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Máté is always in a hurry. He observed that it takes 1.5 minutes to get to the subway when he stands on the moving escalator, while it takes 1 minute to run down the stationary stairs. How long does it take Máté to get down if he can run down the moving escalator?
|
36
|
1/8
|
Given the system of equations for the positive numbers \(x, y, z\):
$$
\left\{\begin{array}{l}
x^{2}+xy+y^{2}=108 \\
y^{2}+yz+z^{2}=16 \\
z^{2}+xz+x^{2}=124
\end{array}\right.
$$
Find the value of the expression \(xy + yz + xz\).
|
48
|
6/8
|
An eight-digit natural number $A$, written in the decimal numeral system, is obtained from a number $B$ by moving the last digit to the first position. It is known that $B$ is coprime with 12 and $B > 44444444$. Find the largest and smallest values among the numbers $A$ that satisfy these conditions. (Two natural numbers are called coprime if they have no common divisors other than one). (8 points)
|
14444446
|
1/8
|
Convert 89 into a binary number.
|
1011001_{(2)}
|
3/8
|
A batch of one hundred parts is subjected to a sampling inspection. The entire batch is considered unacceptable if at least one defective part is found among the four parts that are tested. What is the probability that this batch will not be accepted if it contains $3\%$ defective parts?
|
0.1164
|
3/8
|
Let $r$ be the radius of the circumscribed circle of an arbitrary isosceles triangle, and $\varrho$ be the radius of the inscribed circle. Prove that the distance between the centers of the two circles is
$$
d=\sqrt{r(r-2 \varrho)}
$$
|
\sqrt{r(r-2\varrho)}
|
1/8
|
During the 2013 National Day, a city organized a large-scale group calisthenics performance involving 2013 participants, all of whom were students from the third, fourth, and fifth grades. The students wore entirely red, white, or blue sports uniforms. It was known that the fourth grade had 600 students, the fifth grade had 800 students, and there were a total of 800 students wearing white sports uniforms across all three grades. There were 200 students each wearing red or blue sports uniforms in the third grade, red sports uniforms in the fourth grade, and white sports uniforms in the fifth grade. How many students in the fourth grade wore blue sports uniforms?
|
213
|
7/8
|
Let \( M \) be the point of intersection of the diagonals of a convex quadrilateral \( A B C D \), in which the sides \( A B \), \( A D \), and \( B C \) are equal to each other.
Find the angle \( \angle C M D \), given that \( D M = M C \) and \( \angle C A B \neq \angle D B A \).
|
120
|
1/8
|
Suppose \( a_{0}=a \), and \( a_{n+1}=2a_{n}-n^{2} \) for \( n=0,1,2,\cdots \). If all terms of the sequence \( \{a_{n}\} \) are positive integers, what is the minimum value of \( \{a_{n}\} \)?
|
3
|
6/8
|
Let $f(x)$ be a polynomial of degree 2006 with real coefficients, and let its roots be $r_1,$ $r_2,$ $\dots,$ $r_{2006}.$ There are exactly 1006 distinct values among
\[|r_1|, |r_2|, \dots, |r_{2006}|.\]What is the minimum number of real roots that $f(x)$ can have?
|
6
|
7/8
|
In triangle \( ABC \), side \( AC \) equals \( b \), side \( AB \) equals \( c \), and \( AD \) is the angle bisector such that \( DA = DB \). Find the length of side \( BC \).
|
\sqrt{b()}
|
6/8
|
Given \( a_{n} = \mathrm{C}_{200}^{n} \cdot (\sqrt[3]{6})^{200-n} \cdot \left( \frac{1}{\sqrt{2}} \right)^{n} \) for \( n = 1, 2, \ldots, 95 \), find the number of integer terms in the sequence \(\{a_{n}\}\).
|
15
|
1/8
|
The angles of a triangle are \(\alpha, \beta, \gamma\). What is the maximum possible value of
$$
\sin \alpha \sin \beta \cos \gamma + \sin^{2} \gamma?
$$
|
\frac{9}{8}
|
5/8
|
Five volunteers and two elderly people need to line up in a row, with the two elderly people next to each other but not at the ends. How many different ways can they arrange themselves?
|
960
|
7/8
|
A group of adventurers is showing their loot. It is known that exactly 5 adventurers have rubies; exactly 11 have emeralds; exactly 10 have sapphires; exactly 6 have diamonds. Additionally, it is known that:
- If an adventurer has diamonds, then they have either emeralds or sapphires (but not both simultaneously);
- If an adventurer has emeralds, then they have either rubies or diamonds (but not both simultaneously).
What is the minimum number of adventurers that can be in such a group?
|
16
|
1/8
|
In an isosceles right triangle, the radius of the inscribed circle is 2.
Find the distance from the vertex of the acute angle to the point where the inscribed circle touches the leg opposite to this angle.
|
2\sqrt{7+4\sqrt{2}}
|
1/8
|
Let $a$ and $b$ be real numbers such that $a^5b^8=12$ and $a^8b^{13}=18$ . Find $ab$ .
|
\frac{128}{3}
|
6/8
|
In the diagram, \( S \) lies on \( R T \), \( \angle Q T S = 40^{\circ} \), \( Q S = Q T \), and \( \triangle P R S \) is equilateral. Find the value of \( x \).
|
80
|
1/8
|
A rectangular grazing area 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 $12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $36$ m by $60$ m?
$\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16$
|
(B)12
|
1/8
|
If the scores for innovation capability, innovation value, and innovation impact are $8$ points, $9$ points, and $7$ points, respectively, and the total score is calculated based on the ratio of $5:3:2$ for the three scores, calculate the total score of the company.
|
8.1
|
4/8
|
We say that a function \( f: \mathbb{R}^k \rightarrow \mathbb{R} \) is a metapolynomial if, for some positive integers \( m \) and \( n \), it can be represented in the form
\[ f\left(x_1, \ldots, x_k\right) = \max_{i=1, \ldots, m} \min_{j=1, \ldots, n} P_{i, j}\left(x_1, \ldots, x_k\right) \]
where \( P_{i, j} \) are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.
|
2
|
1/8
|
Prove that for an acute-angled triangle
$$
\frac{1}{l_{a}}+\frac{1}{l_{b}}+\frac{1}{l_{c}} \leq \sqrt{2}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)
$$
|
\frac{1}{l_{}}+\frac{1}{l_{b}}+\frac{1}{l_{}}\le\sqrt{2}(\frac{1}{}+\frac{1}{b}+\frac{1}{})
|
1/8
|
In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced with any of the digits \(0, 2, 4, 7, 8, 9\) (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done?
|
1728
|
2/8
|
Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$ . Find $\angle BCA$ .
|
90
|
1/8
|
What is the distance between the two (non-intersecting) face diagonals on adjacent faces of a unit cube?
|
\frac{\sqrt{3}}{3}
|
7/8
|
Let
\[\mathbf{A} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}.\]Compute $\mathbf{A}^{95}.$
|
\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}
|
5/8
|
The average age of 8 people in a room is 35 years. A 22-year-old person leaves the room. Calculate the average age of the seven remaining people.
|
\frac{258}{7}
|
2/8
|
For positive integers $a>b>1$ , define
\[x_n = \frac {a^n-1}{b^n-1}\]
Find the least $d$ such that for any $a,b$ , the sequence $x_n$ does not contain $d$ consecutive prime numbers.
*V. Senderov*
|
3
|
1/8
|
The year 2013 has arrived, and Xiao Ming's older brother sighed and said, "This is the first year in my life that has no repeated digits." It is known that Xiao Ming's older brother was born in a year that is a multiple of 19. How old is the older brother in 2013?
|
18
|
7/8
|
The sum of the digits in the product of $\overline{A A A A A A A A A} \times \overline{B B B B B B B B B}$.
|
81
|
1/8
|
How many orderings \(\left(a_{1}, \ldots, a_{8}\right)\) of \((1, 2, \ldots, 8)\) exist such that \(a_{1} - a_{2} + a_{3} - a_{4} + a_{5} - a_{6} + a_{7} - a_{8} = 0\)?
|
4608
|
7/8
|
Let \( f(x) = x^3 - x^2 \). For a given value of \( c \), the graph of \( f(x) \), together with the graph of the line \( c + x \), split the plane up into regions. Suppose that \( c \) is such that exactly two of these regions have finite area. Find the value of \( c \) that minimizes the sum of the areas of these two regions.
|
-\frac{11}{27}
|
1/8
|
Let $\triangle ABC$ be an equilateral triangle with side length $55.$ Points $D,$ $E,$ and $F$ lie on $\overline{BC},$ $\overline{CA},$ and $\overline{AB},$ respectively, with $BD = 7,$ $CE=30,$ and $AF=40.$ Point $P$ inside $\triangle ABC$ has the property that \[\angle AEP = \angle BFP = \angle CDP.\] Find $\tan^2(\angle AEP).$
Diagram
[asy] /* Made by MRENTHUSIASM */ size(300); pair A, B, C, D, E, F, P; A = 55*sqrt(3)/3 * dir(90); B = 55*sqrt(3)/3 * dir(210); C = 55*sqrt(3)/3 * dir(330); D = B + 7*dir(0); E = A + 25*dir(C-A); F = A + 40*dir(B-A); P = intersectionpoints(Circle(D,54*sqrt(19)/19),Circle(F,5*sqrt(19)/19))[0]; draw(anglemark(A,E,P,20),red); draw(anglemark(B,F,P,20),red); draw(anglemark(C,D,P,20),red); add(pathticks(anglemark(A,E,P,20), n = 1, r = 0.2, s = 12, red)); add(pathticks(anglemark(B,F,P,20), n = 1, r = 0.2, s = 12, red)); add(pathticks(anglemark(C,D,P,20), n = 1, r = 0.2, s = 12, red)); draw(A--B--C--cycle^^P--E^^P--F^^P--D); dot("$A$",A,1.5*dir(A),linewidth(4)); dot("$B$",B,1.5*dir(B),linewidth(4)); dot("$C$",C,1.5*dir(C),linewidth(4)); dot("$D$",D,1.5*S,linewidth(4)); dot("$E$",E,1.5*dir(30),linewidth(4)); dot("$F$",F,1.5*dir(150),linewidth(4)); dot("$P$",P,1.5*dir(-30),linewidth(4)); label("$7$",midpoint(B--D),1.5*S,red); label("$30$",midpoint(C--E),1.5*dir(30),red); label("$40$",midpoint(A--F),1.5*dir(150),red); [/asy] ~MRENTHUSIASM
|
075
|
1/8
|
In the isosceles trapezoid \( KLMN \), the base \( KN \) is equal to 9, and the base \( LM \) is equal to 5. Points \( P \) and \( Q \) lie on the diagonal \( LN \), with point \( P \) located between points \( L \) and \( Q \), and segments \( KP \) and \( MQ \) perpendicular to the diagonal \( LN \). Find the area of trapezoid \( KLMN \) if \( \frac{QN}{LP} = 5 \).
|
7\sqrt{21}
|
3/8
|
To estimate the consumption of disposable wooden chopsticks, in 1999, a sample of 10 restaurants from a total of 600 high, medium, and low-grade restaurants in a certain county was taken. The daily consumption of disposable chopstick boxes in these restaurants was as follows:
0.6, 3.7, 2.2, 1.5, 2.8, 1.7, 1.2, 2.1, 3.2, 1.0
(1) Estimate the total consumption of disposable chopstick boxes in the county for the year 1999 by calculating the sample (assuming 350 business days per year);
(2) In 2001, another survey on the consumption of disposable wooden chopsticks was conducted in the same manner, and the result was that the average daily use of disposable chopstick boxes in the 10 sampled restaurants was 2.42 boxes. Calculate the average annual growth rate of the consumption of disposable wooden chopsticks for the years 2000 and 2001 (the number of restaurants in the county and the total number of business days in the year remained the same as in 1999);
(3) Under the conditions of (2), if producing a set of desks and chairs for primary and secondary school students requires 0.07 cubic meters of wood, calculate how many sets of student desks and chairs can be produced with the wood used for disposable chopsticks in the county in 2001. The relevant data needed for the calculation are: 100 pairs of chopsticks per box, each pair of chopsticks weighs 5 grams, and the density of the wood used is 0.5×10<sup>3</sup> kg/m<sup>3</sup>;
(4) If you were asked to estimate the amount of wood consumed by disposable chopsticks in your province for a year, how would you use statistical knowledge to do so? Briefly describe in words.
|
7260
|
7/8
|
The altitudes of an acute-angled, scalene triangle \(ABC\) intersect at point \(H\). \(O\) is the circumcenter of triangle \(BHC\). The incenter \(I\) of triangle \(ABC\) lies on the segment \(OA\).
Find the angle \(A\).
|
60
|
1/8
|
A fisherman put the three largest fish, which make up $35\%$ of his daily catch's total weight, into the freezer. The three smallest fish, which together constitute $\frac{5}{13}$ of the remaining part, were taken by the cat, and the rest were cooked for lunch. How many fish did the fisherman catch?
|
10
|
6/8
|
Given \( f_{1}(x) = \{1 - 2x\}, x \in [0,1] \), and \( f_{n}(x) = f(f_{n-1}(x)), x \in [0, 1] \), \( n \geq 2, n \in \mathbb{N} \). Determine the number of solutions to the equation \( f_{2002}(x) = \frac{1}{2} x \).
|
2^{2002}
|
4/8
|
A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths 1,24 , and 3 , and the segment of length 24 is a chord of the circle. Compute the area of the triangle.
|
192
|
3/8
|
In the cells of a $5 \times 5$ table, natural numbers are arranged such that all ten sums of these numbers in the rows and columns of the table are distinct. Find the smallest possible value of the sum of the numbers in the entire table.
|
48
|
1/8
|
Compute the product of all positive integers $b \geq 2$ for which the base $b$ number $111111_{b}$ has exactly $b$ distinct prime divisors.
|
24
|
2/8
|
Given that point \( \mathrm{G} \) is the centroid of \( \triangle ABC \) and point \( \mathrm{P} \) is an interior point of \( \triangle GBC \) (excluding the boundary), if \( AP = \lambda AB + \mu AC \), then what is the range of values for \(\lambda + \mu\)?
|
(\frac{2}{3},1)
|
7/8
|
A triangle \( ABC \) is given. It is known that \( AB=4 \), \( AC=2 \), and \( BC=3 \). The angle bisector of \( \angle BAC \) intersects side \( BC \) at point \( K \). A line passing through point \( B \) parallel to \( AC \) intersects the extension of the bisector \( AK \) at point \( M \). Find \( KM \).
|
2 \sqrt{6}
|
7/8
|
Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the line on which the bases of the original squares were placed?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt{2}\qquad\textbf{(C)}\ \frac{3}{2}\qquad\textbf{(D)}\ \sqrt{2}+\frac{1}{2}\qquad\textbf{(E)}\ 2$
|
\textbf{(D)}\\sqrt{2}+\frac{1}{2}
|
1/8
|
Given that vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=\sqrt{2}$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a}\bot (\overrightarrow{a}-\overrightarrow{b})$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.
|
\frac{\pi}{4}
|
6/8
|
From each vertex of a convex polyhedron, exactly three edges emanate, and at least two of these three edges are equal. Prove that the polyhedron has at least three equal edges.
|
3
|
4/8
|
Find the root of the equation \(169(157-77x)^{2}+100(201-100x)^{2}=26(77x-157)(1000x-2010)\).
|
31
|
7/8
|
There is a mathematical operator "$\odot$" that satisfies the following equations: $2 \odot 4=8, 4 \odot 6=14, 5 \odot 3=13, 8 \odot 7=23$. According to this rule, what is $9 \odot 3=?$
|
21
|
7/8
|
Given a triangle $ABC$, points $C_{1}$, $A_{1}$, and $B_{1}$ are taken on the lines $AB$, $BC$, and $CA$, respectively, with $k$ of them lying on the sides of the triangle and $3-k$ on the extensions of the sides. Let
$$
R=\frac{BA_{1}}{CA_{1}} \cdot \frac{CB_{1}}{AB_{1}} \cdot \frac{AC_{1}}{BC_{1}}
$$
Prove that:
a) The points $A_{1}$, $B_{1}$, and $C_{1}$ are collinear if and only if $R=1$ and $k$ is even (Menelaus' theorem).
b) The lines $AA_{1}$, $BB_{1}$, and $CC_{1}$ are concurrent (intersect at a single point) or are parallel if and only if $R=1$ and $k$ is odd (Ceva's theorem).
|
1
|
2/8
|
The simplest form of $1 - \frac{1}{1 + \frac{a}{1 - a}}$ is:
$\textbf{(A)}\ {a}\text{ if }{a\not= 0} \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ {a}\text{ if }{a\not=-1}\qquad \textbf{(D)}\ {1-a}\text{ with not restriction on }{a}\qquad \textbf{(E)}\ {a}\text{ if }{a\not= 1}$
|
\textbf{(E)}\{}
|
1/8
|
Let $a$ and $b$ be positive real numbers. Find the maximum value of
\[2(a - x)(x + \sqrt{x^2 + b^2})\]in terms of $a$ and $b.$
|
a^2 + b^2
|
4/8
|
How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$
$\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$
|
\textbf{(B)}100
|
1/8
|
Three generous friends, each of whom has candies, redistribute them as follows: Vasya gives part of his candies to Petya and Kolya, thereby doubling the number of candies they have. After that, Petya gives part of his candies to Kolya and Vasya, thereby doubling the number of candies they have as well. Finally, Kolya gives part of his candies to Vasya and Petya, whose counts also double. It turned out that Kolya had 36 candies both at the beginning and at the end. How many candies do the boys have in total?
|
252
|
5/8
|
If a line is perpendicular to a plane, then this line and the plane form a "perpendicular line-plane pair". In a cube, the number of "perpendicular line-plane pairs" formed by a line determined by two vertices and a plane containing four vertices is _________.
|
36
|
2/8
|
The sum of positive numbers \(a, b, c,\) and \(d\) does not exceed 4. Find the maximum value of the expression
\[
\sqrt[4]{a(b+2c)} + \sqrt[4]{b(c+2d)} + \sqrt[4]{c(d+2a)} + \sqrt[4]{d(a+2b)}
\]
|
4\sqrt[4]{3}
|
1/8
|
For some integers $m$ and $n$, the expression $(x+m)(x+n)$ is equal to a quadratic expression in $x$ with a constant term of -12. Which of the following cannot be a value of $m$?
|
5
|
7/8
|
Construction Mayhem University has been on a mission to expand and improve its campus! The university has recently adopted a new construction schedule where a new project begins every two days. Each project will take exactly one more day than the previous one to complete (so the first project takes 3, the second takes 4, and so on.)
Suppose the new schedule starts on Day 1. On which day will there first be at least $10$ projects in place at the same time?
|
51
|
6/8
|
A certain brand specialty store is preparing to hold a promotional event during New Year's Day. Based on market research, the store decides to select 4 different models of products from 2 different models of washing machines, 2 different models of televisions, and 3 different models of air conditioners (different models of different products are different). The store plans to implement a prize sales promotion for the selected products, which involves increasing the price by $150$ yuan on top of the current price. Additionally, if a customer purchases any model of the product, they are allowed 3 chances to participate in a lottery. If they win, they will receive a prize of $m\left(m \gt 0\right)$ yuan each time. It is assumed that the probability of winning a prize each time a customer participates in the lottery is $\frac{1}{2}$.
$(1)$ Find the probability that among the 4 selected different models of products, there is at least one model of washing machine, television, and air conditioner.
$(2)$ Let $X$ be the random variable representing the total amount of prize money obtained by a customer in 3 lottery draws. Write down the probability distribution of $X$ and calculate the mean of $X$.
$(3)$ If the store wants to profit from this promotional plan, what is the maximum amount that the prize money should be less than in order for the plan to be profitable?
|
100
|
5/8
|
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.
How many ways are there to put the cards in the three boxes so that the trick works?
|
12
|
1/8
|
Let $S=\{1, 2, 3, \cdots, 280\}$. Find the smallest positive integer $n$ such that every subset of $S$ with $n$ elements contains 5 pairwise coprime numbers.
|
217
|
3/8
|
Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$.
|
376
|
7/8
|
The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\frac{m+\sqrt{n}}r$, where $m$, $n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0$. Find $m+n+r$.
|
200
|
2/8
|
What is the maximum number of pawns that can be placed on a chessboard (no more than one pawn per square) if: 1) a pawn cannot be placed on the square e4; 2) no two pawns can be placed on squares that are symmetric with respect to the square e4?
|
39
|
1/8
|
Segment $AB$ of length $13$ is the diameter of a semicircle. Points $C$ and $D$ are located on the semicircle but not on segment $AB$ . Segments $AC$ and $BD$ both have length $5$ . Given that the length of $CD$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$ .
|
132
|
7/8
|
Consider the sequence of integers \(\{f(n)\}_{n=1}^{\infty}\) defined by:
- \(f(1)=1\).
- If \(n\) is even, \(f(n)=f(n / 2)\).
- If \(n>1\) is odd and \(f(n-1)\) is odd, then \(f(n)=f(n-1)-1\).
- If \(n>1\) is odd and \(f(n-1)\) is even, then \(f(n)=f(n-1)+1\).
a) Calculate \(f\left(2^{2020}-1\right)\).
b) Prove that \(\{f(n)\}_{n=1}^{\infty}\) is not periodic, that is, there do not exist positive integers \(t\) and \(n_{0}\) such that \(f(n+t)=f(n)\) for any \(n \geq n_{0}\).
|
0
|
6/8
|
Simplify $(2^8 + 4^5)(2^3 - (-2)^2)^{11}$.
|
5368709120
|
5/8
|
Determine the largest of all integers $n$ with the property that $n$ is divisible by all positive integers that are less than $\sqrt[3]{n}$.
|
420
|
6/8
|
In triangle \(ABC\), a median \(AM\) is drawn. Find the angle \(AMC\) if the angles \(BAC\) and \(BCA\) are \(45^\circ\) and \(30^\circ\) respectively.
|
135
|
7/8
|
In the diagram, $\angle PQR = 90^\circ$. A line PS bisects $\angle PQR$, and $\angle PQS = y^\circ$. If $\angle SQR = 2x^\circ$ and $\angle PQS = 2y^\circ$, what is the value of $x + y$?
[asy]
size(100);
draw((0,1)--(0,0)--(1,0));
draw((0,0)--(.9,.47));
draw((0,.1)--(.1,.1)--(.1,0));
label("$P$",(0,1),N); label("$Q$",(0,0),SW); label("$R$",(1,0),E); label("$S$",(.9,.47),NE);
label("$2y^\circ$",(0.15,.2)); label("$2x^\circ$",(.32,-.02),N);
[/asy]
|
45
|
6/8
|
Two jokers are added to a 52 card deck and the entire stack of 54 cards is shuffled randomly. What is the expected number of cards that will be between the two jokers?
|
52 / 3
|
6/8
|
The circumcircle of \( \triangle ABC \) is \(\odot O\). Given that \( \angle C=60^{\circ} \), \( M \) is the midpoint of arc \( AB \), and \( H \) is the orthocenter of \( \triangle ABC \). Prove that \( OM \perp OH \).
|
OM\perpOH
|
1/8
|
Given numbers \(a, b, c\) satisfy \(a b c+a+c-b\). Then the maximum value of the algebraic expression \(\frac{1}{1+a^{2}}-\frac{1}{1+b^{2}}+\frac{1}{1+c^{2}}\) is
|
\frac{5}{4}
|
3/8
|
Let \( p \) and \( q \) be positive integers such that \(\frac{72}{487}<\frac{p}{q}<\frac{18}{121}\). Find the smallest possible value of \( q \).
|
27
|
1/8
|
Let \( I \) be the center of the circle inscribed in the triangle \( ABC \). Suppose that \( CA + AI = BC \). Determine the value of the ratio \(\frac{\widehat{BAC}}{\widehat{CBA}}\).
|
2
|
3/8
|
A certain company, in response to the national call for garbage classification, has launched a project with the support of the research department to innovate technologies. The project processes kitchen waste into reusable chemical products. It is known that the daily processing capacity $x$ (unit: tons) of the company is at least 70 tons and at most 100 tons. The function relationship between the total daily processing cost $y$ (unit: yuan) and the daily processing capacity $x$ can be approximately represented as $y=\frac{1}{2}x^2+40x+3200$, and the selling price of the chemical products obtained from processing 1 ton of kitchen waste is 110 yuan.
$(1)$ For how many tons of daily processing capacity does the company have the lowest average cost per ton of kitchen waste processed? At this point, is the company processing 1 ton of kitchen waste at a loss or profit?
$(2)$ In order for the company to achieve sustainable development, the government has decided to provide financial subsidies to the company, with two subsidy schemes:
① A fixed amount of financial subsidy of 2300 yuan per day;
② A financial subsidy based on the daily processing capacity, with an amount of 30x yuan. If you are the decision-maker of the company, which subsidy scheme would you choose to maximize profit? Why?
|
1800
|
3/8
|
Discussing winter holidays in class, Sasha said: "Now, after flying to Addis Ababa, I have celebrated the New Year in all possible hemispheres of the Earth, except one!"
What is the minimum number of places where Sasha has celebrated the New Year?
Consider the places where Sasha celebrated the New Year as points on a sphere. Points on the boundary of a hemisphere are not considered to belong to that hemisphere.
|
4
|
2/8
|
When one ounce of water is added to a mixture of acid and water, the new mixture is $20\%$ acid. When one ounce of acid is added to the new mixture, the result is $33\frac13\%$ acid. The percentage of acid in the original mixture is
$\textbf{(A)}\ 22\% \qquad \textbf{(B)}\ 24\% \qquad \textbf{(C)}\ 25\% \qquad \textbf{(D)}\ 30\% \qquad \textbf{(E)}\ 33\frac13 \%$
|
\textbf{(C)}\25
|
1/8
|
Given the cubic function $f(x) = x^3 - 9x^2 + 20x - 4$, and the arithmetic sequence $\{a_n\}$ with $a_5 = 3$, find the value of $f(a_1) + f(a_2) + ... + f(a_9)$.
|
18
|
6/8
|
Four siblings inherited a plot of land shaped like a convex quadrilateral. By connecting the midpoints of the opposite sides of the plot, they divided the inheritance into four quadrilaterals. The first three siblings received plots of $360 \, \mathrm{m}^{2}$, $720 \, \mathrm{m}^{2}$, and $900 \, \mathrm{m}^{2}$ respectively. What is the area of the plot received by the fourth sibling?
|
540
|
3/8
|
Given the parabola \( y^2 = 4x \) with its focus at \( F \), a line passing through the focus \( F \) and inclined at an angle \( \theta \left(0 < \theta < \frac{\pi}{2}\right) \) intersects the parabola at points \( A \) and \( B \). Points \( A_O \) and \( B_O \) (where \( O \) is the origin) intersect the directrix at points \( B' \) and \( A' \), respectively. Determine the area of the quadrilateral \( ABB'A' \).
|
\frac{8}{\sin^3(\theta)}
|
1/8
|
Convex quadrilateral $ABCD$ satisfies $\angle{CAB} = \angle{ADB} = 30^{\circ}, \angle{ABD} = 77^{\circ}, BC = CD$ and $\angle{BCD} =n^{\circ}$ for some positive integer $n$ . Compute $n$ .
|
68
|
5/8
|
Find an integer $n$ such that the decimal representation of the number $5^{n}$ contains at least 1968 consecutive zeros.
|
1968
|
1/8
|
Compute
$$
\sum_{\substack{a+b+c=12 \\ a \geq 6, b, c \geq 0}} \frac{a!}{b!c!(a-b-c)!},
$$
where the sum runs over all triples of nonnegative integers \((a, b, c)\) such that \(a+b+c=12\) and \(a \geq 6\).
|
2731
|
3/8
|
The function $f(n)$ defined on the set of natural numbers $\mathbf{N}$ is given by:
$$
f(n)=\left\{\begin{array}{ll}
n-3 & (n \geqslant 1000); \\
f[f(n+7)] & (n < 1000),
\end{array}\right.
$$
What is the value of $f(90)$?
(A) 997;
(B) 998;
(C) 999;
(D) 1000.
|
999
|
1/8
|
Quantities $a$ and $b$ vary inversely. When $a$ is $800$, $b$ is $0.5$. If the product of $a$ and $b$ increases by $200$ when $a$ is doubled, what is $b$ when $a$ is $1600$?
|
0.375
|
4/8
|
Find the coefficient of \(x^5\) in the expansion of \(\left(1+2x+3x^2+4x^3\right)^5\).
|
1772
|
3/8
|
Brown lives on a street with more than 20 but fewer than 500 houses (all houses are numbered sequentially: $1, 2, 3$, etc.). Brown discovered that the sum of all the numbers from the first house to his own house (inclusive) equals half the sum of all the numbers from the first to the last house (inclusive).
What is the number of his house?
|
84
|
7/8
|
The nine points of this grid are equally spaced horizontally and vertically. The distance between two neighboring points is 1 unit. What is the area, in square units, of the region where the two triangles overlap?
[asy]
size(80);
dot((0,0)); dot((0,1));dot((0,2));dot((1,0));dot((1,1));dot((1,2));dot((2,0));dot((2,1));dot((2,2));
draw((0,0)--(2,1)--(1,2)--cycle, linewidth(0.6));
draw((2,2)--(0,1)--(1,0)--cycle, linewidth(0.6));
[/asy]
|
1
|
6/8
|
Given a square region with a side length of 1 meter, and a total of 5120 beans within the square with 4009 beans within the inscribed circle, determine the approximate value of pi rounded to three decimal places.
|
3.13
|
1/8
|
In triangle \(ABC\), side \(BC = 28\). The angle bisector \(BL\) is divided by the intersection point of the angle bisectors of the triangle in the ratio \(4:3\) from the vertex. Find the radius of the circumscribed circle around triangle \(ABC\) if the radius of the inscribed circle is 12.
|
50
|
6/8
|
A bus departed from city $M$ to city $N$ at a speed of 40 km/h. After a quarter of an hour, it encountered a car traveling from city $N$. This car reached city $M$, waited for 15 minutes, and then returned to city $N$, overtaking the bus 20 km from city $N$. Find the distance between cities $M$ and $N$ if the car's speed is 50 km/h.
|
160\,
|
1/8
|
Given a polynomial \( P(x) \) with integer coefficients and known values \( P(2) = 3 \) and \( P(3) = 2 \), what is the maximum number of integer solutions that the equation \( P(x) = x \) can have?
|
0
|
1/8
|
How many natural numbers greater than 10 but less than 100 are relatively prime to 21?
|
51
|
1/8
|
Petya has thought of a 9-digit number obtained by permuting the digits of the number 123456789. Vitya is trying to guess it. To do this, he selects any 9-digit number (possibly with repeating digits and zeros) and tells it to Petya. Petya then responds with the number of digits in Vitya's number that match the digits in the same positions in Petya's number. Can Vitya determine the first digit of Petya's number in no more than 4 attempts? (A match is a digit that is in the same position.)
|
Yes
|
1/8
|
In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled triangles with $\angle AEB=\angle BEC = \angle CED = 45^\circ$ and $AE=32$. Find the length of $CE.$
|
16
|
1/8
|
There are 8 books on a shelf, which include a trilogy that must be selected together. In how many ways can 5 books be selected from this shelf if the order in which the books are selected does not matter?
|
11
|
3/8
|
A regular octagon with a side length of 1 is divided into parallelograms. Prove that among these parallelograms there are at least two rectangles, and the sum of the areas of all rectangles is equal to 2.
|
2
|
1/8
|
Find the smallest positive integer $k$ such that
\[\binom{x+kb}{12} \equiv \binom{x}{12} \pmod{b}\]
for all positive integers $b$ and $x$ . (*Note:* For integers $a,b,c$ we say $a \equiv b \pmod c$ if and only if $a-b$ is divisible by $c$ .)
*Alex Zhu.*
<details><summary>Clarifications</summary>[list=1][*] ${{y}\choose{12}} = \frac{y(y-1)\cdots(y-11)}{12!}$ for all integers $y$ . In particular, ${{y}\choose{12}} = 0$ for $y=1,2,\ldots,11$ .[/list]</details>
|
27720
|
1/8
|
A circle with its center on side $AB$ of triangle $ABC$ touches the other two sides. Find the area of the circle if $a = 13$ cm, $b = 14$ cm, and $c = 15$ cm, where $a$, $b$, and $c$ are the lengths of the sides of the triangle.
|
\frac{3136}{81} \pi
|
6/8
|
Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted to exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$ .
|
21
|
1/8
|
Subsets and Splits
Filtered Answers A-D
Retrieves 100 rows where the answer is a single letter from A to D, providing basic filtering of the dataset.