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 |
|---|---|---|---|---|---|---|
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$ and $(\overrightarrow{a} + \overrightarrow{b}) \perp \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 0.0625 | 2,412.6875 | 1,314 | 2,485.933333 | |
What is the largest integer that is a divisor of \[
(n+1)(n+3)(n+5)(n+7)(n+9)
\]for all positive even integers $n$? | 15 | 0.125 | 8,064.5625 | 7,172.5 | 8,192 | |
Given vectors $\overrightarrow{a}=(\cos x,\sin x)$, $\overrightarrow{b}=( \sqrt {3}\sin x,\sin x)$, where $x\in R$, define the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}- \dfrac {1}{2}$.
(1) Find the smallest positive period of the function $f(x)$;
(2) Find the maximum and minimum values of the function... | -\dfrac{1}{2} | 1 | 4,028.0625 | 4,028.0625 | -1 | |
A cube with a side length of 10 is divided into 1000 smaller cubes with a side length of 1. A number is written in each small cube such that the sum of the numbers in each column of 10 cubes (in any of the three directions) is zero. In one of the small cubes (denoted as \( A \)), the number one is written. Three layers... | -1 | 0 | 7,075.1875 | -1 | 7,075.1875 | |
In the game of projective set, each card contains some nonempty subset of six distinguishable dots. A projective set deck consists of one card for each of the 63 possible nonempty subsets of dots. How many collections of five cards have an even number of each dot? The order in which the cards appear does not matter. | 109368 | We'll first count sets of cards where the order does matter. Suppose we choose the first four cards. Then there is exactly one card that can make each dot appear twice. However, this card could be empty or it could be one of the cards we've already chosen, so we have to subtract for these two cases. First, there are $6... | 0 | 7,952.4375 | -1 | 7,952.4375 |
Dima and Sergey were picking raspberries from a bush that had 900 berries. Dima alternated his actions while picking: he put one berry in the basket, and then he ate the next one. Sergey also alternated: he put two berries in the basket, and then he ate the next one. It is known that Dima picks berries twice as fast as... | 100 | 0.0625 | 7,416.1875 | 2,890 | 7,717.933333 | |
A box of chocolates in the shape of a cuboid was full of chocolates arranged in rows and columns. Míša ate some of them, and the remaining chocolates were rearranged to fill three entire rows completely, except for one space. Míša ate the remaining chocolates from another incomplete row. Then he rearranged the remainin... | 25 | 0.0625 | 7,885.875 | 3,294 | 8,192 | |
A plane is expressed parametrically by
\[\mathbf{v} = \begin{pmatrix} 1 + s - t \\ 2 - s \\ 3 - 2s + 2t \end{pmatrix}.\]Find the equation of the plane. Enter your answer in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$ | 2x + z - 5 = 0 | 0.75 | 4,430.3125 | 3,686.333333 | 6,662.25 | |
Given that sequence {a_n} is an equal product sequence, with a_1=1, a_2=2, and a common product of 8, calculate the sum of the first 41 terms of the sequence {a_n}. | 94 | 0.3125 | 5,453.9375 | 5,115.2 | 5,607.909091 | |
Quadrilateral \(ABCD\) is such that \(\angle BAC = \angle CAD = 60^\circ\) and \(AB + AD = AC\). It is also known that \(\angle ACD = 23^\circ\). How many degrees is \(\angle ABC\)? | 83 | 0.25 | 7,689.6875 | 6,182.75 | 8,192 | |
How many positive integers \( n \) between 10 and 1000 have the property that the sum of the digits of \( n \) is 3? | 9 | We note that the sum of the digits of 1000 is not 3. Every other positive integer in the given range has two or three digits. For the sum of the digits of an integer to be 3, no digit can be greater than 3. If a two-digit integer has sum of digits equal to 3, then its tens digit is 1, 2, or 3. The possible integers are... | 0.5625 | 6,624.8125 | 5,971.111111 | 7,465.285714 |
Factor completely over the set of polynomials with integer coefficients:
\[4(x + 5)(x + 6)(x + 10)(x + 12) - 3x^2.\] | (2x^2 + 35x + 120)(x + 8)(2x + 15) | 0.0625 | 8,090.5 | 6,568 | 8,192 | |
Compute
\[
\log_2 \left( \prod_{a=1}^{2015} \prod_{b=1}^{2015} (1+e^{2\pi i a b/2015}) \right)
\]
Here $i$ is the imaginary unit (that is, $i^2=-1$). | 13725 | The answer is $13725$.
We first claim that if $n$ is odd, then $\prod_{b=1}^{n} (1+e^{2\pi i ab/n}) = 2^{\gcd(a,n)}$. To see this, write $d = \gcd(a,n)$ and $a = da_1$, $n=dn_1$ with $\gcd(a_1,n_1) = 1$. Then
$a_1, 2a_1,\dots,n_1 a_1$ modulo $n_1$ is a permutation of $1,2,\dots,n_1$ modulo $n_1$, and so $\omega^{a_1},\... | 0 | 7,917.3125 | -1 | 7,917.3125 |
Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$? | 4 | 1. **Define the problem**: We need to find the smallest possible value for the sum of the digits of $S$, where $S$ is the sum of two three-digit numbers $a$ and $b$. Each digit among $a$ and $b$ is unique.
2. **Set constraints on $a$ and $b$**: Since $a$ and $b$ are three-digit numbers and all digits are different, we... | 0 | 8,192 | -1 | 8,192 |
Given a regular hexagonal pyramid \( M A B C D E F \). Point \( K \) bisects edge \( B M \). Find the ratio in which the plane \( F E K \) divides edge \( A M \) (at point \( X \)). | 2:1 | 0 | 6,368.6875 | -1 | 6,368.6875 | |
A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of $64. How many values for $x$ are possible? | 5 | To determine the possible values of $x$, the cost of each ticket, we need to consider the total amounts spent by the 9th and 10th graders and find a common ticket price that divides both totals.
1. **Identify the total amounts spent by each grade:**
- 9th graders: $48$
- 10th graders: $64$
2. **Find the greates... | 1 | 1,519.3125 | 1,519.3125 | -1 |
In quadrilateral $ABCD$, $\overrightarrow{AB}=(1,1)$, $\overrightarrow{DC}=(1,1)$, $\frac{\overrightarrow{BA}}{|\overrightarrow{BA}|}+\frac{\overrightarrow{BC}}{|\overrightarrow{BC}|}=\frac{\sqrt{3}\overrightarrow{BD}}{|\overrightarrow{BD}|}$, calculate the area of the quadrilateral. | \sqrt{3} | 0.0625 | 8,130.125 | 7,202 | 8,192 | |
In triangle $ABC$, $M$ is the midpoint of $\overline{BC}$, $AB = 15$, and $AC = 24$. Let $E$ be a point on $\overline{AC}$, and $H$ be a point on $\overline{AB}$, and let $G$ be the intersection of $\overline{EH}$ and $\overline{AM}$. If $AE = 3AH$, find $\frac{EG}{GH}$. | \frac{2}{3} | 0 | 5,601.5 | -1 | 5,601.5 | |
Refer to the diagram, $P$ is any point inside the square $O A B C$ and $b$ is the minimum value of $P O + P A + P B + P C$. Find $b$. | 2\sqrt{2} | 0.625 | 6,786.9375 | 5,943.9 | 8,192 | |
Find the value of $$\frac{\tan 7.5^\circ \cdot \tan 15^\circ}{\tan 15^\circ - \tan 7.5^\circ}$$ + $$\sqrt{3}(\sin^2 7.5^\circ - \cos^2 7.5^\circ)$$. | -\sqrt{2} | 0.625 | 6,774 | 5,923.2 | 8,192 | |
Given \( A \cup B = \left\{a_{1}, a_{2}, a_{3}\right\} \) and \( A \neq B \), where \((A, B)\) and \((B, A)\) are considered different pairs, find the number of such pairs \((A, B)\). | 26 | 0.375 | 6,747.9375 | 5,740 | 7,352.7 | |
Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$:
(i) move the last digit of $a$ to the first position to obtain the numb er $b$;
(ii) square $b$ to obtain the number $c$;
(iii) move the first digit of $c$ to the end to obtain the number $d$.
(All the num... | a = \underbrace{2\dots2}_{n \ge 0}1, \qquad a = 2, \qquad a = 3. |
Given the problem, we want to find all positive integers \( a \) such that the procedure outlined results in \( d(a) = a^2 \). Let's break down the steps of the procedure and solve for \( a \).
### Procedure Analysis
1. **Step (i):** Move the last digit of \( a \) to the first position to obtain the number \( b \).
... | 0 | 8,192 | -1 | 8,192 |
Solve the system of equations $p+3q+r=3$, $p+2q+3r=3$, $p+q+r=2$ for the ordered triple $(p, q, r)$. | \left(\frac{5}{4}, \frac{1}{2}, \frac{1}{4}\right) | We can rewrite the equation in terms of $\ln 2, \ln 3, \ln 5$, to get $3 \ln 2+3 \ln 3+2 \ln 5=\ln 5400=p x+q y+r z=(p+3 q+r) \ln 2+(p+2 q+3 r) \ln 3+(p+q+r) \ln 5$. Consequently, since $p, q, r$ are rational we want to solve the system of equations $p+3 q+r=3, p+2 q+3 r=3, p+q+r=2$, which results in the ordered triple... | 0.875 | 2,873.6875 | 2,113.928571 | 8,192 |
Remove all perfect squares from the sequence of positive integers \(1, 2, 3, \cdots\) to get a new sequence. What is the 2003rd term of this new sequence? | 2048 | 0.8125 | 5,245.75 | 4,580.230769 | 8,129.666667 | |
What is the sum of all the solutions of \( x = |2x - |50-2x|| \)? | \frac{170}{3} | 0 | 5,175.5 | -1 | 5,175.5 | |
The solutions to the system of equations
$\log_{225}x+\log_{64}y=4$
$\log_{x}225-\log_{y}64=1$
are $(x_1,y_1)$ and $(x_2,y_2)$. Find $\log_{30}\left(x_1y_1x_2y_2\right)$. | 12 | Let $A=\log_{225}x$ and let $B=\log_{64}y$.
From the first equation: $A+B=4 \Rightarrow B = 4-A$.
Plugging this into the second equation yields $\frac{1}{A}-\frac{1}{B}=\frac{1}{A}-\frac{1}{4-A}=1 \Rightarrow A = 3\pm\sqrt{5}$ and thus, $B=1\pm\sqrt{5}$.
So, $\log_{225}(x_1x_2)=\log_{225}(x_1)+\log_{225}(x_2)=(3+\sq... | 1 | 4,764.8125 | 4,764.8125 | -1 |
Given the parabola $y^{2}=4x$, a line $l$ passing through its focus $F$ intersects the parabola at points $A$ and $B$ (with point $A$ in the first quadrant), such that $\overrightarrow{AF}=3\overrightarrow{FB}$. A line passing through the midpoint of $AB$ and perpendicular to $l$ intersects the $x$-axis at point $G$. C... | \frac{32\sqrt{3}}{9} | 0 | 6,155.9375 | -1 | 6,155.9375 | |
Let \( S = \{1, 2, \cdots, 2005\} \). Find the minimum value of \( n \) such that any set of \( n \) pairwise coprime elements from \( S \) contains at least one prime number. | 16 | 0.125 | 7,969.9375 | 7,382 | 8,053.928571 | |
Simplify first, then evaluate: $\left(\frac{2}{m-3}+1\right) \div \frac{2m-2}{m^2-6m+9}$, and then choose a suitable number from $1$, $2$, $3$, $4$ to substitute and evaluate. | -\frac{1}{2} | 0.6875 | 3,893.8125 | 3,664.636364 | 4,398 | |
Find \( g(2021) \) if for any real numbers \( x \) and \( y \) the following equality holds:
\[ g(x-y) = 2021(g(x) + g(y)) - 2022xy \] | 2043231 | 0.0625 | 7,962.9375 | 4,527 | 8,192 | |
In triangle $\triangle ABC$, it is known that $\overrightarrow{CD}=2\overrightarrow{DB}$, $P$ is a point on segment $AD$, and satisfies $\overrightarrow{CP}=\frac{1}{2}\overrightarrow{CA}+m\overrightarrow{CB}$. If the area of $\triangle ABC$ is $\sqrt{3}$ and $∠ACB=\frac{π}{3}$, then the minimum value of the length of ... | \sqrt{2} | 0.3125 | 6,849.5625 | 6,134.4 | 7,174.636364 | |
Suppose the state of Georgia uses a license plate format "LLDLLL", and the state of Nebraska uses a format "LLDDDDD". Assuming all 10 digits are equally likely to appear in the numeric positions, and all 26 letters are equally likely to appear in the alpha positions, how many more license plates can Nebraska issue than... | 21902400 | 0 | 4,667.1875 | -1 | 4,667.1875 | |
Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ steps of the same size from the... | 70 | 1. **Define Variables:**
Let $L$ be the length of the ship, $E$ be the length of Emily's step, and $S$ be the length of the ship's step (the distance the ship travels while Emily takes one step).
2. **Set Up Equations:**
- When Emily walks from the back of the ship to the front, she takes $210$ steps, covering a... | 0.9375 | 3,355.1875 | 3,032.733333 | 8,192 |
Given that $a, b > 0$, $2^a = 3^b = m$, and $a, ab, b$ form an arithmetic sequence, find $m$. | \sqrt{6} | 0.9375 | 5,119.5625 | 4,914.733333 | 8,192 | |
Given that $x > 0$, $y > 0$, and ${\!\!}^{2x+2y}=2$, find the minimum value of $\frac{1}{x}+\frac{1}{y}$. | 3 + 2\sqrt{2} | 0 | 3,008.125 | -1 | 3,008.125 | |
In a certain middle school, 500 eighth-grade students took the biology and geography exam. There were a total of 180 students who scored between 80 and 100 points. What is the frequency of this score range? | 0.36 | 0.5625 | 415.75 | 410 | 423.142857 | |
Given point $A$ is on line segment $BC$ (excluding endpoints), and $O$ is a point outside line $BC$, with $\overrightarrow{OA} - 2a \overrightarrow{OB} - b \overrightarrow{OC} = \overrightarrow{0}$, then the minimum value of $\frac{a}{a+2b} + \frac{2b}{1+b}$ is \_\_\_\_\_\_. | 2 \sqrt{2} - 2 | 0.625 | 6,409.3125 | 6,148 | 6,844.833333 | |
Simplify $\dfrac{5+12i}{2-3i}$. Your answer should be of the form $a+bi$, where $a$ and $b$ are both real numbers and written as improper fractions (if necessary). | -2+3i | 1 | 2,029.5 | 2,029.5 | -1 | |
15. Let $a_{n}$ denote the number of ternary strings of length $n$ so that there does not exist a $k<n$ such that the first $k$ digits of the string equals the last $k$ digits. What is the largest integer $m$ such that $3^{m} \mid a_{2023}$ ? | 2022 | 0 | 8,192 | -1 | 8,192 | |
Let $f(x)=-3x^2+x-4$, $g(x)=-5x^2+3x-8$, and $h(x)=5x^2+5x+1$. Express $f(x)+g(x)+h(x)$ as a single polynomial, with the terms in order by decreasing degree. | -3x^2 +9x -11 | 1 | 1,699.625 | 1,699.625 | -1 | |
A right square pyramid with volume $54$ has a base with side length $6.$ The five vertices of the pyramid all lie on a sphere with radius $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 21 | Although I can't draw the exact picture of this problem, but it is quite easy to imagine that four vertices of the base of this pyramid is on a circle (Radius $\frac{6}{\sqrt{2}} = 3\sqrt{2}$). Since all five vertices are on the sphere, the distances of the spherical center and the vertices are the same: $l$. Because o... | 0.875 | 4,318.625 | 3,765.285714 | 8,192 |
Given that Liliane has $30\%$ more cookies than Jasmine and Oliver has $10\%$ less cookies than Jasmine, and the total number of cookies in the group is $120$, calculate the percentage by which Liliane has more cookies than Oliver. | 44.44\% | 0.875 | 4,067.5625 | 3,835.571429 | 5,691.5 | |
Mr. Wang, a math teacher, is preparing to visit a friend. Before leaving, Mr. Wang calls the friend's house, and the phone number is 27433619. After the call, Mr. Wang realizes that this phone number is exactly the product of 4 consecutive prime numbers. What is the sum of these 4 prime numbers? | 290 | 0.4375 | 5,709.9375 | 4,621 | 6,556.888889 | |
Find the numbers \( x \) between 0 and 30 for which the sine of \( x \) degrees equals the sine of \( x \) radians. How many such numbers exist between 30 and 90? | 10 | 0 | 8,192 | -1 | 8,192 | |
Given the line $y=-x+1$ and the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ intersecting at points $A$ and $B$.
(1) If the eccentricity of the ellipse is $\frac{\sqrt{2}}{2}$ and the focal length is $2$, find the length of the line segment $AB$.
(2) If vectors $\overrightarrow{OA}$ and $\overright... | \sqrt{6} | 0.625 | 7,271.3125 | 6,913.9 | 7,867 | |
Let $\triangle ABC$ be an equilateral triangle of side length 1. Let $D,E,F$ be points on $BC,AC,AB$ respectively, such that $\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}$. Let $X,Y,Z$ be on lines $BC,CA,AB$ respectively, such that $XY\perp DE, YZ\perp EF, ZX\perp FD$. Find all possible values of $\frac{1}{[DEF]} + \f... | \frac{97 \sqrt{2} + 40 \sqrt{3}}{15} |
Let \(\triangle ABC\) be an equilateral triangle of side length 1. Let \(D, E, F\) be points on \(BC, AC, AB\) respectively, such that \(\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}\). Let \(X, Y, Z\) be on lines \(BC, CA, AB\) respectively, such that \(XY \perp DE\), \(YZ \perp EF\), \(ZX \perp FD\). We aim to find ... | 0 | 8,192 | -1 | 8,192 |
The numbers \( a, b, c, \) and \( d \) are distinct positive integers chosen from 1 to 10 inclusive. What is the least possible value \(\frac{a}{b}+\frac{c}{d}\) could have?
A) \(\frac{2}{10}\)
B) \(\frac{3}{19}\)
C) \(\frac{14}{45}\)
D) \(\frac{29}{90}\)
E) \(\frac{25}{72}\) | \frac{14}{45} | 0 | 8,192 | -1 | 8,192 | |
The value of \(1 + 0.01 + 0.0001\) is: | 1.0101 | 1 | 3,136.625 | 3,136.625 | -1 | |
There were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Determine the volume, in cubic light years, of the set of all possible locations for a base such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square lig... | \frac{27 \sqrt{6} \pi}{8} | 0 | 6,618.3125 | -1 | 6,618.3125 | |
Let $a$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y)$ satisfy all of the following conditions:
\begin{enumerate}
\item $\frac{a}{2} \le x \le 2a$
\item $\frac{a}{2} \le y \le 2a$
\item $x+y \ge a$
\item $x+a \ge y$
\item $y+a \ge x$
\end{enumerate}
T... | 6 | To solve this problem, we need to analyze the geometric implications of each condition and determine the shape and boundaries of the set $S$.
1. **Understanding the Conditions**:
- $\text{(i) }\frac{a}{2}\le x\le 2a$: This restricts $x$ to the interval from $\frac{a}{2}$ to $2a$.
- $\text{(ii) }\frac{a}{2}\le y\... | 0.0625 | 7,716.5625 | 7,390 | 7,738.333333 |
Determine the number of solutions to the equation
\[\tan (10 \pi \cos \theta) = \cot (10 \pi \sin \theta)\]
where $\theta \in (0, 2 \pi).$ | 56 | 0.125 | 7,872.25 | 6,134 | 8,120.571429 | |
If $C_n^2A_2^2=42$, then $\frac{n!}{3!(n-3)!}=\_\_\_\_\_\_\_\_.$ | 35 | 0.5 | 6,013.5 | 4,485.125 | 7,541.875 | |
Given that the coefficients of the first three terms of the expansion of $(x+ \frac {1}{2})^{n}$ form an arithmetic sequence. Let $(x+ \frac {1}{2})^{n} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots + a_{n}x^{n}$. Find:
(1) The value of $n$;
(2) The value of $a_{5}$;
(3) The value of $a_{0} - a_{1} + a_{2} - a_{3} + \ld... | \frac {1}{256} | 0.3125 | 6,837.9375 | 3,859 | 8,192 | |
What is the volume in cubic inches of a right, rectangular prism with side, front and bottom faces having an area 15 square inches, 10 square inches and 6 square inches, respectively? | 30 | 1 | 1,758.375 | 1,758.375 | -1 | |
Calculate $8 \cdot 5\frac{2}{5} - 3$. | 40.2 | 0 | 482.4375 | -1 | 482.4375 | |
A pebble is shaped as the intersection of a cube of side length 1 with the solid sphere tangent to all of the cube's edges. What is the surface area of this pebble? | \frac{6 \sqrt{2}-5}{2} \pi | Imagine drawing the sphere and the cube. Take a cross section, with a plane parallel to two of the cube's faces, passing through the sphere's center. In this cross section, the sphere looks like a circle, and the cube looks like a square (of side length 1) inscribed in that circle. We can now calculate that the sphere ... | 0 | 8,036.4375 | -1 | 8,036.4375 |
Given that $n$ is a positive integer, and given that $\mathop{\text{lcm}}[24,n]=72$ and $\mathop{\text{lcm}}[n,27]=108$, what is $n$? | 36 | 1 | 3,567 | 3,567 | -1 | |
In the triangle \( \triangle ABC \), it is given that the angles are in the ratio \(\angle A : \angle B : \angle C = 3 : 5 : 10\). Also, it is known that \(\triangle A'B'C \cong \triangle ABC\). What is the ratio \(\angle BCA' : \angle BCB'\)? | 1:4 | 0 | 6,288.8125 | -1 | 6,288.8125 | |
Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2014$, and $a^2 - b^2 + c^2 - d^2 = 2014$. Find the number of possible values of $a$. | 502 | 0.125 | 7,872.125 | 5,633 | 8,192 | |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$? | 60 | 0.25 | 7,413.3125 | 5,077.25 | 8,192 | |
How many four-digit positive integers are divisible by both 12 and 20, but are not divisible by 16? | 113 | 0.9375 | 4,084.8125 | 3,811 | 8,192 | |
A polynomial $P$ of degree 2015 satisfies the equation $P(n)=\frac{1}{n^{2}}$ for $n=1,2, \ldots, 2016$. Find \lfloor 2017 P(2017)\rfloor. | -9 | Let $Q(x)=x^{2} P(x)-1$. Then $Q(n)=n^{2} P(n)-1=0$ for $n=1,2, \ldots, 2016$, and $Q$ has degree 2017 . Thus we may write $$Q(x)=x^{2} P(x)-1=(x-1)(x-2) \ldots(x-2016) L(x)$$ where $L(x)$ is some linear polynomial. Then $Q(0)=-1=(-1)(-2) \ldots(-2016) L(0)$, so $L(0)=-\frac{1}{2016!}$. Now note that $$\begin{aligned} ... | 0 | 7,806.6875 | -1 | 7,806.6875 |
Let $h(4x-1) = 2x + 7$. For what value of $x$ is $h(x) = x$? | 15 | 0.875 | 4,285.6875 | 3,727.642857 | 8,192 | |
Given the set $A=\{x|ax^{2}-x+1=0,a\in\mathbb{R},x\in\mathbb{R}\}$. If the proposition "Set $A$ contains only one element" is true, find the value of $a$. | \frac{1}{4} | 0.625 | 7,668.4375 | 7,354.3 | 8,192 | |
Let \( x = \sqrt{1 + \frac{1}{1^{2}} + \frac{1}{2^{2}}} + \sqrt{1 + \frac{1}{2^{2}} + \frac{1}{3^{2}}} + \cdots + \sqrt{1 + \frac{1}{2012^{2}} + \frac{1}{2013^{2}}} \). Find the value of \( x - [x] \), where \( [x] \) denotes the greatest integer not exceeding \( x \). | \frac{2012}{2013} | 0.9375 | 3,952 | 3,669.333333 | 8,192 | |
I have three distinct mystery novels, three distinct fantasy novels, and three distinct biographies. I'm going on vacation, and I want to take two books of different genres. How many possible pairs can I choose? | 27 | 0.9375 | 573.25 | 565.8 | 685 | |
A and B are playing a guessing game. First, A thinks of a number, denoted as $a$, then B guesses the number A was thinking of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\{0,1,2,…,9\}$. The probability that $|a-b|\leqslant 1$ is __________. | \dfrac{7}{25} | 1 | 4,303.1875 | 4,303.1875 | -1 | |
In the rectangular coordinate system xOy, the parametric equations of the curve C₁ are $$\begin{cases} x=2+2\cos\alpha \\ y=2\sin\alpha \end{cases}$$ (where α is the parameter). Establish a polar coordinate system with the origin of the coordinate system as the pole and the positive semi-axis of the x-axis as the polar... | \sqrt {3} | 0 | 8,137.3125 | -1 | 8,137.3125 | |
In the Cartesian coordinate system $xOy$, it is known that $P$ is a moving point on the graph of the function $f(x) = \ln x$ ($x > 0$). The tangent line $l$ at point $P$ intersects the $x$-axis at point $E$. A perpendicular line to $l$ through point $P$ intersects the $x$-axis at point $F$. Suppose the midpoint of the ... | \dfrac{1}{2}\left(e - \dfrac{1}{e}\right) | 0 | 5,109.5625 | -1 | 5,109.5625 | |
Let \( a \) and \( b \) be integers such that \( ab = 72 \). Find the minimum value of \( a + b \). | -17 | 0.0625 | 4,650.3125 | 1,235 | 4,878 | |
What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $4,5,6,7,8,9$ in one of the six boxes in this addition problem?
[asy]
unitsize(12);
draw((0,0)--(10,0));
draw((-1.5,1.5)--(-1.5,2.5));
draw((-1,2)--(-2,2));
draw((1,1)--(3,1)--(3,3)--(1,3)--cycle);
draw((1,4)--(3,4)... | 1047 | 1. **Identify the problem**: We need to find the smallest sum of two 3-digit numbers formed by the digits 4, 5, 6, 7, 8, and 9, each used exactly once.
2. **Understand the sum of two numbers**: If the two numbers are $\overline{abc}$ and $\overline{def}$, their sum is given by:
\[
100(a+d) + 10(b+e) + (c+f)
\... | 0.5 | 7,269.9375 | 6,347.875 | 8,192 |
Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $s$ be a real number. Assume two of the roots of $f(x)$ are $s + 2$ and $s + 5,$ and two of the roots of $g(x)$ are $s + 4$ and $s + 8.$ Given that:
\[ f(x) - g(x) = 2s \] for all real numbers $x.$ Find $s.$ | 3.6 | 0 | 6,938 | -1 | 6,938 | |
Let $\ell$ and $m$ be two non-coplanar lines in space, and let $P_{1}$ be a point on $\ell$. Let $P_{2}$ be the point on $m$ closest to $P_{1}, P_{3}$ be the point on $\ell$ closest to $P_{2}, P_{4}$ be the point on $m$ closest to $P_{3}$, and $P_{5}$ be the point on $\ell$ closest to $P_{4}$. Given that $P_{1} P_{2}=5... | \frac{\sqrt{39}}{4} | Let $a$ be the answer. By taking the $z$-axis to be the cross product of these two lines, we can let the lines be on the planes $z=0$ and $z=h$, respectively. Then, by projecting onto the $xy$-plane, we get the above diagram. The projected lengths of the first four segments are $\sqrt{25-h^{2}}, \sqrt{9-h^{2}}$, and $\... | 0 | 8,049.6875 | -1 | 8,049.6875 |
Compute the side length of the largest cube contained in the region $\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 25 \text{ and } x \geq 0\}$ of three-dimensional space. | \frac{5 \sqrt{6}}{3} | The given region is a hemisphere, so the largest cube that can fit inside it has one face centered at the origin and the four vertices of the opposite face on the spherical surface. Let the side length of this cube be $s$. Then, the radius of the circle is the hypotenuse of a triangle with side lengths $s$ and $\frac{\... | 0 | 8,187.5 | -1 | 8,187.5 |
Billy and Bobbi each selected a positive integer less than 500. Billy's number is a multiple of 20, and Bobbi's number is a multiple of 30. What is the probability that they selected the same number? Express your answer as a common fraction. | \frac{1}{50} | 0 | 3,033.6875 | -1 | 3,033.6875 | |
Given that $a$, $b$, $c$ are three positive real numbers, and $a(a+b+c)=bc$, find the maximum value of $\frac{a}{b+c}$. | \frac{\sqrt{2} - 1}{2} | 0 | 5,364.4375 | -1 | 5,364.4375 | |
If an irrational number $a$ multiplied by $\sqrt{8}$ is a rational number, write down one possible value of $a$ as ____. | \sqrt{2} | 0.5 | 1,142.75 | 632.125 | 1,653.375 | |
In a polar coordinate system, the polar equation of curve C is $\rho=2\cos\theta+2\sin\theta$. Establish a Cartesian coordinate system with the pole as the origin and the positive x-axis as the polar axis. The parametric equation of line l is $\begin{cases} x=1+t \\ y= \sqrt{3}t \end{cases}$ (t is the parameter). Find ... | \sqrt{7} | 1 | 3,372.875 | 3,372.875 | -1 | |
In the parallelogram \(ABCD\), points \(E\) and \(F\) are located on sides \(AB\) and \(BC\) respectively, and \(M\) is the point of intersection of lines \(AF\) and \(DE\). Given that \(AE = 2BE\) and \(BF = 3CF\), find the ratio \(AM : MF\). | 4:5 | 0.0625 | 6,146.5625 | 3,917 | 6,295.2 | |
Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when $$ \sum_{\substack{2 \leq n \leq 50 \\ \operatorname{gcd}(n, 50)=1}} \phi^{!}(n) $$ is divided by 50 . | 12 | First, $\phi^{!}(n)$ is even for all odd $n$, so it vanishes modulo 2 . To compute the remainder modulo 25 , we first evaluate $\phi^{!}(3)+\phi^{!}(7)+\phi^{!}(9) \equiv 2+5 \cdot 4+5 \cdot 3 \equiv 12$ $(\bmod 25)$. Now, for $n \geq 11$ the contribution modulo 25 vanishes as long as $5 \nmid n$. We conclude the answe... | 0.0625 | 8,004.875 | 8,192 | 7,992.4 |
Cameron writes down the smallest positive multiple of 20 that is a perfect square, the smallest positive multiple of 20 that is a perfect cube, and all the multiples of 20 between them. How many integers are in Cameron's list? | 46 | 0.875 | 3,937 | 3,329.142857 | 8,192 | |
An octopus told me that his underwater cave is $245_{8}$ years old. How many years is this in base ten? | 165 | 0.9375 | 1,777.5 | 1,349.866667 | 8,192 | |
A digital 12-hour clock has a malfunction such that every time it should display a "2", it instead shows a "5". For example, when it is 2:27 PM, the clock incorrectly shows 5:57 PM. What fraction of the day will the clock show the correct time? | \frac{5}{8} | 0.5 | 6,280.6875 | 5,788.125 | 6,773.25 | |
Let $\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$ Find $\mathbf{A}^{20} - 2 \mathbf{A}^{19}.$ |
\begin{pmatrix} 0 & 3 \\ 0 & -1 \end{pmatrix}
| 0.625 | 6,166.5 | 5,430.3 | 7,393.5 | |
A'Niu is riding a horse to cross a river. There are four horses named A, B, C, and D. It takes 2 minutes for horse A to cross the river, 3 minutes for horse B, 7 minutes for horse C, and 6 minutes for horse D. Only two horses can be driven across the river at a time. The question is: what is the minimum number of minut... | 18 | 0.0625 | 7,610.5625 | 6,383 | 7,692.4 | |
Use the Horner's method to compute the value of the polynomial $f(x)=0.5x^{5}+4x^{4}-3x^{2}+x-1$ when $x=3$, and determine the first operation to perform. | 5.5 | 0.1875 | 2,829.125 | 2,044 | 3,010.307692 | |
Xiao Ming set a six-digit passcode for his phone using the numbers $0-9$, but he forgot the last digit. The probability that Xiao Ming can unlock his phone with just one try is ____. | \frac{1}{10} | 0.25 | 1,494.5 | 3,673.5 | 768.166667 | |
A rectangular garden that is $14$ feet wide and $19$ feet long is paved with $2$-foot square pavers. Given that a bug walks from one corner to the opposite corner in a straight line, determine the total number of pavers the bug visits, including the first and the last paver. | 16 | 0.0625 | 6,987.75 | 5,036 | 7,117.866667 | |
What is the value of $m$ if Tobias downloads $m$ apps, each app costs $\$ 2.00$ plus $10 \%$ tax, and he spends $\$ 52.80$ in total on these $m$ apps? | 24 | Since the tax rate is $10 \%$, then the tax on each $\$ 2.00$ app is $\$ 2.00 \times \frac{10}{100}=\$ 0.20$.
Therefore, including tax, each app costs $\$ 2.00+\$ 0.20=\$ 2.20$.
Since Tobias spends $\$ 52.80$ on apps, he downloads $\frac{\$ 52.80}{\$ 2.20}=24$ apps.
Therefore, $m=24$. | 1 | 729.125 | 729.125 | -1 |
Find the sum of the reciprocals of the roots of $x^2-13x+4=0$. | \frac{13}{4} | 1 | 2,124.6875 | 2,124.6875 | -1 | |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $$2\sin^{2} \frac {A+B}{2}+\cos2C=1$$
(1) Find the magnitude of angle $C$;
(2) If vector $$\overrightarrow {m}=(3a，b)$$ and vector $$\overrightarrow {n}=(a，- \frac {b}{3})$$, with $$... | \sqrt {7} | 0 | 3,624.5625 | -1 | 3,624.5625 | |
In the diagram, triangles $ABC$ and $CBD$ are isosceles with $\angle ABC = \angle BAC$ and $\angle CBD = \angle CDB$. The perimeter of $\triangle CBD$ is $18,$ the perimeter of $\triangle ABC$ is $24,$ and the length of $BD$ is $8.$ If $\angle ABC = \angle CBD$, find the length of $AB.$ | 14 | 0.125 | 7,441.125 | 2,779 | 8,107.142857 | |
Given the sequence $a_n$ defined by the piecewise function:
$\begin{eqnarray*} a_n =\left\{ \begin{array}{lr} 11, & \text{if\ }n\ \text{is\ divisible\ by\ }13\ \text{and\ }14;\\ 13, & \text{if\ }n\ \text{is\ divisible\ by\ }14\ \text{and\ }11;\\ 14, & \text{if\ }n\ \text{is\ divisible\ by\ }11\ \text{and\ }13;\\ 0, & ... | 448 | 0.875 | 4,725.875 | 4,539.642857 | 6,029.5 | |
Given that $\triangle ABC$ is an isosceles right triangle with one leg length of $1$, determine the volume of the resulting geometric solid when $\triangle ABC$ is rotated around one of its sides | \frac{\sqrt{2}\pi}{6} | 0 | 7,946.5 | -1 | 7,946.5 | |
Define $H_n = 1+\frac{1}{2}+\cdots+\frac{1}{n}$ . Let the sum of all $H_n$ that are terminating in base 10 be $S$ . If $S = m/n$ where m and n are relatively prime positive integers, find $100m+n$ .
*Proposed by Lewis Chen* | 9920 | 0.1875 | 8,046.9375 | 8,083.666667 | 8,038.461538 | |
Solve for $n$: $5^{2n + 1} = \frac{1}{25}$. Express your answer as a common fraction. | -\frac{3}{2} | 1 | 1,495.25 | 1,495.25 | -1 | |
Connie multiplies a number by 4 and gets 200 as her result. She realizes she should have divided the number by 4 and then added 10 to get the correct answer. Find the correct value of this number. | 22.5 | 0.625 | 5,663.4375 | 4,146.3 | 8,192 | |
Let vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $|\overrightarrow{a}|=|\overrightarrow{b}|=1$, $\overrightarrow{a}\cdot \overrightarrow{b}= \frac{1}{2}$, and $(\overrightarrow{a}- \overrightarrow{c})\cdot(\overrightarrow{b}- \overrightarrow{c})=0$. Then, calculate the maximum value ... | \frac{\sqrt{3}+1}{2} | 0 | 6,998.9375 | -1 | 6,998.9375 | |
Two thousand points are given on a circle. Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. (See figure.) Continue this process until the labels $1,2,3\dot... | 118 | The label $1993$ will occur on the $\frac12(1993)(1994) \pmod{2000}$th point around the circle. (Starting from 1) A number $n$ will only occupy the same point on the circle if $\frac12(n)(n + 1)\equiv \frac12(1993)(1994) \pmod{2000}$.
Simplifying this expression, we see that $(1993)(1994) - (n)(n + 1) = (1993 - n)(199... | 0.0625 | 8,151.5625 | 7,545 | 8,192 |
Points $A$ and $B$ are on a circle of radius $5$ and $AB=6$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$? | \sqrt{10} | 1. **Identify the Geometry and Define Points**:
Let $O$ be the center of the circle with radius $5$. Since $A$ and $B$ are on the circle, $OA = OB = 5$. Let $D$ be the midpoint of $\overline{AB}$. Since $C$ is the midpoint of the minor arc $AB$, $OC$ is perpendicular to $AB$ at $D$.
2. **Calculate $AD$**:
Since... | 0.875 | 4,946.5625 | 4,482.928571 | 8,192 |
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