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 |
|---|---|---|---|---|---|---|
If $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are unit vectors, then find the largest possible value of
\[
\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{a} - \mathbf{d}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{d}\|^2 + \|\mathbf{c} - \mathbf{d}\|^2.
\] | 16 | 0.3125 | 7,203.5625 | 5,205.2 | 8,111.909091 | |
The photographer wants to arrange three boys and three girls in a row such that a boy or a girl could be at each end, and the rest alternate in the middle, calculate the total number of possible arrangements. | 72 | 0.6875 | 5,074.3125 | 3,788.454545 | 7,903.2 | |
A zoo houses five different pairs of animals, each pair consisting of one male and one female. To maintain a feeding order by gender alternation, if the initial animal fed is a male lion, how many distinct sequences can the zookeeper follow to feed all the animals? | 2880 | 0.75 | 4,771.875 | 3,915 | 7,342.5 | |
The bacteria in a jar triple every 20 seconds. After three minutes, there are 275,562 bacteria in the jar. How many were in the jar at the beginning of the experiment? | 14 | 1 | 2,040.5625 | 2,040.5625 | -1 | |
Calculate the area of the smallest square that can completely contain a circle with a radius of 7 units. | 196 | 0.875 | 1,470.1875 | 1,445.285714 | 1,644.5 | |
Compute $(2 \cos 20^\circ + 2i \sin 20^\circ)^6.$ Enter your answer in rectangular form. | -32 + 32i \sqrt{3} | 0 | 5,319.125 | -1 | 5,319.125 | |
On a square \(ABCD\), a line segment \(BE\) is drawn such that point \(E\) lies on the side \(CD\). The perimeter of triangle \(BCE\) is three-quarters of the perimeter of the square \(ABCD\). The ratio of lengths \(CE : CD\) is \(\lambda : 1\). What is the value of \(960 \times \lambda\)? | 720 | 1 | 2,973.375 | 2,973.375 | -1 | |
When $(a-b)^n,n\ge2,ab\ne0$, is expanded by the binomial theorem, it is found that when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals: | 2k+1 | 1. **Substitute $a$ with $kb$:** Given $a = kb$, substitute into $(a-b)^n$:
\[
(a-b)^n = (kb - b)^n = (b(k-1))^n
\]
2. **Apply the Binomial Theorem:** The Binomial Theorem states that
\[
(x+y)^n = \sum_{i=0}^n \binom{n}{i} x^{n-i} y^i
\]
Applying this to $(b(k-1))^n$, we get:
\[
(b(k-1))^n =... | 0.625 | 6,708.8125 | 6,007.8 | 7,877.166667 |
The $5G$ technology is very important to society and the country. From a strategic perspective, the industry generally defines it as the fourth industrial revolution after the steam engine revolution, the electrical revolution, and the computer revolution. A certain technology group produces two core components of $5G$... | 6.684 | 0.6875 | 4,908.5 | 4,298.727273 | 6,250 | |
Let $P$ and $A$ denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Find the largest possible integral value of $\frac{P^{2}}{A}$. | 45 | Assume WLOG that the side lengths of the triangle are pairwise coprime. Then they can be written as $m^{2}-n^{2}, 2mn, m^{2}+n^{2}$ for some coprime integers $m$ and $n$ where $m>n$ and $mn$ is even. Then we obtain $\frac{P^{2}}{A}=\frac{4m(m+n)}{n(m-n)}$. But $n, m-n, m, m+n$ are all pairwise coprime so for this to be... | 0.1875 | 7,544.8125 | 4,740.333333 | 8,192 |
Given \( n = 1990 \), find the value of
\[
\frac{1}{2^{n}}\left(1 - 3 \binom{n}{2} + 3^{2} \binom{n}{4} - 3^{3} \binom{n}{6} + \cdots + 3^{994} \binom{n}{1988} - 3^{995} \binom{n}{1990}\right).
\] | -\frac{1}{2} | 0.75 | 5,193.0625 | 4,744.833333 | 6,537.75 | |
Given that $0 < α < \dfrac {π}{2}$, $\sin α= \dfrac {4}{5}$, and $\tan (α-β)=- \dfrac {1}{3}$, find the value of $\tan β$ and compute the expression $\dfrac {\sin (2β- \dfrac {π}{2})\cdot \sin (β+π)}{ \sqrt {2}\cos (β+ \dfrac {π}{4})}$. | \dfrac {6}{5} | 0.8125 | 5,132.3125 | 4,749.230769 | 6,792.333333 | |
What is $\frac{1}{4}\%$ of 120? Express your answer as a decimal. | .3 | 0.8125 | 1,839.625 | 1,851.384615 | 1,788.666667 | |
The numbers from 1 to 9 are placed in the cells of a \(3 \times 3\) table such that the sum of the numbers on one diagonal equals 7, and the sum on the other diagonal equals 21. What is the sum of the numbers in the five shaded cells? | 25 | 0.0625 | 7,259.625 | 8,192 | 7,197.466667 | |
Let $p,$ $q,$ $r,$ $x,$ $y,$ and $z$ be positive real numbers such that $p + q + r = 2$ and $x + y + z = 1$. Find the maximum value of:
\[\frac{1}{p + q} + \frac{1}{p + r} + \frac{1}{q + r} + \frac{1}{x + y} + \frac{1}{x + z} + \frac{1}{y + z}.\] | \frac{27}{4} | 0.125 | 8,045.625 | 7,021 | 8,192 | |
Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients $2$ and $-2,$ respectively. The graphs of both polynomials pass through the two points $(16,54)$ and $(20,53).$ Find $P(0) + Q(0).$ | 116 | Let $R(x)=P(x)+Q(x).$ Since the $x^2$-terms of $P(x)$ and $Q(x)$ cancel, we conclude that $R(x)$ is a linear polynomial.
Note that \begin{alignat*}{8} R(16) &= P(16)+Q(16) &&= 54+54 &&= 108, \\ R(20) &= P(20)+Q(20) &&= 53+53 &&= 106, \end{alignat*} so the slope of $R(x)$ is $\frac{106-108}{20-16}=-\frac12.$
It follow... | 0.8125 | 4,667.1875 | 4,208.846154 | 6,653.333333 |
Given that $P$ is a point on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. If $\frac{{\overrightarrow{PF_1} \cdot \overrightarrow{PF_2}}}{{|\overrightarrow{PF_1}| \cdot |\overrightarrow{PF_2}|}}=\frac{1}{2}$, then the area of $\triangle F_{1}... | 3\sqrt{3} | 0.6875 | 6,486.75 | 5,711.636364 | 8,192 | |
Compute without using a calculator: $\dfrac{9!}{6!3!}$ | 84 | 1 | 2,586.4375 | 2,586.4375 | -1 | |
[asy] pair A = (0,0), B = (7,4.2), C = (10, 0), D = (3, -5), E = (3, 0), F = (7,0); draw(A--B--C--D--cycle,dot); draw(A--E--F--C,dot); draw(D--E--F--B,dot); markscalefactor = 0.1; draw(rightanglemark(B, A, D)); draw(rightanglemark(D, E, C)); draw(rightanglemark(B, F, A)); draw(rightanglemark(D, C, B)); MP("A",(0,0),W);... | 4.2 | 0 | 4,017.1875 | -1 | 4,017.1875 | |
Let the even function $f(x)$ satisfy $f(x+6) = f(x) + f(3)$ for any $x \in \mathbb{R}$, and when $x \in (-3, -2)$, $f(x) = 5x$. Find the value of $f(201.2)$. | -14 | 0.375 | 7,410.8125 | 6,722 | 7,824.1 | |
Given a triangle \( ABC \) where \( |AB| = |AC| \) and \( \angle BAC = 80^\circ \). Inside the triangle, there is a point \( M \) such that \( \angle MBC = 30^\circ \) and \( \angle MCB = 10^\circ \). Find \( \angle AMC \). | 70 | 0.375 | 7,553.1875 | 6,488.5 | 8,192 | |
Let $p$ and $q$ be positive integers such that \[\frac{3}{5} < \frac{p}{q} < \frac{5}{8}\] and $q$ is as small as possible. What is $p+q$? | 21 | 0.875 | 6,133.4375 | 5,839.357143 | 8,192 | |
If the graph of the function $f(x) = (1-x^2)(x^2+ax+b)$ is symmetric about the line $x = -2$, then the maximum value of $f(x)$ is \_\_\_\_\_\_\_\_. | 16 | 0.5625 | 7,280.5625 | 6,628 | 8,119.571429 | |
Laura constructs a cone for an art project. The cone has a height of 15 inches and a circular base with a diameter of 8 inches. Laura needs to find the smallest cube-shaped box to transport her cone safely to the art gallery. What is the volume of this box, in cubic inches? | 3375 | 0.625 | 4,774.8125 | 4,520.2 | 5,199.166667 | |
Thomas has constant speeds for both running and walking. When a down-escalator is moving, Thomas can run down it in 15 seconds or walk down it in 30 seconds. One day, when the escalator was broken (and stationary), it took Thomas 20 seconds to run down it. How long, in seconds, would it take Thomas to walk down the bro... | 60 | 0.375 | 6,737.875 | 4,326.5 | 8,184.7 | |
A digital clock displays time in a 24-hour format (from 00:00 to 23:59). Find the largest possible sum of the digits in this time display. | 19 | 0 | 7,925.9375 | -1 | 7,925.9375 | |
A positive number $x$ has the property that $x\%$ of $x$ is $4$. What is $x$? | 20 |
We are given that $x\%$ of $x$ is $4$. We need to interpret and solve this equation mathematically.
1. **Understanding the percentage operation**: The expression "$x\%$ of $x$" translates to $x\% \times x$. Since $x\%$ means $x$ percent, which is $\frac{x}{100}$, we can rewrite the expression as:
\[
\frac{x}{10... | 1 | 1,233.125 | 1,233.125 | -1 |
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$. | n = 2^k |
We want to determine all positive integers \( n \) for which there exists an integer \( m \) such that \( 2^n - 1 \mid m^2 + 9 \).
To solve this problem, we start by expressing the divisibility condition explicitly:
\[
2^n - 1 \mid m^2 + 9 \quad \Rightarrow \quad m^2 + 9 = k(2^n - 1) \text{ for some integer } k.
\]
... | 0 | 8,192 | -1 | 8,192 |
Consider the region $A^{}_{}$ in the complex plane that consists of all points $z^{}_{}$ such that both $\frac{z^{}_{}}{40}$ and $\frac{40^{}_{}}{\overline{z}}$ have real and imaginary parts between $0^{}_{}$ and $1^{}_{}$, inclusive. Find the area of $A.$ | 1200 - 200 \pi | 0.125 | 8,030.9375 | 7,920 | 8,046.785714 | |
Christine must buy at least $45$ fluid ounces of milk at the store. The store only sells milk in $200$ milliliter bottles. If there are $33.8$ fluid ounces in $1$ liter, then what is the smallest number of bottles that Christine could buy? (You may use a calculator on this problem.) | 7 | 0.9375 | 3,609.75 | 3,304.266667 | 8,192 | |
The sum of four prime numbers $P,$ $Q,$ $P-Q,$ and $P+Q$ must be expressed in terms of a single letter indicating the property of the sum. | 17 | 0 | 7,798.9375 | -1 | 7,798.9375 | |
The equation \(x^{2}+5x+1=0\) has roots \(x_{1}\) and \(x_{2}\). Find the value of the expression
\[
\left(\frac{x_{1} \sqrt{6}}{1+x_{2}}\right)^{2}+\left(\frac{x_{2} \sqrt{6}}{1+x_{1}}\right)^{2}
\] | 220 | 0.4375 | 7,503.1875 | 6,617.571429 | 8,192 | |
The points $A$ , $B$ , $C$ , $D$ , and $E$ lie in one plane and have the following properties: $AB = 12, BC = 50, CD = 38, AD = 100, BE = 30, CE = 40$ .
Find the length of the segment $ED$ . | 74 | 0.375 | 6,328.1875 | 3,221.833333 | 8,192 | |
Given that the polynomial \(x^2 - kx + 24\) has only positive integer roots, find the average of all distinct possibilities for \(k\). | 15 | 0.9375 | 2,072.3125 | 1,664.333333 | 8,192 | |
Determine, with proof, the smallest positive integer \( n \) with the following property: For every choice of \( n \) integers, there exist at least two whose sum or difference is divisible by 2009. | 1006 | 0.5 | 6,312.0625 | 5,417.75 | 7,206.375 | |
An ellipse has foci at $F_1 = (0,2)$ and $F_2 = (3,0).$ The ellipse intersects the $x$-axis at the origin, and one other point. What is the other point of intersection? | \left( \frac{15}{4}, 0 \right) | 1 | 3,006.25 | 3,006.25 | -1 | |
Let $S$ be the set of integers of the form $2^{x}+2^{y}+2^{z}$, where $x, y, z$ are pairwise distinct non-negative integers. Determine the 100th smallest element of $S$. | 577 | S is the set of positive integers with exactly three ones in its binary representation. The number of such integers with at most $d$ total bits is \binom{d}{3}$, and noting that \binom{9}{3}=84$ and \binom{10}{3}=120$, we want the 16th smallest integer of the form $2^{9}+2^{x}+2^{y}$, where $y<x<9$. Ignoring the $2^{9}... | 0 | 8,192 | -1 | 8,192 |
Let $x_{1}, x_{2}, \ldots, x_{2022}$ be nonzero real numbers. Suppose that $x_{k}+\frac{1}{x_{k+1}}<0$ for each $1 \leq k \leq 2022$, where $x_{2023}=x_{1}$. Compute the maximum possible number of integers $1 \leq n \leq 2022$ such that $x_{n}>0$. | 1010 | Let the answer be $M$. If $M>1011$, there would exist two consecutive positive terms $x_{k}, x_{k+1}$ which contradicts the assumption that $x_{k}+\frac{1}{x_{k+1}}<0$. Thus, $M \leq 1011$. If $M=1011$, then the $2022 x_{i}$ s must alternate between positive and negative. WLOG, assume $x_{2 k-1}>0$ and $x_{2 k}<0$ for ... | 0 | 7,839.375 | -1 | 7,839.375 |
Compute the product of the roots of the equation \[x^3 - 12x^2 + 48x + 28 = 0.\] | -28 | 0.9375 | 4,442.0625 | 4,192.066667 | 8,192 | |
Given that \( x \) and \( y \) are non-zero real numbers and they satisfy \(\frac{x \sin \frac{\pi}{5} + y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5} - y \sin \frac{\pi}{5}} = \tan \frac{9 \pi}{20}\),
1. Find the value of \(\frac{y}{x}\).
2. In \(\triangle ABC\), if \(\tan C = \frac{y}{x}\), find the maximum value of \(\... | \frac{3}{2} | 0.25 | 7,918.3125 | 7,097.25 | 8,192 | |
How many ways are there to arrange the numbers \(\{1,2,3,4,5,6,7,8\}\) in a circle so that every two adjacent elements are relatively prime? Consider rotations and reflections of the same arrangement to be indistinguishable. | 36 | Note that 6 can only be adjacent to 1, 5, and 7, so there are \(\binom{3}{2}=3\) ways to pick its neighbors. Since each of 1, 5, and 7 is relatively prime to every number in \(\{1,2,3,4,5,6,7,8\}\) but itself (and hence can have arbitrary neighbors), without loss of generality suppose we have picked 1 and 5 as neighbor... | 0 | 8,066.9375 | -1 | 8,066.9375 |
Let $r$ and $s$ denote the two real roots of $x^2 - x \sqrt{5} + 1 = 0.$ Then determine $r^8 + s^8.$ | 47 | 0.9375 | 5,524.5625 | 5,346.733333 | 8,192 | |
A polynomial $f \in \mathbb{Z}[x]$ is called splitty if and only if for every prime $p$, there exist polynomials $g_{p}, h_{p} \in \mathbb{Z}[x]$ with $\operatorname{deg} g_{p}, \operatorname{deg} h_{p}<\operatorname{deg} f$ and all coefficients of $f-g_{p} h_{p}$ are divisible by $p$. Compute the sum of all positive i... | \[ 693 \] | We claim that $x^{4}+a x^{2}+b$ is splitty if and only if either $b$ or $a^{2}-4 b$ is a perfect square. (The latter means that the polynomial splits into $(x^{2}-r)(x^{2}-s)$ ). Assuming the characterization, one can easily extract the answer. For $a=16$ and $b=n$, one of $n$ and $64-n$ has to be a perfect square. The... | 0 | 8,066.9375 | -1 | 8,066.9375 |
A positive integer divisor of $10!$ is chosen at random. Calculate the probability that the divisor chosen is a perfect square, expressed as a simplified fraction $\frac{m}{n}$, and find the sum of the numerator and denominator. | 10 | 0.6875 | 2,720.1875 | 2,924.909091 | 2,269.8 | |
A council consists of nine women and three men. During their meetings, they sit around a round table with the women in indistinguishable rocking chairs and the men on indistinguishable stools. How many distinct ways can the nine chairs and three stools be arranged around the round table for a meeting? | 55 | 0.125 | 7,152.9375 | 7,074.5 | 7,164.142857 | |
The cells of a $50 \times 50$ table are colored in $n$ colors such that for any cell, the union of its row and column contains cells of all $n$ colors. Find the maximum possible number of blue cells if
(a) $n=2$
(b) $n=25$. | 1300 | 0.0625 | 7,564.9375 | 7,888 | 7,543.4 | |
A certain type of ray, when passing through a glass plate, attenuates to $\text{a}\%$ of its original intensity for every $1 \mathrm{~mm}$ of thickness. It was found that stacking 10 pieces of $1 \mathrm{~mm}$ thick glass plates results in the same ray intensity as passing through a single $11 \mathrm{~mm}$ thick glass... | 19 | 0.125 | 7,700.625 | 6,695.5 | 7,844.214286 | |
What is the probability that each of 5 different boxes contains exactly 2 fruits when 4 identical pears and 6 different apples are distributed into the boxes? | 0.0074 | 0 | 8,124.3125 | -1 | 8,124.3125 | |
The operation $\#$ is defined as $a \# b = a + \frac{a}{b}$. What is the value of $6 \# 2$? | 9 | 1 | 1,656.5 | 1,656.5 | -1 | |
Let \( n \) be a positive integer. Given a real number \( x \), let \( \lfloor x \rfloor \) be the greatest integer less than or equal to \( x \). For example, \( \lfloor 2.4 \rfloor = 2 \), \( \lfloor 3 \rfloor = 3 \), and \( \lfloor \pi \rfloor = 3 \). Define a sequence \( a_1, a_2, a_3, \ldots \) where \( a_1 = n \)... | 126 | 0 | 8,192 | -1 | 8,192 | |
Let the function $f(x) = (x-3)^3 + x - 1$, and $\{a_n\}$ be an arithmetic sequence with a non-zero common difference. If $f(a_1) + f(a_2) + \ldots + f(a_7) = 14$, then calculate the value of $a_1 + a_2 + \ldots + a_7$. | 21 | 0.5 | 6,479.3125 | 4,766.625 | 8,192 | |
Given points $S$, $A$, $B$, $C$ on the surface of a sphere $O$, $SA \perp$ plane $ABC$, $AB \perp BC$, $SA = AB = 1$, $BC = \sqrt{2}$, calculate the surface area of sphere $O$. | 4\pi | 0.625 | 6,264.8125 | 5,108.5 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it is given that $c\cos B + b\cos C = 3a\cos B$.
$(1)$ Find the value of $\cos B$;
$(2)$ If $\overrightarrow{BA} \cdot \overrightarrow{BC} = 2$, find the minimum value of $b$. | 2\sqrt{2} | 0.8125 | 4,222.6875 | 3,404 | 7,770.333333 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}|=3$, $|\overrightarrow{b}|=2\sqrt{3}$, and $\overrightarrow{a} \perp (\overrightarrow{a}+\overrightarrow{b})$, find the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$. | -3 | 0.75 | 5,025.5625 | 4,435.25 | 6,796.5 | |
Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=28,$ find the smallest possible value of $|w^3+z^3|.$ | 80 | 0.375 | 6,841.9375 | 6,252.5 | 7,195.6 | |
Given a geometric sequence {a_n} satisfies a_1 = 3, and a_1 + a_3 + a_5 = 21, find the value of a_3 + a_5 + a_7. | 42 | 1 | 3,193.3125 | 3,193.3125 | -1 | |
In the diagram below, $\overline{AB}\parallel \overline{CD}$ and $\angle AXF= 118^\circ$. Find $\angle FYD$.
[asy]
unitsize(1inch);
pair A,B,C,D,X,Y,EE,F;
A = (0,0);
B=(1,0);
C = (0,0.8);
D=(1,0.8);
EE = (0.35,-0.3);
F = (0.8,1.1);
draw(EE--F);
draw(A--B);
draw(C--D);
dot(A);
dot(B);
dot(C);
dot(D);
dot(EE);
dot(F);
... | 62^\circ | 0.5625 | 4,527.6875 | 3,530.777778 | 5,809.428571 | |
How many four-digit positive integers are multiples of 7? | 1286 | 0.9375 | 3,285.8125 | 2,958.733333 | 8,192 | |
Lawrence runs \(\frac{d}{2}\) km at an average speed of 8 minutes per kilometre.
George runs \(\frac{d}{2}\) km at an average speed of 12 minutes per kilometre.
How many minutes more did George run than Lawrence? | 104 | 0 | 450.625 | -1 | 450.625 | |
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at $1\!:\!00$ PM and finishes the second task at $2\!:\!40$ PM. When does she finish the third task? | 3:30 PM | 1. **Identify the total time taken for two tasks**: Marie finishes the second task at $2\!:\!40$ PM and she started the first task at $1\!:\!00$ PM. The total time taken for two tasks is from $1\!:\!00$ PM to $2\!:\!40$ PM.
- Convert this time span into minutes:
\[
2\!:\!40\, \text{PM} - 1\!:\!00\, \text{... | 0 | 1,558.625 | -1 | 1,558.625 |
Note that if the product of any two distinct members of {1,16,27} is increased by 9, the result is the perfect square of an integer. Find the unique positive integer $n$ for which $n+9,16n+9,27n+9$ are also perfect squares. | 280 | 0.4375 | 7,561.6875 | 6,751.285714 | 8,192 | |
Given a line $y=-x+1$ and an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), they intersect at points A and B.
(1) If the eccentricity of the ellipse is $\frac{\sqrt{3}}{3}$ and the focal distance is 2, find the length of the segment AB.
(2) (For Liberal Arts students) If segment OA is perpendicular to s... | \sqrt{6} | 0.125 | 7,848.8125 | 6,444.5 | 8,049.428571 | |
On Halloween Casper ate $\frac{1}{3}$ of his candies and then gave $2$ candies to his brother. The next day he ate $\frac{1}{3}$ of his remaining candies and then gave $4$ candies to his sister. On the third day he ate his final $8$ candies. How many candies did Casper have at the beginning? | 57 | Let $x$ represent the total number of candies Casper had at the beginning.
1. **First Day:**
- Casper ate $\frac{1}{3}$ of his candies, so he had $\frac{2}{3}x$ candies left.
- After giving $2$ candies to his brother, he had $\frac{2}{3}x - 2$ candies remaining.
2. **Second Day:**
- Casper ate $\frac{1}{3}$ ... | 0 | 4,057.75 | -1 | 4,057.75 |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a^2 = b(b + c)$. Find the value of $\frac{B}{A}$. | \frac{1}{2} | 0.8125 | 5,204.25 | 4,514.769231 | 8,192 | |
A box contains 5 white balls and 6 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate in color starting with a black ball? | \frac{1}{462} | 0.5625 | 6,993.9375 | 6,062.111111 | 8,192 | |
The storage capacity of two reservoirs, A and B, changes over time. The relationship between the storage capacity of reservoir A (in hundred tons) and time $t$ (in hours) is: $f(t) = 2 + \sin t$, where $t \in [0, 12]$. The relationship between the storage capacity of reservoir B (in hundred tons) and time $t$ (in hours... | 6.721 | 0.25 | 7,120.8125 | 6,018.75 | 7,488.166667 | |
Given that $\cos(\frac{\pi}{6} - \alpha) = \frac{1}{3}$, determine the value of $\sin(\frac{5\pi}{6} - 2\alpha)$. | -\frac{7}{9} | 0.8125 | 6,006.375 | 5,502 | 8,192 | |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \ln \left(\left(e^{x^{2}}-\cos x\right) \cos \left(\frac{1}{x}\right)+\operatorname{tg}\left(x+\frac{\pi}{3}\right)\right)
$$ | \frac{1}{2} \ln (3) | 0 | 4,049.625 | -1 | 4,049.625 | |
Given an equilateral triangle $ABC$ with a circle of radius $3$ tangent to line $AB$ at $B$ and to line $AC$ at $C$, find the area of the circle that passes through vertices $A$, $B$, and $C$. | 36\pi | 0 | 7,122.3125 | -1 | 7,122.3125 | |
Solve the equations:<br/>$(1)2\left(x-1\right)^{2}=1-x$;<br/>$(2)4{x}^{2}-2\sqrt{3}x-1=0$. | \frac{\sqrt{3} - \sqrt{7}}{4} | 0 | 2,661.6875 | -1 | 2,661.6875 | |
Let *Revolution* $(x) = x^3 +Ux^2 +Sx + A$ , where $U$ , $S$ , and $A$ are all integers and $U +S + A +1 = 1773$ . Given that *Revolution* has exactly two distinct nonzero integer roots $G$ and $B$ , find the minimum value of $|GB|$ .
*Proposed by Jacob Xu*
<details><summary>Solution</summary>*Solution.* $\... | 392 | 0.9375 | 4,022.6875 | 3,744.733333 | 8,192 | |
There is a tram with a starting station A and an ending station B. A tram departs from station A every 5 minutes towards station B, completing the journey in 15 minutes. A person starts cycling along the tram route from station B towards station A just as a tram arrives at station B. On his way, he encounters 10 trams ... | 50 | 0.0625 | 7,620.0625 | 6,147 | 7,718.266667 | |
For positive real numbers $a,$ $b,$ and $c,$ compute the maximum value of:
\[\frac{abc(a + b + c)}{(a + b)^3 (b + c)^3}.\] | \frac{1}{8} | 0 | 8,192 | -1 | 8,192 | |
Given the parabola $C$: $y^{2}=2px(p > 0)$ with focus $F$ and passing through point $A(1,-2)$.
$(1)$ Find the equation of the parabola $C$;
$(2)$ Draw a line $l$ through $F$ at an angle of $45^{\circ}$, intersecting the parabola $C$ at points $M$ and $N$, with $O$ being the origin. Calculate the area of $\triangle ... | 2\sqrt{2} | 1 | 4,999.9375 | 4,999.9375 | -1 | |
Find all roots of the equation \((x-a)(x-b)=(x-c)(x-d)\), given that \(a+d=b+c=2016\) and \(a \neq c\) (numerical values are not given). | 1008 | 0.8125 | 5,791.625 | 5,237.692308 | 8,192 | |
Petya inscribed two squares in a right triangle with sides 3, 4, and 5. One vertex of the first square coincides with the right-angle vertex of the triangle, while one side of the second square lies on the hypotenuse. Petya found the side lengths of each square, expressed their ratio as an irreducible fraction, and fou... | 19 | 0 | 7,838.375 | -1 | 7,838.375 | |
Given that the two foci of an ellipse and the endpoints of its minor axis precisely form the four vertices of a square, calculate the eccentricity of the ellipse. | \frac{\sqrt{2}}{2} | 0 | 4,661.25 | -1 | 4,661.25 | |
How many license plates consist of 2 letters followed by 2 digits, if one of the digits must be odd and the other must be even? | 33,\!800 | 0 | 3,091 | -1 | 3,091 | |
Find the minimum value of the expression \(\frac{13 x^{2}+24 x y+13 y^{2}+16 x+14 y+68}{\left(9-x^{2}-8 x y-16 y^{2}\right)^{5 / 2}}\). Round the answer to the nearest hundredth if needed. | 0.26 | 0 | 8,192 | -1 | 8,192 | |
An assortment of 200 pencils is sold through a catalog for $\$19.90$. Shipping is an additional $\$6.95$. Including the charges for both the pencils and the shipping, what is the average cost, in cents, for each pencil? Express your answer rounded to the nearest whole number. | 13 | 0.6875 | 5,270.6875 | 4,172.727273 | 7,686.2 | |
Find all $x$ such that $x^2+5x<6$. Express your answer in interval notation. | (-6, 1) | 1 | 1,543.1875 | 1,543.1875 | -1 | |
Given $A = 30^\circ$ and $B = 60^\circ$, calculate the value of $(1+\tan A)(1+\tan B)$. | 2 + \frac{4\sqrt{3}}{3} | 0 | 6,088.4375 | -1 | 6,088.4375 | |
Arrange the numbers $2011, \sqrt{2011}, 2011^{2}$ in increasing order. | \sqrt{2011}, 2011, 2011^{2} | Since $2011^{2}=4044121$ and $\sqrt{2011} \approx 44.8$, then the list of numbers in increasing order is $\sqrt{2011}, 2011, 2011^{2}$. (If $n$ is a positive integer with $n>1$, then $n^{2}>n$ and $\sqrt{n}<n$, so the list $\sqrt{n}, n, n^{2}$ is always in increasing order.) | 0 | 2,977.375 | -1 | 2,977.375 |
Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, what is the largest integer value of $x$ such that the area of this new tri... | 22 | 0.1875 | 7,793.75 | 7,325.333333 | 7,901.846154 | |
In a kingdom of animals, tigers always tell the truth, foxes always lie, and monkeys sometimes tell the truth and sometimes lie. There are 100 animals of each kind, divided into 100 groups, with each group containing exactly 2 animals of one kind and 1 animal of another kind. After grouping, Kung Fu Panda asked each an... | 76 | 0 | 8,192 | -1 | 8,192 | |
in a right-angled triangle $ABC$ with $\angle C=90$ , $a,b,c$ are the corresponding sides.Circles $K.L$ have their centers on $a,b$ and are tangent to $b,c$ ; $a,c$ respectively,with radii $r,t$ .find the greatest real number $p$ such that the inequality $\frac{1}{r}+\frac{1}{t}\ge p(\frac{1}{a}+\frac{1}{b... | \sqrt{2} + 1 | 0 | 7,157.25 | -1 | 7,157.25 | |
Let $x, y, z$ be real numbers satisfying $$\begin{aligned} 2 x+y+4 x y+6 x z & =-6 \\ y+2 z+2 x y+6 y z & =4 \\ x-z+2 x z-4 y z & =-3 \end{aligned}$$ Find $x^{2}+y^{2}+z^{2}$. | 29 | We multiply the first, second, and third equations by $\frac{1}{2},-\frac{1}{2}$, and -1 , respectively, then add the three resulting equations. This gives $x y+x z+y z=-2$. Doing the same with the coefficients $-1,2$, and 3 gives $x+y+z=5$, from which $(x+y+z)^{2}=25$. So $x^{2}+y^{2}+z^{2}=25-2 \cdot-2=29$. | 0 | 8,192 | -1 | 8,192 |
During the first year, ABC's stock price starts at $ \$100 $ and increases $ 100\% $. During the second year, its stock price goes down $ 25\% $ from its price at the end of the first year. What is the price of the stock, in dollars, at the end of the second year? | \$150 | 1 | 1,374.9375 | 1,374.9375 | -1 | |
Given an ellipse $C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, whose left and right foci are $F_{1}$ and $F_{2}$ respectively, and the top vertex is $B$. If the perimeter of $\triangle BF_{1}F_{2}$ is $6$, and the distance from point $F_{1}$ to the line $BF_{2}$ is $b$.
$(1)$ Find the equation of ell... | 14 | 0.3125 | 7,683.375 | 6,564.4 | 8,192 | |
The mean of $5,8$ and $17$ is equal to the mean of $12$ and $y$. What is the value of $y$? | 8 | 1 | 1,118.875 | 1,118.875 | -1 | |
The areas of three squares are 16, 49 and 169. What is the average (mean) of their side lengths? | 8 | 1 | 1,611 | 1,611 | -1 | |
An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.) | 33 | 0.5 | 7,271.6875 | 6,351.375 | 8,192 | |
From the integers 1 to 2020, there are a total of 1616 integers that are not multiples of 5. These 1616 numbers need to be divided into groups (each group may have a different number of elements), such that the difference (larger number minus smaller number) between any two numbers in the same group is a prime number. ... | 404 | 0 | 8,192 | -1 | 8,192 | |
Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times? | \frac{1}{7} | 1. **Understanding the Problem**: Tom, Dick, and Harry each flip a fair coin repeatedly until they get a head. We need to find the probability that all three stop flipping their coins after the same number of flips.
2. **Modeling the Coin Flips**: The probability that a single person flips his first head on the $n$-th... | 1 | 3,539.3125 | 3,539.3125 | -1 |
Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function that satisfies
\[ f(1) = 2, \]
\[ f(2) = 1, \]
\[ f(3n) = 3f(n), \]
\[ f(3n + 1) = 3f(n) + 2, \]
\[ f(3n + 2) = 3f(n) + 1. \]
Find how many integers \( n \leq 2014 \) satisfy \( f(n) = 2n \). | 127 | 0 | 8,188.375 | -1 | 8,188.375 | |
Positive integers $a$, $b$, and $c$ are such that $a<b<c$. Consider the system of equations
\[
2x + y = 2022 \quad \text{and} \quad y = |x-a| + |x-b| + |x-c|
\]
Determine the minimum value of $c$ such that the system has exactly one solution. | 1012 | 0.125 | 7,987.1875 | 6,553.5 | 8,192 | |
Let \( p, q, r, s, t, u, v, \) and \( w \) be real numbers such that \( pqrs = 16 \) and \( tuvw = 25 \). Find the minimum value of
\[
(pt)^2 + (qu)^2 + (rv)^2 + (sw)^2.
\] | 400 | 0 | 6,297.5625 | -1 | 6,297.5625 | |
The height of a ball when it is thrown off a cliff can be represented by the equation $h=45-7t-6t^2$, where $t$ is time in seconds. In how many seconds will the ball reach a height of 25 feet? | \frac43 | 1 | 3,714.875 | 3,714.875 | -1 | |
In a school, a dodgeball tournament was held. Each game was between two teams. A win awarded 15 points, a draw awarded 11 points, and a loss awarded no points. Each team played every other team exactly once. At the end of the tournament, it turned out that the total number of points scored was 1151. How many teams part... | 12 | 0.625 | 6,154.625 | 4,932.2 | 8,192 | |
The area of the rectangular region is | .088 m^2 | To find the area of a rectangular region, we use the formula:
\[ \text{Area} = \text{length} \times \text{width} \]
Given the dimensions of the rectangle are 0.4 meters and 0.22 meters, we calculate the area as follows:
1. Convert the decimal numbers to fractions to facilitate the multiplication:
\[ 0.4 = \frac{4}{... | 0 | 7,764.8125 | -1 | 7,764.8125 |
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