Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
28,100 | Xiao Zhang and Xiao Zhao can only take on the first two roles, while the other three can take on any of the four roles, calculate the total number of different selection schemes. | 48 | 3.90625 |
28,101 | There are 10 cards, labeled from 1 to 10. Three cards denoted by $ a,\ b,\ c\ (a > b > c)$ are drawn from the cards at the same time.
Find the probability such that $ \int_0^a (x^2 \minus{} 2bx \plus{} 3c)\ dx \equal{} 0$ . | 1/30 | 8.59375 |
28,102 | In △ABC, the sides opposite to angles A, B, C are a, b, c, respectively. Given cosB= $\frac {2}{5}$, and sinAcosB - (2c - cosA)•sinB = 0.
(1) Find the value of b;
(2) Find the maximum value of the perimeter of △ABC. | \frac { \sqrt {30}}{6} + \frac {1}{2} | 0 |
28,103 | What is the smallest positive value of $x$ such that $x + 3956$ results in a palindrome? | 48 | 0.78125 |
28,104 | Mindy is attempting to solve the quadratic equation by completing the square: $$100x^2+80x-144 = 0.$$ She rewrites the given quadratic equation in the form $$(dx + e)^2 = f,$$ where \(d\), \(e\), and \(f\) are integers and \(d > 0\). What are the values of \(d + e + f\)? | 174 | 17.96875 |
28,105 | On a grid, there are three locations $A, B, C$, with each small square having a side length of 100 meters. If roads are to be built along the grid lines to connect the three locations, what is the minimum total length of the roads in meters? | 1000 | 3.125 |
28,106 | Given that children enter at a discounted rate, half that of an adult ticket, and the total cost for $6$ adult tickets and $5$ child tickets amounts to $32.50$, calculate the total cost for $10$ adult tickets and $8$ child tickets. | 53.50 | 0 |
28,107 | In the equation $\frac{1}{(\;\;\;)} + \frac{4}{(\;\;\;)} + \frac{9}{(\;\;\;\;)} = 1$, fill in the three brackets in the denominators with a positive integer, respectively, such that the equation holds true. The minimum value of the sum of these three positive integers is $\_\_\_\_\_\_$. | 36 | 36.71875 |
28,108 | The sequence $\{a_n\}$ satisfies $a_{n+1} + (-1)^n a_n = 2n - 1$. Find the sum of the first 60 terms of $\{a_n\}$. | 1830 | 25 |
28,109 | In a plane Cartesian coordinate system, a point whose x and y coordinates are both integers is called a "lattice point." How many lattice points are there inside and on the boundaries of the triangle formed by the line $7x + 11y = 77$ and the coordinate axes? | 49 | 8.59375 |
28,110 | In the Cartesian coordinate system, A and B are points moving on the x-axis and y-axis, respectively. If the circle C with AB as its diameter is tangent to the line $3x+y-4=0$, then the minimum area of circle C is \_\_\_\_\_\_. | \frac {2}{5}\pi | 0 |
28,111 | Find the smallest positive integer $n$ for which $315^2-n^2$ evenly divides $315^3-n^3$ .
*Proposed by Kyle Lee* | 90 | 50 |
28,112 | In the polar coordinate system, the curve $\rho=4\sin \left( \theta- \frac{\pi}{3} \right)$ is symmetric about what axis? | \frac{5\pi}{6} | 0 |
28,113 | If
\[1 \cdot 1992 + 2 \cdot 1991 + 3 \cdot 1990 + \dots + 1991 \cdot 2 + 1992 \cdot 1 = 1992 \cdot 996 \cdot y,\]
compute the integer \(y\). | 664 | 3.125 |
28,114 | Let $k$ be a real number such that the product of real roots of the equation $$ X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0 $$ is $-2013$ . Find the sum of the squares of these real roots. | 4027 | 4.6875 |
28,115 | Given functions \(f(x)\) and \(g(x)\) are defined on \(\mathbb{R}\) with \(g(x)\neq 0\), and \(f(x)g'(x) > f'(x)g(x)\). It is also given that \(f(x)=a^{x}\cdot g(x)\) where \(a > 0, a\neq 1\), and \(\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{5}{2}\). Considering the finite sequence \(\left\{\frac{f(n)}{g(n)}\right\}_{n=1,2,\ldots,10}\), find the probability that the sum of the first \(k\) terms is greater than \(\frac{15}{16}\) for any positive integer \(k\) with \(1 \leq k \leq 10\). | \frac{3}{5} | 31.25 |
28,116 | Given that $k$ is a composite number, and $1 < k < 100$, when the sum of the digits of $k$ is a prime number, this prime number is called the "derived prime" of $k$.
(1) If the "derived prime" of $k$ is 2, then $k=$
(2) Let set $A=\{P(k) | P(k)$ is the "derived prime" of $k\}$, $B=\{k | P(k)$ is the "derived prime" of $k\}$, then the number of elements in the set $A \cup B$ is | 30 | 6.25 |
28,117 | The following diagram shows equilateral triangle $\vartriangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\vartriangle ABC$ a distance $\frac18 AB$ along a side of $\vartriangle ABC$ in the directions from $A$ to $B$ , from $B$ to $C$ , and from $C$ to $A$ . The shaded region inside all four of the triangles has area $300$ . Find the area of $\vartriangle ABC$ .
 | 768 | 4.6875 |
28,118 | How many three-eighths are there in $8\frac{5}{3} - 3$? | 17\frac{7}{9} | 14.0625 |
28,119 | Given vectors $\overrightarrow{a}=(2\cos\omega x,-2)$ and $\overrightarrow{b}=(\sqrt{3}\sin\omega x+\cos\omega x,1)$, where $\omega\ \ \gt 0$, and the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}+1$. The distance between two adjacent symmetric centers of the graph of $f(x)$ is $\frac{\pi}{2}$.
$(1)$ Find $\omega$;
$(2)$ Given $a$, $b$, $c$ are the opposite sides of the three internal angles $A$, $B$, $C$ of scalene triangle $\triangle ABC$, and $f(A)=f(B)=\sqrt{3}$, $a=\sqrt{2}$, find the area of $\triangle ABC$. | \frac{3-\sqrt{3}}{4} | 20.3125 |
28,120 | The product of the digits of 4321 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 18? | 24 | 3.90625 |
28,121 | Given a set $A_n = \{1, 2, 3, \ldots, n\}$, define a mapping $f: A_n \rightarrow A_n$ that satisfies the following conditions:
① For any $i, j \in A_n$ with $i \neq j$, $f(i) \neq f(j)$;
② For any $x \in A_n$, if the equation $x + f(x) = 7$ has $K$ pairs of solutions, then the mapping $f: A_n \rightarrow A_n$ is said to contain $K$ pairs of "good numbers." Determine the number of such mappings for $f: A_6 \rightarrow A_6$ that contain 3 pairs of good numbers. | 40 | 57.03125 |
28,122 | Given the ten digits 0, 1, 2, 3, …, 9 and the imaginary unit i, determine the total number of distinct imaginary numbers that can be formed. | 90 | 2.34375 |
28,123 | What is the smallest positive integer with exactly 12 positive integer divisors? | 60 | 88.28125 |
28,124 | A certain product in a shopping mall sells an average of 30 items per day, with a profit of 50 yuan per item. In order to reduce inventory quickly, the mall decides to take appropriate price reduction measures. After investigation, it was found that for each item, for every 1 yuan reduction in price, the mall can sell an additional 2 items per day. Let $x$ represent the price reduction per item. Based on this rule, please answer:<br/>$(1)$ The daily sales volume of the mall increases by ______ items, and the profit per item is ______ yuan (expressed in algebraic expressions containing $x$);<br/>$(2)$ Under the same conditions as above, in normal sales situations, how much should each item be reduced in price so that the mall's daily profit reaches 2100 yuan? | 20 | 82.03125 |
28,125 | Let $ a,b,c,d>0$ for which the following conditions:: $a)$ $(a-c)(b-d)=-4$ $b)$ $\frac{a+c}{2}\geq\frac{a^{2}+b^{2}+c^{2}+d^{2}}{a+b+c+d}$ Find the minimum of expression $a+c$ | 4\sqrt{2} | 0.78125 |
28,126 | There are 2 dimes of Chinese currency, how many ways can they be exchanged into coins (1 cent, 2 cents, and 5 cents)? | 28 | 4.6875 |
28,127 | Solve the equation: $2\left(x-1\right)^{2}=x-1$. | \frac{3}{2} | 7.03125 |
28,128 | What is the area of the portion of the circle defined by the equation $x^2 + 6x + y^2 = 50$ that lies below the $x$-axis and to the left of the line $y = x - 3$? | \frac{59\pi}{4} | 4.6875 |
28,129 | Let \( a_1, a_2, \ldots, a_{2020} \) be the roots of the polynomial
\[ x^{2020} + x^{2019} + \cdots + x^2 + x - 2022 = 0. \]
Compute
\[ \sum_{n = 1}^{2020} \frac{1}{1 - a_n}. \] | 2041210 | 2.34375 |
28,130 | Amelia and Blaine are playing a modified game where they toss their respective coins. Amelia's coin lands on heads with a probability of $\frac{3}{7}$, and Blaine's lands on heads with a probability of $\frac{1}{3}$. They begin their game only after observing at least one head in a simultaneous toss of both coins. Once the game starts, they toss coins alternately with Amelia starting first, and the player to first toss heads twice consecutively wins the game. What is the probability that Amelia wins the game?
A) $\frac{1}{2}$
B) $\frac{9}{49}$
C) $\frac{2401}{6875}$
D) $\frac{21609}{64328}$ | \frac{21609}{64328} | 10.9375 |
28,131 | Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. | 10 | 0 |
28,132 | In $\triangle ABC$, $AC=5 \sqrt {2}$, $\cos C= \frac {3}{5}$, $B= \frac {\pi}{4}$.
(1) Find the length of $AB$;
(2) Find the area of $\triangle ABC$, denoted as $S_{\triangle ABC}$. | 28 | 75.78125 |
28,133 | How many whole numbers between 1 and 2000 do not contain the digits 1 or 2? | 511 | 3.90625 |
28,134 | Given the function $f(x)=ax^{3}+2bx^{2}+3cx+4d$, where $a,b,c,d$ are real numbers, $a < 0$, and $c > 0$, is an odd function, and when $x\in[0,1]$, the range of $f(x)$ is $[0,1]$. Find the maximum value of $c$. | \frac{\sqrt{3}}{2} | 0 |
28,135 | In a large bag of decorative ribbons, $\frac{1}{4}$ are yellow, $\frac{1}{3}$ are purple, $\frac{1}{8}$ are orange, and the remaining 45 ribbons are silver. How many of the ribbons are orange? | 19 | 21.09375 |
28,136 | The water tank in the diagram below is in the shape of an inverted right circular cone. The radius of its base is 8 feet, and its height is 64 feet. The water in the tank is $40\%$ of the tank's capacity. The height of the water in the tank can be written in the form $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers, and $b$ is not divisible by a perfect cube greater than 1. What is $a+b$? | 66 | 2.34375 |
28,137 | Ten circles of diameter 1 are arranged in the first quadrant of a coordinate plane. Five circles are in the base row with centers at $(0.5, 0.5)$, $(1.5, 0.5)$, $(2.5, 0.5)$, $(3.5, 0.5)$, $(4.5, 0.5)$, and the remaining five directly above the first row with centers at $(0.5, 1.5)$, $(1.5, 1.5)$, $(2.5, 1.5)$, $(3.5, 1.5)$, $(4.5, 1.5)$. Let region $\mathcal{S}$ be the union of these ten circular regions. Line $m,$ with slope $-2$, divides $\mathcal{S}$ into two regions of equal area. Line $m$'s equation can be expressed in the form $px=qy+r$, where $p, q,$ and $r$ are positive integers whose greatest common divisor is 1. Find $p^2+q^2+r^2$. | 30 | 0.78125 |
28,138 | Given the sequence $\{a\_n\}$, where $a\_1=1$, $a\_2=3$, and $2nS\_n=(n+1)S\_{n+1}+(n-1)S\_{n-1}$ ($n\geqslant 2,n\in N$), find $S\_{30}$. | \frac{34}{5} | 0 |
28,139 | Given an odd function defined on $\mathbb{R}$, when $x > 0$, $f(x)=x^{2}+2x-1$.
(1) Find the value of $f(-3)$;
(2) Find the analytic expression of the function $f(x)$. | -14 | 36.71875 |
28,140 | Given $|m|=3$, $|n|=2$, and $m<n$, find the value of $m^2+mn+n^2$. | 19 | 23.4375 |
28,141 | In $\triangle ABC$, the sides opposite to angles A, B, and C are $a$, $b$, and $c$, respectively. Given the equation $$2b\cos A - \sqrt{3}c\cos A = \sqrt{3}a\cos C$$.
(1) Find the value of angle A;
(2) If $\angle B = \frac{\pi}{6}$, and the median $AM = \sqrt{7}$ on side $BC$, find the area of $\triangle ABC$. | \sqrt{3} | 42.1875 |
28,142 | In the diagram below, the circle with center $A$ is congruent to and tangent to the circle with center $B$ . A third circle is tangent to the circle with center $A$ at point $C$ and passes through point $B$ . Points $C$ , $A$ , and $B$ are collinear. The line segment $\overline{CDEFG}$ intersects the circles at the indicated points. Suppose that $DE = 6$ and $FG = 9$ . Find $AG$ .
[asy]
unitsize(5);
pair A = (-9 sqrt(3), 0);
pair B = (9 sqrt(3), 0);
pair C = (-18 sqrt(3), 0);
pair D = (-4 sqrt(3) / 3, 10 sqrt(6) / 3);
pair E = (2 sqrt(3), 4 sqrt(6));
pair F = (7 sqrt(3), 5 sqrt(6));
pair G = (12 sqrt(3), 6 sqrt(6));
real r = 9sqrt(3);
draw(circle(A, r));
draw(circle(B, r));
draw(circle((B + C) / 2, 3r / 2));
draw(C -- D);
draw(" $6$ ", E -- D);
draw(E -- F);
draw(" $9$ ", F -- G);
dot(A);
dot(B);
label(" $A$ ", A, plain.E);
label(" $B$ ", B, plain.E);
label(" $C$ ", C, W);
label(" $D$ ", D, dir(160));
label(" $E$ ", E, S);
label(" $F$ ", F, SSW);
label(" $G$ ", G, N);
[/asy] | 9\sqrt{19} | 0 |
28,143 | The maximum value of the function $f(x) = \frac{\frac{1}{6} \cdot (-1)^{1+ C_{2x}^{x}} \cdot A_{x+2}^{5}}{1+ C_{3}^{2} + C_{4}^{2} + \ldots + C_{x-1}^{2}}$ ($x \in \mathbb{N}$) is ______. | -20 | 0.78125 |
28,144 | Calculate the product of $1101_2 \cdot 111_2$. Express your answer in base 2. | 1100111_2 | 0 |
28,145 | Given an obtuse triangle \(ABC\) with obtuse angle \(C\). Points \(P\) and \(Q\) are marked on its sides \(AB\) and \(BC\) respectively, such that \(\angle ACP = CPQ = 90^\circ\). Find the length of segment \(PQ\) if it is known that \(AC = 25\), \(CP = 20\), and \(\angle APC = \angle A + \angle B\). | 16 | 9.375 |
28,146 | Given several numbers, one of them, $a$ , is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$ . This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called *good* if there are $m$ or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number. | 667 | 4.6875 |
28,147 | Evaluate: $6 - 5\left[7 - (\sqrt{16} + 2)^2\right] \cdot 3.$ | -429 | 33.59375 |
28,148 | Solve the equations:<br/>$(1)x^{2}-4x-1=0$;<br/>$(2)\left(x+3\right)^{2}=x+3$. | -2 | 1.5625 |
28,149 | Let $ABCD$ be a rectangle with side lengths $AB = CD = 5$ and $BC = AD = 10$ . $W, X, Y, Z$ are points on $AB, BC, CD$ and $DA$ respectively chosen in such a way that $WXYZ$ is a kite, where $\angle ZWX$ is a right angle. Given that $WX = WZ = \sqrt{13}$ and $XY = ZY$ , determine the length of $XY$ . | \sqrt{65} | 2.34375 |
28,150 | There are real numbers $a, b, c, d$ such that for all $(x, y)$ satisfying $6y^2 = 2x^3 + 3x^2 + x$ , if $x_1 = ax + b$ and $y_1 = cy + d$ , then $y_1^2 = x_1^3 - 36x_1$ . What is $a + b + c + d$ ? | 90 | 39.84375 |
28,151 | Given the function $f(x) = \frac{x}{\ln x}$, and $g(x) = f(x) - mx (m \in \mathbb{R})$,
(I) Find the interval of monotonic decrease for function $f(x)$.
(II) If function $g(x)$ is monotonically decreasing on the interval $(1, +\infty)$, find the range of the real number $m$.
(III) If there exist $x_1, x_2 \in [e, e^2]$ such that $m \geq g(x_1) - g'(x_2)$ holds true, find the minimum value of the real number $m$. | \frac{1}{2} - \frac{1}{4e^2} | 0 |
28,152 | Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 500 | 8.59375 |
28,153 | A shepherd uses 15 segments of fencing, each 2 meters long, to form a square or rectangular sheep pen with one side against a wall. What is the maximum area of the sheep pen in square meters? | 112 | 23.4375 |
28,154 |
It is planned to establish an additional channel for exchanging stereo audio signals (messages) for daily reporting communication sessions between two working sites of a deposit. Determine the required bandwidth of this channel in kilobits, considering that the sessions will be conducted for no more than 51 minutes. The requirements for a mono signal per second are given below:
- Sampling rate: 63 Hz
- Sampling depth: 17 bits
- Metadata volume: 47 bytes for every 5 kilobits of audio | 2.25 | 0 |
28,155 | Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. Find the ratio of the area of quadrilateral $A_2A_5B_2B_5$ (shaded in the picture to the right) to the area of the entire 10-sided polygon.
[asy]
size(8cm);
defaultpen(fontsize(10pt));
pair A_2=(-0.4382971011,5.15554989475), B_4=(-2.1182971011,-0.0149584477027), B_5=(-4.8365942022,8.3510997895), A_3=(0.6,8.3510997895), B_1=(2.28,13.521608132), A_4=(3.96,8.3510997895), B_2=(9.3965942022,8.3510997895), A_5=(4.9982971011,5.15554989475), B_3=(6.6782971011,-0.0149584477027), A_1=(2.28,3.18059144705);
filldraw(A_2--A_5--B_2--B_5--cycle,rgb(.8,.8,.8));
draw(B_1--A_4^^A_4--B_2^^B_2--A_5^^A_5--B_3^^B_3--A_1^^A_1--B_4^^B_4--A_2^^A_2--B_5^^B_5--A_3^^A_3--B_1,linewidth(1.2)); draw(A_1--A_2--A_3--A_4--A_5--cycle);
pair O = (A_1+A_2+A_3+A_4+A_5)/5;
label(" $A_1$ ",A_1, 2dir(A_1-O));
label(" $A_2$ ",A_2, 2dir(A_2-O));
label(" $A_3$ ",A_3, 2dir(A_3-O));
label(" $A_4$ ",A_4, 2dir(A_4-O));
label(" $A_5$ ",A_5, 2dir(A_5-O));
label(" $B_1$ ",B_1, 2dir(B_1-O));
label(" $B_2$ ",B_2, 2dir(B_2-O));
label(" $B_3$ ",B_3, 2dir(B_3-O));
label(" $B_4$ ",B_4, 2dir(B_4-O));
label(" $B_5$ ",B_5, 2dir(B_5-O));
[/asy] | \frac{1}{2} | 6.25 |
28,156 | Given the polynomial $$Q(x) = \left(1 + x + x^2 + \ldots + x^{20}\right)^2 - x^{20},$$ find the sum $$\beta_1 + \beta_2 + \beta_6$$ where the complex zeros of $Q(x)$ are written in the form, $\beta_k=r_k[\cos(2\pi\beta_k)+i\sin(2\pi\beta_k)]$, with $0<\beta_1\le\beta_2\le\ldots\le\beta_{41}<1$ and $r_k>0$. | \frac{3}{7} | 11.71875 |
28,157 | Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and passing through the point $P(0, \sqrt{5})$, with an eccentricity of $\frac{2}{3}$, and $A$ being a moving point on the line $x=4$.
- (I) Find the equation of the ellipse $C$;
- (II) Point $B$ is on the ellipse $C$, satisfying $OA \perpendicular OB$, find the minimum length of segment $AB$. | \sqrt{21} | 2.34375 |
28,158 | A mathematical demonstration showed that there were distinct positive integers such that $97^4 + 84^4 + 27^4 + 3^4 = m^4$. Calculate the value of $m$. | 108 | 1.5625 |
28,159 | We denote by gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd $(4, 6, 8)=2$ and gcd $(12, 15)=3$ .) Suppose that positive integers $a, b, c$ satisfy the following four conditions: $\bullet$ gcd $(a, b, c)=1$ , $\bullet$ gcd $(a, b + c)>1$ , $\bullet$ gcd $(b, c + a)>1$ , $\bullet$ gcd $(c, a + b)>1$ .
a) Is it possible that $a + b + c = 2015$ ?
b) Determine the minimum possible value that the sum $a+ b+ c$ can take. | 30 | 51.5625 |
28,160 | A rectangular piece of paper $A B C D$ is folded and flattened such that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ landing on side $A B$. Given that $\angle 1 = 22^{\circ}$, find $\angle 2$. | 44 | 0 |
28,161 | Given a triangle $\triangle ABC$ with its three interior angles $A$, $B$, and $C$ satisfying: $$A+C=2B, \frac {1}{\cos A}+ \frac {1}{\cos C}=- \frac { \sqrt {2}}{\cos B}$$, find the value of $$\cos \frac {A-C}{2}$$. | \frac { \sqrt {2}}{2} | 0 |
28,162 | What is the smallest positive integer with exactly 16 positive divisors? | 384 | 0 |
28,163 | A rectangular prism with dimensions 1 cm by 1 cm by 2 cm has a dot marked in the center of the top face (1 cm by 2 cm face). It is sitting on a table, which is 1 cm by 2 cm face. The prism is rolled over its shorter edge (1 cm edge) on the table, without slipping, and stops once the dot returns to the top. Find the length of the path followed by the dot in terms of $\pi$. | 2\pi | 50.78125 |
28,164 | The height $BD$ of the acute-angled triangle $ABC$ intersects with its other heights at point $H$. Point $K$ lies on segment $AC$ such that the angle $BKH$ is maximized. Find $DK$ if $AD = 2$ and $DC = 3$. | \sqrt{6} | 0.78125 |
28,165 | On a luxurious ocean liner, 3000 adults consisting of men and women embark on a voyage. If 55% of the adults are men and 12% of the women as well as 15% of the men are wearing sunglasses, determine the total number of adults wearing sunglasses. | 409 | 0 |
28,166 | $2.46\times 8.163\times (5.17+4.829)$ is approximately equal to what value? | 200 | 2.34375 |
28,167 | How many distinct four-digit positive integers have a digit product equal to 18? | 48 | 0 |
28,168 | In the xy-plane, consider a right triangle $ABC$ with the right angle at $C$. The hypotenuse $AB$ is of length $50$. The medians through vertices $A$ and $B$ are described by the lines $y = x + 5$ and $y = 2x + 2$, respectively. Determine the area of triangle $ABC$. | 500 | 2.34375 |
28,169 | Initially, the numbers 1 and 2 are written at opposite positions on a circle. Each operation consists of writing the sum of two adjacent numbers between them. For example, the first operation writes two 3's, and the second operation writes two 4's and two 5's. After each operation, the sum of all the numbers becomes three times the previous total. After sufficient operations, find the sum of the counts of the numbers 2015 and 2016 that are written. | 2016 | 0 |
28,170 | The Wolf and the three little pigs wrote a detective story "The Three Little Pigs-2", and then, together with Little Red Riding Hood and her grandmother, a cookbook "Little Red Riding Hood-2". The publisher gave the fee for both books to the pig Naf-Naf. He took his share and handed the remaining 2100 gold coins to the Wolf. The fee for each book is divided equally among its authors. How much money should the Wolf take for himself? | 700 | 10.15625 |
28,171 | Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} | 98.4375 |
28,172 | The houses on the south side of Crazy Street are numbered in increasing order starting at 1 and using consecutive odd numbers, except that odd numbers that contain the digit 3 are missed out. What is the number of the 20th house on the south side of Crazy Street? | 59 | 53.90625 |
28,173 | How many two-digit numbers have digits whose sum is a prime number? | 31 | 0 |
28,174 | Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. | 26 | 0.78125 |
28,175 | Given that $(2x)_((-1)^{5}=a_0+a_1x+a_2x^2+...+a_5x^5$, find:
(1) $a_0+a_1+...+a_5$;
(2) $|a_0|+|a_1|+...+|a_5|$;
(3) $a_1+a_3+a_5$;
(4) $(a_0+a_2+a_4)^2-(a_1+a_3+a_5)^2$. | -243 | 12.5 |
28,176 | Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of
\[8a + 3b + 5c.\] | \frac{\sqrt{373}}{6} | 77.34375 |
28,177 | Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | 27 | 8.59375 |
28,178 | Given the sequence $\{a\_n\}(n=1,2,3,...,2016)$, circle $C\_1$: $x^{2}+y^{2}-4x-4y=0$, circle $C\_2$: $x^{2}+y^{2}-2a_{n}x-2a_{2017-n}y=0$. If circle $C\_2$ bisects the circumference of circle $C\_1$, then the sum of all terms in the sequence $\{a\_n\}$ is $\_\_\_\_\_\_$. | 4032 | 71.09375 |
28,179 | Let $\triangle ABC$ be a right triangle with $\angle ABC = 90^\circ$, and let $AB = 10\sqrt{21}$ be the hypotenuse. Point $E$ lies on $AB$ such that $AE = 10\sqrt{7}$ and $EB = 20\sqrt{7}$. Let $F$ be the foot of the altitude from $C$ to $AB$. Find the distance $EF$. Express $EF$ in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$. | 31 | 5.46875 |
28,180 | How many non-empty subsets $S$ of $\{1, 2, 3, \ldots, 10\}$ satisfy the following two conditions?
1. No two consecutive integers belong to $S$.
2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$. | 59 | 0 |
28,181 | Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing this edge. | \frac{9 \sqrt{3}}{4} | 26.5625 |
28,182 | For a nonnegative integer $n$, let $r_7(3n)$ represent the remainder when $3n$ is divided by $7$. Determine the $22^{\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\le 4~.$$ | 29 | 5.46875 |
28,183 | Find the greatest common divisor of $8!$ and $(6!)^2.$ | 5760 | 76.5625 |
28,184 | A district in a city is laid out in an $11 \times 11$ grid. Every day, a sprinkler truck departs from the bottom-left corner $A(0,0)$ and travels along the streets to reach the top-right corner $B(10,10)$. At each intersection, the driver randomly chooses a direction, as long as it does not deviate from the shortest path. One day, the street from $(9,9)$ to $(10,9)$ is blocked due to an accident, but the driver is not aware of this at the time of departure. What is the probability that the sprinkler truck can still reach $B$ without any issues? | 1 - \frac{\binom{18}{9}}{\binom{20}{10}} | 0 |
28,185 | Evaluate $|\omega^2 + 4\omega + 34|$ if $\omega = 5 + 3i$. | \sqrt{6664} | 0 |
28,186 | From the numbers 2, 3, 4, 5, 6, 7, 8, 9, two different numbers are selected to be the base and the exponent of a logarithm, respectively. How many different logarithmic values can be formed? | 52 | 8.59375 |
28,187 | Let $ABC$ be a triangle with side lengths $AB=6, AC=7,$ and $BC=8.$ Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC.$ $\tfrac{[ABH']}{[ACH']}$ can be expressed as $\frac{p}{q}$ . Find $p+q$ .
*2022 CCA Math Bonanza Individual Round #14* | 251 | 0 |
28,188 | What is the smallest positive integer with exactly 12 positive integer divisors? | 72 | 0 |
28,189 | Let \( g_{1}(x) = \sqrt{4 - x} \), and for integers \( n \geq 2 \), define \[ g_{n}(x) = g_{n-1}\left(\sqrt{(n+1)^2 - x}\right). \] Find the largest \( n \) (denote this as \( M \)) for which the domain of \( g_n \) is nonempty. For this value of \( M \), if the domain of \( g_M \) consists of a single point \( \{d\} \), compute \( d \). | -589 | 0 |
28,190 | A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$ , $ 2$ , ..., $ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$ . What is maximum value of minimum sum in column (with minimal sum)? | 24 | 1.5625 |
28,191 | Given that $15^{-1} \equiv 31 \pmod{53}$, find $38^{-1} \pmod{53}$, as a residue modulo 53. | 22 | 67.96875 |
28,192 | Given that there are 20 cards numbered from 1 to 20 on a table, and Xiao Ming picks out 2 cards such that the number on one card is 2 more than twice the number on the other card, find the maximum number of cards Xiao Ming can pick. | 12 | 14.0625 |
28,193 | Given the piecewise function $f(x)= \begin{cases} x+2 & (x\leq-1) \\ x^{2} & (-1<x<2) \\ 2x & (x\geq2)\end{cases}$, if $f(x)=3$, determine the value of $x$. | \sqrt{3} | 54.6875 |
28,194 | The graph of the equation $10x + 270y = 2700$ is drawn on graph paper where each square represents one unit in each direction. A second line defined by $x + y = 10$ also passes through the graph. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below both graphs and entirely in the first quadrant? | 50 | 3.125 |
28,195 | A rectangular pasture is to be fenced off on three sides using part of a 100 meter rock wall as the fourth side. Fence posts are to be placed every 15 meters along the fence including at the points where the fence meets the rock wall. Given the dimensions of the pasture are 36 m by 75 m, find the minimum number of posts required. | 14 | 11.71875 |
28,196 | In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$ - and $y$ - axes, and call a set of points $nice$ if no two of them have the same $x$ - or $y$ - coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert, Ernie then chooses a set $E$ of $n$ points in the coordinate plane such that $B\cup E$ is a nice set with $2016+n$ points. Bert returns and then miraculously notices that there does not exist a standard rectangle that contains at least two points in $B$ and no points in $E$ in its interior. For a given nice set $B$ that Bert chooses, define $f(B)$ as the smallest positive integer $n$ such that Ernie can find a nice set $E$ of size $n$ with the aforementioned properties. Help Bert determine the minimum and maximum possible values of $f(B)$ .
*Yannick Yao* | 2015 | 39.0625 |
28,197 | Two lines with slopes $\dfrac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$. | 8.625 | 0 |
28,198 | A function is given by
$$
f(x)=\ln (a x+b)+x^{2} \quad (a \neq 0).
$$
(1) If the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$ is $y=x$, find the values of $a$ and $b$.
(2) If $f(x) \leqslant x^{2}+x$ always holds, find the maximum value of $ab$. | \frac{e}{2} | 20.3125 |
28,199 | The inclination angle $\alpha$ of the line $l: \sqrt{3}x+3y+1=0$ is $\tan^{-1}\left( -\frac{\sqrt{3}}{3} \right)$. Calculate the value of the angle $\alpha$. | \frac{5\pi}{6} | 71.09375 |
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