problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 0.125 | 7,408.75 | 6,012.5 | 7,608.214286 | |
Given a quadratic function $f\left(x\right)=ax^{2}+bx+c$, where $f\left(0\right)=1$, $f\left(1\right)=0$, and $f\left(x\right)\geqslant 0$ for all real numbers $x$. <br/>$(1)$ Find the analytical expression of the function $f\left(x\right)$; <br/>$(2)$ If the maximum value of the function $g\left(x\right)=f\left(x\righ... | m = 2 | 0.1875 | 7,134.5625 | 6,488.333333 | 7,283.692308 | |
In the figure with circle $Q$, angle $KAT$ measures 42 degrees. What is the measure of minor arc $AK$ in degrees? [asy]
import olympiad; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
draw(unitcircle);
draw(dir(84)--(-1,0)--(1,0));
dot("$A$",(-1,0),W); dot("$K$",dir(84),NNE); dot("$T$",(1,0),E); dot("$Q$",(0,0),S)... | 96 | 0.9375 | 4,364 | 4,108.8 | 8,192 | |
Given \(0<\theta<\pi\), a complex number \(z_{1}=1-\cos \theta+i \sin \theta\) and \(z_{2}=a^{2}+a i\), where \(a \in \mathbb{R}\), it is known that \(z_{1} z_{2}\) is a pure imaginary number, and \(\bar{a}=z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}\). Determine the value of \(\theta\) when \(\bar{a}\) is a negative real number... | \frac{\pi}{2} | 0.0625 | 8,192 | 8,192 | 8,192 | |
During a space experiment conducted by astronauts, they must implement a sequence of 6 procedures. Among them, Procedure A can only occur as the first or the last step, and Procedures B and C must be adjacent when conducted. How many different sequences are there to arrange the experiment procedures? | 96 | 0.9375 | 4,591.25 | 4,351.2 | 8,192 | |
When the number $2^{1000}$ is divided by $13$, the remainder in the division is | 3 | 1. **Application of Fermat's Little Theorem**: Fermat's Little Theorem states that if $p$ is a prime number and $a$ is an integer not divisible by $p$, then $a^{p-1} \equiv 1 \pmod{p}$. Here, $p = 13$ and $a = 2$, so we have:
\[
2^{12} \equiv 1 \pmod{13}
\]
2. **Reduction of Exponent Modulo 12**: Since $2^{12... | 0.875 | 3,911.25 | 3,299.714286 | 8,192 |
If you flip a fair coin 1000 times, what is the expected value of the product of the number of heads and the number of tails? | 249750 | We solve the problem for $n$ coins. We want to find $$E(n)=\sum_{k=0}^{n} \frac{1}{2^{n}}\binom{n}{k} k(n-k)$$ We present three methods for evaluating this sum. Method 1: Discard the terms $k=0, k=n$. Since $\binom{n}{k} k(n-k)=n(n-1)\binom{n-2}{k-1}$ by the factorial definition, we may rewrite the sum as $$E(n)=\frac{... | 0.6875 | 5,140.375 | 3,753.272727 | 8,192 |
Consider a quadrilateral ABCD inscribed in a circle with radius 300 meters, where AB = BC = AD = 300 meters, and CD being the side of unknown length. Determine the length of side CD. | 300 | 0 | 8,004.0625 | -1 | 8,004.0625 | |
Point \(D\) is the midpoint of the hypotenuse \(AB\) in the right triangle \(ABC\) with legs measuring 3 and 4.
Find the distance between the centers of the inscribed circles of triangles \(ACD\) and \(BCD\). | \frac{5 \sqrt{13}}{12} | 0 | 6,011.1875 | -1 | 6,011.1875 | |
How many times does the digit 9 appear in the list of all integers from 1 to 1000? | 300 | 0.375 | 7,285.3125 | 5,774.166667 | 8,192 | |
Find $2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot 16^{\frac{1}{16}} \dotsm.$ | 4 | 0.8125 | 4,253.5 | 3,774.076923 | 6,331 | |
Square $ABCD$ has sides of length 2. Set $S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.
| 86 | 0 | 8,192 | -1 | 8,192 | |
Find the constant term in the expansion of $\left(1+x+\frac{1}{x^{2}}\right)^{10}$. | 4351 | 0.3125 | 7,693.1875 | 6,624.4 | 8,179 | |
Consider the cards $A, 2, \cdots, J, Q, K$ as the numbers $1, 2, \cdots, 11, 12, 13$. If we take the 13 cards of spades and 13 cards of hearts together and randomly draw 2 cards, what is the probability that the two cards are of the same suit and the product of the two numbers is a perfect square? | 2/65 | 0.25 | 7,607.875 | 5,855.5 | 8,192 | |
Select two distinct numbers simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6\}$. What is the probability that the smaller one divides the larger one and both numbers are either both even or both odd? | \frac{4}{15} | 0.5625 | 6,137.8125 | 4,898.333333 | 7,731.428571 | |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% | 0.3125 | 3,799 | 3,075.6 | 4,127.818182 | |
Evaluate $\log_3\frac{1}{\sqrt3}$. | -\frac12 | 1 | 1,963.125 | 1,963.125 | -1 | |
How many 11 step paths are there from point A to point D, which pass through points B and C in that order? Assume the grid layout permits only right and down steps, where B is 2 right and 2 down steps from A, and C is 1 right and 3 down steps from B, and finally, D is 3 right and 1 down step from C. | 96 | 0.5 | 7,476.6875 | 6,970.875 | 7,982.5 | |
The perimeter of a rectangle is 30 inches. One side of the rectangle is fixed at 7 inches. What is the number of square inches in the maximum possible area for this rectangle? | 56 | 0.5625 | 5,640.5625 | 4,814.777778 | 6,702.285714 | |
What is the coefficient of $x^4$ when
$$x^5 - 4x^4 + 6x^3 - 7x^2 + 2x - 1$$
is multiplied by
$$3x^4 - 2x^3 + 5x - 8$$
and combining the similar terms? | 27 | 0 | 2,099.8125 | -1 | 2,099.8125 | |
Given vectors $a=(1,1)$ and $b=(2,t)$, find the value of $t$ such that $|a-b|=a·b$. | \frac{-5 - \sqrt{13}}{2} | 0 | 3,262.375 | -1 | 3,262.375 | |
The increasing sequence of positive integers $a_1,$ $a_2,$ $a_3,$ $\dots$ has the property that
\[a_{n + 2} = a_{n + 1} + a_n\]for all $n \ge 1.$ If $a_7 = 120,$ then find $a_8.$ | 194 | 0.9375 | 3,656.875 | 3,354.533333 | 8,192 | |
Given a tesseract (4-dimensional hypercube), calculate the sum of the number of edges, vertices, and faces. | 72 | 0.4375 | 6,471.1875 | 5,339.714286 | 7,351.222222 | |
Consider a set of 150 cards numbered from 1 to 150. Each card is randomly placed in a box. A card is selected at random from this box. What is the probability that the number on the card is a multiple of 4, 5, or 6? Express your answer as a common fraction. | \frac{7}{15} | 0.75 | 3,457.3125 | 3,497.5 | 3,336.75 | |
Given a geometric sequence with positive terms $\{a_n\}$ and a common ratio of $2$, if $a_ma_n=4a_2^2$, then the minimum value of $\frac{2}{m}+ \frac{1}{2n}$ equals \_\_\_\_\_\_. | \frac{3}{4} | 1 | 3,248.1875 | 3,248.1875 | -1 | |
Given a quadratic polynomial \( P(x) \). It is known that the equations \( P(x) = x - 2 \) and \( P(x) = 1 - x / 2 \) each have exactly one root. What is the discriminant of \( P(x) \)? | -\frac{1}{2} | 0.4375 | 7,344.5 | 6,254.857143 | 8,192 | |
Given a rectangular painting that measures $12$ inches high and $16$ inches wide, and is placed in a frame that forms a border three inches wide on all sides, find the area of the border, in square inches. | 204 | 0.75 | 4,974.6875 | 3,988 | 7,934.75 | |
Let \( f(x) \) be a function with the property that \( f(x) + f\left(\frac{x-1}{3x-2}\right) = x \) for all real numbers \( x \) other than \( \frac{2}{3} \). What is the sum \( f(0) + f(1) + f(2) \)? | \frac{87}{40} | 0.5625 | 7,507.0625 | 6,974.333333 | 8,192 | |
Define the function $f: \mathbb{R} \rightarrow \mathbb{R}$ by $$f(x)= \begin{cases}\frac{1}{x^{2}+\sqrt{x^{4}+2 x}} & \text { if } x \notin(-\sqrt[3]{2}, 0] \\ 0 & \text { otherwise }\end{cases}$$ The sum of all real numbers $x$ for which $f^{10}(x)=1$ can be written as $\frac{a+b \sqrt{c}}{d}$, where $a, b, c, d$ are ... | 932 | If $x \in(-\sqrt[3]{2}, 0]$, it is evidently not a solution, so let us assume otherwise. Then, we find $$f(x)=\frac{\sqrt{x^{4}+2 x}-x^{2}}{2 x}$$ which implies that $x f(x)^{2}+x^{2} f(x)-1 / 2=0$, by reverse engineering the quadratic formula. Therefore, if $x>0, f(x)$ is the unique positive real $t$ so that $x t^{2}+... | 0 | 8,192 | -1 | 8,192 |
In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\angle A = 37^\circ$, $\angle D = 53^\circ$, and $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $MN$. | 504 | Obviously, these angles are random--the only special thing about them is that they add up to 90. So we might as well let the given angles equal 45 and 45, and now the answer is trivially $\boxed{504}$. (The trapezoid is isosceles, and you see two 45-45-90 triangles;from there you can get the answer.) | 0.5 | 7,056.0625 | 6,308.25 | 7,803.875 |
Let $a$, $b$, $c$ be positive numbers, and $a+b+9c^2=1$. Find the maximum value of $\sqrt{a}+ \sqrt{b}+ \sqrt{3}c$. | \frac{\sqrt{21}}{3} | 0 | 7,741.1875 | -1 | 7,741.1875 | |
Given $f(x) = e^{-x}$, calculate the limit $$\lim_{\Delta x \to 0} \frac{f(1 + \Delta x) - f(1 - 2\Delta x)}{\Delta x}$$. | -\frac{3}{e} | 0.6875 | 6,178.1875 | 5,262.818182 | 8,192 | |
You have $7$ red shirts, $8$ green shirts, $10$ pairs of pants, $10$ blue hats, $10$ red hats, and $5$ scarves (each distinct). How many outfits can you make consisting of one shirt, one pair of pants, one hat, and one scarf, without having the same color of shirts and hats? | 7500 | 0.3125 | 7,249.3125 | 6,902 | 7,407.181818 | |
In \\(\triangle ABC\\), \\(AB=BC\\), \\(\cos B=-\dfrac{7}{18}\\). If an ellipse with foci at points \\(A\\) and \\(B\\) passes through point \\(C\\), find the eccentricity of the ellipse. | \dfrac{3}{8} | 0.75 | 4,499.5 | 3,268.666667 | 8,192 | |
There are two values of $k$ for which the equation $x^{2}+2kx+7k-10=0$ has two equal real roots (that is, has exactly one solution for $x$). What is the sum of these values of $k$? | 7 | The equation $x^{2}+2kx+7k-10=0$ has two equal real roots precisely when the discriminant of this quadratic equation equals 0. The discriminant, $\Delta$, equals $\Delta=(2k)^{2}-4(1)(7k-10)=4k^{2}-28k+40$. For the discriminant to equal 0, we have $4k^{2}-28k+40=0$ or $k^{2}-7k+10=0$ or $(k-2)(k-5)=0$. Thus, $k=2$ or $... | 1 | 1,904 | 1,904 | -1 |
Suppose the mean of one set consisting of seven numbers is 18, and the mean of another set consisting of eight numbers is 16. What is the mean of all fifteen numbers combined? | \frac{254}{15} | 0.9375 | 2,979.25 | 2,940.533333 | 3,560 | |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $c= \sqrt {7}$, $C= \frac {\pi}{3}$.
(1) If $2\sin A=3\sin B$, find $a$ and $b$;
(2) If $\cos B= \frac {3 \sqrt {10}}{10}$, find the value of $\sin 2A$. | \frac {3-4 \sqrt {3}}{10} | 0 | 7,382.6875 | -1 | 7,382.6875 | |
Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$ .
Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$ | 22 | 0.25 | 7,684.875 | 7,100.75 | 7,879.583333 | |
Given that the function $f(x)$ satisfies: $4f(x)f(y)=f(x+y)+f(x-y)$ $(x,y∈R)$ and $f(1)= \frac{1}{4}$, find $f(2014)$. | - \frac{1}{4} | 0.75 | 6,675.125 | 6,169.5 | 8,192 | |
An infinite geometric series has a sum of 2020. If the first term, the third term, and the fourth term form an arithmetic sequence, find the first term. | 1010(1+\sqrt{5}) | 0.8125 | 5,123 | 4,414.769231 | 8,192 | |
Cutting equilateral triangle $BFC$ out of square $ABCD$ and translating it to the left of the square creates the shaded figure $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. What is the perimeter, in inches, of $ABFCDE$?
[asy]
filldraw((0,0)--(sqrt(3),1)--(sqrt(3)+2,1)--(2,0)--(sqrt(3)+2,-1)--(sqrt(3),-1)--(0... | 72 | 0.3125 | 7,477.625 | 7,046.2 | 7,673.727273 | |
In diagram square $ABCD$, four triangles are removed resulting in rectangle $PQRS$. Two triangles at opposite corners ($SAP$ and $QCR$) are isosceles with each having area $120 \text{ m}^2$. The other two triangles ($SDR$ and $BPQ$) are right-angled at $D$ and $B$ respectively, each with area $80 \text{ m}^2$. What is ... | 4\sqrt{15} | 0 | 8,192 | -1 | 8,192 | |
Let $a_n= \frac {1}{n}\sin \frac {n\pi}{25}$, and $S_n=a_1+a_2+\ldots+a_n$. Find the number of positive terms among $S_1, S_2, \ldots, S_{100}$. | 100 | 0 | 8,192 | -1 | 8,192 | |
A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has an altitude of $24$ centimeters, the area of its lower base is $225\pi$ sq cm and the area of its upper base is $25\pi$ sq cm, what is the altitude of the small cone that was cut off? [asy]s... | 12 | 1 | 2,047.75 | 2,047.75 | -1 | |
Evaluate $\log_{\sqrt{6}} (216\sqrt{6})$. | 7 | 1 | 2,282.375 | 2,282.375 | -1 | |
Given points $A(-2, 0)$ and $B(0, 2)$, $k$ is a constant, and $M$, $N$ are two distinct points on the circle ${x^2} + {y^2} + kx = 0$. $P$ is a moving point on the circle ${x^2} + {y^2} + kx = 0$. If $M$ and $N$ are symmetric about the line $x - y - 1 = 0$, find the maximum area of $\triangle PAB$. | 3 + \sqrt{2} | 0.0625 | 8,192 | 8,192 | 8,192 | |
Let $1 = x_{1} < x_{2} < \dots < x_{k} = n$ denote the sequence of all divisors $x_{1}, x_{2} \dots x_{k}$ of $n$ in increasing order. Find the smallest possible value of $n$ such that $$ n = x_{1}^{2} + x_{2}^{2} +x_{3}^{2} + x_{4}^{2}. $$ *Proposed by Justin Lee* | 130 | 0.0625 | 8,155.6875 | 7,611 | 8,192 | |
Scientists found a fragment of an ancient mechanics manuscript. It was a piece of a book where the first page was numbered 435, and the last page was numbered with the same digits, but in some different order. How many sheets did this fragment contain? | 50 | 0.3125 | 6,564.1875 | 5,205 | 7,182 | |
Place each of the digits 4, 5, 6, and 7 in exactly one square to make the smallest possible product. The grid placement is the same as described before. | 2622 | 0.3125 | 7,389.1875 | 6,397.8 | 7,839.818182 | |
In the diagram, a road network between the homes of five friends is shown. The shortest distance by road from Asya to Galia is 12 km, from Galia to Borya is 10 km, from Asya to Borya is 8 km, from Dasha to Galia is 15 km, and from Vasya to Galia is 17 km. What is the shortest distance by road from Dasha to Vasya? | 18 | 0 | 5,538.1875 | -1 | 5,538.1875 | |
As shown in the diagram, \( D \), \( E \), and \( F \) are points on the sides \( BC \), \( CA \), and \( AB \) of \(\triangle ABC\), respectively, and \( AD \), \( BE \), \( CF \) intersect at point \( G \). Given that the areas of \(\triangle BDG\), \(\triangle CDG\), and \(\triangle AEG\) are 8, 6, and 14 respective... | 63 | 0 | 8,192 | -1 | 8,192 | |
Two counterfeit coins of equal weight are mixed with $8$ identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the $10$ coins. A second pair is selected at random without replacement ... | \frac{15}{19} | To solve this problem, we need to calculate the probability that all four selected coins are genuine, given that the combined weight of the first pair is equal to the combined weight of the second pair.
#### Step 1: Define the events
Let $A$ be the event that all four selected coins are genuine.
Let $B$ be the event t... | 0.375 | 6,945.4375 | 5,374.666667 | 7,887.9 |
How many integers between 1000 and 2000 have all three of the numbers 15, 20 and 25 as factors? | 3 | 1 | 4,261.875 | 4,261.875 | -1 | |
Let $a,b,$ and $c$ be real numbers such that \[a + b + c = 2\]and \[a^2+b^2+c^2=12.\]What is the difference between the maximum and minimum possible values of $c$? | \tfrac{16}3 | 0 | 3,008.75 | -1 | 3,008.75 | |
$\frac{2^1+2^0+2^{-1}}{2^{-2}+2^{-3}+2^{-4}}$ equals | 8 | 1. **Evaluate the Numerator**:
\[
2^1 + 2^0 + 2^{-1} = 2 + 1 + \frac{1}{2} = 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}
\]
2. **Evaluate the Denominator**:
\[
2^{-2} + 2^{-3} + 2^{-4} = \frac{1}{4} + \frac{1}{8} + \frac{1}{16}
\]
To add these fractions, find a common denominator, ... | 1 | 3,179.875 | 3,179.875 | -1 |
Given $(2x-1)^{2015} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots + a_{2015}x^{2015}$ ($x \in \mathbb{R}$), evaluate the expression $\frac {1}{2}+ \frac {a_{2}}{2^{2}a_{1}}+ \frac {a_{3}}{2^{3}a_{1}}+\ldots+ \frac {a_{2015}}{2^{2015}a_{1}}$. | \frac {1}{4030} | 0.25 | 7,618.1875 | 5,896.75 | 8,192 | |
In the center of a square, there is a police officer, and in one of the vertices, there is a gangster. The police officer can run throughout the whole square, while the gangster can only run along its sides. It is known that the ratio of the maximum speed of the police officer to the maximum speed of the gangster is: 0... | 1/3 | 0.125 | 7,987.0625 | 7,303.5 | 8,084.714286 | |
Last year a bicycle cost $160 and a cycling helmet $40. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is: | 6\% | 1. **Calculate the new cost of the bicycle:**
The original cost of the bicycle is $160. The cost increased by $5\%$. Therefore, the increase in cost is:
\[
160 \times 0.05 = 8
\]
Adding this to the original cost gives the new cost:
\[
160 + 8 = 168
\]
2. **Calculate the new cost of the helmet:*... | 1 | 2,148.9375 | 2,148.9375 | -1 |
Given the vertex of angle α is at the origin of the coordinate system, its initial side coincides with the non-negative half-axis of the x-axis, and its terminal side passes through the point (-√3,2), find the value of tan(α - π/6). | -3\sqrt{3} | 0.9375 | 4,398.5625 | 4,145.666667 | 8,192 | |
A certain product in a shopping mall sells an average of 70 items per day, with a profit of $50 per item. In order to reduce inventory quickly, the mall decides to take appropriate price reduction measures. After investigation, it was found that for each item, for every $1 decrease in price, the mall can sell an additi... | 12 | 1 | 6,057.25 | 6,057.25 | -1 | |
For real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right]=1994\). | 312 | 0.1875 | 7,724.0625 | 5,987 | 8,124.923077 | |
Ellen wants to color some of the cells of a $4 \times 4$ grid. She wants to do this so that each colored cell shares at least one side with an uncolored cell and each uncolored cell shares at least one side with a colored cell. What is the largest number of cells she can color? | 12 | 0 | 8,046.0625 | -1 | 8,046.0625 | |
In the expansion of $({\frac{1}{x}-\sqrt{x}})^{10}$, determine the coefficient of $x^{2}$. | 45 | 0.9375 | 3,626.25 | 3,321.866667 | 8,192 | |
The orchestra has more than 150 members but fewer than 300 members. When they line up in rows of 6 there are two extra people; when they line up in rows of 8 there are four extra people; and when they line up in rows of 9 there are six extra people. How many members are in the orchestra? | 212 | 0 | 8,192 | -1 | 8,192 | |
Jeremy's friend Steven will pay him $\dfrac{11}{2}$ dollars for every room he cleans. Yesterday, Jeremy cleaned $\dfrac{7}{3}$ rooms. How many dollars does Steven owe Jeremy? Leave your answer in fraction form. | \frac{77}{6} | 1 | 1,219.375 | 1,219.375 | -1 | |
What is the largest prime factor of $3328$? | 13 | 0.9375 | 2,481.0625 | 2,100.333333 | 8,192 | |
A graph has $ 30$ vertices, $ 105$ edges and $ 4822$ unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph. | 22 | 0 | 8,192 | -1 | 8,192 | |
Expand $(x+10)(2y+10)$. | 2xy + 10x + 20y + 100 | 1 | 1,927.125 | 1,927.125 | -1 | |
Let \( p(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( a, b, c, d \) are constants, and given that \( p(1) = 1993 \), \( p(2) = 3986 \), and \( p(3) = 5979 \). Calculate \( \frac{1}{4} [p(11) + p(-7)] \). | 5233 | 0.5625 | 6,381.8125 | 5,848.888889 | 7,067 | |
Marisa has two identical cubical dice labeled with the numbers \(\{1,2,3,4,5,6\}\). However, the two dice are not fair, meaning that they can land on each face with different probability. Marisa rolls the two dice and calculates their sum. Given that the sum is 2 with probability 0.04, and 12 with probability 0.01, the... | 28 | Let \(p_{i}\) be the probability that the dice lands on the number \(i\). The problem gives that \(p_{1}^{2}=0.04, p_{6}^{2}=0.01\), so we have \(p_{1}=0.2, \quad p_{6}=0.1, \quad p_{2}+p_{3}+p_{4}+p_{5}=0.7\). We are asked to maximize $$2\left(p_{1} p_{6}+p_{2} p_{5}+p_{3} p_{4}\right)=2\left(0.02+p_{2} p_{5}+p_{3} p_... | 0.0625 | 8,116.8125 | 7,205 | 8,177.6 |
Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{23}{36} | 0 | 7,489.5625 | -1 | 7,489.5625 | |
Let $g_0(x) = x + |x - 150| - |x + 150|$, and for $n \geq 1$, let $g_n(x) = |g_{n-1}(x)| - 2$. For how many values of $x$ is $g_{100}(x) = 0$? | 299 | 0 | 8,192 | -1 | 8,192 | |
Let $S$ denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when $S$ is divided by $1000$.
| 680 | 0.125 | 6,734.25 | 4,093 | 7,111.571429 | |
[b]Problem Section #1
a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$ , and y. Find the greatest possible value of: $x + y$ .
<span style="color:red">NOTE: There is a high chance that this problems was copied.</span> | 761 | 0.125 | 7,976.3125 | 7,007 | 8,114.785714 | |
The left focus of the hyperbola $C$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \, (a > 0, b > 0)$ is $F$. If the symmetric point $A$ of $F$ with respect to the line $\sqrt{3}x + y = 0$ is a point on the hyperbola $C$, then the eccentricity of the hyperbola $C$ is \_\_\_\_\_\_. | \sqrt{3} + 1 | 0 | 6,045.5625 | -1 | 6,045.5625 | |
Given the expansion of the binomial $({x+\frac{a}{{\sqrt{x}}}})^n$ where $n\in{N^*}$, in the expansion, ___, ___. Given the following conditions: ① the ratio of the binomial coefficients of the second term to the third term is $1:4$; ② the sum of all coefficients is $512$; ③ the $7$th term is a constant term. Choose tw... | -48 | 0.4375 | 6,883.8125 | 5,507.857143 | 7,954 | |
If $x$ and $y$ are positive integers less than $20$ for which $x + y + xy = 76$, what is the value of $x + y$? | 16 | 1 | 1,716.75 | 1,716.75 | -1 | |
Let $\triangle ABC$ be a right triangle with $B$ as the right angle. A circle with diameter $BC$ intersects side $AB$ at point $E$ such that $BE = 6$. If $AE = 3$, find the length of $EC$. | 12 | 0 | 8,192 | -1 | 8,192 | |
Evaluate $\log_2 (4^2)$. | 4 | 1 | 1,891.3125 | 1,891.3125 | -1 | |
In the diagram, point $E$ lies on line segment $AB$, and triangles $AED$ and $BEC$ are isosceles. Also, $\angle DEC$ is twice $\angle ADE$. What is the measure of $\angle EBC$ in degrees? [asy]
import olympiad;
import math;
size(7cm);
// Draw triangles
pair a = (-2 * Cos(70), 0);
pair b = (1, 0);
pair c = dir(30);
pa... | 75^\circ | 0.125 | 7,351 | 6,501.5 | 7,472.357143 | |
(Answer in numbers)
From 5 different storybooks and 4 different math books, 4 books are to be selected and given to 4 students, one book per student. How many different ways are there to:
(1) Select 2 storybooks and 2 math books?
(2) Ensure one specific storybook and one specific math book are among the selected?... | 1080 | 0.375 | 4,106.5625 | 5,421 | 3,317.9 | |
Four siblings inherited a plot of land shaped like a convex quadrilateral. By connecting the midpoints of the opposite sides of the plot, they divided the inheritance into four quadrilaterals. The first three siblings received plots of $360 \, \mathrm{m}^{2}$, $720 \, \mathrm{m}^{2}$, and $900 \, \mathrm{m}^{2}$ respec... | 540 | 0.125 | 7,744.5625 | 6,384 | 7,938.928571 | |
A circle with a radius of 2 units rolls around the inside of a triangle with sides 9, 12, and 15 units. The circle is always tangent to at least one side of the triangle. Calculate the total distance traveled by the center of the circle when it returns to its starting position. | 24 | 0 | 8,025 | -1 | 8,025 | |
Regular hexagon $ABCDEF$ is divided into six smaller equilateral triangles, such as $\triangle ABG$, shown in boldface in the diagram. By connecting every other vertex, we obtain a larger equilateral triangle $\triangle ACE$, also shown in boldface. Compute the ratio $[\triangle ABG]/[\triangle ACE]$. [asy]
size(150)... | \frac{1}{3} | 0.75 | 6,433.9375 | 5,847.916667 | 8,192 | |
Consider all 4-digit palindromes that can be written as $\overline{abba}$, where $a$ is non-zero and $b$ ranges from 1 to 9. Calculate the sum of the digits of the sum of all such palindromes. | 36 | 0.75 | 5,250.375 | 4,547.083333 | 7,360.25 | |
Calculate the sum: $\dfrac{2}{100} + \dfrac{5}{1000} + \dfrac{8}{10000} + \dfrac{6}{100000}$. | 0.02586 | 0.625 | 5,132.4375 | 3,296.7 | 8,192 | |
Given that the domains of functions f(x) and g(x) are both $\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, $g(2) = 4$, calculate the value of $\sum _{k=1}^{22}f(k)$. | -24 | 0.125 | 7,889.25 | 6,293 | 8,117.285714 | |
Mrs. Novel has five different Batman comic books, six different X-Men comic books, and four different Calvin and Hobbes comic books. All of the Batman comic books are grouped together, all of the X-men comic books are grouped together, and all of the Calvin and Hobbes comic books are grouped together. In how many diffe... | 12,441,600 | 0 | 3,056.5 | -1 | 3,056.5 | |
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand? | \frac{47}{256} | 1. **Understanding the Problem:**
We need to find the probability that no two adjacent people will stand after flipping their coins around a circular table with 8 people. Each person flips a fair coin, resulting in either heads (stand) or tails (sit).
2. **Total Outcomes:**
Each person has 2 possible outcomes (h... | 0.1875 | 7,815.4375 | 7,011 | 8,001.076923 |
A cone has a volume of $12288\pi$ cubic inches and the vertex angle of the vertical cross section is 60 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. [asy]
import markers;
size(150);
import geometry;
draw(scale(1,.2)*arc((0,0),1,0,180),dashed);
draw(scale(1,.2)*arc((0,... | 48.0 | 0.6875 | 5,221.875 | 4,406 | 7,016.8 | |
Find the sum of all integers $k$ such that $\binom{23}{4} + \binom{23}{5} = \binom{24}{k}$. | 24 | 0.875 | 3,338.125 | 2,644.714286 | 8,192 | |
Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. If the polar equation of curve $C$ is $\rho\cos^2\theta-4\sin\theta=0$, and the polar coordin... | \frac{\sqrt{6}}{6} | 0 | 5,764 | -1 | 5,764 | |
For how many values of $n$ with $3 \leq n \leq 12$ can a Fano table be created? | 3 | First, we calculate the number of pairs that can be formed from the integers from 1 to $n$. One way to form a pair is to choose one number to be the first item of the pair ($n$ choices) and then a different number to be the second item of the pair ($n-1$ choices). There are $n(n-1)$ ways to choose these two items in th... | 0.0625 | 7,401.0625 | 4,410 | 7,600.466667 |
A parallelogram has its diagonals making an angle of \(60^{\circ}\) with each other. If two of its sides have lengths 6 and 8, find the area of the parallelogram. | 14\sqrt{3} | 0.625 | 6,158.3125 | 5,594.6 | 7,097.833333 | |
In right triangle $GHI$, we have $\angle G = 30^\circ$, $\angle H = 90^\circ$, and $HI = 12$. Find $GH$ to the nearest tenth. | 20.8 | 1 | 2,419.0625 | 2,419.0625 | -1 | |
Let $\left\{x_{n}\right\}$ be a sequence of positive real numbers. If $x_{1}=\frac{3}{2}$ and $x_{n+1}^{2}-x_{n}^{2}=\frac{1}{(n+2)^{2}}-\frac{1}{n^{2}}$ for all positive integers $n$, find $x_{1}+x_{2}+\cdots+x_{2009}$. | \frac{4040099}{2010} | 0 | 8,156.3125 | -1 | 8,156.3125 | |
Given that \(\alpha\), \(\beta\), and \(\gamma\) are acute angles such that \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\), find the minimum value of \(\tan \alpha \cdot \tan \beta \cdot \tan \gamma\). | 2\sqrt{2} | 0.3125 | 7,704.25 | 6,631.2 | 8,192 | |
How many ways are there to put 7 balls into 4 boxes if the balls are indistinguishable and the boxes are also indistinguishable? | 11 | 0.375 | 6,853.5 | 5,258.166667 | 7,810.7 | |
The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron? | 45 | 0 | 8,192 | -1 | 8,192 | |
Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard? | 481 | 1. **Understanding the Checkerboard Pattern**: The checkerboard is $31 \times 31$ with alternating red and black squares. Each corner square is black. Since $31$ is odd, each row and each column starts and ends with a black square.
2. **Counting Black Squares in a $30 \times 30$ Board**:
- A $30 \times 30$ board (... | 0.875 | 4,524.9375 | 4,001.071429 | 8,192 |
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