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 the function $f(x)=\cos x\cdot\sin \left(x+ \frac {\pi}{3}\right)- \sqrt {3}\cos ^{2}x+ \frac { \sqrt {3}}{4}$, $x\in\mathbb{R}$.
(I) Find the smallest positive period of $f(x)$.
(II) Find the maximum and minimum values of $f(x)$ on the closed interval $\left[- \frac {\pi}{4}, \frac {\pi}{4}\right]$. | - \frac {1}{2} | 0.5 | 6,551.0625 | 6,955 | 6,147.125 | |
A right triangle has one angle measuring $30^\circ$. This triangle shares its hypotenuse with a second triangle that is also right-angled. The two triangles together form a quadrilateral. If the other acute angle in the second triangle is $45^\circ$, find the area of the quadrilateral given that the hypotenuse common t... | \frac{25\sqrt{3} + 50}{2} | 0 | 8,065.1875 | -1 | 8,065.1875 | |
Circle $A$ is tangent to circle $B$ at one point, and the center of circle $A$ lies on the circumference of circle $B$. The area of circle $A$ is $16\pi$ square units. Find the area of circle $B$. | 64\pi | 0 | 6,711.375 | -1 | 6,711.375 | |
Solve the equation $\frac{n!}{2}=k!+l!$ in natural numbers $n$, $k$, and $l$, where $n! = 1 \cdot 2 \cdots n$. In your answer, indicate 0 if there are no solutions, specify the value of $n$ if there is only one solution, or provide the sum of all $n$ values if there are multiple solutions. Recall that a solution is a t... | 10 | 0.0625 | 7,682.0625 | 6,950 | 7,730.866667 | |
Person A and Person B each shoot at a target once. The probability of Person A hitting the target is $\dfrac{2}{3}$, and the probability of Person B hitting the target is $\dfrac{4}{5}$. Calculate the probability that exactly one person hits the target. | \dfrac{86}{225} | 0 | 2,053.125 | -1 | 2,053.125 | |
In a right triangle PQR with right angle at P, suppose $\sin Q = 0.6$. If the length of QP is 15, what is the length of QR? | 25 | 0 | 2,465.8125 | -1 | 2,465.8125 | |
In triangle \( ABC \), \( AB = 33 \), \( AC = 21 \), and \( BC = m \), where \( m \) is a positive integer. If point \( D \) can be found on \( AB \) and point \( E \) can be found on \( AC \) such that \( AD = DE = EC = n \), where \( n \) is a positive integer, what must the value of \( m \) be? | 30 | 0.0625 | 8,053 | 6,382 | 8,164.4 | |
How many solutions does the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ have in the closed interval $[0,\pi]$? | 2 | To solve the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ over the interval $[0,\pi]$, we analyze the behavior of both sides of the equation.
#### Step 1: Evaluate the functions at critical points
We start by evaluating the functions at the endpoints and the midpoint of the in... | 0.3125 | 8,104.75 | 7,912.8 | 8,192 |
Given that the terminal side of angle $\alpha$ passes through the fixed point $P$ on the function $y=\log _{a}(x-3)+2$, find the value of $\sin 2\alpha+\cos 2\alpha$. | \frac{7}{5} | 0.3125 | 6,934.0625 | 4,166.6 | 8,192 | |
In the sequence ${a_{n}}$, $a_{1}=1$, $a_{n+2}+(-1)^{n}a_{n}=1$. Let $s_{n}$ be the sum of the first $n$ terms of the sequence ${a_{n}}$. Find $s_{100}$ = \_\_\_\_\_\_. | 1300 | 0.375 | 7,419.9375 | 6,133.166667 | 8,192 | |
A cooperative receives apple and grape juice in identical containers and produces an apple-grape drink in identical cans. One container of apple juice is enough for exactly 6 cans of the drink, and one container of grape juice is enough for exactly 10 cans. When the recipe of the drink was changed, one container of app... | 15 | 0.0625 | 6,962 | 4,334 | 7,137.2 | |
Determine the product of all constants $t$ such that the quadratic $x^2 + tx - 24$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. | 5290000 | 0.875 | 5,593.3125 | 5,222.071429 | 8,192 | |
From 6 sprinters, 4 are to be selected to participate in a 4×100 m relay. If among them, Athlete A cannot run the first leg, and Athlete B cannot run the fourth leg, how many different ways are there to form the team? | 252 | 0.25 | 8,114.9375 | 7,883.75 | 8,192 | |
Petya wants to color some cells of a $6 \times 6$ square so that there are as many vertices as possible that belong to exactly three colored squares. What is the maximum number of such vertices he can achieve? | 25 | 0 | 8,178.75 | -1 | 8,178.75 | |
Given the fraction $\frac{987654321}{2^{30}\cdot 5^6}$, determine the minimum number of digits to the right of the decimal point required to express this fraction as a decimal. | 30 | 0.1875 | 7,610.125 | 5,088.666667 | 8,192 | |
There exists a constant $c,$ so that among all chords $\overline{AB}$ of the parabola $y = x^2$ passing through $C = (0,c),$
\[t = \frac{1}{AC^2} + \frac{1}{BC^2}\]is a fixed constant. Find the constant $t.$
[asy]
unitsize(1 cm);
real parab (real x) {
return(x^2);
}
pair A, B, C;
A = (1.7,parab(1.7));
B = (-1,pa... | 4 | 0.625 | 6,880.0625 | 6,092.9 | 8,192 | |
Count all the distinct anagrams of the word "YOANN". | 60 | 0.0625 | 689.4375 | 811 | 681.333333 | |
When a die is thrown twice in succession, the numbers obtained are recorded as $a$ and $b$, respectively. The probability that the line $ax+by=0$ and the circle $(x-3)^2+y^2=3$ have no points in common is ______. | \frac{2}{3} | 0.625 | 6,641.6875 | 6,054.2 | 7,620.833333 | |
Abby, Bart, Cindy and Damon weigh themselves in pairs. Together Abby and Bart weigh 260 pounds, Bart and Cindy weigh 245 pounds, and Cindy and Damon weigh 270 pounds. How many pounds do Abby and Damon weigh together? | 285 | 1 | 2,477.5625 | 2,477.5625 | -1 | |
Consider an $8 \times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e. it cannot stay in the same square). What is the expected number of minutes until the rook reaches the upper right corner? | 70 | Let the expected number of minutes it will take the rook to reach the upper right corner from the top or right edges be $E_{e}$, and let the expected number of minutes it will take the rook to reach the upper right corner from any other square be $E_{c}$. Note that this is justified because the expected time from any s... | 0 | 8,192 | -1 | 8,192 |
How many degrees are in the measure of the smaller angle that is formed by the hour-hand and minute-hand of a clock when it is 5 o'clock? | 150^\circ | 1 | 2,532.375 | 2,532.375 | -1 | |
On grid paper, a step-like right triangle was drawn with legs equal to 6 cells. Then all the grid lines inside the triangle were traced. What is the maximum number of rectangles that can be found in this drawing? | 126 | 0 | 8,192 | -1 | 8,192 | |
Solve the equation: $4x^2 - (x^2 - 2x + 1) = 0$. | -1 | 0 | 2,364.25 | -1 | 2,364.25 | |
15. If \( a = 1.69 \), \( b = 1.73 \), and \( c = 0.48 \), find the value of
$$
\frac{1}{a^{2} - a c - a b + b c} + \frac{2}{b^{2} - a b - b c + a c} + \frac{1}{c^{2} - a c - b c + a b}.
$$ | 20 | 0.4375 | 6,879 | 5,574.857143 | 7,893.333333 | |
What is the value of $2-(-2)^{-2}$? | \frac{7}{4} | 1. **Identify the expression to simplify:**
\[
2 - (-2)^{-2}
\]
2. **Simplify the exponentiation and negative sign:**
\[
(-2)^{-2} = \left(\frac{1}{-2}\right)^2
\]
Since squaring eliminates the negative sign:
\[
\left(\frac{1}{-2}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
\]
3... | 1 | 1,859.6875 | 1,859.6875 | -1 |
A rectangular table of size \( x \) cm \( \times 80 \) cm is covered with identical sheets of paper of size 5 cm \( \times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed 1 cm higher and 1 cm to the right of the previous one. The last sheet is adjacent to the top-right ... | 77 | 0.4375 | 5,311.9375 | 3,842.428571 | 6,454.888889 | |
What is the smallest positive integer $n$ such that $\frac{n}{n+50}$ is equal to a terminating decimal? | 14 | 0.625 | 5,877.0625 | 4,792.3 | 7,685 | |
Consider a $10\times10$ checkerboard with alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard? | 140 | 0.625 | 6,875.9375 | 6,178.4 | 8,038.5 | |
Evaluate the product $\frac{1}{2}\cdot\frac{4}{1}\cdot\frac{1}{8}\cdot\frac{16}{1} \dotsm \frac{1}{16384}\cdot\frac{32768}{1}$. | 256 | 0 | 7,383.6875 | -1 | 7,383.6875 | |
Suppose that $a,b,c$ are real numbers such that $a < b < c$ and $a^3-3a+1=b^3-3b+1=c^3-3c+1=0$ . Then $\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a}$ can be written as $\frac pq$ for relatively prime positive integers $p$ and $q$ . Find $100p+q$ .
*Proposed by Michael Ren* | 301 | 0 | 8,063.8125 | -1 | 8,063.8125 | |
Given the expression \(\frac{a}{b}+\frac{c}{d}+\frac{e}{f}\), where each letter is replaced by a different digit from \(1, 2, 3, 4, 5,\) and \(6\), determine the largest possible value of this expression. | 9\frac{5}{6} | 0 | 8,124.75 | -1 | 8,124.75 | |
If $f(1)=5$, $f(2)=8$ and $f(x)=ax+bx+2$, what is the value of $f(3)$? | 11 | 1 | 2,408.25 | 2,408.25 | -1 | |
Given a rhombus with diagonals of length $12$ and $30$, find the radius of the circle inscribed in this rhombus. | \frac{90\sqrt{261}}{261} | 0 | 4,119.6875 | -1 | 4,119.6875 | |
Fill each cell in the given grid with a number from 1 to 4 so that no number repeats within any row or column. Each "L" shaped block spans two rows and two columns. The numbers inside the circles on the line indicate the sum of the numbers in the two adjacent cells (as shown in the provided example, where the third row... | 2143 | 0 | 8,177.125 | -1 | 8,177.125 | |
Xibing is a local specialty in Haiyang, with a unique flavor, symbolizing joy and reunion. Person A and person B went to the market to purchase the same kind of gift box filled with Xibing at the same price. Person A bought $2400$ yuan worth of Xibing, which was $10$ boxes less than what person B bought for $3000$ yuan... | 50 | 0.3125 | 3,566.375 | 2,786.6 | 3,920.818182 | |
A man walked a certain distance at a constant rate. If he had gone $\frac{1}{2}$ mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone $\frac{1}{2}$ mile per hour slower, he would have been $2\frac{1}{2}$ hours longer on the road. The distance in miles he walked was | 15 | 1. **Set up the equations based on the problem statement:**
Let $x$ be the man's usual speed in miles per hour, and $t$ be the usual time in hours it takes him to walk the distance. The distance he walks can be represented as $d = xt$.
2. **Equation for increased speed:**
If the man walks at a speed of $x + \fra... | 1 | 2,443.9375 | 2,443.9375 | -1 |
Let \( d = \overline{xyz} \) be a three-digit number that cannot be divisible by 10. If the sum of \( \overline{xyz} \) and \( \overline{zyx} \) is divisible by \( c \), find the largest possible value of this integer \( d \). | 979 | 0.1875 | 7,918.8125 | 7,381.666667 | 8,042.769231 | |
The number of values of $x$ satisfying the equation
\[\frac {2x^2 - 10x}{x^2 - 5x} = x - 3\]is: | 0 | 1. **Identify the domain of the equation**: The given equation is
\[
\frac{2x^2 - 10x}{x^2 - 5x} = x - 3.
\]
We first note that the denominator $x^2 - 5x$ must not be zero to avoid division by zero. Factoring out $x$, we get:
\[
x(x - 5) \neq 0.
\]
Therefore, $x \neq 0$ and $x \neq 5$.
2. **Sim... | 1 | 4,288.0625 | 4,288.0625 | -1 |
We know about a convex pentagon that each side is parallel to one of its diagonals. What can be the ratio of the length of a side to the length of the diagonal parallel to it? | \frac{\sqrt{5} - 1}{2} | 0 | 8,007.4375 | -1 | 8,007.4375 | |
In triangle $PQR,$ $PQ = 4,$ $PR = 9,$ $QR = 10,$ and a point $S$ lies on $\overline{QR}$ such that $\overline{PS}$ bisects $\angle QPR.$ Find $\cos \angle QPS.$ | \sqrt{\frac{23}{48}} | 0 | 6,218.3125 | -1 | 6,218.3125 | |
A pyramid is constructed using twenty cubical blocks: the first layer has 10 blocks arranged in a square, the second layer contains 6 blocks arranged in a larger square centered on the 10, the third layer has 3 blocks arranged in a triangle, and finally one block sits on top of the third layer. Each block in layers 2, ... | 54 | 0 | 8,192 | -1 | 8,192 | |
In \( \triangle ABC \), \( AB = 4 \), \( BC = 7 \), \( CA = 5 \). Let \(\angle BAC = \alpha\). Find the value of \( \sin^6 \frac{\alpha}{2} + \cos^6 \frac{\alpha}{2} \). | 7/25 | 1 | 4,397.5625 | 4,397.5625 | -1 | |
The school now introduces a new color, silver, for the flag design. Crestview's school colors are now purple, gold, and silver. The students are designing a flag using three solid-colored horizontal stripes. Using one, two, or all three of the school colors, how many different flags are possible if adjacent stripes may... | 27 | 0.875 | 4,838.6875 | 4,359.642857 | 8,192 | |
What is the sum of all numbers $q$ which can be written in the form $q=\frac{a}{b}$ where $a$ and $b$ are positive integers with $b \leq 10$ and for which there are exactly 19 integers $n$ that satisfy $\sqrt{q}<n<q$? | 777.5 | Suppose that a number $q$ has the property that there are exactly 19 integers $n$ with $\sqrt{q}<n<q$. Suppose that these 19 integers are $m, m+1, m+2, \ldots, m+17, m+18$. Then $\sqrt{q}<m<m+1<m+2<\cdots<m+17<m+18<q$. This tells us that $q-\sqrt{q}>(m+18)-m=18$ because $q-\sqrt{q}$ is as small as possible when $q$ is ... | 0 | 8,192 | -1 | 8,192 |
Given \( 1991 = 2^{\alpha_{1}} + 2^{\alpha_{2}} + \cdots + 2^{\alpha_{n}} \), where \( \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} \) are distinct non-negative integers, find the sum \( \alpha_{1} + \alpha_{2} + \cdots + \alpha_{n} \). | 43 | 0.5625 | 5,591.1875 | 4,711.777778 | 6,721.857143 | |
Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$ . | 83 | 0.75 | 6,610.625 | 6,083.5 | 8,192 | |
At Jefferson High School, there are 500 students enrolled. One hundred twenty students are in the orchestra, 190 are in band, and 220 are in chorus. If only 400 students are in orchestra, band, and/or chorus, how many students are in exactly two of these groups? | 130 | 0.125 | 7,575.625 | 6,229 | 7,768 | |
Each principal of Lincoln High School serves exactly one $3$-year term. What is the maximum number of principals this school could have during an $8$-year period? | 4 | To determine the maximum number of principals that can serve during an 8-year period at Lincoln High School, where each principal serves a 3-year term, we need to consider how the terms can overlap with the 8-year period.
1. **Understanding the Term Length**: Each principal serves for exactly 3 years.
2. **Maximizing... | 0 | 7,814.5625 | -1 | 7,814.5625 |
In right triangle $ABC$ with right angle $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. Points $O_1$ and $O_2$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_1$ is tangent to the hypotenuse and to the extension of... | 737 | Let the radius of the circle be $r$. It can be seen that $\Delta FHO_{1}$ and $\Delta O_{2}GJ$ are similar to $\Delta ACB$, and the length of the hypotenuses are $\frac{17}{8}r$ and $\frac {17}{15}r$, respectively. Then, the entire length of $HJ$ is going to be $(\frac{17}{8}+\frac{17}{15}+2)r = \frac{631}{120}r$. The ... | 0 | 8,192 | -1 | 8,192 |
In the quadrilateral \(ABCD\), it is known that \(AB = BD\), \(\angle ABD = \angle DBC\), and \(\angle BCD = 90^\circ\). On the segment \(BC\), there is a point \(E\) such that \(AD = DE\). What is the length of segment \(BD\) if it is known that \(BE = 7\) and \(EC = 5\)? | 17 | 0.0625 | 7,930.0625 | 5,437 | 8,096.266667 | |
A line is parameterized by a parameter $t,$ so that the vector on the line at $t = 2$ is $\begin{pmatrix} 1 \\ 4 \end{pmatrix},$ and the vector on the line at $t = 3$ is $\begin{pmatrix} 3 \\ -4 \end{pmatrix}.$ Find the vector on the line at $t = -7.$ | \begin{pmatrix} -17 \\ 76 \end{pmatrix} | 0.9375 | 4,496.6875 | 4,250.333333 | 8,192 | |
A sequence of integers is defined as follows: $a_i = i$ for $1 \le i \le 5,$ and
\[a_i = a_1 a_2 \dotsm a_{i - 1} - 1\]for $i > 5.$ Evaluate $a_1 a_2 \dotsm a_{2011} - \sum_{i = 1}^{2011} a_i^2.$ | -1941 | 0.0625 | 7,860.0625 | 8,192 | 7,837.933333 | |
Given an increasing geometric sequence $\{a_{n}\}$ with a common ratio greater than $1$ such that $a_{2}+a_{4}=20$, $a_{3}=8$.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ Let $b_{m}$ be the number of terms of $\{a_{n}\}$ in the interval $\left(0,m\right]\left(m\in N*\right)$. Find the sum of the first... | 480 | 0.9375 | 5,275.375 | 5,286.333333 | 5,111 | |
Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 6x^2 + 11x = 12$.
Compute $\frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b}$. | -\frac{23}{12} | 0.625 | 6,791.125 | 5,950.6 | 8,192 | |
What is the base ten equivalent of $12345_{6}$? | 1865 | 1 | 1,753.25 | 1,753.25 | -1 | |
A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have? | 84 | To solve this problem, we need to understand the structure of the solid after the smaller cubes are removed from each corner of the larger cube.
1. **Original Cube Characteristics**:
- The original cube has a side length of $3$.
- It has $8$ vertices (corners).
- It has $12$ edges.
- It has $6$ faces.
2. ... | 0 | 7,671.375 | -1 | 7,671.375 |
A huge number $y$ is given by $2^33^24^65^57^88^39^{10}11^{11}$. What is the smallest positive integer that, when multiplied with $y$, results in a product that is a perfect square? | 110 | 0.0625 | 2,167.25 | 2,945 | 2,115.4 | |
Determine the value of $l$ for which
\[\frac{9}{x + y + 1} = \frac{l}{x + z - 1} = \frac{13}{z - y + 2}.\] | 22 | 0 | 7,655 | -1 | 7,655 | |
We wrote the numbers from 1 to 2009 on a piece of paper. In the second step, we also wrote down twice each of these numbers on the paper, and then we erased the numbers that appeared twice. We repeat this process such that, in the $i$-th step, we write $i$ times each of the numbers from 1 to 2009 on the paper and then... | 2009 | 0.5 | 7,487.875 | 6,783.75 | 8,192 | |
Let \(a\), \(b\), \(c\), and \(d\) be distinct positive integers such that \(a+b\), \(a+c\), and \(a+d\) are all odd and are all squares. Let \(L\) be the least possible value of \(a + b + c + d\). What is the value of \(10L\)? | 670 | 0.0625 | 8,076 | 6,336 | 8,192 | |
In triangle $ABC, \angle A=2 \angle C$. Suppose that $AC=6, BC=8$, and $AB=\sqrt{a}-b$, where $a$ and $b$ are positive integers. Compute $100 a+b$. | 7303 | Let $x=AB$, and $\angle C=\theta$, then $\angle A=2 \theta$ and $\angle B=180-3 \theta$. Extend ray $BA$ to $D$ so that $AD=AC$. We know that $\angle CAD=180-2 \theta$, and since $\triangle ADC$ is isosceles, it follows that $\angle ADC=\angle ACD=\theta$, and so $\angle DCB=2 \theta=\angle BAC$, meaning that $\triangl... | 0.75 | 5,748.0625 | 4,933.416667 | 8,192 |
If the direction vector of line $l$ is $\overrightarrow{d}=(1,\sqrt{3})$, then the inclination angle of line $l$ is ______. | \frac{\pi}{3} | 0 | 1,335.75 | -1 | 1,335.75 | |
In right triangle $ABC$ with $\angle BAC = 90^\circ$, we have $AB = 15$ and $BC = 17$. Find $\tan A$ and $\sin A$. | \frac{8}{17} | 0 | 5,995.5625 | -1 | 5,995.5625 | |
If the function $f(x)=x^{2}-m\cos x+m^{2}+3m-8$ has a unique zero, then the set of real numbers $m$ that satisfy this condition is \_\_\_\_\_\_. | \{2\} | 0 | 8,192 | -1 | 8,192 | |
$M$ is an $8 \times 8$ matrix. For $1 \leq i \leq 8$, all entries in row $i$ are at least $i$, and all entries on column $i$ are at least $i$. What is the minimum possible sum of the entries of $M$ ? | 372 | Let $s_{n}$ be the minimum possible sum for an $n$ by $n$ matrix. Then, we note that increasing it by adding row $n+1$ and column $n+1$ gives $2 n+1$ additional entries, each of which has minimal size at least $n+1$. Consequently, we obtain $s_{n+1}=s_{n}+(2 n+1)(n+1)=s_{n}+2 n^{2}+3 n+1$. Since $s_{0}=0$, we get that ... | 0.4375 | 6,751.9375 | 5,449.142857 | 7,765.222222 |
Write any natural number on a piece of paper, and rotate the paper 180 degrees. If the value remains the same, such as $0$, $11$, $96$, $888$, etc., we call such numbers "神马数" (magical numbers). Among all five-digit numbers, how many different "magical numbers" are there? | 60 | 0.0625 | 7,100.5625 | 4,304 | 7,287 | |
What is the value of $x$ if \begin{align*}x &= y+5,\\
y &= z+10,\\
z &= w+20,\\
\text{and }\qquad w &= 80?
\end{align*} | 115 | 1 | 1,790.125 | 1,790.125 | -1 | |
An ellipse satisfies the property that a light ray emitted from one focus of the ellipse, after reflecting off the ellipse, will pass through the other focus. Consider a horizontally placed elliptical billiards table that satisfies the equation $\frac{x^2}{16} + \frac{y^2}{9} = 1$. Let points A and B correspond to its ... | 16 | 0 | 7,557.75 | -1 | 7,557.75 | |
Compute $\displaystyle \sum_{n=2}^\infty \sum_{k=1}^{n-1} \frac{k}{2^{n+k}}$. | \frac{4}{9} | 0.375 | 6,623.5 | 5,026.5 | 7,581.7 | |
In a book, the pages are numbered from 1 through $n$. When summing the page numbers, one page number was mistakenly added three times instead of once, resulting in an incorrect total sum of $2046$. Identify the page number that was added three times. | 15 | 1 | 3,558.875 | 3,558.875 | -1 | |
What is the product of all real numbers that are doubled when added to their reciprocals? | -1 | 1 | 1,458.25 | 1,458.25 | -1 | |
Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(8, 0)$. A horizontal line with equation $y=t$ intersects line segment $ \overline{AB} $ at $T$ and line segment $ \overline{AC} $ at $U$, forming $\triangle ATU$ with area 13.5. Compute $t$. | 2 | 1 | 3,743.0625 | 3,743.0625 | -1 | |
To reach the Solovyov family's dacha from the station, one must first travel 3 km on the highway and then 2 km on a path. Upon arriving at the station, the mother called her son Vasya at the dacha and asked him to meet her on his bicycle. They started moving towards each other at the same time. The mother walks at a co... | 800 | 0 | 8,192 | -1 | 8,192 | |
The expression $(5 \times 5)+(5 \times 5)+(5 \times 5)+(5 \times 5)+(5 \times 5)$ is equal to what? | 125 | The given sum includes 5 terms each equal to $(5 \times 5)$. Thus, the given sum is equal to $5 \times(5 \times 5)$ which equals $5 \times 25$ or 125. | 1 | 1,051.125 | 1,051.125 | -1 |
How many distinct sequences of four letters can be made from the letters in EQUALS if each sequence must begin with L, end with Q, and no letter can appear in a sequence more than once? | 12 | 1 | 1,843.125 | 1,843.125 | -1 | |
Factor $x^2+4x+4-81x^4$ into two quadratic polynomials with integer coefficients. Submit your answer in the form $(ax^2+bx+c)(dx^2+ex+f)$, with $a<d$. | (-9x^2+x+2)(9x^2+x+2) | 0.4375 | 7,642.3125 | 6,935.571429 | 8,192 | |
Elon Musk's Starlink project belongs to his company SpaceX. He plans to use tens of thousands of satellites to provide internet services to every corner of the Earth. A domestic company also plans to increase its investment in the development of space satellite networks to develop space internet. It is known that the r... | 50 | 0.25 | 7,337.125 | 5,436.25 | 7,970.75 | |
Find the largest positive integer solution of the equation $\left\lfloor\frac{N}{3}\right\rfloor=\left\lfloor\frac{N}{5}\right\rfloor+\left\lfloor\frac{N}{7}\right\rfloor-\left\lfloor\frac{N}{35}\right\rfloor$. | 65 | For $N$ to be a solution, it is necessary that $\frac{N-2}{3}+\frac{N-34}{35} \leq \frac{N}{5}+\frac{N}{7}$, which simplifies to $N \leq 86$. However, if $N \geq 70$, then $N \leq 59$, contradicting $N \geq 70$. It follows that $N$ must be at most 69. Checking for $N \leq 69$, we find that when $N=65$, the equation hol... | 0 | 8,192 | -1 | 8,192 |
Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 5y^2 = 6,$ as shown below. Find $r.$
[asy]
size(7cm);
draw(scale(sqrt(6), sqrt(6)/sqrt(5))* unitcircle);
draw((0,-1.5)--(0,1.7),EndArrow);
draw((-3,0)--(3,0),EndArrow);
draw(Circle( (sqrt(0.96),0), sqrt(0.96) ));... | \frac{2\sqrt6}{5} | 0 | 5,495.5 | -1 | 5,495.5 | |
For transportation between points located hundreds of kilometers apart on the Earth's surface, people of the future will likely dig straight tunnels through which capsules will travel frictionlessly under the influence of Earth's gravity. Let points \( A, B, \) and \( C \) lie on the same meridian, with the surface dis... | 42 | 0 | 8,176.1875 | -1 | 8,176.1875 | |
Find all solutions to $aabb=n^4-6n^3$ , where $a$ and $b$ are non-zero digits, and $n$ is an integer. ( $a$ and $b$ are not necessarily distinct.) | 6655 | 0.5625 | 7,500.8125 | 7,001.888889 | 8,142.285714 | |
Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.
a) How much time will Petya need to defi... | \frac{29}{120} | 0 | 6,214.8125 | -1 | 6,214.8125 | |
The coefficient of $x^2$ in the expansion of $(x-1) - (x-1)^2 + (x-1)^3 - (x-1)^4 + (x-1)^5$ is ____. | -20 | 0.5625 | 7,008.6875 | 6,088.333333 | 8,192 | |
From a large bottle containing 1 liter of alcohol, 1/3 liter of alcohol is poured out, an equal amount of water is added and mixed thoroughly. Then, 1/3 liter of the mixture is poured out, an equal amount of water is added and mixed thoroughly again. Finally, 1/3 liter of the mixture is poured out once more, and an equ... | 8/27 | 0.875 | 3,297.75 | 2,598.571429 | 8,192 | |
For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$? | 16 | 1. **Understanding the repeating base-$k$ representation**: Given that $0.\overline{23}_k = 0.232323..._k$, we interpret this as the infinite series:
\[
0.232323..._k = \frac{2}{k} + \frac{3}{k^2} + \frac{2}{k^3} + \frac{3}{k^4} + \cdots
\]
2. **Converting the series into a single fraction**: We can split the... | 0.875 | 3,516.125 | 2,848.142857 | 8,192 |
The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term? | $-\frac{2\sqrt{3}}{3}$ | 0 | 4,349 | -1 | 4,349 | |
In the quadrilateral \(ABCD\), the lengths of the sides \(BC\) and \(CD\) are 2 and 6, respectively. The points of intersection of the medians of triangles \(ABC\), \(BCD\), and \(ACD\) form an equilateral triangle. What is the maximum possible area of quadrilateral \(ABCD\)? If necessary, round the answer to the neare... | 29.32 | 0 | 8,192 | -1 | 8,192 | |
For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1) = 1$, $c(2n) = c(n)$, and $c(2n+1) = (-1)^n c(n)$. Find the value of \[ \sum_{n=1}^{2013} c(n) c(n+2). \] | -1 | Note that \begin{align*} c(2k+1)c(2k+3) &= (-1)^k c(k) (-1)^{k+1} c(k+1) \\ &= -c(k)c(k+1) \\ &= -c(2k)c(2k+2). \end{align*} It follows that $\sum_{n=2}^{2013} c(n)c(n+2) = \sum_{k=1}^{1006} (c(2k)c(2k+2)+c(2k+1)c(2k+3)) = 0$, and so the desired sum is $c(1)c(3) = -1$. | 0.0625 | 7,908.75 | 4,036 | 8,166.933333 |
The line $ax+2by=1$ intersects the circle $x^{2}+y^{2}=1$ at points $A$ and $B$ (where $a$ and $b$ are real numbers), and $\triangle AOB$ is a right-angled triangle ($O$ is the origin). The maximum distance between point $P(a,b)$ and point $Q(0,0)$ is ______. | \sqrt{2} | 0.1875 | 7,993.5625 | 7,133.666667 | 8,192 | |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/29 of the original integer. | 725 | 0.8125 | 6,039.25 | 5,542.461538 | 8,192 | |
The probability that Kim has a math test today is $\frac{4}{7}$. What is the probability that Kim does not have a math test today? Express your answer as a common fraction. | \frac{3}{7} | 1 | 1,070.6875 | 1,070.6875 | -1 | |
Given a function $f(x) = \cos x \sin \left( x + \frac{\pi}{3} \right) - \sqrt{3} \cos^2 x + \frac{\sqrt{3}}{4}$, where $x \in \mathbb{R}$,
(1) Find the smallest positive period of $f(x)$ and the interval where $f(x)$ is monotonically decreasing;
(2) Find the maximum and minimum values of $f(x)$ in the closed interval $... | -\frac{1}{2} | 0.75 | 7,489.75 | 7,348.583333 | 7,913.25 | |
What is the smallest number with three different prime factors, none of which can be less than 10? | 2431 | 0.75 | 4,354.4375 | 3,075.25 | 8,192 | |
Given the function $f(x)=x^{2-m}$ defined on the interval $[-3-m,m^{2}-m]$, which is an odd function, find $f(m)=$____. | -1 | 0.75 | 5,112.8125 | 4,168.333333 | 7,946.25 | |
Form a three-digit number using the digits 0, 1, 2, 3. Repeating digits is not allowed.
① How many three-digit numbers can be formed?
② If the three-digit numbers from ① are sorted in ascending order, what position does 230 occupy?
③ If repeating digits is allowed, how many of the formed three-digit numbers are d... | 16 | 0.25 | 7,689.625 | 6,193.75 | 8,188.25 | |
For how many integer values of $m$ ,
(i) $1\le m \le 5000$ (ii) $[\sqrt{m}] =[\sqrt{m+125}]$ Note: $[x]$ is the greatest integer function | 72 | 0.375 | 7,416.0625 | 6,283.833333 | 8,095.4 | |
Given plane vectors $\vec{a}, \vec{b}, \vec{c}$ that satisfy the following conditions: $|\vec{a}| = |\vec{b}| \neq 0$, $\vec{a} \perp \vec{b}$, $|\vec{c}| = 2 \sqrt{2}$, and $|\vec{c} - \vec{a}| = 1$, determine the maximum possible value of $|\vec{a} + \vec{b} - \vec{c}|$. | 3\sqrt{2} | 0.0625 | 8,177.375 | 7,958 | 8,192 | |
Simplify: $-{-\left[-|-1|^2\right]^3}^4$. | -1 | 0.375 | 5,931.1875 | 4,893.5 | 6,553.8 | |
Given $\triangle ABC$ with $AC=1$, $\angle ABC= \frac{2\pi}{3}$, $\angle BAC=x$, let $f(x)= \overrightarrow{AB} \cdot \overrightarrow{BC}$.
$(1)$ Find the analytical expression of $f(x)$ and indicate its domain;
$(2)$ Let $g(x)=6mf(x)+1$ $(m < 0)$, if the range of $g(x)$ is $\left[- \frac{3}{2},1\right)$, find the va... | - \frac{5}{2} | 0.1875 | 8,067.6875 | 8,042.333333 | 8,073.538462 | |
Given the function $f(x)=2x^{3}-ax^{2}+1$ $(a\in\mathbb{R})$ has exactly one zero in the interval $(0,+\infty)$, find the sum of the maximum and minimum values of $f(x)$ on the interval $[-1,1]$. | -3 | 0.5625 | 6,439.125 | 5,075.777778 | 8,192 |
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