of the function $y=\frac{x}{{{e}^{x}}}$ on the interval $[0,2]$.
A) When $x=1$, $y=\frac{1}{e}$
B) When $x=2$, $y=\frac{2}{{{e}^{2}}}$
C) When $x=0$, $y=0$
D) When $x=\frac{1}{2}$, $y=\frac{1}{2\sqrt{e}}$ | \frac{1}{e} | 0 | 1,778.0625 | -1 | 1,778.0625 | |
Given that the odometer initially displays $12321$ miles, and after $4$ hours, the next higher palindrome is displayed, calculate the average speed in miles per hour during this $4$-hour period. | 25 | 1 | 3,027.3125 | 3,027.3125 | -1 | |
Consider the sequence $(a_k)_{k\ge 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k\ge 1$, if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then
\[a_{k+1} = \frac{m + 18}{n+19}.\]Determine the sum of all positive integers $j$ such that the rational number $a_j$ ... | 59 | We know that $a_{1}=\tfrac{t}{t+1}$ when $t=2020$ so $1$ is a possible value of $j$. Note also that $a_{2}=\tfrac{2038}{2040}=\tfrac{1019}{1020}=\tfrac{t}{t+1}$ for $t=1019$. Then $a_{2+q}=\tfrac{1019+18q}{1020+19q}$ unless $1019+18q$ and $1020+19q$ are not relatively prime which happens when $q+1$ divides $18q+1019$ o... | 0 | 7,857.8125 | -1 | 7,857.8125 |
There are 10 different natural numbers, their sum is 604, and these 10 numbers have the same sum of digits. What is the largest number among these 10 numbers? $\qquad | 109 | 0.3125 | 7,708.6875 | 6,645.4 | 8,192 |
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