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 |
|---|---|---|---|---|---|---|
Three integers from the list $1,2,4,8,16,20$ have a product of 80. What is the sum of these three integers? | 25 | The three integers from the list whose product is 80 are 1, 4, and 20, since $1 \times 4 \times 20=80$. The sum of these integers is $1+4+20=25$.
(Since 80 is a multiple of 5 and 20 is the only integer in the list that is a multiple of 5, then 20 must be included in the product. This leaves two integers to choose, an... | 0.9375 | 3,642.75 | 3,339.466667 | 8,192 |
Karl the old shoemaker made a pair of boots and sent his son Hans to the market to sell them for 25 talers. At the market, two people, one missing his left leg and the other missing his right leg, approached Hans and asked to buy one boot each. Hans agreed and sold each boot for 12.5 talers.
When Hans came home and t... | 25 | 0.125 | 7,154.5625 | 7,100 | 7,162.357143 | |
Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane so that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$ . What largest possible number of these angles can be equal to $90^\circ$ ?
*Proposed by Anton Trygub* | 2,042,220 | 0 | 8,192 | -1 | 8,192 | |
A fair 6-sided die is rolled. What is the probability that the number rolled is a divisor of 6? | \dfrac23 | 1 | 1,321.3125 | 1,321.3125 | -1 | |
The surface area of a sphere with edge lengths 3, 4, and 5 on the rectangular solid is what? | 50\pi | 0.6875 | 4,237.3125 | 4,578.545455 | 3,486.6 | |
Find the measure of angle \( B \widehat{A} D \), given that \( D \widehat{A C}=39^{\circ} \), \( A B = A C \), and \( A D = B D \). | 47 | 0.1875 | 7,310.875 | 4,574.666667 | 7,942.307692 | |
Given the parabola $C: y^2 = 2px \ (0 < p < 4)$ with focus $F$, and a moving point $P$ on $C$. Let $A(4, 0)$ and $B(p, \sqrt{2}p)$ be such that the minimum value of $|PA|$ is $\sqrt{15}$. Find the value of $|BF|$. | \frac{9}{2} | 0.75 | 3,855.125 | 3,727.25 | 4,238.75 | |
$n$ mushroom pickers went into the forest and brought back a total of 200 mushrooms (possibly, some of the pickers did not bring any mushrooms home). A boy named Petya, upon learning this, stated: "Some two of them must have brought the same number of mushrooms!" What is the smallest $n$ for which Petya is certainly ri... | 21 | 0.4375 | 6,606.3125 | 5,895.714286 | 7,159 | |
What is the maximum number of finite roots that the equation
$$
\left|x - a_{1}\right| + \ldots + |x - a_{50}| = \left|x - b_{1}\right| + \ldots + |x - b_{50}|
$$
can have, where $a_{1}, a_{2}, \ldots, a_{50}, b_{1}, b_{2}, \ldots, b_{50}$ are distinct numbers? | 49 | 0 | 7,955.375 | -1 | 7,955.375 | |
Aunt Anna is $42$ years old. Caitlin is $5$ years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? | 17 | 1. **Determine Brianna's Age:**
Given that Aunt Anna is $42$ years old and Brianna is half as old as Aunt Anna, we calculate Brianna's age as follows:
\[
\text{Brianna's age} = \frac{1}{2} \times \text{Aunt Anna's age} = \frac{1}{2} \times 42 = 21 \text{ years}
\]
2. **Determine Caitlin's Age:**
Caitlin... | 0 | 843.375 | -1 | 843.375 |
The base of the quadrangular pyramid \( M A B C D \) is a parallelogram \( A B C D \). Given that \( \overline{D K} = \overline{K M} \) and \(\overline{B P} = 0.25 \overline{B M}\), the point \( X \) is the intersection of the line \( M C \) and the plane \( A K P \). Find the ratio \( M X: X C \). | 3 : 4 | 0.125 | 5,986.375 | 4,396 | 6,213.571429 | |
The arithmetic mean of nine numbers is 54. If two numbers $u$ and $v$ are added to the list, the mean of the eleven-member list becomes 66. What is the mean of $u$ and $v$? | 120 | 1 | 2,073 | 2,073 | -1 | |
Calculate the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.
\[ y = \ln \frac{5}{2 x}, \quad \sqrt{3} \leq x \leq \sqrt{8} \] | 1 + \frac{1}{2} \ln \frac{3}{2} | 0.375 | 6,467.875 | 5,656 | 6,955 | |
Acute angles \( A \) and \( B \) of a triangle satisfy the equation \( \tan A - \frac{1}{\sin 2A} = \tan B \) and \( \cos^2 \frac{B}{2} = \frac{\sqrt{6}}{3} \). Determine the value of \( \sin 2A \). | \frac{2\sqrt{6} - 3}{3} | 0 | 8,098 | -1 | 8,098 | |
A positive integer $N$ greater than $1$ is described as special if in its base- $8$ and base- $9$ representations, both the leading and ending digit of $N$ are equal to $1$ . What is the smallest special integer in decimal representation?
*Proposed by Michael Ren* | 793 | 0.1875 | 7,945.75 | 6,878.666667 | 8,192 | |
In a regular tetrahedron with edge length $2\sqrt{6}$, the total length of the intersection between the sphere with center $O$ and radius $\sqrt{3}$ and the surface of the tetrahedron is ______. | 8\sqrt{2}\pi | 0.3125 | 7,296.5 | 6,194.8 | 7,797.272727 | |
In triangle $ABC$, $AB = 12$, $AC = 10$, and $BC = 16$. The centroid $G$ of triangle $ABC$ divides each median in the ratio $2:1$. Calculate the length $GP$, where $P$ is the foot of the perpendicular from point $G$ to side $BC$. | \frac{\sqrt{3591}}{24} | 0 | 6,940.3125 | -1 | 6,940.3125 | |
An isosceles triangle $DEF$ has $DE = DF = 5\sqrt{3}$, and a circle with radius $3\sqrt{3}$ is tangent to line $DE$ at $E$ and to line $DF$ at $F$. What is the area of the circle that passes through vertices $D$, $E$, and $F$?
A) $63\pi$
B) $48\pi$
C) $72\pi$
D) $36\pi$
E) $54\pi$ | 48\pi | 0 | 8,192 | -1 | 8,192 | |
The number of integer solutions for the inequality \( |x| < 3 \pi \) is ( ). | 19 | 1 | 2,762.4375 | 2,762.4375 | -1 | |
How many different real numbers $x$ satisfy the equation $(x^{2}-5)^{2}=16$? | 4 | 1. **Rewrite the given equation**: We start with the equation \[(x^2-5)^2 = 16.\]
2. **Simplify the equation**: We can simplify this equation by taking the square root of both sides, remembering to consider both the positive and negative roots:
\[x^2 - 5 = \pm 4.\]
3. **Solve for \(x^2\)**: This gives us two separ... | 1 | 1,682.4375 | 1,682.4375 | -1 |
How many ways can change be made for a quarter using standard U.S. coins? (Don't count "1 quarter" as making change for a quarter.) | 12 | 0.6875 | 6,080.5 | 5,177.272727 | 8,067.6 | |
Let \( n \) be a fixed integer, \( n \geqslant 2 \).
(a) Determine the minimal constant \( c \) such that the inequality
$$
\sum_{1 \leqslant i < j \leqslant n} x_i x_j \left(x_i^2 + x_j^2\right) \leqslant c \left( \sum_{1 \leqslant i \leqslant n} x_i \right)^4
$$
holds for all non-negative real numbers \( x_1, x_2, \... | \frac{1}{8} | 0 | 8,192 | -1 | 8,192 | |
Given $f(x) = 2\cos^{2}x + \sqrt{3}\sin2x + a$, where $a$ is a real constant, find the value of $a$, given that the function has a minimum value of $-4$ on the interval $\left[0, \frac{\pi}{2}\right]$. | -4 | 0.625 | 6,447.875 | 6,566.5 | 6,250.166667 | |
Given a triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. If $a=2$, $A= \frac{\pi}{3}$, and $\frac{\sqrt{3}}{2} - \sin(B-C) = \sin 2B$, find the area of $\triangle ABC$. | \frac{2\sqrt{3}}{3} | 0 | 7,611.4375 | -1 | 7,611.4375 | |
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ For example, $\tau (1)=1$ and $\tau(6) =4.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 2005$ with $S(n)$ odd, and let $b$ denote ... | 25 | Let $\Delta n$ denote the sum $1+2+3+ \dots +n-1+n$. We can easily see from the fact "It is well-known that $\tau(n)$ is odd if and only if $n$ is a perfect square.", that
$a = (2^2-1^2) + (4^2-3^2) \dots (44^2 - 43^2) = (2+1)(2-1)+(4+3)(4-3) \dots (44+43)(44-43) = 1+2+3...44 = \Delta 44$.
$b = 3^2-2^2+5^2-4^2...2006-4... | 0.25 | 7,518.375 | 6,470.25 | 7,867.75 |
Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$
if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$. | $0 \leq \frac{y-x}{x+4y} \leq 4$ |
Given the equation \( x^2y^2 + xy + 1 = 3y^2 \), we need to determine the largest and smallest values of the fraction \( F = \frac{y-x}{x+4y} \).
### Step 1: Analyze the Given Equation
To simplify the problem, we first explore the given equation:
\[
x^2y^2 + xy + 1 = 3y^2
\]
Rearranging terms, we get:
\[
x^2y^2 + xy... | 0 | 8,183.375 | -1 | 8,183.375 |
The first term of a sequence is \( a_{1} = 1 \), and each subsequent term is defined by
\[ a_{n+1} = 1 + \frac{n}{a_{n}}, \quad n = 1, 2, 3, \ldots \]
Does the following limit exist? If it exists, determine it.
\[ \lim_{n \rightarrow \infty} \left(a_{n} - \sqrt{n}\right) \] | \frac{1}{2} | 0.125 | 8,073.375 | 7,243 | 8,192 | |
Given $\alpha \in \left(0, \frac{\pi}{2}\right)$, $\beta \in \left(\frac{\pi}{2}, \pi\right)$, $\cos\beta = -\frac{1}{3}$, $\sin(\alpha + \beta) = \frac{7}{9}$.
(1) Find the value of $\tan \frac{\beta}{2}$.
(2) Find the value of $\sin\alpha$. | \frac{1}{3} | 0.8125 | 4,945.8125 | 4,196.692308 | 8,192 | |
Using $600$ cards, $200$ of them having written the number $5$ , $200$ having a $2$ , and the other $200$ having a $1$ , a student wants to create groups of cards such that the sum of the card numbers in each group is $9$ . What is the maximum amount of groups that the student may create? | 100 | 0 | 8,192 | -1 | 8,192 | |
What is the sum of the positive solutions to \( 2x^2 - x \lfloor x \rfloor = 5 \), where \( \lfloor x \rfloor \) is the largest integer less than or equal to \( x \)? | \frac{3 + \sqrt{41} + 2\sqrt{11}}{4} | 0 | 7,433.5 | -1 | 7,433.5 | |
Simplify the expression $(-\frac{1}{343})^{-2/3}$. | 49 | 0.9375 | 3,956.8125 | 3,674.466667 | 8,192 | |
What is the value of $0.\overline{789}-0.\overline{456}-0.\overline{123}?$ Express your answer as a fraction in lowest terms. | \frac{70}{333} | 1 | 3,171.1875 | 3,171.1875 | -1 | |
A triangle with side lengths 8, 13, and 17 has an incircle. The side length of 8 is divided by the point of tangency into segments \( r \) and \( s \), with \( r < s \). Find the ratio \( r : s \). | 1: 3 | 0.9375 | 4,317.625 | 4,361.066667 | 3,666 | |
The increasing [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) $3, 15, 24, 48, \ldots\,$ consists of those [positive](https://artofproblemsolving.com/wiki/index.php/Positive) multiples of 3 that are one less than a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfect_square). Wha... | 063 | 0 | 7,142.875 | -1 | 7,142.875 | |
How many of the divisors of $8!$ are larger than $7!$? | 7 | 0.1875 | 7,478.1875 | 5,492 | 7,936.538462 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b\cos C=3a\cos B-c\cos B$, $\overrightarrow{BA}\cdot \overrightarrow{BC}=2$, find the area of $\triangle ABC$. | 2\sqrt{2} | 0.875 | 5,011.9375 | 4,557.642857 | 8,192 | |
The domain of the function \( f(x) \) is \( (0,1) \), and the function is defined as follows:
\[
f(x)=\begin{cases}
x, & \text{if } x \text{ is an irrational number}, \\
\frac{p+1}{q}, & \text{if } x=\frac{p}{q}, \; p, q \in \mathbf{N}^{*}, \; (p, q) = 1, \; p < q.
\end{cases}
\]
Find the maximum value of \( f(x) \) ... | 16/17 | 0 | 7,884.875 | -1 | 7,884.875 | |
A triangle \(A B C\) is considered. Point \(F\) is the midpoint of side \(A B\). Point \(S\) lies on the ray \(A C\) such that \(C S = 2 A C\). In what ratio does the line \(S F\) divide side \(B C\)? | 2:3 | 0 | 6,377 | -1 | 6,377 | |
In a certain business district parking lot, temporary parking is charged by time period. The charging standard is: a charge of 6 yuan for parking not exceeding 1 hour per car, and for the part exceeding 1 hour, a charge of 8 yuan per hour (parts of an hour are rounded up to the next hour). Now, two people, A and B, par... | \frac{1}{4} | 0.0625 | 7,234.3125 | 4,720 | 7,401.933333 | |
In every acyclic graph with 2022 vertices we can choose $k$ of the vertices such that every chosen vertex has at most 2 edges to chosen vertices. Find the maximum possible value of $k$ . | 1517 | 0 | 8,066.9375 | -1 | 8,066.9375 | |
When $n$ is a positive integer, $n! = n \times (n - 1) \times \ldots \times 2 \times 1$ is defined as the factorial of $n$ (for example, $10! = 10 \times 9 \times \ldots \times 2 \times 1 = 3,628,800$). Then, how many zeros are there at the end of $2010!$? | 501 | 1 | 2,007.0625 | 2,007.0625 | -1 | |
A sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the inscribed cube? Express your answer in simplest radical form. | 81\sqrt{3} | 1 | 2,274.625 | 2,274.625 | -1 | |
A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial?
(A) $\frac{1 + i \sqrt{11}}{2}$
(B) $\frac{1 + i}{2}$
(C) $\frac{1}{2} + i$
(D) $1 + \frac{i}{2}$
(E) $\frac{1 + i \sqrt{13}}{... | \text{(A)} | 0 | 6,767.25 | -1 | 6,767.25 | |
Calculate the integral $\int_{2}^{7}(x-3)^{2} d x$.
a) Using the substitution $z=x-3$.
b) Using the substitution $z=(x-3)^{2}$. | \frac{65}{3} | 0.9375 | 5,117.6875 | 4,912.733333 | 8,192 | |
If $a$ and $b$ are positive real numbers such that $a \cdot 2^{b}=8$ and $a^{b}=2$, compute $a^{\log _{2} a} 2^{b^{2}}$. | 128 | Taking $\log _{2}$ of both equations gives $\log _{2} a+b=3$ and $b \log _{2} a=1$. We wish to find $a^{\log _{2} a} 2^{b^{2}}$; taking $\log _{2}$ of that gives $\left(\log _{2} a\right)^{2}+b^{2}$, which is equal to $\left(\log _{2} a+b\right)^{2}-2 b \log _{2} a=3^{2}-2=7$. Hence, our answer is $2^{7}=128$. | 1 | 4,761.625 | 4,761.625 | -1 |
John needs to pay 2010 dollars for his dinner. He has an unlimited supply of 2, 5, and 10 dollar notes. In how many ways can he pay? | 20503 | Let the number of 2,5 , and 10 dollar notes John can use be $x, y$, and $z$ respectively. We wish to find the number of nonnegative integer solutions to $2 x+5 y+10 z=2010$. Consider this equation $\bmod 2$. Because $2 x, 10 z$, and 2010 are even, $5 y$ must also be even, so $y$ must be even. Now consider the equation ... | 0.6875 | 6,139.8125 | 5,256.181818 | 8,083.8 |
Billy Bones has two coins - a gold one and a silver one. One of them is symmetric, and the other is not. It is not known which coin is not symmetric, but it is given that the non-symmetric coin lands heads with a probability of $p = 0.6$.
Billy Bones flipped the gold coin, and it landed heads immediately. Then Billy B... | 0.6 | 0 | 5,120 | -1 | 5,120 | |
How many three-digit perfect cubes are divisible by $9?$ | 2 | 1 | 2,588.25 | 2,588.25 | -1 | |
Given an ellipse $M$ with its axes of symmetry being the coordinate axes, and its eccentricity is $\frac{\sqrt{2}}{2}$, and one of its foci is at $(\sqrt{2}, 0)$.
$(1)$ Find the equation of the ellipse $M$;
$(2)$ Suppose a line $l$ intersects the ellipse $M$ at points $A$ and $B$, and a parallelogram $OAPB$ is forme... | \frac{\sqrt{2}}{2} | 0 | 7,856.0625 | -1 | 7,856.0625 | |
Add 75.892 to 34.5167 and then multiply the sum by 2. Round the final result to the nearest thousandth. | 220.817 | 0.25 | 341.0625 | 333 | 343.75 | |
The $8 \times 18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$? | 6 | 1. **Understanding the problem**: We are given an $8 \times 18$ rectangle that is cut into two congruent hexagons. These hexagons are rearranged to form a square. We need to find the value of $y$, which is a dimension in the hexagon.
2. **Area of the rectangle**: The area of the rectangle is calculated as:
\[
\t... | 0.75 | 5,429.6875 | 5,707.833333 | 4,595.25 |
The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.
[asy]draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); draw((36,33)--(69,33)... | 260 | Call the squares' side lengths from smallest to largest $a_1,\ldots,a_9$, and let $l,w$ represent the dimensions of the rectangle.
The picture shows that \begin{align*} a_1+a_2 &= a_3\\ a_1 + a_3 &= a_4\\ a_3 + a_4 &= a_5\\ a_4 + a_5 &= a_6\\ a_2 + a_3 + a_5 &= a_7\\ a_2 + a_7 &= a_8\\ a_1 + a_4 + a_6 &= a_9\\ a_6 + a... | 0.125 | 8,058.4375 | 7,123.5 | 8,192 |
For positive integers $n$, let $f(n)$ be the product of the digits of $n$. Find the largest positive integer $m$ such that $$\sum_{n=1}^{\infty} \frac{f(n)}{m\left\lfloor\log _{10} n\right\rfloor}$$ is an integer. | 2070 | We know that if $S_{\ell}$ is the set of all positive integers with $\ell$ digits, then $$\begin{aligned} & \sum_{n \in S_{\ell}} \frac{f(n)}{k^{\left\lfloor\log _{10}(n)\right\rfloor}}=\sum_{n \in S_{\ell}} \frac{f(n)}{k^{\ell-1}}=\frac{(0+1+2+\ldots+9)^{\ell}}{k^{\ell-1}}= \\ & 45 \cdot\left(\frac{45}{k}\right)^{\ell... | 0 | 8,192 | -1 | 8,192 |
What is the largest even integer that cannot be written as the sum of two odd composite numbers? | 38 | Let $n$ be an integer that cannot be written as the sum of two odd composite numbers. If $n>33$, then $n-9,n-15,n-21,n-25,n-27,$ and $n-33$ must all be prime (or $n-33=1$, which yields $n=34=9+25$ which does not work). Thus $n-9,n-15,n-21,n-27,$ and $n-33$ form a prime quintuplet. However, only one prime quintuplet exi... | 0.125 | 7,625 | 6,380 | 7,802.857143 |
In a \(10 \times 10\) grid (where the sides of the cells have a unit length), \(n\) cells are selected, and a diagonal is drawn in each of them with an arrow pointing in one of two directions. It turns out that for any two arrows, either the end of one coincides with the beginning of the other, or the distance between ... | 48 | 0 | 8,093.375 | -1 | 8,093.375 | |
Determine the number of pairs of positive integers $x,y$ such that $x\le y$ , $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$ . | 16384 | 0.4375 | 7,663.375 | 6,983.714286 | 8,192 | |
Given that $F_{1}$ and $F_{2}$ are the two foci of the ellipse $\frac {x^{2}}{16}+ \frac {y^{2}}{9}=1$, and a line passing through point $F_{2}$ intersects the ellipse at points $A$ and $B$. If $|AB|=5$, calculate $|AF_{1}|+|BF_{1}|$. | 11 | 0.75 | 5,457.5625 | 4,546.083333 | 8,192 | |
Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number \( k \), she calls a placement of nonzero real numbers on the \( 2^{2019} \) vertices of the hypercube \( k \)-harmonic if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to \( k \) times the number on thi... | 2040200 | 0.3125 | 7,216.6875 | 6,725.2 | 7,440.090909 | |
One of the angles in a triangle is $120^{\circ}$, and the lengths of the sides form an arithmetic progression. Find the ratio of the lengths of the sides of the triangle. | 3 : 5 : 7 | 1 | 3,377.8125 | 3,377.8125 | -1 | |
Find the minimum value of the distance $|AB|$ where point $A$ is the intersection of the line $y=a$ and the line $y=2x+2$, and point $B$ is the intersection of the line $y=a$ and the curve $y=x+\ln x$. | \frac{3}{2} | 0.8125 | 4,908.5 | 4,364.461538 | 7,266 | |
An athlete's heart beats an average of 150 times per minute while running. How many times does the athlete's heart beat during a 26-mile race if the athlete runs at a pace of 5 minutes per mile? | 19500 | 1 | 1,662.6875 | 1,662.6875 | -1 | |
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that
\[\begin{aligned} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7 &= 1 \\
4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7 &= 12 \\
9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7 &= 123. \end{aligned}\]Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$. | 334 | 0.625 | 5,985.3125 | 5,048.9 | 7,546 | |
Given the proportion 3:5 = 6:10, if 3 is changed to 12, determine the new value of 5. | 20 | 0.875 | 436.4375 | 446.642857 | 365 | |
Suppose that $x^{10} + x + 1 = 0$ and $x^100 = a_0 + a_1x +... + a_9x^9$ . Find $a_5$ . | -252 | 0.0625 | 7,787.875 | 3,904 | 8,046.8 | |
A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. Find the probability that the roots of the polynomial
\[x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2\]are all real. | \frac{18}{19} | 0.875 | 5,465.625 | 5,076.142857 | 8,192 | |
John has cut out these two polygons made out of unit squares. He joins them to each other to form a larger polygon (but they can't overlap). Find the smallest possible perimeter this larger polygon can have. He can rotate and reflect the cut out polygons. | 18 | 0 | 7,450.625 | -1 | 7,450.625 | |
How many five-digit numbers divisible by 3 are there that include the digit 6? | 12504 | 0.25 | 7,622 | 6,222.25 | 8,088.583333 | |
Three identical rods each have a piece broken off at a random point. What is the probability that the three resulting pieces can form a triangle?
| 1/2 | 0 | 7,739.8125 | -1 | 7,739.8125 | |
Five integers have an average of 69. The middle integer (the median) is 83. The most frequently occurring integer (the mode) is 85. The range of the five integers is 70. What is the second smallest of the five integers? | 77 | 0.8125 | 4,604 | 3,776 | 8,192 | |
A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and that each number rolled is a prime number? Express your answer as a common fraction. | \frac{1}{6561} | 0.6875 | 2,519 | 2,284.090909 | 3,035.8 | |
Simplify first, then evaluate: $\left(\dfrac{a+2}{a^{2}-2a}+\dfrac{8}{4-a^{2}}\right)\div \dfrac{a^{2}-4}{a}$, where $a$ satisfies the equation $a^{2}+4a+1=0$. | \dfrac{1}{3} | 0.875 | 4,081.0625 | 3,493.785714 | 8,192 | |
For a transatlantic flight, three flight attendants are selected by lot from 20 girls competing for these positions. Seven of them are blondes, and the rest are brunettes. What is the probability that among the three chosen flight attendants there will be at least one blonde and at least one brunette? | 0.718 | 0 | 5,090.125 | -1 | 5,090.125 | |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively, and $\sqrt{3}c\sin A = a\cos C$.
$(I)$ Find the value of $C$;
$(II)$ If $c=2a$ and $b=2\sqrt{3}$, find the area of $\triangle ABC$. | \frac{\sqrt{15} - \sqrt{3}}{2} | 0 | 5,678.25 | -1 | 5,678.25 | |
A cylindrical glass is half full of lemonade. The ratio of lemon juice to water in the lemonade is 1:11. If the glass is 6 inches tall and has a diameter of 2 inches, what is the volume of lemon juice in the glass? Express your answer as a decimal to the nearest hundredth. | .79 | 1 | 2,412 | 2,412 | -1 | |
The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers $1,2,3$. Then $A+B=$ \begin{tabular}{|c|c|c|}\hline 1 & &\\ \hline & 2 & A\\ \hline & & B\\ \hline\end{tabular} | 4 | 1. **Fill in the first row**: We start with the given number in the top left corner, which is $1$. Since each row and column must contain each of the numbers $1, 2, 3$, the middle cell in the top row cannot be $1$. It also cannot be $2$ because there is already a $2$ in the middle column. Therefore, the middle cell in ... | 0.5 | 5,483.375 | 3,448.625 | 7,518.125 |
Are there integers $m$ and $n$ such that
\[5m^2 - 6mn + 7n^2 = 1985 \ ?\] | \text{No} |
To determine whether there are integers \( m \) and \( n \) such that
\[
5m^2 - 6mn + 7n^2 = 1985,
\]
we begin by analyzing the quadratic form. We can rewrite the equation as:
\[
5m^2 - 6mn + 7n^2.
\]
First, let's complete the square with respect to \( m \) in the expression \( 5m^2 - 6mn \):
1. Factor out the co... | 0.3125 | 6,262.25 | 5,227.8 | 6,732.454545 |
Let \( a, b, c \) be prime numbers such that \( a^5 \) divides \( b^2 - c \), and \( b + c \) is a perfect square. Find the minimum value of \( abc \). | 1958 | 0 | 8,192 | -1 | 8,192 | |
Consider two right-angled triangles, ABC and DEF. Triangle ABC has a right angle at C with AB = 10 cm and BC = 7 cm. Triangle DEF has a right angle at F with DE = 3 cm and EF = 4 cm. If these two triangles are arranged such that BC and DE are on the same line segment and point B coincides with point D, what is the area... | 29 | 0 | 7,455.25 | -1 | 7,455.25 | |
Given vectors $\vec{a}$ and $\vec{b}$ satisfy $|\vec{a}|=|\vec{b}|=2$, and $\vec{b}$ is perpendicular to $(2\vec{a}+\vec{b})$, find the angle between vector $\vec{a}$ and $\vec{b}$. | \frac{2\pi}{3} | 0 | 1,790.4375 | -1 | 1,790.4375 | |
In $\triangle ABC$ lines $CE$ and $AD$ are drawn so that $\dfrac{CD}{DB}=\dfrac{3}{1}$ and $\dfrac{AE}{EB}=\dfrac{3}{2}$. Let $r=\dfrac{CP}{PE}$ where $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals:
[asy] size(8cm); pair A = (0, 0), B = (9, 0), C = (3, 6); pair D = (7.5, 1.5), E = (6.5, 0); pair P = in... | 5 | 0 | 6,656.875 | -1 | 6,656.875 | |
If the moving point $P$ is on the line $y=x+1$, and the moving point $Q$ is on the curve $x^{2}=-2y$, calculate the minimum value of $|PQ|$. | \frac{\sqrt{2}}{4} | 0 | 6,635.0625 | -1 | 6,635.0625 | |
In an ${8}$ × ${8}$ squares chart , we dig out $n$ squares , then we cannot cut a "T"shaped-5-squares out of the surplus chart .
Then find the mininum value of $n$ . | 32 | 0 | 8,137.8125 | -1 | 8,137.8125 | |
In a right triangle $DEF$ with $\angle D = 90^\circ$, we have $DE = 8$ and $DF = 15$. Find $\cos F$. | \frac{15}{17} | 0.875 | 2,530.6875 | 2,487.428571 | 2,833.5 | |
Given that the arithmetic square root of $m$ is $3$, and the square roots of $n$ are $a+4$ and $2a-16$.
$(1)$ Find the values of $m$ and $n$.
$(2)$ Find $\sqrt[3]{{7m-n}}$. | -1 | 0.75 | 2,113.9375 | 1,960.25 | 2,575 | |
On the sides of triangle \(ABC\), points were marked: 10 on side \(AB\), 11 on side \(BC\), and 12 on side \(AC\). None of the vertices of the triangle were marked. How many triangles with vertices at the marked points exist? | 4951 | 0.3125 | 5,576.3125 | 3,481.6 | 6,528.454545 | |
For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ | 10 | To solve this problem, we need to understand the function $f_j(n)$ and its behavior as $j$ increases. We start by analyzing the function $f_1(n)$, which is defined as twice the number of positive integer divisors of $n$. We then recursively apply $f_1$ to its own outputs to determine $f_j(n)$ for $j \geq 2$.
#### Step... | 0 | 8,008.5625 | -1 | 8,008.5625 |
For $a>0$ , denote by $S(a)$ the area of the part bounded by the parabolas $y=\frac 12x^2-3a$ and $y=-\frac 12x^2+2ax-a^3-a^2$ .
Find the maximum area of $S(a)$ . | \frac{8\sqrt{2}}{3} | 0 | 7,313.5 | -1 | 7,313.5 | |
What is the remainder when (99)(101) is divided by 9? | 0 | 1 | 1,640.875 | 1,640.875 | -1 | |
$2020$ positive integers are written in one line. Each of them starting with the third is divisible by previous and by the sum of two previous numbers. What is the smallest value the last number can take?
A. Gribalko | 2019! |
Given the problem, we have a sequence of \(2020\) positive integers, say \(a_1, a_2, \ldots, a_{2020}\). Each term in the sequence starting with the third term (\(a_i\) for \(i \geq 3\)) is divisible by its preceding term and the sum of its two immediate predecessors. Formally, this can be expressed as:
\[
a_i \text{... | 0.0625 | 8,187.4375 | 8,119 | 8,192 |
What is the sum of the first 9 positive multiples of 5? | 225 | Since $1+2+3+4+5+6+7+8+9=45$ then $5+10+15+\cdots+40+45=5(1+2+3+\cdots+8+9)=5(45)=225$ | 1 | 686.125 | 686.125 | -1 |
Find the smallest integer satisfying the following conditions:
$\bullet$ I. The sum of the squares of its digits is $85$.
$\bullet$ II. Each digit is larger than the one on its left.
What is the product of the digits of this integer? | 18 | 0.25 | 7,506.9375 | 6,553.75 | 7,824.666667 | |
An enterprise has four employees participating in a vocational skills assessment. Each employee can draw any one of the four available assessment projects to participate in. Calculate the probability that exactly one project is not selected. | \frac{9}{16} | 0.4375 | 7,231.75 | 6,355.285714 | 7,913.444444 | |
What is the area, in square units, of the interior region formed by the lines $y = 2x - 4, y = -3x +16$ and the $y$-axis? | 40 | 1 | 3,012.875 | 3,012.875 | -1 | |
Find the mean of all solutions for $x$ when $x^3 + 3x^2 - 10x = 0$. | -1 | 1 | 1,597.0625 | 1,597.0625 | -1 | |
Consider the set $E = \{5, 6, 7, 8, 9\}$ . For any partition ${A, B}$ of $E$ , with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$ . Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $... | 17 | 0.25 | 8,036.0625 | 7,568.25 | 8,192 | |
For a positive number $x$, define $f(x)=\frac{2x}{x+1}$. Calculate: $f(\frac{1}{101})+f(\frac{1}{100})+f(\frac{1}{99})+\ldots +f(\frac{1}{3})+f(\frac{1}{2})+f(1)+f(2)+f(3)+\ldots +f(99)+f(100)+f(101)$. | 201 | 0.4375 | 6,265.625 | 5,858.285714 | 6,582.444444 | |
Let $a$ , $b$ , $c$ , $d$ , $e$ be positive reals satisfying \begin{align*} a + b &= c a + b + c &= d a + b + c + d &= e.\end{align*} If $c=5$ , compute $a+b+c+d+e$ .
*Proposed by Evan Chen* | 40 | 1 | 2,444.8125 | 2,444.8125 | -1 | |
(15) Given the following propositions:
(1) "If $x > 2$, then $x > 0$" - the negation of the proposition
(2) "For all $a \in (0, +\infty)$, the function $y = a^x$ is strictly increasing on its domain" - the negation
(3) "$π$ is a period of the function $y = \sin x$" or "$2π$ is a period of the function $y = \sin 2x$"... | (2)(3) | 0 | 4,540.375 | -1 | 4,540.375 | |
Some mice live in three neighboring houses. Last night, every mouse left its house and moved to one of the other two houses, always taking the shortest route. The numbers in the diagram show the number of mice per house, yesterday and today. How many mice used the path at the bottom of the diagram?
A 9
B 11
C 12
D 16
E... | 11 | 0 | 7,302.6875 | -1 | 7,302.6875 | |
In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $B$ and $D$ coincide, forming the pentagon $ABEFC$. What is the length of segment $EF$? Express your answer in simplest radical form. | 4\sqrt{5} | 0 | 5,254.9375 | -1 | 5,254.9375 |
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