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 |
|---|---|---|---|---|---|---|
What number should be removed from the list \[1,2,3,4,5,6,7,8,9,10,11\] so that the average of the remaining numbers is $6.1$? | 5 | 1. **Calculate the total sum of the numbers in the list**: The list contains the first 11 natural numbers, so the sum can be calculated using the formula for the sum of an arithmetic series:
\[
\text{Sum} = \frac{n}{2} \times (\text{first term} + \text{last term})
\]
where $n$ is the number of terms. Here, ... | 1 | 2,149.875 | 2,149.875 | -1 |
Human beings have discovered a habitable planet and soon after, they find 10 more habitable planets. Of these 11, only 5 are deemed ``Earth-like'' in their resources and the rest are deemed ``Mars-like'' since they lack many important resources. Assume that planets like Earth take up 2 units of colonization, while thos... | 100 | 0.75 | 4,443.5625 | 3,194.083333 | 8,192 | |
What is the smallest positive integer with exactly 16 positive divisors? | 120 | 0.875 | 5,795.0625 | 5,452.642857 | 8,192 | |
Real numbers $x$ and $y$ have an arithmetic mean of 18 and a geometric mean of $\sqrt{92}$. Find $x^2+y^2$. | 1112 | 1 | 1,778.1875 | 1,778.1875 | -1 | |
Given the imaginary unit $i$, let $z=1+i+i^{2}+i^{3}+\ldots+i^{9}$, then $|z|=$______. | \sqrt {2} | 0 | 3,572.4375 | -1 | 3,572.4375 | |
Given the parabola C: x² = 2py (p > 0), draw a line l: y = 6x + 8, which intersects the parabola C at points A and B. Point O is the origin, and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$. A moving circle P has its center on the parabola C and passes through a fixed point D(0, 4). If the moving circle P inters... | \sqrt{2} - 1 | 0.625 | 6,752.0625 | 5,951.5 | 8,086.333333 | |
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) with its left focus at F and the eccentricity $e = \frac{\sqrt{2}}{2}$, the line segment cut by the ellipse from the line passing through F and perpendicular to the x-axis has length $\sqrt{2}$.
(Ⅰ) Find the equation of the ellipse.
(Ⅱ) A line... | \frac{3}{2} | 0.625 | 7,489.625 | 7,178 | 8,009 | |
The residents of an accommodation need to pay the rent for the accommodation. If each of them contributes $10 \mathrm{Ft}$, the amount collected falls $88 \mathrm{Ft}$ short of the rent. However, if each of them contributes $10.80 \mathrm{Ft}$, then the total amount collected exceeds the rent by $2.5 \%$. How much shou... | 10.54 | 0.0625 | 5,875.5 | 6,971 | 5,802.466667 | |
Of the $500$ balls in a large bag, $80\%$ are red and the rest are blue. How many of the red balls must be removed so that $75\%$ of the remaining balls are red? | 100 | 1. **Determine the initial number of red and blue balls:**
Given that $80\%$ of the $500$ balls are red, we calculate the number of red balls as:
\[
0.8 \times 500 = 400 \text{ red balls}
\]
The remaining balls are blue, so:
\[
500 - 400 = 100 \text{ blue balls}
\]
2. **Set up the equation for ... | 1 | 2,589.375 | 2,589.375 | -1 |
Given a sequence $\{a_n\}$ that satisfies $a_1=33$ and $a_{n+1}-a_n=2n$, find the minimum value of $\frac {a_{n}}{n}$. | \frac {21}{2} | 0.8125 | 5,551.8125 | 4,942.538462 | 8,192 | |
A trapezium is given with parallel bases having lengths $1$ and $4$ . Split it into two trapeziums by a cut, parallel to the bases, of length $3$ . We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trap... | 15 | 0 | 8,192 | -1 | 8,192 | |
Point $P$ moves on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1$, and points $A$ and $B$ move respectively on the circles $x^2 + (y-4)^2 = 16$ and $x^2 + (y+4)^2 = 4$. The maximum value of $PA + PB$ is \_\_\_\_\_\_. | 16 | 0.1875 | 7,665.9375 | 5,386.333333 | 8,192 | |
An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:
(a) the selection of four red marbles;
(b) the selection of one white and three red marbles;
(c) the selection of one white, one blue, and two red marbles; and
(... | 21 | 1. **Define Variables and Equations**:
Let $r$, $w$, $b$, and $g$ represent the number of red, white, blue, and green marbles, respectively, and let $n$ be the total number of marbles. Thus, we have:
\[ r + w + b + g = n \]
2. **Calculate the Probabilities**:
- The number of ways to select four red marbles:
... | 0.5625 | 6,724.3125 | 5,582.777778 | 8,192 |
Points $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C$: $\frac{x^{2}}{2}+y^{2}=1$, respectively. Point $N$ is the top vertex of the ellipse $C$. If a moving point $M$ satisfies $|\overrightarrow{MN}|^{2}=2\overrightarrow{MF_{1}}\cdot\overrightarrow{MF_{2}}$, then the maximum value of $|\overrightarro... | 6+\sqrt{10} | 0.625 | 6,637.3125 | 5,704.5 | 8,192 | |
A blind box refers to a toy box where consumers cannot know the specific product style in advance. A certain brand has launched two blind box sets. Set $A$ contains $4$ different items, including a small rabbit toy. Set $B$ contains $2$ different items, with a $50\%$ chance of getting a small rabbit toy.
$(1)$ Individ... | \frac{1}{3} | 0.0625 | 6,470.4375 | 6,822 | 6,447 | |
In $\triangle ABC$, $\tan A = \frac{1}{4}$ and $\tan B = \frac{3}{5}$.
(1) Find the measure of angle $C$;
(2) If the shortest side length of $\triangle ABC$ is $\sqrt{2}$, find the area of $\triangle ABC$. | \frac{3}{2} | 0.9375 | 4,949.3125 | 4,733.133333 | 8,192 | |
Cátia leaves school every day at the same time and rides her bicycle home. When she pedals at $20 \mathrm{~km} / \mathrm{h}$, she arrives home at $16 \mathrm{~h} 30 \mathrm{~m}$. If she pedals at $10 \mathrm{~km} / \mathrm{h}$, she arrives home at $17 \mathrm{~h} 15 \mathrm{~m}$. At what speed should she pedal to arriv... | 12 | 0.125 | 7,468.75 | 5,118 | 7,804.571429 | |
For all real numbers $x$ and $y$, define the mathematical operation $\star$ such that the following conditions apply: $x\ \star\ 0 = x+1, x\ \star\ y = y\ \star\ x$, and $(x + 2)\ \star\ y = (x\ \star\ y) + y + 2$. What is the value of $7\ \star\ 3$? | 24 | 0 | 7,958.25 | -1 | 7,958.25 | |
What is the area of the portion of the circle defined by the equation $x^2 + 6x + y^2 = 50$ that lies below the $x$-axis and to the left of the line $y = x - 3$? | \frac{59\pi}{4} | 0 | 8,192 | -1 | 8,192 | |
A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 8 units has its center at $(18, 0)$. A line tangent to both circles intersects the $x$-axis at point $(x, 0)$ to the right of the origin. Find the value of $x$. Express your answer as a common fraction. | \frac{54}{11} | 0.9375 | 5,133.0625 | 4,929.133333 | 8,192 | |
Given that point $M(x\_0, y\_0)$ moves on the circle $x^{2}+y^{2}=4$, $N(4,0)$, and point $P(x,y)$ is the midpoint of segment $MN$.
(1) Find the trajectory equation of point $P(x,y)$;
(2) Find the maximum and minimum distances from point $P(x,y)$ to the line $3x+4y-86=0$. | 15 | 0.9375 | 4,211.625 | 4,256.466667 | 3,539 | |
Given vectors $\overrightarrow {a}$ = (4, 3) and $\overrightarrow {b}$ = (-1, 2), with $\overrightarrow {m}$ = $\overrightarrow {a}$ - $λ \overrightarrow {b}$ and $\overrightarrow {n}$ = 2$\overrightarrow {a}$ + $\overrightarrow {b}$, find the values of $λ$ such that $\overrightarrow {m}$ is perpendicular to $\overrigh... | -\frac{1}{2} | 0.8125 | 4,129.875 | 3,192.461538 | 8,192 | |
How many different counting numbers will each leave a remainder of 5 when divided into 47? | 5 | 1 | 2,543.9375 | 2,543.9375 | -1 | |
Rationalize the denominator: $$\frac{1}{\sqrt[3]{2}+\sqrt[3]{16}}$$ | \frac{\sqrt[3]{4}}{6} | 0 | 4,103.75 | -1 | 4,103.75 | |
Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$. | 23 | 0.75 | 4,388.9375 | 4,203.166667 | 4,946.25 | |
Given the function $f(x)=3\sin(2x-\frac{π}{3})-2\cos^{2}(x-\frac{π}{6})+1$, the graph of function $f(x)$ is shifted to the left by $\frac{π}{6}$ units, resulting in the graph of function $g(x)$. Find $\sin (2x_{1}+2x_{2})$, where $x_{1}$ and $x_{2}$ are the two roots of the equation $g(x)=a$ in the interval $[0,\frac{π... | -\frac{3}{5} | 0.1875 | 7,452.1875 | 7,788.333333 | 7,374.615385 | |
Wei has designed a logo for his new company using circles and a large square, as shown. Each circle is tangent to two sides of the square and its two adjacent circles. If he wishes to create a version of this logo that is 20 inches on each side, how many square inches will be shaded?
[asy]
size(100);
draw((0,0)--(4,... | 400 - 100\pi | 0.9375 | 4,334.375 | 4,077.2 | 8,192 | |
How many positive four-digit integers of the form $\_\_35$ are divisible by 35? | 13 | 0.4375 | 6,868.8125 | 6,097.714286 | 7,468.555556 | |
In trapezoid $ABCD$ with $AD\parallel BC$ , $AB=6$ , $AD=9$ , and $BD=12$ . If $\angle ABD=\angle DCB$ , find the perimeter of the trapezoid. | 39 | 0.1875 | 7,745.9375 | 6,604.333333 | 8,009.384615 | |
Denote by $P(n)$ the greatest prime divisor of $n$. Find all integers $n\geq 2$ for which \[P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor\] | 3 |
We seek all integers \( n \geq 2 \) such that the greatest prime divisor of \( n \), denoted \( P(n) \), together with the integer part of the square root of \( n \), satisfies the equation:
\[
P(n) + \lfloor \sqrt{n} \rfloor = P(n+1) + \lfloor \sqrt{n+1} \rfloor.
\]
**Step 1: Understand the structure of the equatio... | 0 | 8,192 | -1 | 8,192 |
For any real number $x$, the symbol $\lfloor x \rfloor$ represents the integer part of $x$, which is the greatest integer not exceeding $x$. This function, $\lfloor x \rfloor$, is called the "floor function". Calculate the sum $\lfloor \log_3 1 \rfloor + \lfloor \log_3 2 \rfloor + \lfloor \log_3 3 \rfloor + \lfloor \lo... | 857 | 0.9375 | 4,912.4375 | 4,693.8 | 8,192 | |
Calculate the probability of the Alphas winning given the probability of the Reals hitting 0, 1, 2, 3, or 4 singles. | \frac{224}{243} | The probability of the Reals hitting 0 singles is $\left(\frac{2}{3}\right)^{3}$. The probability of the Reals hitting exactly 1 single is $\binom{3}{2} \cdot\left(\frac{2}{3}\right)^{3} \cdot \frac{1}{3}$, since there are 3 spots to put the two outs (the last spot must be an out, since the inning has to end on an out)... | 0 | 8,111.125 | -1 | 8,111.125 |
A box contains $12$ ping-pong balls, of which $9$ are new and $3$ are old. Three balls are randomly drawn from the box for use, and then returned to the box. Let $X$ denote the number of old balls in the box after this process. What is the value of $P(X = 4)$? | \frac{27}{220} | 0 | 4,602.1875 | -1 | 4,602.1875 | |
Let the set of three-digit numbers composed of \\(0\\), \\(1\\), \\(2\\), and \\(3\\) without repeating digits be \\(A\\). If a number is randomly selected from \\(A\\), the probability that the number is exactly even is. | \dfrac{5}{9} | 0.8125 | 5,653 | 5,067.076923 | 8,192 | |
The new individual income tax law has been implemented since January 1, 2019. According to the "Individual Income Tax Law of the People's Republic of China," it is known that the part of the actual wages and salaries (after deducting special, additional special, and other legally determined items) obtained by taxpayers... | 9720 | 0 | 7,463.5625 | -1 | 7,463.5625 | |
Distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=BC=CD=1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle? | 3 | 1. **Identify Possible Bases and Heights**:
- The points $A$, $B$, $C$, and $D$ are collinear with equal distances between consecutive points, i.e., $AB = BC = CD = 1$.
- Points $E$ and $F$ are also collinear on a different line parallel to the line containing $A$, $B$, $C$, and $D$, with $EF = 1$.
- The heig... | 0.3125 | 7,559.9375 | 6,590.2 | 8,000.727273 |
How many 6-digit numbers have at least two zeros? | 73,314 | 0 | 6,241.5625 | -1 | 6,241.5625 | |
Pentagon $ANDD'Y$ has $AN \parallel DY$ and $AY \parallel D'N$ with $AN = D'Y$ and $AY = DN$ . If the area of $ANDY$ is 20, the area of $AND'Y$ is 24, and the area of $ADD'$ is 26, the area of $ANDD'Y$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . ... | 71 | 0 | 8,118.125 | -1 | 8,118.125 | |
The vertex of a parabola is \( O \) and its focus is \( F \). When a point \( P \) moves along the parabola, find the maximum value of the ratio \( \left|\frac{P O}{P F}\right| \). | \frac{2\sqrt{3}}{3} | 0 | 5,972.25 | -1 | 5,972.25 | |
On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57\textsc{am}$, and the sunset as $8:15\textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set? | $5:21\textsc{pm}$ | 1. **Convert the length of daylight into a time format**: The length of daylight is given as 10 hours and 24 minutes. This can be represented as $10:24$.
2. **Convert the sunrise time into a 24-hour format**: The sunrise time is given as $6:57\textsc{am}$. In 24-hour time format, this remains $6:57$.
3. **Add the len... | 0 | 4,733.9375 | -1 | 4,733.9375 |
Solve for $n$: $0.03n + 0.08(20 + n) = 12.6$. | 100 | 0.9375 | 2,767.6875 | 2,406.066667 | 8,192 | |
Given \( x_{1}, x_{2}, x_{3} \in [0, 12] \),
\[ x_{1} x_{2} x_{3} = \left(\left(12 - x_{1}\right)\left(12 - x_{2}\right)\left(12 - x_{3}\right)\right)^{2}. \]
Find the maximum value of \( f = x_{1} x_{2} x_{3} \). | 729 | 0 | 8,192 | -1 | 8,192 | |
A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken than on the first round. How many people we... | 10 | 0.0625 | 3,846.8125 | 4,057 | 3,832.8 | |
Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and $S_n=n^2$ ($n\in\mathbb{N}^*$).
1. Find $a_n$;
2. The function $f(n)$ is defined as $$f(n)=\begin{cases} a_{n} & \text{, $n$ is odd} \\ f(\frac{n}{2}) & \text{, $n$ is even}\end{cases}$$, and $c_n=f(2^n+4)$ ($n\in\mathbb{N}^*$), find the su... | \frac{9}{2} | 0 | 8,192 | -1 | 8,192 | |
A circular sheet of iron with a radius of 6 has a sector removed, which is $\frac{1}{6}$ of the original area. The remaining part is rolled into the lateral surface of a cone. The volume of the cone is \_\_\_\_\_\_. | \frac{25\sqrt{11}}{3}\pi | 0 | 2,632.4375 | -1 | 2,632.4375 | |
A man has $2.73 in pennies, nickels, dimes, quarters and half dollars. If he has an equal number of coins of each kind, then the total number of coins he has is | 15 | Let $x$ represent the number of each type of coin the man has. Since he has pennies, nickels, dimes, quarters, and half dollars, and he has an equal number of each, we can set up the following equation to represent the total value of his coins in cents:
- Pennies contribute $1x$ cents.
- Nickels contribute $5x$ cents.... | 1 | 1,400.5 | 1,400.5 | -1 |
Given that both $α$ and $β$ are acute angles, and $\cos(α + β) = \frac{\sin α}{\sin β}$, find the maximum value of $\tan α$. | \frac{\sqrt{2}}{4} | 0 | 7,208.5625 | -1 | 7,208.5625 | |
Given that $a$, $b$, and $c$ are nonzero real numbers, find all possible values of the expression
\[\frac{a}{|a|} + \frac{b}{|b|} + \frac{c}{|c|} + \frac{abc}{|abc|}.\]Enter all possible values, separated by commas. | 4, 0, -4 | 0 | 5,375 | -1 | 5,375 | |
Find the difference between the largest integer solution of the equation \(\lfloor \frac{x}{3} \rfloor = 102\) and the smallest integer solution of the equation \(\lfloor \frac{x}{3} \rfloor = -102\). | 614 | 0.9375 | 3,221.375 | 3,344.333333 | 1,377 | |
What is the smallest positive integer \(n\) such that \(\frac{n}{n+75}\) is equal to a terminating decimal? | 50 | 0 | 7,675.5 | -1 | 7,675.5 | |
Given that the internal angles $A$ and $B$ of $\triangle ABC$ satisfy $\frac{\sin B}{\sin A} = \cos(A+B)$, find the maximum value of $\tan B$. | \frac{\sqrt{2}}{4} | 0 | 6,615.625 | -1 | 6,615.625 | |
Two similar right triangles have areas of 6 square inches and 150 square inches. The length of the hypotenuse of the smaller triangle is 5 inches. What is the sum of the lengths of the legs of the larger triangle? | 35 | 0.9375 | 2,741.6875 | 2,378.333333 | 8,192 | |
If $3^p + 3^4 = 90$, $2^r + 44 = 76$, and $5^3 + 6^s = 1421$, what is the product of $p$, $r$, and $s$? | 40 | We are given three equations involving exponents and asked to find the product of $p$, $r$, and $s$.
1. **Solving for $p$:**
\[
3^p + 3^4 = 90
\]
We know that $3^4 = 81$. Substituting this into the equation, we get:
\[
3^p + 81 = 90
\]
Subtracting 81 from both sides:
\[
3^p = 9
\]
S... | 1 | 1,405.6875 | 1,405.6875 | -1 |
Rectangle ABCD has dimensions AB=CD=4 and BC=AD=8. The rectangle is rotated 90° clockwise about corner D, then rotated 90° clockwise about the corner C's new position after the first rotation. What is the length of the path traveled by point A?
A) $8\sqrt{5}\pi$
B) $4\sqrt{5}\pi$
C) $2\sqrt{2}\pi$
D) $4\sqrt{2}\p... | 4\sqrt{5}\pi | 0 | 7,891.9375 | -1 | 7,891.9375 | |
The integer points $(x, y)$ in the first quadrant satisfy $x + y > 8$ and $x \leq y \leq 8$. Determine the number of such integer points $(x, y)$. | 20 | 0.875 | 5,644.25 | 5,280.285714 | 8,192 | |
Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy? | 14 | Let $x$ be the total number of apples Bridget bought. According to the problem, Bridget first gave half of the apples to Ann. This means that Bridget was left with $\frac{x}{2}$ apples.
Next, Bridget gave Cassie 3 apples. After giving these apples to Cassie, the number of apples Bridget had left is:
\[
\frac{x}{2} - 3... | 1 | 681.8125 | 681.8125 | -1 |
During intervals, students played table tennis. Any two students played against each other no more than one game. At the end of the week, it turned out that Petya played half, Kolya played a third, and Vasya played a fifth of all the games played during the week. How many games could have been played during the week if... | 30 | 0.0625 | 8,167.125 | 8,192 | 8,165.466667 | |
A standard deck of 52 cards is divided into 4 suits, with each suit containing 13 cards. Two of these suits are red, and the other two are black. The deck is shuffled, placing the cards in random order. What is the probability that the first three cards drawn from the deck are all the same color? | \frac{40}{85} | 0 | 4,525.8125 | -1 | 4,525.8125 | |
Find the set consisting of all real values of $x$ such that the three numbers $2^{x}, 2^{x^{2}}, 2^{x^{3}}$ form a non-constant arithmetic progression (in that order). | \varnothing | The empty set, $\varnothing$. Trivially, $x=0,1$ yield constant arithmetic progressions; we show that there are no other possibilities. If these numbers do form a progression, then, by the AM-GM (arithmetic mean-geometric mean) inequality, $$2 \cdot 2^{x^{2}}=2^{x}+2^{x^{3}} \geq 2 \sqrt{2^{x} \cdot 2^{x^{3}}} \Rightar... | 0 | 8,192 | -1 | 8,192 |
There is one odd integer \(N\) between 400 and 600 that is divisible by both 5 and 11. What is the sum of the digits of \(N\)? | 18 | If \(N\) is divisible by both 5 and 11, then \(N\) is divisible by \(5 \times 11=55\). This is because 5 and 11 have no common divisor larger than 1. Therefore, we are looking for a multiple of 55 between 400 and 600 that is odd. One way to find such a multiple is to start with a known multiple of 55, such as 550, whic... | 1 | 2,353.1875 | 2,353.1875 | -1 |
Let $x, y, z$ be real numbers satisfying $$\frac{1}{x}+y+z=x+\frac{1}{y}+z=x+y+\frac{1}{z}=3$$ The sum of all possible values of $x+y+z$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$. | 6106 | The equality $\frac{1}{x}+y+z=x+\frac{1}{y}+z$ implies $\frac{1}{x}+y=x+\frac{1}{y}$, so $x y=-1$ or $x=y$. Similarly, $y z=-1$ or $y=z$, and $z x=-1$ or $z=x$. If no two elements multiply to -1 , then $x=y=z$. which implies $2 x+\frac{1}{x}=3$ and so $(x, y, z) \in$ $\left\{(1,1,1),\left(\frac{1}{2}, \frac{1}{2}, \fra... | 0.4375 | 5,938 | 6,230.857143 | 5,710.222222 |
Simplify: $|{-3^2+4}|$ | 5 | 0.9375 | 1,022.4375 | 1,038.133333 | 787 | |
Simplify $\frac{\sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{3}}{\sqrt{6}} \cdot \frac{\sqrt{4}}{\sqrt{7}}$ and rationalize the denominator of the resulting fraction. | \frac{2\sqrt{35}}{35} | 0 | 2,705.875 | -1 | 2,705.875 | |
If the shortest distance from a point on the ellipse $\frac{y^2}{16} + \frac{x^2}{9} = 1$ to the line $y = x + m$ is $\sqrt{2}$, find the minimum value of $m$. | -7 | 0.25 | 7,644.0625 | 6,000.25 | 8,192 | |
The solutions to $z^4 = -16i$ can be expressed in the form
\begin{align*}
z_1 &= r_1 (\cos \theta_1 + i \sin \theta_1), \\
z_2 &= r_2 (\cos \theta_2 + i \sin \theta_2), \\
z_3 &= r_3 (\cos \theta_3 + i \sin \theta_3), \\
z_4 &= r_4 (\cos \theta_4 + i \sin \theta_4),
\end{align*}where $r_k > 0$ and $0^\circ \le \theta_k... | 810^\circ | 0.875 | 4,467.25 | 3,935.142857 | 8,192 | |
There are 7 volunteers to be arranged for community service activities on Saturday and Sunday, with 3 people arranged for each day, calculate the total number of different arrangements. | 140 | 0.25 | 5,624.5625 | 4,061 | 6,145.75 | |
Consider a sequence $F_0=2$ , $F_1=3$ that has the property $F_{n+1}F_{n-1}-F_n^2=(-1)^n\cdot2$ . If each term of the sequence can be written in the form $a\cdot r_1^n+b\cdot r_2^n$ , what is the positive difference between $r_1$ and $r_2$ ?
| \frac{\sqrt{17}}{2} | 0 | 7,780.9375 | -1 | 7,780.9375 | |
How many values of $x$, $-19<x<98$, satisfy $\cos^2 x + 2\sin^2 x = 1?$ (Note: $x$ is measured in radians.) | 38 | 1 | 2,944.75 | 2,944.75 | -1 | |
Xiao Ming, Xiao Hong, and Xiao Gang are three people whose ages are three consecutive even numbers. Their total age is 48 years old. What is the youngest age? What is the oldest age? | 18 | 0.5625 | 456.625 | 473.333333 | 435.142857 | |
Let $P(x)$ be a polynomial with degree 2008 and leading coefficient 1 such that $P(0)=2007, P(1)=2006, P(2)=2005, \ldots, P(2007)=0$. Determine the value of $P(2008)$. You may use factorials in your answer. | 2008!-1 | Consider the polynomial $Q(x)=P(x)+x-2007$. The given conditions tell us that $Q(x)=0$ for $x=0,1,2, \ldots, 2007$, so these are the roots of $Q(x)$. On the other hand, we know that $Q(x)$ is also a polynomial with degree 2008 and leading coefficient 1 . It follows that $Q(x)=x(x-1)(x-2)(x-3) \cdots(x-2007)$. Thus $$P(... | 0.75 | 5,067.9375 | 4,149.166667 | 7,824.25 |
What is the minimum total number of boxes that Carley could have bought if each treat bag contains exactly 1 chocolate, 1 mint, and 1 caramel, and chocolates come in boxes of 50, mints in boxes of 40, and caramels in boxes of 25? | 17 | Suppose that Carley buys $x$ boxes of chocolates, $y$ boxes of mints, and $z$ boxes of caramels. In total, Carley will then have $50x$ chocolates, $40y$ mints, and $25z$ caramels. Since $50x=40y=25z$, dividing by 5 gives $10x=8y=5z$. The smallest possible value of $10x$ which is a multiple of both 10 and 8 is 40. In th... | 0.3125 | 2,555.9375 | 1,081.8 | 3,226 |
Let $A B C$ be a triangle that satisfies $A B=13, B C=14, A C=15$. Given a point $P$ in the plane, let $P_{A}, P_{B}, P_{C}$ be the reflections of $A, B, C$ across $P$. Call $P$ good if the circumcircle of $P_{A} P_{B} P_{C}$ intersects the circumcircle of $A B C$ at exactly 1 point. The locus of good points $P$ enclos... | \frac{4225}{64} \pi | By the properties of reflection, the circumradius of $P_{A} P_{B} P_{C}$ equals the circumradius of $A B C$. Therefore, the circumcircle of $P_{A} P_{B} P_{C}$ must be externally tangent to the circumcircle of $A B C$. Now it's easy to see that the midpoint of the 2 centers of $A B C$ and $P_{A} P_{B} P_{C}$ lies on th... | 0.5 | 7,641.5 | 7,091 | 8,192 |
For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 20 | 0.625 | 5,704.5 | 4,212 | 8,192 | |
Evaluate the expression
\[
\frac{121 \left( \frac{1}{13} - \frac{1}{17} \right)
+ 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{
11 \left( \frac{1}{13} - \frac{1}{17} \right)
+ 13 \left( \frac{1}{17} - \frac{1}{11} \right) + 17 \left( \f... | 41 | 0.6875 | 6,033.5 | 5,206.727273 | 7,852.4 | |
Ms. Johnson awards bonus points to students in her class whose test scores are above the median. The class consists of 81 students. What is the maximum number of students who could receive bonus points? | 40 | 0.8125 | 413.5 | 425 | 363.666667 | |
A subset $M$ of $\{1, 2, . . . , 2006\}$ has the property that for any three elements $x, y, z$ of $M$ with $x < y < z$ , $x+ y$ does not divide $z$ . Determine the largest possible size of $M$ . | 1004 | 0 | 8,192 | -1 | 8,192 | |
In a particular game, each of $4$ players rolls a standard $6$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the pr... | \frac{41}{144} | We start by defining the events and variables:
- Let $H_1$ be the outcome of Hugo's first roll.
- Let $A_1, B_1, C_1$ be the outcomes of the first rolls of the other three players, respectively.
- Let $W = H$ denote the event that Hugo wins the game.
We are asked to find the probability that Hugo's first roll was a $5... | 0 | 7,788.3125 | -1 | 7,788.3125 |
You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1 , the value of each blue card is equal to twice the number of red cards, and the val... | 168 | If there are $B$ blue cards, then each red card contributes $1+2 B$ points (one for itself and two for each blue card) and each yellow card contributes $3 B$ points. Thus, if $B>1$, it is optimal to change all red cards to yellow cards. When $B=0$, the maximum number of points is 15 . When $B=1$, the number of points i... | 0.4375 | 7,691.75 | 7,048.571429 | 8,192 |
Let $X,$ $Y,$ and $Z$ be points such that $\frac{XZ}{XY} = \frac{ZY}{XY} = \frac{1}{2}.$ If $Y = (1, 7)$, $Z = (-1, -7)$, then what is the sum of the coordinates of $X$? | -24 | 0.8125 | 3,515.5625 | 2,436.384615 | 8,192 | |
The area of triangle $ABC$ is $2 \sqrt{3}$, side $BC$ is equal to $1$, and $\angle BCA = 60^{\circ}$. Point $D$ on side $AB$ is $3$ units away from point $B$, and $M$ is the intersection point of $CD$ with the median $BE$. Find the ratio $BM: ME$. | 3 : 5 | 0 | 7,957.25 | -1 | 7,957.25 | |
Given the functions $f(x)=x-\frac{1}{x}$ and $g(x)=2a\ln x$.
(1) When $a\geqslant -1$, find the monotonically increasing interval of $F(x)=f(x)-g(x)$;
(2) Let $h(x)=f(x)+g(x)$, and $h(x)$ has two extreme values $({{x}_{1}},{{x}_{2}})$, where ${{x}_{1}}\in (0,\frac{1}{3}]$, find the minimum value of $h({{x}_{1}})-h({{... | \frac{20\ln 3-16}{3} | 0.0625 | 8,192 | 8,192 | 8,192 | |
Given $\left(x+y\right)^{2}=1$ and $\left(x-y\right)^{2}=49$, find the values of $x^{2}+y^{2}$ and $xy$. | -12 | 1 | 1,853.5 | 1,853.5 | -1 | |
Three rays emanate from a single point and form pairs of angles of $60^{\circ}$. A sphere with a radius of one unit touches all three rays. Calculate the distance from the center of the sphere to the initial point of the rays. | \sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Two is $10 \%$ of $x$ and $20 \%$ of $y$. What is $x - y$? | 10 | 1. **Translate the percentages into equations:**
- Given that two is $10\%$ of $x$, we can write this as:
\[
2 = 0.10 \times x
\]
- Similarly, two is $20\%$ of $y$, which can be written as:
\[
2 = 0.20 \times y
\]
2. **Solve for $x$ and $y$:**
- From the equation $2 = 0.10 \times... | 1 | 1,533.5 | 1,533.5 | -1 |
Given a right triangle $ABC$. On the extension of the hypotenuse $BC$, a point $D$ is chosen such that the line $AD$ is tangent to the circumcircle $\omega$ of triangle $ABC$. The line $AC$ intersects the circumcircle of triangle $ABD$ at point $E$. It turns out that the angle bisector of $\angle ADE$ is tangent to the... | 1:2 | 0.0625 | 7,965.25 | 7,191 | 8,016.866667 | |
Let \( S = \left\{\left(s_{1}, s_{2}, \cdots, s_{6}\right) \mid s_{i} \in \{0, 1\}\right\} \). For any \( x, y \in S \) where \( x = \left(x_{1}, x_{2}, \cdots, x_{6}\right) \) and \( y = \left(y_{1}, y_{2}, \cdots, y_{6}\right) \), define:
(1) \( x = y \) if and only if \( \sum_{i=1}^{6}\left(x_{i} - y_{i}\right)^{2} ... | 32 | 0.5625 | 7,120.625 | 6,287.333333 | 8,192 | |
If $x + \frac{1}{x} = 5,$ then compute the value of
\[(x - 2)^2 + \frac{25}{(x - 2)^2}.\] | 11 | 0.75 | 6,944.8125 | 6,529.083333 | 8,192 | |
Let the function $y=f\left(x\right)$ have domain $D$, and all points on its graph be above the line $y=t$. If the function $f\left(x\right)=\left(x-t\right)e^{x}$ has the domain $R$ and is a "$\left(-\infty ,+\infty \right)-t$ function", determine the largest integer value of the real number $t$. | -1 | 0.4375 | 7,979.1875 | 7,705.571429 | 8,192 | |
Given the arithmetic sequence {a<sub>n</sub>} satisfies a<sub>3</sub> − a<sub>2</sub> = 3, a<sub>2</sub> + a<sub>4</sub> = 14.
(I) Find the general term formula for {a<sub>n</sub>};
(II) Let S<sub>n</sub> be the sum of the first n terms of the geometric sequence {b<sub>n</sub>}. If b<sub>2</sub> = a<sub>2</sub>, b<sub>... | -86 | 0.5625 | 6,579.6875 | 6,608.555556 | 6,542.571429 | |
By joining four identical trapezoids, each with equal non-parallel sides and bases measuring 50 cm and 30 cm, we form a square with an area of 2500 cm² that has a square hole in the middle. What is the area, in cm², of each of the four trapezoids? | 400 | 0.0625 | 7,881.5 | 3,224 | 8,192 | |
Given that $(1-3x)^6 = a + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5 + a_6x^6$, find the total sum of elements in all subsets containing 2 elements of the set $\{a_1, a_2, a_3, a_4, a_5, a_6\}$. | 315 | 0.875 | 4,354.8125 | 3,806.642857 | 8,192 | |
There is a ten-digit number. From left to right:
- Its first digit indicates how many zeros are in the number.
- Its second digit indicates how many ones are in the number.
- Its third digit indicates how many twos are in the number.
- $\cdots \cdots$
- Its tenth digit indicates how many nines are in the number.
Find ... | 6210001000 | 0.375 | 7,385.1875 | 6,040.5 | 8,192 | |
Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, and $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit some citie... | 16 | 0 | 8,166 | -1 | 8,166 | |
Given a box containing $30$ red balls, $22$ green balls, $18$ yellow balls, $15$ blue balls, and $10$ black balls, determine the minimum number of balls that must be drawn from the box to guarantee that at least $12$ balls of a single color will be drawn. | 55 | 0.875 | 4,075.6875 | 4,101.928571 | 3,892 | |
Using systematic sampling to select a sample of size \\(20\\) from \\(160\\) students, the \\(160\\) students are numbered from \\(1\\) to \\(160\\) and evenly divided into \\(20\\) groups (\\(1～8\\), \\(9～16\\), ..., \\(153～160\\)). If the number drawn from the \\(16th\\) group is \\(123\\), then the number of the ind... | 11 | 0.5625 | 6,077.6875 | 4,433.222222 | 8,192 | |
In the equation on the right, each Chinese character represents one of the ten digits from 0 to 9. The same character represents the same digit, and different characters represent different digits. What is the four-digit number represented by "数学竞赛"? | 1962 | 0 | 7,835.75 | -1 | 7,835.75 | |
There are unique integers $a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}$ such that
\[\frac {5}{7} = \frac {a_{2}}{2!} + \frac {a_{3}}{3!} + \frac {a_{4}}{4!} + \frac {a_{5}}{5!} + \frac {a_{6}}{6!} + \frac {a_{7}}{7!}\]where $0\leq a_{i} < i$ for $i = 2,3,\ldots,7$. Find $a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7}$. | 9 | 1. **Start by multiplying both sides by 7** to clear the denominators:
\[
5 = \frac{7}{2} a_2 + \frac{7}{6} a_3 + \frac{7}{24} a_4 + \frac{7}{120} a_5 + \frac{7}{720} a_6 + \frac{7}{5040} a_7
\]
Simplifying each term:
\[
5 = 3.5 a_2 + 1.1667 a_3 + 0.2917 a_4 + 0.0583 a_5 + 0.0097 a_6 + 0.0014 a_7
\... | 0.5 | 6,688.0625 | 5,635.875 | 7,740.25 |
Find the difference between the sum of the numbers $3$, $-4$, and $-5$ and the sum of their absolute values. | -18 | 1 | 1,241.125 | 1,241.125 | -1 | |
Wendy has 180 feet of fencing. She needs to enclose a rectangular space with an area that is ten times its perimeter. If she uses up all her fencing material, how many feet is the largest side of the enclosure? | 60 | 1 | 1,765.0625 | 1,765.0625 | -1 | |
The product of two whole numbers is 24. The smallest possible sum of these two numbers is: | 10 | 1 | 1,864.8125 | 1,864.8125 | -1 |
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