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 $a,b,c$ satisfy the system of equations \begin{align*}b + c &= 12-3a \\
a+c &= -14 - 3b \\
a+b &= 7 - 3c,
\end{align*} what is $2a + 2b + 2c$? | 2 | 1 | 2,766.125 | 2,766.125 | -1 | |
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$? | 20 | 1. **Calculate the total amount paid by all three:**
Tom paid $105, Dorothy paid $125, and Sammy paid $175. Therefore, the total amount paid is:
\[
105 + 125 + 175 = 405
\]
2. **Determine the amount each should have paid:**
Since they agreed to split the costs evenly, each should have paid:
\[
\fr... | 1 | 1,373.875 | 1,373.875 | -1 |
Define the operation "" such that $ab = a^2 + 2ab - b^2$. Let the function $f(x) = x2$, and the equation $f(x) = \lg|x + 2|$ (where $x \neq -2$) has exactly four distinct real roots $x_1, x_2, x_3, x_4$. Find the value of $x_1 + x_2 + x_3 + x_4$. | -8 | 0.375 | 7,194.8125 | 6,123.833333 | 7,837.4 | |
Let $R = (8,6)$. The lines whose equations are $8y = 15x$ and $10y = 3x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\overline{PQ}$. The length of $PQ$ equals $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 67 | [asy] pointpen = black; pathpen = black+linewidth(0.7); pair R = (8,6), P = (32,60)/7, Q= (80,24)/7; D((0,0)--MP("x",(13,0),E),EndArrow(6)); D((0,0)--MP("y",(0,10),N),EndArrow(6)); D((0,0)--(10/(15/8),10),EndArrow(6)); D((0,0)--(13,13 * 3/10),EndArrow(6)); D(D(MP("P",P,NW))--D(MP("Q",Q),SE),linetype("4 4")); D(MP("R",R... | 1 | 3,021.0625 | 3,021.0625 | -1 |
What is the sum of the solutions of the equation $(3x+5)(2x-9) = 0$? Express your answer as a common fraction. | \frac{17}{6} | 1 | 1,673 | 1,673 | -1 | |
Consider that Henry's little brother now has 10 identical stickers and 5 identical sheets of paper. How many ways can he distribute all the stickers on the sheets of paper, if only the number of stickers on each sheet matters and no sheet can remain empty? | 126 | 0 | 7,638.25 | -1 | 7,638.25 | |
Squares $S_1$ and $S_2$ are inscribed in right triangle $ABC$, as shown in the figures below. Find $AC + CB$ if area $(S_1) = 441$ and area $(S_2) = 440$. | 462 | Let $\tan\angle ABC = x$. Now using the 1st square, $AC=21(1+x)$ and $CB=21(1+x^{-1})$. Using the second square, $AB=\sqrt{440}(1+x+x^{-1})$. We have $AC^2+CB^2=AB^2$, or \[441(x^2+x^{-2}+2x+2x^{-1}+2)=440(x^2+x^{-2}+2x+2x^{-1}+3).\] Rearranging and letting $u=x+x^{-1} \Rightarrow u^2 - 2 = x^2 + x^{-2}$ gives us $u^2+... | 0 | 8,192 | -1 | 8,192 |
If $y = 2x$ and $z = 2y$, then $x + y + z$ equals | $7x$ | Given the equations:
1. \( y = 2x \)
2. \( z = 2y \)
We need to find the expression for \( x + y + z \).
**Step 1:** Substitute the value of \( y \) from equation 1 into equation 2:
\[ z = 2y = 2(2x) = 4x \]
**Step 2:** Substitute the values of \( y \) and \( z \) into the expression \( x + y + z \):
\[ x + y + z = ... | 0 | 3,004.125 | -1 | 3,004.125 |
In the number \(2 * 0 * 1 * 6 * 0 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0,1,2,3,4,5,6,7,8\) (digits can repeat) such that the resulting 10-digit number is divisible by 18. How many ways can this be done? | 32805 | 0 | 7,530.625 | -1 | 7,530.625 | |
A parabola, given by the equation $y^{2}=2px (p > 0)$, has a focus that lies on the line $l$. This line intersects the parabola at two points, $A$ and $B$. A circle with the chord $AB$ as its diameter has the equation $(x-3)^{2}+(y-2)^{2}=16$. Find the value of $p$. | p = 2 | 0.375 | 7,205.3125 | 5,590.5 | 8,174.2 | |
Koalas absorb only $25\%$ of the fiber they eat. A koala absorbed 10.5 ounces of fiber in one day. How many ounces of fiber did he eat that day? | 42 | 1 | 911.6875 | 911.6875 | -1 | |
Given that the plane containing $\triangle PAD$ is perpendicular to the plane containing rectangle $ABCD$, and $PA = PD = AB = 2$, with $\angle APD = 60^\circ$. If points $P, A, B, C, D$ all lie on the same sphere, find the surface area of this sphere. | \frac{28}{3}\pi | 0.25 | 6,507.75 | 5,636.5 | 6,798.166667 | |
When rolling a certain unfair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6, the probability of obtaining face $F$ is greater than $1/6$, the probability of obtaining the face opposite is less than $1/6$, the probability of obtaining any one of the other four faces is $1/6$, and the sum of the numbers on oppos... | 29 | We have that the cube probabilities to land on its faces are $\frac{1}{6}$, $\frac{1}{6}$, $\frac{1}{6}$, $\frac{1}{6}$ ,$\frac{1}{6}+x$ ,$\frac{1}{6}-x$ we also know that the sum could be 7 only when the faces in each of the two tosses are opposite hence the probability to get a 7 is: \[4 \cdot \left(\frac{1}{6} \righ... | 0.625 | 5,264.625 | 3,912.6 | 7,518 |
It is known that $\tan\alpha$ and $\tan\beta$ are the two roots of the equation $x^2+6x+7=0$, and $\alpha, \beta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. What is the value of $\alpha + \beta$? | - \frac{3\pi}{4} | 0.875 | 4,700.4375 | 4,658.571429 | 4,993.5 | |
Given a point $P$ on a triangular piece of paper $ABC,\,$ consider the creases that are formed in the paper when $A, B,\,$ and $C\,$ are folded onto $P.\,$ Let us call $P$ a fold point of $\triangle ABC\,$ if these creases, which number three unless $P$ is one of the vertices, do not intersect. Suppose that $AB=36, AC=... | 597 | 0 | 8,175.25 | -1 | 8,175.25 | |
The Quill and Scroll is a stationery shop. Its stock and sales for May are listed in the table shown. What percent of its sales were not pens or pencils? \begin{tabular}{|l|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{Item}&\textbf{$\%$~of May Sales}\\\hline
Pens&38\\\hline
Pencils&35\\\hline
Other&?\\\hline
\end{tabular} | 27\% | 1 | 1,032.4375 | 1,032.4375 | -1 | |
For a given list of three numbers, the operation "changesum" replaces each number in the list with the sum of the other two. For example, applying "changesum" to \(3,11,7\) gives \(18,10,14\). Arav starts with the list \(20,2,3\) and applies the operation "changesum" 2023 times. What is the largest difference between t... | 18 | 0.375 | 7,516.1875 | 6,495.5 | 8,128.6 | |
Let $l$ some line, that is not parallel to the coordinate axes. Find minimal $d$ that always exists point $A$ with integer coordinates, and distance from $A$ to $l$ is $\leq d$ | \frac{1}{2\sqrt{2}} | 0 | 7,962.375 | -1 | 7,962.375 | |
A quadrilateral \(ABCD\) is inscribed in a circle with a diameter of 1, where \(\angle D\) is a right angle and \(AB = BC\).
Find the area of quadrilateral \(ABCD\) if its perimeter is \(\frac{9\sqrt{2}}{5}\). | \frac{8}{25} | 0 | 8,192 | -1 | 8,192 | |
In a row of 10 chairs, Mary and James each choose their seats at random but are not allowed to sit in the first or the last chair (chairs #1 and #10). What is the probability that they do not sit next to each other? | \frac{3}{4} | 0.5 | 6,480.375 | 4,768.75 | 8,192 | |
Investment funds A, B, and C claim that they can earn profits of 200%, 300%, and 500% respectively in one year. Tommy has $90,000 and plans to invest in these funds. However, he knows that only one of these funds can achieve its claim while the other two will close down. He has thought of an investment plan which can g... | 30000 | 0 | 7,523.5 | -1 | 7,523.5 | |
A rectangular cuboid \(A B C D-A_{1} B_{1} C_{1} D_{1}\) has \(A A_{1} = 2\), \(A D = 3\), and \(A B = 251\). The plane \(A_{1} B D\) intersects the lines \(C C_{1}\), \(C_{1} B_{1}\), and \(C_{1} D_{1}\) at points \(L\), \(M\), and \(N\) respectively. What is the volume of tetrahedron \(C_{1} L M N\)? | 2008 | 0.1875 | 8,080.5625 | 7,597.666667 | 8,192 | |
A point $Q$ is chosen inside $\triangle DEF$ such that lines drawn through $Q$ parallel to $\triangle DEF$'s sides decompose it into three smaller triangles $u_1$, $u_2$, and $u_3$, which have areas $3$, $12$, and $15$ respectively. Determine the area of $\triangle DEF$. | 30 | 0 | 8,073.6875 | -1 | 8,073.6875 | |
Suppose that $a * b$ means $3a-b.$ What is the value of $x$ if $2 * (5 * x)=1$ | 10 | 1. **Interpret the operation $*$**: Given that $a * b = 3a - b$, we need to find $x$ such that $2 * (5 * x) = 1$.
2. **Evaluate $5 * x$**: According to the operation definition, $5 * x = 3 \cdot 5 - x = 15 - x$.
3. **Substitute $5 * x$ into the equation**: Now substitute $15 - x$ for $(5 * x)$ in the equation $2 * (5... | 1 | 1,797.5 | 1,797.5 | -1 |
$(1)$ Calculate: $2^{-1}+|\sqrt{6}-3|+2\sqrt{3}\sin 45^{\circ}-\left(-2\right)^{2023}\cdot (\frac{1}{2})^{2023}$.
$(2)$ Simplify and then evaluate: $\left(\frac{3}{a+1}-a+1\right) \div \frac{{{a}^{2}}-4}{{{a}^{2}}+2a+1}$, where $a$ takes a suitable value from $-1$, $2$, $3$ for evaluation. | -4 | 1 | 4,450.6875 | 4,450.6875 | -1 | |
A circle with center P and radius 4 inches is tangent at D to a circle with center Q, located at a 45-degree angle from P. If point Q is on the smaller circle, what is the area of the shaded region? Express your answer in terms of $\pi$. | 48\pi | 0.0625 | 5,937.5625 | 3,076 | 6,128.333333 | |
Given an arithmetic sequence $\{a_n\}$, it is known that $\frac {a_{11}}{a_{10}} + 1 < 0$, and the sum of the first $n$ terms of the sequence, $S_n$, has a maximum value. Find the maximum value of $n$ for which $S_n > 0$. | 19 | 0.1875 | 7,856.25 | 6,594.333333 | 8,147.461538 | |
Two circles of radius \( r \) touch each other. Additionally, each of them is externally tangent to a third circle of radius \( R \) at points \( A \) and \( B \) respectively.
Find the radius \( r \), given that \( AB = 12 \) and \( R = 8 \). | 24 | 0.125 | 7,455.8125 | 6,663 | 7,569.071429 | |
Evaluate $16^{7/4}$. | 128 | 1 | 3,062.5625 | 3,062.5625 | -1 | |
If the average of six data points $a_1, a_2, a_3, a_4, a_5, a_6$ is $\bar{x}$, and the variance is 0.20, what is the variance of the seven data points $a_1, a_2, a_3, a_4, a_5, a_6, \bar{x}$? | \frac{6}{35} | 0.875 | 4,043.1875 | 3,653.285714 | 6,772.5 | |
The sum of the first $n$ terms of the arithmetic sequences ${a_n}$ and ${b_n}$ are $S_n$ and $T_n$ respectively. If $$\frac {S_{n}}{T_{n}}= \frac {2n+1}{3n+2}$$, find the value of $$\frac {a_{3}+a_{11}+a_{19}}{b_{7}+b_{15}}$$. | \frac{129}{130} | 0.5 | 5,915.6875 | 5,120.875 | 6,710.5 | |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}(1-\ln (1+\sqrt[3]{x}))^{\frac{x}{\sin ^{4} \sqrt[3]{x}}}
$$ | e^{-1} | 0 | 6,867.5 | -1 | 6,867.5 | |
There are 100 chips numbered from 1 to 100 placed in the vertices of a regular 100-gon in such a way that they follow a clockwise order. In each move, it is allowed to swap two chips located at neighboring vertices if their numbers differ by at most $k$. What is the smallest value of $k$ for which, by a series of such ... | 50 | 0.125 | 8,070.375 | 7,219 | 8,192 | |
Given the sequence $\left\{a_{n}\right\}$ satisfying: $a_{1}=1, a_{2}=2, a_{2k+1}=\frac{a_{2k}^{2}}{a_{2k-1}}$, and $a_{2k+2}=2a_{2k+1}-a_{2k}$ for $k \in \mathbf{N}^{*}$, find the last two digits of $a_{2022}$. | 32 | 0.4375 | 7,047.3125 | 5,575.571429 | 8,192 | |
Let $b = \pi/2010$. Find the smallest positive integer $m$ such that
\[2[\cos(b)\sin(b) + \cos(4b)\sin(2b) + \cos(9b)\sin(3b) + \cdots + \cos(m^2b)\sin(mb)]\]
is an integer. | 67 | 0 | 7,954.3125 | -1 | 7,954.3125 | |
Let $A B C$ be a triangle with $A B=2, C A=3, B C=4$. Let $D$ be the point diametrically opposite $A$ on the circumcircle of $A B C$, and let $E$ lie on line $A D$ such that $D$ is the midpoint of $\overline{A E}$. Line $l$ passes through $E$ perpendicular to $\overline{A E}$, and $F$ and $G$ are the intersections of t... | \frac{1024}{45} | Using Heron's formula we arrive at $[A B C]=\frac{3 \sqrt{15}}{4}$. Now invoking the relation $[A B C]=\frac{a b c}{4 R}$ where $R$ is the circumradius of $A B C$, we compute $R^{2}=\left(\frac{2 \cdot 3}{[A B C]^{2}}\right)=$ $\frac{64}{15}$. Now observe that $\angle A B D$ is right, so that $B D E F$ is a cyclic quad... | 0.1875 | 8,041.0625 | 7,387 | 8,192 |
Given the function $f(x)=2\sqrt{3}\sin ^{2}x+2\sin x\cos x-\sqrt{3}$, where $x\in\left[ \frac{\pi}{3}, \frac{11\pi}{24}\right]$.
(1) Find the range of the function $f(x)$.
(2) Suppose that the lengths of two sides of an acute-angled triangle $ABC$ are the maximum and minimum values of the function $f(x)$, respectivel... | \sqrt{2} | 0.375 | 7,827.3125 | 7,219.5 | 8,192 | |
Given that $b$ is an odd multiple of 9, find the greatest common divisor of $8b^2 + 81b + 289$ and $4b + 17$. | 17 | 0 | 7,567.5625 | -1 | 7,567.5625 | |
When you simplify $\sqrt[3]{24a^4b^6c^{11}}$, what is the sum of the exponents of the variables that are outside the radical? | 6 | 0.625 | 3,210.75 | 2,332.6 | 4,674.333333 | |
Consider three squares: $PQRS$, $TUVW$, and $WXYZ$, where each side of the squares has length $s=1$. $S$ is the midpoint of $WY$, and $R$ is the midpoint of $WU$. Calculate the ratio of the area of the shaded quadrilateral $PQSR$ to the sum of the areas of the three squares.
A) $\frac{1}{12}$
B) $\frac{1}{6}$
C) $\frac... | \frac{1}{12} | 0 | 8,192 | -1 | 8,192 | |
One can holds $12$ ounces of soda, what is the minimum number of cans needed to provide a gallon ($128$ ounces) of soda? | 11 | To find the minimum number of cans needed to provide at least one gallon (128 ounces) of soda, where each can holds 12 ounces, we need to calculate the smallest integer $n$ such that $12n \geq 128$.
1. **Calculate the exact number of cans needed if there were no remainder:**
\[
\frac{128}{12} \approx 10.67
\]... | 1 | 1,993.0625 | 1,993.0625 | -1 |
The random variable $X$ follows a normal distribution $N(1,4)$. Given that $P(X \geqslant 2) = 0.2$, calculate the probability that $0 \leqslant X \leqslant 1$. | 0.3 | 0.4375 | 6,715.375 | 6,056.714286 | 7,227.666667 | |
The circumference of a circle is 100. The diameter of this circle is equal to: | $\frac{100}{\pi}$ | 0 | 1,826.375 | -1 | 1,826.375 | |
If the line $l_{1}:ax+2y+6=0$ is parallel to the line $l_{2}:x+\left(a-1\right)y+\left(a^{2}-1\right)=0$, determine the value of $a$. | -1 | 0.4375 | 6,801.8125 | 7,122.142857 | 6,552.666667 | |
If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk? | \frac{x(x+2)(x+5)}{(x+1)(x+3)} | 1. **Understanding the problem**: We are given that $x$ cows produce $x+1$ cans of milk in $x+2$ days. We need to find out how many days it will take for $x+3$ cows to produce $x+5$ cans of milk.
2. **Calculate the daily milk production per cow**:
- The daily production per cow can be calculated by dividing the to... | 1 | 4,650.3125 | 4,650.3125 | -1 |
For how many real numbers $a^{}_{}$ does the quadratic equation $x^2 + ax^{}_{} + 6a=0$ have only integer roots for $x^{}_{}$? | 10 | Let $x^2 + ax + 6a = (x - s)(x - r)$. Vieta's yields $s + r = - a, sr = 6a$. \begin{eqnarray*}sr + 6s + 6r &=& 0\\ sr + 6s + 6r + 36 &=& 36\\ (s + 6)(r + 6) &=& 36 \end{eqnarray*}
Without loss of generality let $r \le s$.
The possible values of $(r + 6,s + 6)$ are: $( - 36, - 1),( - 18, - 2),( - 12, - 3),( - 9, - 4),(... | 0.6875 | 6,311.5625 | 5,456.818182 | 8,192 |
Kim earned scores of 86, 82, and 89 on her first three mathematics examinations. She is expected to increase her average score by at least 2 points with her fourth exam. What is the minimum score Kim must achieve on her fourth exam to meet this target? | 94 | 0.9375 | 596.6875 | 594.866667 | 624 | |
(In this question, 12 points) During a shooting training session, the probabilities of a shooter hitting the 10, 9, 8, and 7 rings are 0.21, 0.23, 0.25, and 0.28, respectively. Calculate the probability that the shooter in a single shot:
(1) Hits either the 10 or 7 ring;
(2) Scores below 7 rings. | 0.03 | 0.4375 | 4,414.5 | 3,950.428571 | 4,775.444444 | |
Given that the four vertices of the tetrahedron $P-ABC$ are all on the surface of a sphere with radius $3$, and $PA$, $PB$, $PC$ are mutually perpendicular, find the maximum value of the lateral surface area of the tetrahedron $P-ABC$. | 18 | 0.5625 | 7,112.875 | 6,273.555556 | 8,192 | |
What is the intersection of the lines given by $2y=-x+3$ and $-y=5x+1$? Enter the answer as an ordered pair. | \left(-\frac{5}{9}, \frac{16}{9}\right) | 1 | 1,972.125 | 1,972.125 | -1 | |
A $150$-gon $Q_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $150$ vertices equals $3000$. The midpoints of the sides of $Q_1$ form a second $150$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $150$-gon, $Q_3$. Find the sum of the $x$-coordinates of the vertices of $Q... | 3000 | 0.8125 | 5,006.8125 | 4,324.769231 | 7,962.333333 | |
The number \(abcde\) has five distinct digits, each different from zero. When this number is multiplied by 4, the result is a five-digit number \(edcba\), which is the reverse of \(abcde\). What is the value of \(a + b + c + d + e\)? | 27 | 0.0625 | 8,188.875 | 8,142 | 8,192 | |
What is the product of the prime numbers less than 20? | 9699690 | 0.9375 | 1,633.625 | 1,699.266667 | 649 | |
Given a triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, such that $b^2 + c^2 - a^2 = \sqrt{3}bc$.
(1) If $\tan B = \frac{\sqrt{6}}{12}$, find $\frac{b}{a}$;
(2) If $B = \frac{2\pi}{3}$ and $b = 2\sqrt{3}$, find the length of the median on side $BC$. | \sqrt{7} | 0.375 | 7,356.8125 | 6,049 | 8,141.5 | |
The largest prime factor of 101101101101 is a four-digit number $N$. Compute $N$. | 9901 | Note that $$\begin{aligned} 101101101101 & =101 \cdot 1001001001 \\ & =101 \cdot 1001 \cdot 1000001 \\ & =101 \cdot 1001 \cdot\left(100^{3}+1\right) \\ & =101 \cdot 1001 \cdot(100+1)\left(100^{2}-100+1\right) \\ & =101 \cdot 1001 \cdot 101 \cdot 9901 \\ & =101^{2} \cdot 1001 \cdot 9901 \\ & =(7 \cdot 11 \cdot 13) \cdot... | 0.125 | 8,179.1875 | 8,089.5 | 8,192 |
Given a set with three elements, it can be represented as $\{a, \frac{b}{a}, 1\}$ and also as $\{a^2, a+b, 0\}$. Find the value of $a^{2013} + b^{2013}$ \_\_\_\_\_\_. | -1 | 0.8125 | 4,654.0625 | 3,837.615385 | 8,192 | |
You can arrange 15 balls in the shape of a triangle, but you cannot arrange 96 balls in the shape of a square (missing one ball). Out of how many balls, not exceeding 50, can you arrange them both in the shape of a triangle and a square? | 36 | 0.25 | 3,942.125 | 4,521.25 | 3,749.083333 | |
The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overlin... | 594 | 0 | 8,192 | -1 | 8,192 | |
A function \( f(n) \) defined for positive integers satisfies:
\[ f(n) = \begin{cases}
n - 3 & \text{if } n \geq 1000 \\
f[f(n + 7)] & \text{if } n < 1000
\end{cases} \]
Determine \( f(90) \). | 999 | 0 | 8,192 | -1 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $b\sin 2C=c\sin B$.
1. Find angle $C$.
2. If $\sin \left(B-\frac{\pi }{3}\right)=\frac{3}{5}$, find the value of $\sin A$. | \frac{4\sqrt{3}-3}{10} | 0 | 6,259.1875 | -1 | 6,259.1875 | |
Find the distance between the midpoints of the non-parallel sides of different bases of a regular triangular prism, each of whose edges is 2. | \sqrt{5} | 0 | 7,745.875 | -1 | 7,745.875 | |
Find the radius of the circle with the equation $x^2 - 8x + y^2 - 10y + 34 = 0$. | \sqrt{7} | 1 | 1,966.3125 | 1,966.3125 | -1 | |
Let $A$ be a positive integer which is a multiple of 3, but isn't a multiple of 9. If adding the product of each digit of $A$ to $A$ gives a multiple of 9, then find the possible minimum value of $A$ . | 138 | 0.0625 | 8,129 | 7,184 | 8,192 | |
For a rational number $r$ , its *period* is the length of the smallest repeating block in its decimal expansion. for example, the number $r=0.123123123...$ has period $3$ . If $S$ denotes the set of all rational numbers of the form $r=\overline{abcdefgh}$ having period $8$ , find the sum of all elements in $S... | 50000000 | 0 | 8,192 | -1 | 8,192 | |
The function $f(x)$ is of the form $a x^{2}+b x+c$ for some integers $a, b$, and $c$. Given that $\{f(177883), f(348710), f(796921), f(858522)\} = \{1324754875645,1782225466694,1984194627862,4388794883485\}$ compute $a$. | 23 | We first match the outputs to the inputs. To start, we observe that since $a \geq 0$ (since the answer to the problem is nonnegative), we must either have $f(858522) \approx 4.39 \cdot 10^{12}$ or $f(177883) \approx 4.39 \cdot 10^{12}$. However, since 858522 is relatively close to 796921, the first case is unrealistic,... | 0 | 8,192 | -1 | 8,192 |
A bullet was fired perpendicular to a moving express train with a speed of \( c = 60 \frac{\text{km}}{\text{hr}} \). The bullet pierced a windowpane on both sides of the car. How are the two holes positioned relative to each other if the bullet's speed was \( c' = 40 \frac{\text{m}}{\text{sec}} \) and the width of the ... | 1.667 | 0 | 7,881.375 | -1 | 7,881.375 | |
Find the area in the first quadrant bounded by the hyperbola $x^{2}-y^{2}=1$, the $x$-axis, and the line $3 x=4 y$. | \frac{\ln 7}{4} | Convert to polar coordinates: the hyperbola becomes $$1=r^{2}\left(\cos ^{2} \theta-\sin ^{2} \theta\right)=r^{2} \cos (2 \theta)$$ so, letting $\alpha:=\arctan (3 / 4)$, the area is $$S:=\int_{0}^{\alpha} \frac{r^{2}}{2} d \theta=\frac{1}{2} \int_{0}^{\alpha} \sec (2 \theta) d \theta=\left.\frac{1}{4} \ln |\sec (2 \th... | 0.0625 | 8,192 | 8,192 | 8,192 |
Let $S$ be a subset of $\{1,2,3,...,100\}$ such that no pair of distinct elements in $S$ has a product divisible by $5$. What is the maximum number of elements in $S$? | 80 | 0.3125 | 7,077.6875 | 5,474.8 | 7,806.272727 | |
The operation \( \otimes \) is defined by \( a \otimes b = \frac{a}{b} + \frac{b}{a} \). What is the value of \( 4 \otimes 8 \)? | \frac{5}{2} | From the given definition, \( 4 \otimes 8 = \frac{4}{8} + \frac{8}{4} = \frac{1}{2} + 2 = \frac{5}{2} \). | 0.875 | 2,656.375 | 2,426.428571 | 4,266 |
Farmer Yang has a \(2015 \times 2015\) square grid of corn plants. One day, the plant in the very center of the grid becomes diseased. Every day, every plant adjacent to a diseased plant becomes diseased. After how many days will all of Yang's corn plants be diseased? | 2014 | 0.125 | 6,396.9375 | 7,719 | 6,208.071429 | |
The number of scalene triangles having all sides of integral lengths, and perimeter less than $13$ is: | 3 | To find the number of scalene triangles with integral side lengths and a perimeter less than 13, we need to consider the properties of scalene triangles and the triangle inequality theorem. A scalene triangle has all sides of different lengths, and the sum of the lengths of any two sides must be greater than the length... | 0.0625 | 8,046.8125 | 5,869 | 8,192 |
You are given that $3^{400}\equiv 1\pmod{1000}$. What are the last three digits of $3^{12000}$? | 001 | 0.9375 | 5,090.0625 | 4,883.266667 | 8,192 | |
Find the least positive integer $n$ , such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$ .
- There is a real number $\xi$ with $P(\xi)=... | 2014 | 0.0625 | 7,925.125 | 8,192 | 7,907.333333 | |
Enlarge each edge of a graph by four times its original size. This is equivalent to enlarging the graph by a scale of \_\_\_\_\_\_. | 4:1 | 0 | 660.1875 | -1 | 660.1875 | |
Let $m$ and $n$ satisfy $mn=4$ and $m+n=5$. What is $|m-n|$? | 3 | 1 | 1,662.875 | 1,662.875 | -1 | |
The graph of $y=ax^2+bx+c$ is given below, where $a$, $b$, and $c$ are integers. Find $a$.
[asy]
size(140);
Label f;
f.p=fontsize(4);
xaxis(-3,3,Ticks(f, 1.0));
yaxis(-4,4,Ticks(f, 1.0));
real f(real x)
{
return -2x^2+4x+1;
}
draw(graph(f,-.7,2.7),linewidth(1),Arrows(6));
[/asy] | -2 | 0.9375 | 3,141.1875 | 2,804.466667 | 8,192 | |
Find the maximum of
\[\sqrt{x + 27} + \sqrt{13 - x} + \sqrt{x}\]for $0 \le x \le 13.$ | 11 | 0.3125 | 7,584.8125 | 6,624.4 | 8,021.363636 | |
Given that $a>0$, the minimum value of the function $f(x) = e^{x-a} - \ln(x+a) - 1$ $(x>0)$ is 0. Determine the range of values for the real number $a$. | \{\frac{1}{2}\} | 0 | 8,069.8125 | -1 | 8,069.8125 | |
Given:
$$ \frac{ \left( \frac{1}{3} \right)^2 + \left( \frac{1}{4} \right)^2 }{ \left( \frac{1}{5} \right)^2 + \left( \frac{1}{6} \right)^2} = \frac{37x}{73y} $$
Express $\sqrt{x} \div \sqrt{y}$ as a common fraction. | \frac{75 \sqrt{73}}{6 \sqrt{61} \sqrt{37}} | 0 | 8,192 | -1 | 8,192 | |
Given the function $f(x)=\sin (2x+ \frac {π}{3})$, it is shifted right by $\frac {2π}{3}$ units, and then the resulting function's graph has each point's horizontal coordinate changed to twice its original value while the vertical coordinate remains unchanged, yielding the function $y=g(x)$. Calculate the area enclosed... | \frac{3}{2} | 0.8125 | 4,515.1875 | 4,132.076923 | 6,175.333333 | |
Let the set $I = \{1,2,3,4,5\}$. Choose two non-empty subsets $A$ and $B$ from $I$. How many different ways are there to choose $A$ and $B$ such that the smallest number in $B$ is greater than the largest number in $A$? | 49 | 0.0625 | 7,997.9375 | 5,816 | 8,143.4 | |
Square $ABCD$ has sides of length 4. Set $T$ is the set of all line segments that have length 4 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $T$ enclose a region whose area to the nearest hundredth is $m$. Find $100m$. | 343 | 0 | 8,192 | -1 | 8,192 | |
Although I am certain that my clock is 5 minutes fast, it is actually 10 minutes slow. On the other hand, my friend's clock is really 5 minutes fast, even though he thinks it is correct. We scheduled a meeting for 10 o'clock and plan to arrive on time. Who will arrive first? After how much time will the other arrive? | 20 | 0 | 5,526.1875 | -1 | 5,526.1875 | |
Simplify: $\sqrt{50} + \sqrt{18}$ . Express your answer in simplest radical form. | 8\sqrt{2} | 1 | 1,289.125 | 1,289.125 | -1 | |
A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft ... | 4.5 |
Let's analyze the problem step by step:
1. **Define Variables:**
- Let $t$ be the time it takes for the power boat to travel from dock $A$ to dock $B$.
- Let $r$ be the speed of the river current (and also the speed of the raft, since it drifts with the current).
- Let $p$ be the speed of the power boat rela... | 0.0625 | 4,736.5 | 6,093 | 4,646.066667 |
On Friday, a snowboard originally priced at $\$100$ was discounted $50\%$. On Monday, that sale price was reduced by $30\%$. In dollars, what is the price of the snowboard after the Monday reduction? | 35 | 1 | 1,796.6875 | 1,796.6875 | -1 | |
Let $x$ and $y$ be non-negative real numbers that sum to 1. Compute the number of ordered pairs $(a, b)$ with $a, b \in\{0,1,2,3,4\}$ such that the expression $x^{a} y^{b}+y^{a} x^{b}$ has maximum value $2^{1-a-b}$. | 17 | Let $f(x, y)=x^{a} y^{b}+y^{a} x^{b}$. Observe that $2^{1-a-b}$ is merely the value of $f\left(\frac{1}{2}, \frac{1}{2}\right)$, so this value is always achievable. We claim (call this result $(*)$ ) that if $(a, b)$ satisfies the condition, so does $(a+1, b+1)$. To see this, observe that if $f(x, y) \leq 2^{1-a-b}$, t... | 0 | 8,192 | -1 | 8,192 |
In triangle $ABC$, $\cos(2A-B)+\sin(A+B)=2$ and $AB=4$. What is $BC$? | 2 | 1 | 2,027.625 | 2,027.625 | -1 | |
Let \( M_{n} = \left\{ 0 . \overline{a_{1} a_{2} \cdots a_{n}} \mid a_{i} \ \text{is either 0 or 1 for} \ i=1,2, \cdots, n-1, \ a_{n}=1 \right\} \). \( T_{n} \) is the number of elements in \( M_{n} \) and \( S_{n} \) is the sum of all elements in \( M_{n} \). Find \( \lim_{n \rightarrow \infty} \frac{S_{n}}{T_{n}} \). | 1/18 | 0.3125 | 7,692.1875 | 6,793.4 | 8,100.727273 | |
Many of the students in M. Gamache's class brought a skateboard or a bicycle to school yesterday. The ratio of the number of skateboards to the number of bicycles was $7:4$. There were 12 more skateboards than bicycles. How many skateboards and bicycles were there in total? | 44 | Since the ratio of the number of skateboards to the number of bicycles was $7:4$, then the numbers of skateboards and bicycles can be written in the form $7k$ and $4k$ for some positive integer $k$. Since the difference between the numbers of skateboards and bicycles is 12, then $7k - 4k = 12$ and so $3k = 12$ or $k = ... | 1 | 1,170.9375 | 1,170.9375 | -1 |
Let the number $x$ . Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent of the number $x$ (for example, $x\cdot x=x^{2}$ , $x^{2}\cdot x^{2}=x^{4}$ , $x^{4}: x=x^{3}$ , etc). Determine the minimal number of operations needed for calculating ... | 17 | 0 | 8,192 | -1 | 8,192 | |
Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by $A$. How many sev... | 192 | 0.625 | 6,199.25 | 5,003.6 | 8,192 | |
The sum of two fractions is $\frac{11}{12}$ and their product is $\frac{1}{6}$. What is the lesser of the two fractions? Express your answer as a common fraction. | \frac{1}{4} | 1 | 2,814.5 | 2,814.5 | -1 | |
Four of the eight vertices of a cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron. | \sqrt{3} | 1. **Assume the side length of the cube**: Let the side length of the cube be $s = 1$.
2. **Identify the vertices of the tetrahedron**: Four vertices of a cube that form a regular tetrahedron can be chosen such that they are not all on the same face and no three are collinear on an edge. For example, if we label the v... | 0.9375 | 4,266.8125 | 4,257.8 | 4,402 |
What is the number of radians in the smaller angle formed by the hour and minute hands of a clock at 3:40? Express your answer as a decimal rounded to three decimal places. | 2.278 | 0 | 5,378.25 | -1 | 5,378.25 | |
This pattern is made from toothpicks. If the pattern is continued by adding two toothpicks to the previous stage, how many toothpicks are used to create the figure for the $15^{th}$ stage?
[asy]draw((0,0)--(7.5,13)--(-7.5,13)--cycle);
draw((0,0)--(-15,0)--(-7.5,13)--cycle);
label("stage 2",(-4,0),S);
draw((-23,0)--(-3... | 31 | 0.875 | 4,533.25 | 4,393.142857 | 5,514 | |
Let x; y; z be real numbers, satisfying the relations
$x \ge 20$
$y \ge 40$
$z \ge 1675$
x + y + z = 2015
Find the greatest value of the product P = $xy z$ | \frac{721480000}{27} |
Given the conditions:
\[
x \geq 20, \quad y \geq 40, \quad z \geq 1675
\]
and the equation:
\[
x + y + z = 2015
\]
we need to find the greatest value of the product \( P = xyz \).
### Step 1: Analyze the Variables
We express \( z \) in terms of \( x \) and \( y \):
\[
z = 2015 - x - y
\]
Given the constraints ... | 0 | 8,019.75 | -1 | 8,019.75 |
In a building with 10 mailboxes, a distributor places a flyer in 5 of the mailboxes. Later, another distributor also places a flyer in 5 of the mailboxes. What is the probability that at least 8 mailboxes receive a flyer? | 1/2 | 0.125 | 8,035.625 | 6,941 | 8,192 | |
Find the smallest positive integer that is both an integer power of 7 and is not a palindrome. | 2401 | 0.1875 | 2,712.3125 | 2,879.666667 | 2,673.692308 | |
What is the smallest positive integer that has exactly eight distinct positive divisors, where all divisors are powers of prime numbers? | 24 | 0.125 | 6,170.625 | 3,534.5 | 6,547.214286 |
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