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 |
|---|---|---|---|---|---|---|
Define $\varphi^{k}(n)$ as the number of positive integers that are less than or equal to $n / k$ and relatively prime to $n$. Find $\phi^{2001}\left(2002^{2}-1\right)$. (Hint: $\phi(2003)=2002$.) | 1233 | $\varphi^{2001}\left(2002^{2}-1\right)=\varphi^{2001}(2001 \cdot 2003)=$ the number of $m$ that are relatively prime to both 2001 and 2003, where $m \leq 2003$. Since $\phi(n)=n-1$ implies that $n$ is prime, we must only check for those $m$ relatively prime to 2001, except for 2002, which is relatively prime to $2002^{... | 0.125 | 7,007.625 | 6,181 | 7,125.714286 |
Five students from a certain class participated in a speech competition and the order of appearance was determined by drawing lots, under the premise that student A must appear before student B. Calculate the probability of students A and B appearing adjacent to each other. | \frac{2}{5} | 0.5 | 6,934.25 | 5,676.5 | 8,192 | |
Calculate $7 \cdot 9\frac{2}{5}$. | 65\frac{4}{5} | 0.625 | 905.3125 | 1,054.1 | 657.333333 | |
Determine all pairs of positive integers $(m,n)$ such that $(1+x^n+x^{2n}+\cdots+x^{mn})$ is divisible by $(1+x+x^2+\cdots+x^{m})$ . | \(\gcd(m+1, n) = 1\) | Denote the first and larger polynomial to be $f(x)$ and the second one to be $g(x)$ . In order for $f(x)$ to be divisible by $g(x)$ they must have the same roots. The roots of $g(x)$ are the (m+1)th roots of unity, except for 1. When plugging into $f(x)$ , the root of unity is a root of $f(x)$ if and only if the terms ... | 0 | 8,192 | -1 | 8,192 |
The sequence $\{a_n\}$ satisfies $a_1=1$, $na_{n+1}=(n+1)a_n+n(n+1)$, and $b_n=a_n\cos \frac {2n\pi}{3}$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Calculate $S_{24}$. | 304 | 0.6875 | 6,483.5625 | 5,707 | 8,192 | |
Find all solutions to the equation $\sqrt{5+2z} = 11$. | 58 | 1 | 1,233.3125 | 1,233.3125 | -1 | |
Two circles have radius $2$ and $3$ , and the distance between their centers is $10$ . Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$ .
(A *common external tangent* is a tangent line to two circles such that th... | 24 | 0.25 | 7,711.375 | 6,269.5 | 8,192 | |
$\frac{2}{25}=$ | .08 | To convert the fraction $\frac{2}{25}$ into a decimal, we can multiply the numerator and the denominator by a number that makes the denominator a power of 10, which simplifies the division process.
1. **Choosing the multiplier**: We choose 4 because multiplying 25 by 4 gives 100, which is a power of 10.
\[
25 \t... | 0.9375 | 2,203.1875 | 1,803.933333 | 8,192 |
Given \( x, y, z \in \mathbb{Z}_{+} \) with \( x \leq y \leq z \), how many sets of solutions satisfy the equation \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2}\) ? | 10 | 0.0625 | 8,168.8125 | 7,821 | 8,192 | |
Given the function $f(x)={(3\ln x-x^{2}-a-2)}^{2}+{(x-a)}^{2}$ $(a\in \mathbb{R})$, determine the value of the real number $a$ such that the inequality $f(x)\leqslant 8$ has solutions for $x$. | -1 | 0.0625 | 8,167.25 | 7,796 | 8,192 | |
In the polar coordinate system, the polar equation of curve \\(C\\) is given by \\(\rho = 6\sin \theta\\). The polar coordinates of point \\(P\\) are \\((\sqrt{2}, \frac{\pi}{4})\\). Taking the pole as the origin and the positive half-axis of the \\(x\\)-axis as the polar axis, a Cartesian coordinate system is establis... | 3\sqrt{2} | 0.5625 | 6,869 | 6,109.555556 | 7,845.428571 | |
A coordinate system and parametric equations problem (4-4):
In the rectangular coordinate system $x0y$, the parametric equations of line $l$ are given by $\begin{cases} x = \frac{1}{2}t \ y = \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}t \end{cases}$, where $t$ is the parameter. If we establish a polar coordinate system wi... | \frac{\sqrt{10}}{2} | 0 | 5,362.4375 | -1 | 5,362.4375 | |
Inside the triangle \(ABC\), a point \(M\) is taken such that \(\angle MBA = 30^\circ\) and \(\angle MAB = 10^\circ\). Find \(\angle AMC\) if \(\angle ACB = 80^\circ\) and \(AC = BC\). | 70 | 0.25 | 7,389.875 | 5,917.5 | 7,880.666667 | |
Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the tre... | 4 | 1. **Calculate the time spent on each day using the formula $d = rt$, where $d$ is distance, $r$ is rate, and $t$ is time.**
- **Monday:** The rate is $5$ mph and the distance is $2$ miles.
\[
t = \frac{d}{r} = \frac{2}{5} \text{ hours}
\]
- **Wednesday:** The rate is $3$ mph and the distance is $... | 1 | 1,893.3125 | 1,893.3125 | -1 |
Given $\sin\alpha= \frac{1}{2}+\cos\alpha$, and $\alpha\in(0, \frac{\pi}{2})$, then $\sin2\alpha= \_\_\_\_\_\_$, $\cos2\alpha= \_\_\_\_\_\_$. | -\frac{\sqrt{7}}{4} | 0 | 3,969.6875 | -1 | 3,969.6875 | |
Let $r$ be the speed in miles per hour at which a wheel, $13$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{3}$ of a second, the speed $r$ is increased by $6$ miles per hour. Find $r$.
A) 10
B) 11
C) 12
D) 13
E) 14 | 12 | 0 | 8,026.8125 | -1 | 8,026.8125 | |
Evaluate the sum $$\lceil\sqrt{10}\rceil + \lceil\sqrt{11}\rceil + \lceil\sqrt{12}\rceil + \cdots + \lceil\sqrt{40}\rceil$$ | 170 | 0.4375 | 5,758.4375 | 5,012.571429 | 6,338.555556 | |
The power function $f(x)=(m^{2}+2m-2)x^{m}$ is a decreasing function on $(0,+\infty)$. Find the value of the real number $m$. | -3 | 0 | 8,192 | -1 | 8,192 | |
Person A and person B independently attempt to decrypt a password. Their probabilities of successfully decrypting the password are $\dfrac{1}{3}$ and $\dfrac{1}{4}$, respectively. Calculate:
$(1)$ The probability that exactly one of them decrypts the password.
$(2)$ If the probability of decrypting the password needs... | 17 | 0.3125 | 5,103.25 | 4,380.6 | 5,431.727273 | |
Suppose $n$ is a positive integer and $d$ is a single digit in base 10. Find $n$ if
$\frac{n}{810}=0.d25d25d25\ldots$ | 750 | Write out these equations:
$\frac{n}{180} = \frac{d25}{999}$
$\frac{n}{30} = \frac{d25}{37}$
$37n = 30(d25)$
Thus $n$ divides 25 and 30. The only solution for this under 1000 is $\boxed{750}$.
-jackshi2006 | 1 | 3,708.25 | 3,708.25 | -1 |
How many different collections of 9 letters are there? A letter can appear multiple times in a collection. Two collections are equal if each letter appears the same number of times in both collections. | \binom{34}{9} | We put these collections in bijections with binary strings of length 34 containing 9 zeroes and 25 ones. Take any such string - the 9 zeroes will correspond to the 9 letters in the collection. If there are $n$ ones before a zero, then that zero corresponds to the $(n+1)$ st letter of the alphabet. This scheme is an inj... | 0 | 7,236.375 | -1 | 7,236.375 |
A large urn contains $100$ balls, of which $36 \%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72 \%$? (No red balls are to be removed.) | 36 | 1. **Calculate the initial number of red and blue balls:**
Given that $36\%$ of the balls are red, and there are $100$ balls in total, the number of red balls is:
\[
0.36 \times 100 = 36 \text{ red balls}
\]
The rest of the balls are blue, so the number of blue balls is:
\[
100 - 36 = 64 \text{ blu... | 0 | 2,662.4375 | -1 | 2,662.4375 |
$\frac{(3!)!}{3!} = $ | 120 | 1. **Calculate $3!$:**
\[
3! = 3 \times 2 \times 1 = 6
\]
2. **Calculate $(3!)!$:**
\[
(3!)! = (6)!
\]
3. **Expand $(6)!$:**
\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1
\]
4. **Simplify the expression $\frac{(3!)!}{3!}$:**
\[
\frac{(3!)!}{3!} = \frac{6!}{6}
\]
\[
... | 1 | 1,515.4375 | 1,515.4375 | -1 |
Given a quadratic function $f(x) = ax^2 + bx + c$ (where $a$, $b$, and $c$ are constants). If the solution set of the inequality $f(x) \geq 2ax + b$ is $\mathbb{R}$ (the set of all real numbers), then the maximum value of $\frac{b^2}{a^2 + c^2}$ is __________. | 2\sqrt{2} - 2 | 0.5 | 7,021.1875 | 5,989.5 | 8,052.875 | |
A scalene triangle has side lengths which are prime numbers and the length of its perimeter is also prime. What is its smallest possible perimeter? | 23 | 0.75 | 5,681.1875 | 5,496.25 | 6,236 | |
For the set $E=\{a_1, a_2, \ldots, a_{100}\}$, define a subset $X=\{a_1, a_2, \ldots, a_n\}$, and its "characteristic sequence" as $x_1, x_2, \ldots, x_{100}$, where $x_1=x_{10}=\ldots=x_n=1$. The rest of the items are 0. For example, the "characteristic sequence" of the subset $\{a_2, a_3\}$ is $0, 1, 0, 0, \ldots, 0$... | 17 | 0.8125 | 4,524.4375 | 3,886.153846 | 7,290.333333 | |
What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \frac{1}{2}$? | 10 | 1. **Identify the expression and substitute $a$:**
Given the expression $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$, we need to evaluate it at $a = \frac{1}{2}$.
2. **Calculate $a^{-1}$:**
Since $a^{-1}$ is the reciprocal of $a$, when $a = \frac{1}{2}$, we have:
\[
a^{-1} = \left(\frac{1}{2}\right)^{-1} = 2.
\... | 1 | 1,814.6875 | 1,814.6875 | -1 |
Two identical squares, \(A B C D\) and \(P Q R S\), have side length 12. They overlap to form the 12 by 20 rectangle \(A Q R D\). What is the area of the shaded rectangle \(P B C S\)? | 48 | 0.375 | 6,942 | 5,038.5 | 8,084.1 | |
Given an $8 \times 6$ grid, consider a triangle with vertices at $D=(2,1)$, $E=(7,1)$, and $F=(5,5)$. Determine the fraction of the grid covered by this triangle. | \frac{5}{24} | 1 | 2,932.3125 | 2,932.3125 | -1 | |
Point \(P\) is inside an equilateral \(\triangle ABC\) such that the measures of \(\angle APB, \angle BPC, \angle CPA\) are in the ratio 5:6:7. Determine the ratio of the measures of the angles of the triangle formed by \(PA, PB, PC\) (in increasing order). | 2: 3: 4 | 0.0625 | 8,096.25 | 7,095 | 8,163 | |
Given that $x \sim N(-1,36)$ and $P(-3 \leqslant \xi \leqslant -1) = 0.4$, calculate $P(\xi \geqslant 1)$. | 0.1 | 0.1875 | 7,431.875 | 5,397.333333 | 7,901.384615 | |
\( \mathrm{n} \) is a positive integer not greater than 100 and not less than 10, and \( \mathrm{n} \) is a multiple of the sum of its digits. How many such \( \mathrm{n} \) are there? | 24 | 0 | 8,161.3125 | -1 | 8,161.3125 | |
Find all values of \( a \) for which the system
\[
\left\{
\begin{array}{l}
x^{2} + 4y^{2} = 1 \\
x + 2y = a
\end{array}
\right.
\]
has a unique solution. If necessary, round your answer to two decimal places. If there are no solutions, answer with 0. | -1.41 | 0 | 6,220.625 | -1 | 6,220.625 | |
Let $x,$ $y,$ $z$ be nonnegative real numbers. Let
\begin{align*}
A &= \sqrt{x + 2} + \sqrt{y + 5} + \sqrt{z + 10}, \\
B &= \sqrt{x + 1} + \sqrt{y + 1} + \sqrt{z + 1}.
\end{align*}Find the minimum value of $A^2 - B^2.$ | 36 | 0.0625 | 8,192 | 8,192 | 8,192 | |
What is the smallest positive integer representable as the sum of the cubes of three positive integers in two different ways? | 251 | 0 | 8,167 | -1 | 8,167 | |
Design a computer operation program:
1. Initial values \( x = 3 \), \( y = 0 \).
2. \( x = x + 2 \).
3. \( y = y + x \).
4. If \( y \geqslant 10000 \), proceed to (5); otherwise, go back to (2).
5. Print \( x \).
6. Stop running. What will be the printed result when this program is executed? | 201 | 0.3125 | 7,713.6875 | 7,608.2 | 7,761.636364 | |
Determine how many triangles can be formed using the vertices of a regular hexadecagon (a 16-sided polygon). | 560 | 0.5625 | 5,149.25 | 2,782.666667 | 8,192 | |
A three-digit number is composed of three different non-zero digits in base ten. When divided by the sum of these three digits, the smallest quotient value is what? | 10.5 | 0 | 8,192 | -1 | 8,192 | |
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? | \frac{3}{16} |
To solve this problem, we need to determine the probability that a continuous stripe encircles the cube. We will analyze the problem using the approach described in Solution 1, which is both clear and concise.
1. **Total Possible Stripe Combinations**:
Each face of the cube has two possible orientations for the st... | 0.0625 | 7,852.8125 | 5,719 | 7,995.066667 |
Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$. | 750 | We can use complementary counting. We can choose a five-element subset in ${14\choose 5}$ ways. We will now count those where no two numbers are consecutive. We will show a bijection between this set, and the set of 10-element strings that contain 5 $A$s and 5 $B$s, thereby showing that there are ${10\choose 5}$ such s... | 0.9375 | 4,812.125 | 4,586.8 | 8,192 |
Given that the function $y = f(x)$ is an even function defined on $\mathbb{R}$, and when $x \geq 0$, $f(x) = \log_2(x+2) - 3$. Find the values of $f(6)$ and $f(f(0))$. | -1 | 1 | 2,190.4375 | 2,190.4375 | -1 | |
Person A starts traveling from point A to point B. Persons B and C start traveling from point B to point A. After person A has traveled 50 kilometers, persons B and C start traveling from point B. Person A and person B meet at point C, and person A and person C meet at point D. It is known that the speed of person A is... | 130 | 0.0625 | 7,351.25 | 3,053 | 7,637.8 | |
A number is randomly selected from the interval $[-π, π]$. Calculate the probability that the value of the function $y = \cos x$ falls within the range $[-\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}]$. | \frac{2}{3} | 0.25 | 7,337 | 5,868.25 | 7,826.583333 | |
Let $A B C D$ be an isosceles trapezoid with parallel bases $A B=1$ and $C D=2$ and height 1. Find the area of the region containing all points inside $A B C D$ whose projections onto the four sides of the trapezoid lie on the segments formed by $A B, B C, C D$ and $D A$. | \frac{5}{8} | Let $E, F$, be the projections of $A, B$ on $C D$. A point whose projections lie on the sides must be contained in the square $A B F E$. Furthermore, the point must lie under the perpendicular to $A D$ at $A$ and the perpendicular to $B C$ at $B$, which have slopes $\frac{1}{2}$ and $-\frac{1}{2}$. The area of the desi... | 0 | 8,192 | -1 | 8,192 |
Given the function $$f(x)=2\sin(wx+\varphi+ \frac {\pi}{3})+1$$ where $|\varphi|< \frac {\pi}{2}$ and $w>0$, is an even function, and the distance between two adjacent axes of symmetry of the function $f(x)$ is $$\frac {\pi}{2}$$.
(1) Find the value of $$f( \frac {\pi}{8})$$.
(2) When $x\in(-\frac {\pi}{2}, \frac {... | 2\pi | 0.5 | 7,161 | 6,131.25 | 8,190.75 | |
In triangle \( \triangle ABC \), \( \angle BAC = 60^{\circ} \). The angle bisector of \( \angle BAC \), \( AD \), intersects \( BC \) at point \( D \). Given that \( \overrightarrow{AD} = \frac{1}{4} \overrightarrow{AC} + t \overrightarrow{AB} \) and \( AB = 8 \), find the length of \( AD \). | 6 \sqrt{3} | 0.8125 | 4,681.375 | 3,871.230769 | 8,192 | |
Rationalize the denominator of $\frac{1+\sqrt{3}}{1-\sqrt{3}}$. When you write your answer in the form $A+B\sqrt{C}$, where $A$, $B$, and $C$ are integers, what is $ABC$? | 6 | 1 | 2,279.4375 | 2,279.4375 | -1 | |
All lines with equation $ax+by=c$ such that $a,b,c$ form an arithmetic progression pass through a common point. What are the coordinates of that point? | (-1,2) | 1. **Identify the Arithmetic Progression (AP) Relationship**: Given that $a$, $b$, and $c$ form an arithmetic progression, we can express $b$ and $c$ in terms of $a$ and a common difference $d$. Thus, $b = a + d$ and $c = a + 2d$.
2. **Substitute in Line Equation**: Substitute $b$ and $c$ into the line equation $ax + ... | 1 | 2,008.875 | 2,008.875 | -1 |
What is the total area, in square units, of the four triangular faces of a right, square-based pyramid that has base edges measuring 6 units and lateral edges measuring 5 units? | 48 | 0.9375 | 3,154.0625 | 2,818.2 | 8,192 | |
Let $N=\overline{5 A B 37 C 2}$, where $A, B, C$ are digits between 0 and 9, inclusive, and $N$ is a 7-digit positive integer. If $N$ is divisible by 792, determine all possible ordered triples $(A, B, C)$. | $(0,5,5),(4,5,1),(6,4,9)$ | First, note that $792=2^{3} \times 3^{2} \times 11$. So we get that $$\begin{gathered} 8|N \Rightarrow 8| \overline{7 C 2} \Rightarrow 8 \mid 10 C+6 \Rightarrow C=1,5,9 \\ 9|N \Rightarrow 9| 5+A+B+3+7+C+2 \Rightarrow A+B+C=1,10,19 \\ 11|N \Rightarrow 11| 5-A+B-3+7-C+2 \Rightarrow-A+B-C=-11,0 \end{gathered}$$ Adding the... | 0 | 7,062.25 | -1 | 7,062.25 |
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? | \frac{3}{16} |
1. **Total Possible Stripe Combinations**:
Each face of the cube has two possible orientations for the stripe. Since there are six faces, the total number of stripe combinations is calculated as:
\[
2^6 = 64
\]
2. **Counting Favorable Outcomes**:
To have a continuous stripe encircling the cube, we can ... | 0 | 8,143.875 | -1 | 8,143.875 |
Diane has one 1-cent stamp, two identical 2-cent stamps, and so on, up to nine identical 9-cent stamps. In how many different arrangements can Diane paste exactly 10 cents worth of postage in a row across the top of an envelope? (Note, however, that simply rotating or inverting a stamp, or exchanging the positions of t... | 88 | 0 | 8,192 | -1 | 8,192 | |
A point \(A_{1}\) is taken on the side \(AC\) of triangle \(ABC\), and a point \(C_{1}\) is taken on the extension of side \(BC\) beyond point \(C\). The length of segment \(A_{1}C\) is 85% of the length of side \(AC\), and the length of segment \(BC_{1}\) is 120% of the length of side \(BC\). What percentage of the ar... | 102 | 0.1875 | 7,859.375 | 7,112.666667 | 8,031.692308 | |
Write 1 as a sum of 4 distinct unit fractions. | \frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{42} | $\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{42}$ | 0.625 | 6,815.75 | 6,152.7 | 7,920.833333 |
Jason rolls four fair standard six-sided dice. He looks at the rolls and decides to either reroll all four dice or keep two and reroll the other two. After rerolling, he wins if and only if the sum of the numbers face up on the four dice is exactly $9.$ Jason always plays to optimize his chances of winning. What is the... | \frac{1}{18} | 0 | 8,192 | -1 | 8,192 | |
Circle $\Gamma$ is the incircle of $\triangle ABC$ and is also the circumcircle of $\triangle XYZ$. The point $X$ is on $\overline{BC}$, point $Y$ is on $\overline{AB}$, and the point $Z$ is on $\overline{AC}$. If $\angle A=40^\circ$, $\angle B=60^\circ$, and $\angle C=80^\circ$, what is the measure of $\angle AYX$? | 120^\circ | 0.0625 | 8,037.5625 | 7,561 | 8,069.333333 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $\overrightarrow{m}=(b,a-2c)$, $\overrightarrow{n}=(\cos A-2\cos C,\cos B)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
1. Find the value of $\frac{\sin C}{\sin A}$;
2. If $a=2, |m|=3 \sqrt {5}$, find th... | \frac{3\sqrt{15}}{4} | 0 | 4,704 | -1 | 4,704 | |
What is the largest prime factor of $1337$? | 191 | 1 | 1,816.75 | 1,816.75 | -1 | |
A battery of three guns fired a volley, and two shells hit the target. Find the probability that the first gun hit the target, given that the probabilities of hitting the target by the first, second, and third guns are $p_{1}=0,4$, $p_{2}=0,3$, and $p_{3}=0,5$, respectively. | 20/29 | 0.375 | 6,601.1875 | 4,661.833333 | 7,764.8 | |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors, if $|\overrightarrow{a}-2\overrightarrow{b}|=\sqrt{3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ____. | \frac{1}{3}\pi | 0 | 1,945.375 | -1 | 1,945.375 | |
Find the largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$. | \frac{1}{4} | 1 | 2,366.5625 | 2,366.5625 | -1 | |
Let \( m \) and \( n \) be positive integers satisfying
\[ m n^{2} + 876 = 4 m n + 217 n. \]
Find the sum of all possible values of \( m \). | 93 | 0.1875 | 7,618.6875 | 5,134.333333 | 8,192 | |
Let $m \ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1 \le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial
\[q(x) = c_3x^3+c_2x^2+c_1x+c_0\]such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c... | 11 | We start by defining a transformation for each $a_i$:
\[ b_i = \begin{cases}
a_i & \text{if } 1 \leq a_i \leq \frac{m-1}{2}, \\
a_i - m & \text{if } \frac{m-1}{2} + 1 \leq a_i \leq m - 1, \\
0 & \text{if } a_i = m
\end{cases} \]
This transformation maps each $a_i$ to $b_i$ such that $b_i$ ranges from $-\frac{m-1}{2}$... | 0.8125 | 5,755.625 | 5,193.384615 | 8,192 |
Inside a convex 13-sided polygon, there are 200 points such that no three of these 213 points (including the vertices of the polygon) lie on the same line. The polygon is divided into triangles, each vertex of which is any three of the given 213 points. What is the maximum number of triangles that could result? | 411 | 0.9375 | 5,167.0625 | 5,138.133333 | 5,601 | |
Given a set of data $x_1, x_2, x_3, \ldots, x_n$ with a mean of 2 and a variance of 3, calculate the mean and variance of the data set $2x_1+5, 2x_2+5, 2x_3+5, \ldots, 2x_n+5$ respectively. | 12 | 1 | 1,997.9375 | 1,997.9375 | -1 | |
Ninety-nine children are standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If two balls end up with the same child, one of these balls is irrevocably lost. What is the minimum time required for the children to have only one ball le... | 98 | 0.0625 | 7,986.125 | 8,192 | 7,972.4 | |
Randomly select $3$ out of $6$ small balls with the numbers $1$, $2$, $3$, $4$, $5$, and $6$, which are of the same size and material. The probability that exactly $2$ of the selected balls have consecutive numbers is ____. | \frac{3}{5} | 0.0625 | 7,658.25 | 4,699 | 7,855.533333 | |
Let $T = (2+i)^{20} - (2-i)^{20}$, where $i = \sqrt{-1}$. Find $|T|$. | 19531250 | 0 | 8,192 | -1 | 8,192 | |
The graph of the equation \[ x^2 + 4y^2 - 10x + 56y = k\]is a non-degenerate ellipse if and only if $k > a.$ What is $a?$ | -221 | 1 | 2,569.1875 | 2,569.1875 | -1 | |
Mrs. Riley revised her data after realizing that there was an additional score bracket and a special bonus score for one of the brackets. Recalculate the average percent score for the $100$ students given the updated table:
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Stude... | 80.2 | 0.125 | 746.9375 | 633 | 763.214286 | |
For how many integer values of $n$ between 1 and 2000 inclusive does the decimal representation of $\frac{n}{2940}$ terminate? | 13 | 0.375 | 7,496.875 | 6,458.333333 | 8,120 | |
If the function $f(x)$ is monotonic in its domain $(-\infty, +\infty)$, and for any real number $x$, it satisfies $f(f(x)+e^{x})=1-e$, where $e$ is the base of the natural logarithm, determine the value of $f(\ln 2)$. | -1 | 0.8125 | 3,558.75 | 3,181.538462 | 5,193.333333 | |
Suppose that $a$ and $b$ are positive integers such that $(a+bi)^2 = 3+4i$. What is $a+bi$? | 2 + i | 1 | 1,615 | 1,615 | -1 | |
In rectangle $ABCD$, side $AB$ measures $8$ units and side $BC$ measures $4$ units. Points $F$ and $G$ are on side $CD$ such that segment $DF$ measures $2$ units and segment $GC$ measures $2$ units, and lines $AF$ and $BG$ intersect at $E$. What is the area of triangle $AEB$? | 32 | 0.4375 | 7,068.9375 | 5,625 | 8,192 | |
The figure shows a square in the interior of a regular hexagon. The square and regular hexagon share a common side. What is the degree measure of $\angle ABC$? [asy]
size(150);
pair A, B, C, D, E, F, G, H;
A=(0,.866);
B=(.5,1.732);
C=(1.5,1.732);
D=(2,.866);
E=(1.5,0);
F=(.5,0);
G=(.5,1);
H=(1.5,1);
draw(A--B);
draw(B... | 45 | 0.125 | 6,604.625 | 6,896 | 6,563 | |
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}$? | 279 | To solve this problem, we need to understand the relationship between the powers of $5$ and $2$ and how they are distributed between $5^n$ and $5^{n+1}$.
1. **Understanding the relationship between $5^n$ and $2^m$:**
We know that $5^{867}$ is between $2^{2013}$ and $2^{2014}$. This gives us a way to compare the gro... | 0 | 8,192 | -1 | 8,192 |
A set of three numbers has both a mean and median equal to 4. If the smallest number in the set is 1, what is the range of the set of numbers? | 6 | 1 | 1,630.5 | 1,630.5 | -1 | |
The graphs \( y = 2 \cos 3x + 1 \) and \( y = - \cos 2x \) intersect at many points. Two of these points, \( P \) and \( Q \), have \( x \)-coordinates between \(\frac{17 \pi}{4}\) and \(\frac{21 \pi}{4}\). The line through \( P \) and \( Q \) intersects the \( x \)-axis at \( B \) and the \( y \)-axis at \( A \). If \... | \frac{361\pi}{8} | 0.1875 | 7,600.5625 | 6,102 | 7,946.384615 | |
(1) Given $\cos(15°+\alpha) = \frac{15}{17}$, with $\alpha \in (0°, 90°)$, find the value of $\sin(15°-\alpha)$.
(2) Given $\cos\alpha = \frac{1}{7}$, $\cos(\alpha-\beta) = \frac{13}{14}$, and $0 < \beta < \alpha < \frac{\pi}{2}$, find the value of $\beta$. | \frac{\pi}{3} | 0.4375 | 7,175.625 | 5,868.857143 | 8,192 | |
How many even natural-number factors does $n = 2^2 \cdot 3^1 \cdot 7^2$ have? | 12 | 0.9375 | 3,369.6875 | 3,048.2 | 8,192 | |
Given that P and Q are points on the graphs of the functions $2x-y+6=0$ and $y=2\ln x+2$ respectively, find the minimum value of the line segment |PQ|. | \frac{6\sqrt{5}}{5} | 0 | 7,181.25 | -1 | 7,181.25 | |
Given the equation $5^{12} = \frac{5^{90/x}}{5^{50/x} \cdot 25^{30/x}}$, find the value of $x$ that satisfies this equation. | -\frac{5}{3} | 1 | 1,992.4375 | 1,992.4375 | -1 | |
In circle $O$, $\overline{EB}$ is a diameter and the line $\overline{DC}$ is parallel to $\overline{EB}$. The line $\overline{AB}$ intersects the circle again at point $F$ such that $\overline{AB}$ is parallel to $\overline{ED}$. If angles $AFB$ and $ABF$ are in the ratio 3:2, find the degree measure of angle $BCD$. | 72 | 0.0625 | 7,779.5 | 6,372 | 7,873.333333 | |
Circle $\omega$ is inscribed in rhombus $H M_{1} M_{2} T$ so that $\omega$ is tangent to $\overline{H M_{1}}$ at $A, \overline{M_{1} M_{2}}$ at $I, \overline{M_{2} T}$ at $M$, and $\overline{T H}$ at $E$. Given that the area of $H M_{1} M_{2} T$ is 1440 and the area of $E M T$ is 405 , find the area of $A I M E$. | 540 | First, from equal tangents, we know that $T E=T M$. As the sides of a rhombus are also equal, this gives from SAS similarity that $E M T \sim T H M_{2}$. Further, the ratio of their areas is $\frac{405}{1440 / 2}=\frac{9}{16}$. This means that $T E=T M=\frac{3}{4} H T$. Then, we get that $M M_{2}=M I$, so $M_{2} M I \s... | 0 | 8,192 | -1 | 8,192 |
In a scalene triangle, the lengths of the medians $A N$ and $B P$ are 3 and 6, respectively, and the area is $3 \sqrt{15}$. The length of the third median $C M$ is | $3 \sqrt{6}$ | 0 | 5,422.1875 | -1 | 5,422.1875 | |
In triangle $ABC$, a midline $MN$ that connects the sides $AB$ and $BC$ is drawn. A circle passing through points $M$, $N$, and $C$ touches the side $AB$, and its radius is equal to $\sqrt{2}$. The length of side $AC$ is 2. Find the sine of angle $ACB$. | \frac{1}{2} | 0.0625 | 8,192 | 8,192 | 8,192 | |
A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (... | \frac{5}{8} | To solve this problem, we define $P_{(x,y)}$ as the probability that the frog's sequence of jumps ends on a vertical side of the square when starting from the point $(x,y)$. We will use symmetry and recursive relations to find $P_{(1,2)}$.
#### Step 1: Symmetry Analysis
Due to the symmetry of the problem about the lin... | 0 | 8,192 | -1 | 8,192 |
When I saw Eleonora, I found her very pretty. After a brief trivial conversation, I told her my age and asked how old she was. She answered:
- When you were as old as I am now, you were three times older than me. And when I will be three times older than I am now, together our ages will sum up to exactly a century.
... | 15 | 0 | 6,110.3125 | -1 | 6,110.3125 | |
How many ways are there to put 4 distinguishable balls into 2 distinguishable boxes? | 16 | 0.9375 | 2,789.8125 | 2,429.666667 | 8,192 | |
Given \( f(x) = x^2 + 3x + 2 \) and \( S = \{0, 1, 2, 3, \cdots, 100\} \), if \( a \in S \) and \( f(a) \) is divisible by 6, how many such \( a \) exist? | 67 | 0.9375 | 4,645.125 | 4,566.4 | 5,826 | |
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? | 225 | 0.125 | 7,174.0625 | 7,567 | 7,117.928571 | |
Let \( a \) be a nonzero real number. In the Cartesian coordinate system \( xOy \), the quadratic curve \( x^2 + ay^2 + a^2 = 0 \) has a focal distance of 4. Determine the value of \( a \). | \frac{1 - \sqrt{17}}{2} | 0 | 6,968.5625 | -1 | 6,968.5625 | |
Let $S$ be a set of positive integers satisfying the following two conditions: - For each positive integer $n$, at least one of $n, 2 n, \ldots, 100 n$ is in $S$. - If $a_{1}, a_{2}, b_{1}, b_{2}$ are positive integers such that $\operatorname{gcd}\left(a_{1} a_{2}, b_{1} b_{2}\right)=1$ and $a_{1} b_{1}, a_{2} b_{2} \... | 396 | The optimal value of $r$ is $\frac{1}{252}$. This is attained by letting $S$ be the set of integers $n$ for which $\nu_{2}(n) \equiv 4 \bmod 5$ and $\nu_{3}(n) \equiv 1 \bmod 2$. Let $S$ be a set of positive integers satisfying the two conditions. For each prime $p$, let $A_{p}=\left\{\nu_{p}(n)\right.$ : $n \in S\}$. ... | 0 | 8,192 | -1 | 8,192 |
Let $ABC$ be a triangle with $AB = 5$ , $AC = 8$ , and $BC = 7$ . Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$ . Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\overline{ID}$ and $\overline{BC}$ . Suppose $DE = \frac{a\sqrt{b}}{... | 13 | 0.875 | 5,564 | 5,188.571429 | 8,192 | |
The moisture content of freshly cut grass is $60\%$, and the moisture content of hay is $15\%$. How much hay will be obtained from one ton of freshly cut grass? | 470.588 | 0 | 4,592.0625 | -1 | 4,592.0625 | |
Let $(a_1,a_2,a_3,\ldots,a_{14})$ be a permutation of $(1,2,3,\ldots,14)$ where $a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7$ and $a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13} < a_{14}$. An example of such a permutation is $(7,6,5,4,3,2,1,8,9,10,11,12,13,14)$. Determine the number of such permutations. | 1716 | 0.1875 | 7,103.3125 | 5,023.666667 | 7,583.230769 | |
Let \(ABCD\) be a quadrilateral inscribed in a circle with center \(O\). Let \(P\) denote the intersection of \(AC\) and \(BD\). Let \(M\) and \(N\) denote the midpoints of \(AD\) and \(BC\). If \(AP=1\), \(BP=3\), \(DP=\sqrt{3}\), and \(AC\) is perpendicular to \(BD\), find the area of triangle \(MON\). | 3/4 | 0.3125 | 7,998.8125 | 7,625 | 8,168.727273 | |
In a bus, there are single and double seats. In the morning, 13 people were sitting in the bus, and there were 9 completely free seats. In the evening, 10 people were sitting in the bus, and there were 6 completely free seats. How many seats are there in the bus? | 16 | 0.1875 | 1,172.3125 | 554 | 1,315 | |
Find the smallest positive integer \( n \) such that \( n(n+1)(n+2) \) is divisible by 247. | 37 | 0.125 | 8,129.0625 | 7,688.5 | 8,192 | |
Consider the following diagram showing a rectangular grid of dots consisting of 3 rows and 4 columns. How many rectangles can be formed in this grid? | 60 | 0.625 | 6,438.875 | 5,888.2 | 7,356.666667 |
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