problem
stringlengths 10
7.44k
| answer
stringlengths 1
270
| difficulty
stringclasses 8
values |
|---|---|---|
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?
|
382
|
2/8
|
Let $ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\angle OIA=90^{\circ}$. Given that $AI=97$ and $BC=144$, compute the area of $\triangle ABC$.
|
14040
|
1/8
|
$M$ is a moving point on the circle $x^{2}+y^{2}-6 x-8 y=0$, $O$ is the origin, and $N$ is a point on the ray $O M$. If $|O M| \cdot |O N| = 150$, find the equation of the locus of point $N$.
|
3x+4y=75
|
1/8
|
In rectangle \(ABCD\), \(AD = a\) and \(AB = b\) (with \(b > a\)). The rectangle is folded such that point \(A\) coincides with point \(C\), creating fold line \(MN\). Then, the half-plane \(D A M N\) is opened to form a dihedral angle of \(57^\circ\) with the half-plane \(M N C B\). What is the angle between line \(AC\) and line \(MN\)?
|
90
|
1/8
|
To meet market demand, a supermarket purchased a brand of zongzi before the arrival of the Dragon Boat Festival on May 5th. The cost of each box is $40. The supermarket stipulates that the selling price of each box must not be less than $45. Based on past sales experience, it was found that when the selling price is set at $45 per box, 700 boxes can be sold per day. For every $1 increase in the selling price per box, 20 fewer boxes are sold per day.
$(1)$ Find the functional relationship between the daily sales volume $y$ (boxes) and the selling price per box $x$ (in dollars).
$(2)$ At what price per box should the selling price be set to maximize the daily profit $P$ (in dollars)? What is the maximum profit?
$(3)$ To stabilize prices, the relevant management department has set a maximum selling price of $58 per box for this type of zongzi. If the supermarket wants to make a profit of at least $6000 per day, how many boxes of zongzi must be sold per day at least?
|
440
|
7/8
|
Determine all six-digit numbers \( p \) that satisfy the following properties:
(1) \( p, 2p, 3p, 4p, 5p, 6p \) are all six-digit numbers;
(2) Each of the six-digit numbers' digits is a permutation of \( p \)'s six digits.
|
142857
|
5/8
|
Determine the value of the infinite product $(3^{1/4})(9^{1/16})(27^{1/64})(81^{1/256}) \dotsm$ plus 2, the result in the form of "$\sqrt[a]{b}$ plus $c$".
|
\sqrt[9]{81} + 2
|
5/8
|
Consider a sphere inscribed in a right cone with the base radius of 10 cm and height of 40 cm. The radius of the inscribed sphere can be expressed as $b\sqrt{d} - b$ cm. Determine the value of $b+d$.
|
19.5
|
1/8
|
A traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each person told the traveler whether the neighbor to their right was truthful or deceitful. Based on these statements, the traveler was able to determine what fraction of the villagers are truthful. Determine this fraction.
|
1/2
|
1/8
|
The road from home to school takes Sergei 30 minutes. One day, on his way, he remembered that he had forgotten his pen at home. Sergei knew that if he continued to the school at the same speed, he would arrive there 9 minutes before the bell rings, but if he returned home for the pen and then continued to school at the same speed, he would be 11 minutes late for the lesson. What fraction of the way did he cover?
|
\frac{1}{3}
|
2/8
|
A lemming starts at a corner of a rectangular area measuring 8 meters by 15 meters. It dashes diagonally across the rectangle towards the opposite corner for 11.3 meters. Then the lemming makes a $90^{\circ}$ right turn and sprints upwards for 3 meters. Calculate the average of the shortest distances to each side of the rectangle.
|
5.75
|
7/8
|
Find the biggest real number $ k$ such that for each right-angled triangle with sides $ a$ , $ b$ , $ c$ , we have
\[ a^{3}\plus{}b^{3}\plus{}c^{3}\geq k\left(a\plus{}b\plus{}c\right)^{3}.\]
|
\frac{3\sqrt{2} - 4}{2}
|
3/8
|
Find the length of the curve defined by the parametric equations $\left\{\begin{array}{l}x=2 \cos ^{2} \theta \\ y=3 \sin ^{2} \theta\end{array}\right.$ where $\theta$ is the parameter.
|
\sqrt{13}
|
4/8
|
Given the function $f(x) = x^3 - 3x^2 - 9x + 1$, find the intervals of monotonicity and the extrema of $f(x)$.
|
-26
|
1/8
|
A sports team's members have unique numbers taken from the integers 1 to 100. If no member's number is the sum of the numbers of any two other members, nor is it twice the number of any other member, what is the maximum number of people this team can have?
|
50
|
2/8
|
A right-angled triangle has sides of lengths 6, 8, and 10. A circle is drawn so that the area inside the circle but outside this triangle equals the area inside the triangle but outside the circle. The radius of the circle is closest to:
|
2.8
|
2/8
|
Consider the two hands of an analog clock, each of which moves with constant angular velocity. Certain positions of these hands are possible (e.g. the hour hand halfway between the 5 and 6 and the minute hand exactly at the 6), while others are impossible (e.g. the hour hand exactly at the 5 and the minute hand exactly at the 6). How many different positions are there that would remain possible if the hour and minute hands were switched?
|
143
|
1/8
|
The sequences \(\left(a_{n}\right)\) and \(\left(b_{n}\right)\) are defined by the conditions \(a_{1}=1\), \(b_{1}=2\), \(a_{n+1}=\frac{1+a_{n}+a_{n} h_{n}}{b_{n}}\), and \(b_{n+1}=\frac{1+b_{n}+a_{n} h_{n}}{a_{n}}\). Prove that \(a_{2008} < 5\).
|
a_{2008}<5
|
1/8
|
Let there be $2n$ positive reals $a_1,a_2,...,a_{2n}$ . Let $s = a_1 + a_3 +...+ a_{2n-1}$ , $t = a_2 + a_4 + ... + a_{2n}$ , and $x_k = a_k + a_{k+1} + ... + a_{k+n-1}$ (indices are taken modulo $2n$ ). Prove that $$ \frac{s}{x_1}+\frac{t}{x_2}+\frac{s}{x_3}+\frac{t}{x_4}+...+\frac{s}{x_{2n-1}}+\frac{t}{x_{2n}}>\frac{2n^2}{n+1} $$
|
\frac{2n^2}{n+1}
|
1/8
|
Juan takes a number, adds 3 to it, squares the result, then multiplies the answer by 2, subtracts 3 from the result, and finally divides that number by 2. If his final answer is 49, what was the original number?
|
\sqrt{\frac{101}{2}} - 3
|
1/8
|
(1) Prove that \( f(x)=\sin |x| \) is not a periodic function.
(2) Prove that \( y=\sin \sqrt[3]{x} \) is not a periodic function.
|
\sin\sqrt[3]{x}
|
6/8
|
13 children sat around a circular table and agreed that boys would lie to girls but tell the truth to each other. Conversely, girls would lie to boys but tell the truth to each other. One of the children told their right-hand neighbor, "The majority of us are boys." That child then told their right-hand neighbor, "The majority of us are girls," and the third child told their right-hand neighbor, "The majority of us are boys," and so on, continuing in this pattern until the last child told the first child, "The majority of us are boys." How many boys were at the table?
|
7
|
5/8
|
A and B bought the same number of sheets of stationery. A put 1 sheet of stationery into each envelope and had 40 sheets of stationery left after using all the envelopes. B put 3 sheets of stationery into each envelope and had 40 envelopes left after using all the sheets of stationery. How many sheets of stationery did they each buy?
|
120
|
6/8
|
A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube?
|
90
|
7/8
|
Grace is trying to solve the following equation by completing the square: $$36x^2-60x+25 = 0.$$ She successfully rewrites the equation in the form: $$(ax + b)^2 = c,$$ where $a$, $b$, and $c$ are integers and $a > 0$. What is the value of $a + b + c$?
|
26
|
1/8
|
Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.
|
130
|
5/8
|
Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, point $P$ is on the edge $BC$, and point $Q$ is the midpoint of the edge $CC_{1}$. If the plane passing through points $A$, $P$, and $Q$ intersects the cube to form a pentagon, what is the range of values for $B P$?
|
(\frac{1}{2},1)
|
2/8
|
For his birthday, Piglet baked a big cake weighing 10 kg and invited 100 guests. Among them was Winnie-the-Pooh, who has a weakness for sweets. The birthday celebrant announced the cake-cutting rule: the first guest cuts themselves a piece of cake equal to \(1\%\) of the remaining cake, the second guest cuts themselves a piece of cake equal to \(2\%\) of the remaining cake, the third guest cuts themselves a piece of cake equal to \(3\%\) of the remaining cake, and so on. Which position in the queue should Winnie-the-Pooh take to get the largest piece of cake?
|
10
|
4/8
|
Three squares, $ABCD$, $EFGH$, and $GHIJ$, each have side length $s$. Point $C$ is located at the midpoint of side $HG$, and point $D$ is located at the midpoint of side $EF$. The line segment $AJ$ intersects the line segment $GH$ at point $X$. Determine the ratio of the area of the shaded region formed by triangle $AXD$ and trapezoid $JXCB$ to the total area of the three squares.
|
\frac{1}{3}
|
1/8
|
Determine the largest constant \( C \) such that for all real numbers \( x_1, x_2, \ldots, x_6 \), the inequality
\[
\left(x_1 + x_2 + \cdots + x_6\right)^2 \geq C \cdot \left(x_1 \left(x_2 + x_3\right) + x_2 \left(x_3 + x_4\right) + \cdots + x_6 \left(x_1 + x_2\right) \right)
\]
holds.
Also, find for this \( C \) all \( x_1, x_2, \ldots, x_6 \) for which equality holds.
|
3
|
1/8
|
A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have?
|
9
|
7/8
|
If an internal point of an $n$-sided prism is connected to all its vertices, $n$ quadrangular pyramids with a common vertex at this point will be formed, with their bases being the lateral faces of the prism. Find the ratio of the sum of the volumes of these pyramids to the volume of the given prism.
|
\frac{2}{3}
|
3/8
|
Let the sequence of positive numbers $\{a_{n}\}$ be defined by
$$
\begin{array}{l}
a_{1}=\sqrt{2}-1, \\
a_{n+1}=\frac{2 n+1}{S_{n}+S_{n+1}+2} \quad (n=1,2, \cdots),
\end{array}
$$
where $S_{n}$ represents the sum of the first $n$ terms of the sequence $a_{n}$. Determine the general term of the sequence.
|
a_{n}=\sqrt{n^{2}+1}-\sqrt{(n-1)^{2}+1}
|
1/8
|
Let $\mathcal{P}_{1}, \mathcal{P}_{2}, \mathcal{P}_{3}$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the $\mathcal{P}_{i}$. In other words, find the maximum number of points that can lie on two or more of the parabolas $\mathcal{P}_{1}, \mathcal{P}_{2}, \mathcal{P}_{3}$.
|
12
|
3/8
|
Among the first 1500 positive integers, there are n whose hexadecimal representation contains only numeric digits. What is the sum of the digits of n?
|
23
|
7/8
|
A certain product in a shopping mall sells an average of 70 items per day, with a profit of $50 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 decrease 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 ______ dollars. (Express using algebraic expressions involving $x$)<br/>$(2)$ With the above conditions unchanged, how much should the price of each item be reduced so that the mall's daily profit reaches $3572$ dollars.
|
12
|
7/8
|
A 0-1 sequence of length $2^k$ is given. Alice can pick a member from the sequence, and reveal it (its place and its value) to Bob. Find the largest number $s$ for which Bob can always pick $s$ members of the sequence, and guess all their values correctly.
Alice and Bob can discuss a strategy before the game with the aim of maximizing the number of correct guesses of Bob. The only information Bob has is the length of the sequence and the member of the sequence picked by Alice.
|
k+1
|
1/8
|
Given that in $\triangle ABC$, $BD:DC = 3:2$ and $AE:EC = 3:4$, and the area of $\triangle ABC$ is 1, find the area of $\triangle BMD$.
|
\frac{4}{15}
|
4/8
|
Find all triples $m$ ; $n$ ; $l$ of natural numbers such that $m + n = gcd(m; n)^2$ ; $m + l = gcd(m; l)^2$ ; $n + l = gcd(n; l)^2$ :
*S. Tokarev*
|
(2,2,2)
|
2/8
|
Prove that every convex polygon with $2n$ sides has at least $n$ diagonals that are not parallel to any of its sides!
|
n
|
2/8
|
Suppose Xiao Ming's family subscribes to a newspaper. The delivery person may deliver the newspaper to Xiao Ming's home between 6:30 and 7:30 in the morning. Xiao Ming's father leaves for work between 7:00 and 8:00 in the morning. What is the probability that Xiao Ming's father can get the newspaper before leaving home?
|
\frac{7}{8}
|
5/8
|
James writes down three integers. Alex picks some two of those integers, takes the average of them, and adds the result to the third integer. If the possible final results Alex could get are 42, 13, and 37, what are the three integers James originally chose?
|
-20, 28, 38
|
1/8
|
A $6$ -inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches.
|
12
|
4/8
|
Pasha knows the speed of his motorboat. He calculated that it would take him 20 minutes to go from the pier to the bridge and back. However, he forgot to account for the river's current. How many minutes will the actual round trip take if the speed of the current is exactly 1/3 of the motorboat's speed? (Both the motorboat's speed and the current's speed are constant.)
|
22.5\,
|
1/8
|
Add $452_8$ and $167_8$ in base $8$, then subtract $53_8$ from the result.
|
570_8
|
1/8
|
Given that a certain middle school has 3500 high school students and 1500 junior high school students, if 70 students are drawn from the high school students, calculate the total sample size $n$.
|
100
|
7/8
|
Given two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an acute angle between them, and satisfying $|\overrightarrow{a}|= \frac{8}{\sqrt{15}}$, $|\overrightarrow{b}|= \frac{4}{\sqrt{15}}$. If for any $(x,y)\in\{(x,y)| |x \overrightarrow{a}+y \overrightarrow{b}|=1, xy > 0\}$, it holds that $|x+y|\leqslant 1$, then the minimum value of $\overrightarrow{a} \cdot \overrightarrow{b}$ is \_\_\_\_\_\_.
|
\frac{8}{15}
|
3/8
|
The value of $(\sqrt{1+\sqrt{1+\sqrt{1}}})^{4}$ is:
(a) $\sqrt{2}+\sqrt{3}$;
(b) $\frac{1}{2}(7+3 \sqrt{5})$;
(c) $1+2 \sqrt{3}$;
(d) 3 ;
(e) $3+2 \sqrt{2}$.
|
3 + 2 \sqrt{2}
|
1/8
|
The kindergarten received flashcards for reading: some say "MA", and the others say "NYA". Each child took three cards to form words. It turned out that 20 children can form the word "MAMA" from their cards, 30 children can form the word "NYANYA", and 40 children can form the word "MANYA". How many children have all three cards the same?
|
10
|
5/8
|
Even natural numbers \( a \) and \( b \) are such that the greatest common divisor (GCD) of \( a \) and \( b \) plus the least common multiple (LCM) of \( a \) and \( b \) equals \( 2^{23} \). How many distinct values can the LCM of \( a \) and \( b \) take?
|
22
|
6/8
|
Given that Ms. Demeanor's class consists of 50 students, more than half of her students bought crayons from the school bookstore, each buying the same number of crayons, with each crayon costing more than the number of crayons bought by each student, and the total cost for all crayons was $19.98, determine the cost of each crayon in cents.
|
37
|
1/8
|
The four-digit number $\overline{a b c d}$ is divisible by 3, and $a, b, c$ are permutations of three consecutive integers. How many such four-digit numbers are there?
|
184
|
5/8
|
How many unique numbers can you get by multiplying two or more distinct members of the set $\{1,2,3,5,7\}$ together?
|
11
|
1/8
|
Given that sequence {a_n} is an equal product sequence, with a_1=1, a_2=2, and a common product of 8, calculate the sum of the first 41 terms of the sequence {a_n}.
|
94
|
5/8
|
Given the set \( A = \{1, 2, 3, \ldots, 104\} \), and \( S \) is a subset of \( A \). If for any \( x \in S \), neither \( x-1 \in S \) nor \( x+1 \in S \), then \( x \) is called an isolated point of \( S \). How many 5-element subsets of \( A \) have no isolated points?
|
10000
|
1/8
|
Mrs. Riley revised her data after realizing that there was an additional score bracket and a special bonus score for one of the brackets. Recalculate the average percent score for the $100$ students given the updated table:
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
100&5\\\hline
95&12\\\hline
90&20\\\hline
80&30\\\hline
70&20\\\hline
60&8\\\hline
50&4\\\hline
40&1\\\hline
\end{tabular}
Furthermore, all students scoring 95% receive a 5% bonus, which effectively makes their score 100%.
|
80.2
|
7/8
|
Line $l$ in the coordinate plane has equation $3x-5y+40=0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20,20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k?$
$\textbf{(A) } 10 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 25 \qquad \textbf{(E) } 30$
|
\textbf{(B)}15
|
1/8
|
Given a parabola $y = ax^2 + bx + c$ ($a \neq 0$) with its axis of symmetry on the left side of the y-axis, where $a, b, c \in \{-3, -2, -1, 0, 1, 2, 3\}$. Let the random variable $X$ represent the value of $|a-b|$. Calculate the expected value $E(X)$.
|
\frac{8}{9}
|
7/8
|
For a given list of three numbers, the operation "changesum" replaces each number in the list with the sum of the other two. For example, applying "changesum" to \(3,11,7\) gives \(18,10,14\). Arav starts with the list \(20,2,3\) and applies the operation "changesum" 2023 times. What is the largest difference between two of the three numbers in his final list?
A 17
B 18
C 20
D 2021
E 2023
|
18
|
1/8
|
Determine all integers \( n > 3 \) such that there exist \( n \) points \( A_1, A_2, \cdots, A_n \) in the plane and real numbers \( r_1, r_2, \cdots, r_n \) satisfying:
(1) Any three points among \( A_1, A_2, \cdots, A_n \) are not collinear.
(2) For each triplet \( i, j, k \) (with \( 1 \leq j < k \leq n \)), the area of triangle \( \triangle A_i A_j A_k \) is equal to \( r_i + r_j + r_k \).
|
4
|
1/8
|
How many ways are there to choose integers \(a, b,\) and \(c\) with \(a < b < c\) from the list \(1, 5, 8, 21, 22, 27, 30, 33, 37, 39, 46, 50\) so that the product \(abc\) is a multiple of 12?
|
76
|
1/8
|
Let $ABC$ be an acute-angled triangle and $\Gamma$ be a circle with $AB$ as diameter intersecting $BC$ and $CA$ at $F ( \not= B)$ and $E (\not= A)$ respectively. Tangents are drawn at $E$ and $F$ to $\Gamma$ intersect at $P$ . Show that the ratio of the circumcentre of triangle $ABC$ to that if $EFP$ is a rational number.
|
2
|
1/8
|
We consider 2021 lines in the plane, no two of which are parallel and no three of which are concurrent. Let E be the set of their intersection points. We want to assign a color to each point in E such that any two points on the same line, whose connecting segment contains no other point of E, have different colors. What is the minimum number of colors needed to achieve such a coloring?
|
3
|
1/8
|
Archimedes already knew that a sphere occupies exactly \(\frac{2}{3}\) of the volume of the cylinder into which it is inscribed (the sphere touches the walls, bottom, and lid of the cylinder). A cylindrical package contains 5 spheres stacked on top of each other. Find the ratio of the empty space to the occupied space in this package.
|
1:2
|
1/8
|
A square is inscribed in a segment whose arc is $60^\circ$. Find the area of the square if the radius of the circle is $2\sqrt{3} + \sqrt{17}$.
|
1
|
3/8
|
Consider a standard twelve-hour clock whose hour and minute hands move continuously. Let $m$ be an integer, with $1 \leq m \leq 720$ . At precisely $m$ minutes after 12:00, the angle made by the hour hand and minute hand is exactly $1^\circ$ .
Determine all possible values of $m$ .
|
458
|
2/8
|
On the sides $AB, BC,$ and $CA$ of an equilateral triangle $ABC$, points $D, E,$ and $F$ are chosen respectively such that $DE \parallel AC$ and $DF \parallel BC$. Find the angle between the lines $AE$ and $BF$.
|
60
|
4/8
|
In a 2-dimensional Cartesian coordinate system, there are 16 lattice points \((i, j)\) where \(0 \leq i \leq 3\) and \(0 \leq j \leq 3\). If \(n\) points are selected from these 16 points, determine the minimum value of \(n\) such that there always exist four points which are the vertices of a square.
|
11
|
1/8
|
On a spherical planet with an equator length of 1, it is planned to build \( N \) ring roads, each of which will follow a circumference of length 1. Then, several trains will be launched on each road. All trains will travel on the roads at the same positive constant speed, never stopping and never colliding. What is the maximum possible total length of all trains under these conditions? Consider trains as arcs of zero thickness, excluding the end points. Solve the problem for the cases: a) \( N = 3 \); b) \( N = 4 \).
|
2
|
3/8
|
The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?
$\begin{tabular}{rr}&\ \texttt{6 4 1}\\ &\texttt{8 5 2}\\ &+\texttt{9 7 3}\\ \hline &\texttt{2 4 5 6}\end{tabular}$
$\text{(A)}\ 4\qquad\text{(B)}\ 5\qquad\text{(C)}\ 6\qquad\text{(D)}\ 7\qquad\text{(E)}\ 8$
|
\textbf{(D)}\7
|
1/8
|
Find the smallest real number \( a \) such that for any non-negative real numbers \( x \), \( y \), and \( z \) that sum to 1, the following inequality holds:
$$
a(x^{2} + y^{2} + z^{2}) + xyz \geq \frac{a}{3} + \frac{1}{27}.
$$
|
\frac{2}{9}
|
2/8
|
Let \(\mathcal{G}\) be the set of all points \((x, y)\) in the Cartesian plane such that \(0 \leq y \leq 8\) and
\[ (x-3)^{2} + 31 = (y-4)^{2} + 8 \sqrt{y(8-y)}. \]
There exists a unique line \(\ell\) with a negative slope that is tangent to \(\mathcal{G}\) and passes through the point \((0,4)\). Suppose \(\ell\) is tangent to \(\mathcal{G}\) at a unique point \(P\). Find the coordinates \((\alpha, \beta)\) of \(P\).
|
(\frac{12}{5},\frac{8}{5})
|
1/8
|
Find the smallest positive integer $n$ that satisfies the following two properties:
1. $n$ has exactly 144 distinct positive divisors.
2. Among the positive divisors of $n$, there are ten consecutive integers.
|
110880
|
1/8
|
In rectangle $PQRS$, $PQ = 150$. Let $T$ be the midpoint of $\overline{PS}$. Given that line $PT$ and line $QT$ are perpendicular, find the greatest integer less than $PS$.
|
212
|
1/8
|
Inside triangle \(ABC\), a point \(P\) is chosen such that \(\angle PAB : \angle PAC = \angle PCA : \angle PCB = \angle PBC : \angle PBA = x\). Prove that \(x = 1\).
|
1
|
6/8
|
$p$ is a prime number. For which numbers $a$ will the solution of the congruence $a x \equiv 1 (\bmod p)$ be the number $a$ itself?
|
\equiv\1\pmod{p}
|
2/8
|
$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$ , for all $ n \in N$ . Let $ r_n$ be the average of $ g_n(x)$ 's roots. If $ r_{19}\equal{}99$ , find $ r_{99}$ .
|
99
|
5/8
|
Calculate the areas of the regions bounded by the curves given in polar coordinates.
$$
r=2 \cos 6 \phi
$$
|
2\pi
|
5/8
|
Carly takes three steps to walk the same distance as Jim walks in four steps. Each of Carly's steps covers 0.5 metres. How many metres does Jim travel in 24 steps?
(A) 16
(B) 9
(C) 36
(D) 12
(E) 18
|
9
|
1/8
|
Given that $\triangle ABC$ is an acute triangle, vector $\overrightarrow{m}=(\cos (A+ \frac{\pi}{3}),\sin (A+ \frac{\pi}{3}))$, $\overrightarrow{n}=(\cos B,\sin B)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(I) Find the value of $A-B$;
(II) If $\cos B= \frac{3}{5}$, and $AC=8$, find the length of $BC$.
|
4\sqrt{3}+3
|
7/8
|
In acute $\triangle ABC,$ let $I$ denote the incenter and suppose that line $AI$ intersects segment $BC$ at a point $D.$ Given that $AI=3, ID=2,$ and $BI^2+CI^2=64,$ compute $BC^2.$ *Proposed by Kyle Lee*
|
\frac{272}{3}
|
2/8
|
Xiao Ming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, Xiao Ming took the subway first and then transferred to the bus, taking a total of 40 minutes to reach school, with the transfer process taking 6 minutes. How many minutes did Xiao Ming take the bus that day?
|
10
|
1/8
|
Given that the random variable $ξ$ follows a normal distribution $N(1,σ^{2})$, and $P(ξ\leqslant 4)=0.79$, determine the value of $P(-2\leqslant ξ\leqslant 1)$.
|
0.29
|
7/8
|
In triangle \( ABC \), the sides \( AC = 14 \) and \( AB = 6 \) are known. A circle with center \( O \), constructed on side \( AC \) as the diameter, intersects side \( BC \) at point \( K \). It turns out that \( \angle BAK = \angle ACB \). Find the area of triangle \( BOC \).
|
21
|
2/8
|
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
[asy]
size(100);
pair A, B, C, D, E, F;
A = (0,0);
B = (1,0);
C = (2,0);
D = rotate(60, A)*B;
E = B + D;
F = rotate(60, A)*C;
draw(Circle(A, 0.5));
draw(Circle(B, 0.5));
draw(Circle(C, 0.5));
draw(Circle(D, 0.5));
draw(Circle(E, 0.5));
draw(Circle(F, 0.5));
[/asy]
|
12
|
1/8
|
In equilateral triangle \(ABC\), a circle \(\omega\) is drawn such that it is tangent to all three sides of the triangle. A line is drawn from \(A\) to point \(D\) on segment \(BC\) such that \(AD\) intersects \(\omega\) at points \(E\) and \(F\). If \(EF = 4\) and \(AB = 8\), determine \(|AE - FD|\).
|
\frac{4\sqrt{5}}{5}
|
2/8
|
Find the smallest integer $n$ such that each subset of $\{1,2,\ldots, 2004\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$ .
|
337
|
7/8
|
The function \( f(x)=\frac{\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)+2 x^{2}+x}{2 x^{2}+\cos x} \) has a maximum value and a minimum value. Find the sum of these maximum and minimum values.
|
2
|
5/8
|
Let \( M \) and \( N \) be the midpoints of the sides \( CD \) and \( DE \) of a regular hexagon \( ABCDEF \). Let \( P \) be the intersection point of the segments \( AM \) and \( BN \).
a) Find the angle between the lines \( AM \) and \( BN \).
b) Prove that \( S_{\mathrm{ABP}} = S_{\mathrm{MDNP}} \).
|
60
|
7/8
|
Given the line $y=ax$ intersects the circle $C:x^2+y^2-2ax-2y+2=0$ at points $A$ and $B$, and $\Delta ABC$ is an equilateral triangle, then the area of circle $C$ is __________.
|
6\pi
|
7/8
|
Alice and Bob are playing a game where Alice declares, "My number is 36." Bob has to choose a number such that all the prime factors of Alice's number are also prime factors of his, but with the condition that the exponent of at least one prime factor in Bob's number is strictly greater than in Alice's. What is the smallest possible number Bob can choose?
|
72
|
4/8
|
In a set of five numbers, the average of two of the numbers is 12 and the average of the other three numbers is 7. The average of all five numbers is:
(A) $8 \frac{1}{3}$
(B) $8 \frac{1}{2}$
(C) 9
(D) $8 \frac{3}{4}$
(E) $9 \frac{1}{2}$
|
9
|
1/8
|
In a table consisting of $n$ rows and $m$ columns, numbers are written such that the sum of the elements in each row is 1248, and the sum of the elements in each column is 2184. Find the numbers $n$ and $m$ for which the expression $2n - 3m$ takes the smallest possible natural value. In the answer, specify the value of $n + m$.
|
143
|
1/8
|
Jia and Yi are dividing 999 playing cards numbered 001, 002, 003, ..., 998, 999. All the cards whose numbers have all three digits not greater than 5 belong to Jia; cards whose numbers have one or more digits greater than 5 belong to Yi.
(1) How many cards does Jia get?
(2) What is the sum of the numbers on all the cards Jia gets?
|
59940
|
6/8
|
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t\,$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t\,$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
|
163
|
1/8
|
In a country with 100 cities, between any two cities there is either no connection, an air connection, or a railway connection (both cannot exist simultaneously). It is known that if two cities are connected to a third city by railway, then there is an air route between them, and if two cities are connected to a third city by air route, then there is a railway between them. Due to a natural disaster, all air flights in the country were canceled. The government decided to set up humanitarian aid centers in some of the cities, ensuring that from each city, it is possible to travel to such a center. Prove that it is necessary to open at least 20 such centers.
|
20
|
1/8
|
Find the area of an equilateral triangle inscribed in a square with side length \(a\), where one of the vertices of the triangle coincides with a vertex of the square.
|
^2(2\sqrt{3}-3)
|
3/8
|
For which smallest natural number \( k \) is the expression \( 2018 \cdot 2019 \cdot 2020 \cdot 2021 + k \) a square of a natural number?
|
1
|
5/8
|
Given the function $f(x)= \sqrt {2}\cos (x+ \frac {\pi}{4})$, after translating the graph of $f(x)$ by the vector $\overrightarrow{v}=(m,0)(m > 0)$, the resulting graph exactly matches the function $y=f′(x)$. The minimum value of $m$ is \_\_\_\_\_\_.
|
\frac {3\pi}{2}
|
7/8
|
Let $ABC$ be an isosceles triangle with \( A \) as the vertex angle and \( D \) the foot of the internal bisector from \( B \). It is given that \( BC = AD + DB \). What is the measure of \( \widehat{BAC} \)?
|
100
|
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.