problem
stringlengths 18
2.25k
| gt
stringlengths 1
173
|
|---|---|
[ [Volume helps solve the problem]
Does a triangular pyramid exist whose heights are 1, 2, 3, and 6?
#
|
No
|
The polynomial $f(x)=x^3-4\sqrt{3}x^2+13x-2\sqrt{3}$ has three real roots, $a$, $b$, and $c$. Find
$$\max\{a+b-c,a-b+c,-a+b+c\}.$$
|
2\sqrt{3} + 2\sqrt{2}
|
91. (8th grade) Find all rectangles that can be cut into 13 equal squares.
|
1\times13
|
In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has two
lanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane,
and Sammy on the outer. They will race for one complete lap, measured by the inner track.
What is the square of the distance between Willy and Sammy's starting positions so that they will both race
the same distance? Assume that they are of point size and ride perfectly along their respective lanes
|
265 - 132\sqrt{3}
|
2.17. The base of a right prism is a rhombus. The areas of the diagonal sections of this prism are $P$ and $Q$. Find the lateral surface area of the prism.
|
2\sqrt{P^{2}+Q^{2}}
|
5. $S$ is the set of rational numbers $r$ where $0<r<1$, and $r$ has a repeating decimal expansion of the form $0 . a b c a b c a b c \cdots$ $=0 . \dot{a} b \dot{c}, a, b, c$ are not necessarily distinct. Among the elements of $S$ that can be written as a fraction in simplest form, how many different numerators are there?
|
660
|
142. What is the sum of the exterior angles at the four vertices of a convex quadrilateral?
|
360
|
The expression $\circ \ 1\ \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
[i]Proposed by Aaron Lin[/i]
|
1006
|
56.
Suppose the brother Alice met said: "I am lying today or I am Trulala." Could it be determined in this case which of the brothers it was?
|
Trulala
|
B2. What is $12 \%$ of the number $\sqrt[3]{(\sqrt{5}+2)(\sqrt{5}-2)}+2^{-\log _{10} 0.01}$? Write down the answer.
|
\frac{3}{5}
|
7.2. On his birthday, Nikita decided to treat his classmates and took 4 bags of candies to school, each containing the same number of candies. He gave out 20 candies to his classmates, which was more than $60 \%$ but less than $70 \%$ of all the candies he had. How many candies did Nikita have in total?
|
32
|
Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.
|
\angle CED = 45^\circ
|
7. Each cell of a $3 \times 3$ square has a number written in it. Unfortunately the numbers are not visible because they are covered in ink. However, the sum of the numbers in each row and the sum of the numbers in two of the columns are all known, as shown by the arrows on the diagram. What is the sum of the numbers in the third column?
A 41
B 43
C 44
D 45
E 47
|
43
|
4. Bivariate function
$$
\begin{array}{l}
f(x, y) \\
=\sqrt{\cos 4 x+7}+\sqrt{\cos 4 y+7}+ \\
\quad \sqrt{\cos 4 x+\cos 4 y-8 \sin ^{2} x \cdot \sin ^{2} y+6}
\end{array}
$$
The maximum value is . $\qquad$
|
6\sqrt{2}
|
## Task B-1.3.
Marko spent 100 euros after receiving his salary. Five days later, he won $\frac{1}{4}$ of the remaining amount from his salary in a lottery and spent another 100 euros. After fifteen days, he received $\frac{1}{4}$ of the amount he had at that time and spent another 100 euros. In the end, he had 800 euros more than he had at the moment he received his salary. What was Marko's salary?
|
2100
|
Example 1 If numbers $1,2, \cdots, 14$ are taken in ascending order as $a_{1}, a_{2}, a_{3}$, such that both $a_{2}-$ $a_{1} \geqslant 3$ and $a_{3}-a_{2} \geqslant 3$ are satisfied, then the total number of different ways to select the numbers is $\qquad$.
(1989 National High School League Question)
|
120
|
8. (4 points) A triangle is divided into 1000 triangles. What is the maximum number of different points at which the vertices of these triangles can be located?
|
1002
|
1. (2004 National College Entrance Examination Chongqing Paper) The solution set of the inequality $x+\frac{2}{x+1}>2$ is
A. $(-1,0) \cup(1,+\infty)$
B. $(-\cdots,-1) \cup(0,1)$
C. $(-1,0) \cup(0,1)$
D. $(-\infty,-1) \cup(1,+\infty)$
|
A
|
5. (7 points) Three schoolgirls entered a store. Anya bought 2 pens, 7 pencils, and 1 notebook, Varya - 5 pens, 6 pencils, and 5 notebooks, Sasha - 8 pens, 4 pencils, and 9 notebooks. They all paid equally, but one of them used a discount when paying. Who? (Explain your answer).
|
Varya
|
Task 2. Mom baked a cake for Anya's birthday, which weighs an integer number of grams. Before decorating it, Mom weighed the cake on digital scales that round the weight to the nearest ten grams (if the weight ends in 5, the scales round it down). The result was 1440 g. When Mom decorated the cake with identical candles, the number of which was equal to Anya's age, the scales showed 1610 g. It is known that the weight of each candle is an integer number of grams, and if you put one candle on the scales, they will show 40 g. How old can Anya be? List all possible answers and explain why there are no others. (20 points)
|
4
|
2. Let the 20 vertices of a regular 20-sided polygon inscribed in the unit circle in the complex plane correspond to the complex numbers $z_{1}, z_{2}, \cdots, z_{20}$. Then the number of distinct points corresponding to $z_{1}^{2015}, z_{2}^{2015}, \cdots, z_{20}^{2015}$ is $\qquad$
|
4
|
Example 7 (2006 Zhejiang Province High School Mathematics Summer Camp Problem) If the solution set of the inequality about $x$
$$
a x^{2}-|x+1|+2 a<0
$$
is an empty set, then the range of values for $a$ is $\qquad$ .
|
[\frac{\sqrt{3}+1}{4},+\infty)
|
2. The octahedron $A B C D E F$ has a square $A B C D$ as its base, while the line $E F$ is normal to the plane determined by the square $A B C D$ and passes through its center. It is known that the sphere that touches all faces of the octahedron and the sphere that touches all lateral edges (i.e., $E A, E B, E C, E D, F A, F B, F C$ and $F D$) have the same center, and that the sphere that touches the lateral edges has a surface area that is $50\%$ larger. Find the ratio $\frac{E F}{A B}$.
|
\sqrt{2}
|
Example 1 As shown in Figure 1, two utility poles $A B$ and $C D$ are erected on a construction site. They are $15 \mathrm{~m}$ apart, and points $A$ and $C$ are $4 \mathrm{~m}$ and $6 \mathrm{~m}$ above the ground, respectively. Steel wires are stretched from these points to points $E$ and $D$, and $B$ and $F$ on the ground to secure the poles. The height of the intersection point $P$ of the steel wires $A D$ and $B C$ above the ground is $\qquad$ $\mathrm{m}$.
(2000, National Junior High School Mathematics Competition)
|
\frac{12}{5}
|
2. Find the length of the side of a square whose area is numerically equal to its perimeter.
|
4
|
1 Positive integer $n$ satisfies the following property: when $n$ different odd numbers are chosen from $1,2, \cdots, 100$, there must be two whose sum is 102. Find the minimum value of $n$.
|
27
|
Five children had dinner. Chris ate more than Max. Brandon ate less than Kayla. Kayla ate less than Max but more than Tanya. Which child ate the second most?
(A) Brandon
(B) Chris
(C) Kayla
(D) Max
(E) Tanya
|
D
|
## 4. Magic Table
In each cell of the table in the picture, one of the numbers $11, 22, 33, 44, 55$, 66, 77, 88, and 99 is written. The sums of the three numbers in the cells of each row and each column are equal. If the number 33 is written in the central cell of the table, what is the sum of the numbers written in the four colored cells?
Result: $\quad 198$
|
198
|
6. The range of the function $f(x)=\sqrt{3 x-6}+\sqrt{3-x}$ is
$\qquad$
|
[1,2]
|
Problem 11.3. A natural number $n$ is called interesting if $2 n$ is a perfect square, and $15 n$ is a perfect cube. Find the smallest interesting number.
Answer: 1800.
|
1800
|
Zamyatin $B$.
Vladimir wants to make a set of cubes of the same size and write one digit on each face of each cube so that he can use these cubes to form any 30-digit number. What is the smallest number of cubes he will need? (The digits 6 and 9 do not turn into each other when flipped.)
#
|
50
|
3. Let the complex numbers be
$$
z_{1}=\sin \alpha+2 \mathrm{i}, z_{2}=1+\cos \alpha \cdot \mathrm{i}(\alpha \in \mathbf{R}) \text {. }
$$
Then the minimum value of $f=\frac{13-\left|z_{1}+\mathrm{i} z_{2}\right|^{2}}{\left|z_{1}-\mathrm{i} z_{2}\right|}$ is . $\qquad$
|
2
|
6.370 Solve the equation $\sqrt[4]{x^{4}+x-2}+\sqrt{x^{4}+x-2}=6$ given that $\boldsymbol{x}>\mathbf{0}$.
|
2
|
Problem 4. In the isosceles trapezoid $ABCD$ with leg length 10, one of the bases is twice as large as the other. If one of the angles of the trapezoid is $75^{\circ}$, calculate its area.
|
75
|
2. Calculate: $(1234+2341+3412+4123) \div(1+2+3+4)=$
|
1111
|
4. (3 points) On the Island of Misfortune, there live knights who always tell the truth, and liars who always lie. One day, $n$ islanders gathered in a room.
The first one said: "Exactly every second person in this room is a liar."
The second one said: "Exactly every third person in this room is a liar."
and so on
The person with number $n$ said: "Exactly every ( $n_{33} 1$ )-th person in this room is a liar."
How many people could have been in the room, given that not all of them are liars?
|
2
|
## Task 6/82
Given the inequality $|x|+|y| \leq n$ with $n \in N, x ; y \in G$ (where $G$ denotes the set of integers). Determine the number of ordered solution pairs $(x ; y)$.
|
n^{2}+(n+1)^{2}
|
14.3.16 ** (1) For what integers $n>2$, are there $n$ consecutive positive integers, where the largest number is a divisor of the least common multiple of the other $n-1$ numbers?
(2) For what integers $n>2$, is there exactly one set of positive integers with the above property?
|
4
|
Given is the function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}$ that satisfies the properties:
(i) $f(p)=1$ for all prime numbers $p$,
(ii) $f(x y)=y f(x)+x f(y)$ for all $x, y \in \mathbb{Z}_{>0}$.
Determine the smallest $n \geq 2016$ with $f(n)=n$.
|
3125
|
4. Given that $\left\{a_{n}\right\}$ is a geometric sequence, $a_{2}=2, a_{5}=\frac{1}{4}$, then the range of $a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{n} a_{n+1}\left(n \in \mathbf{N}^{*}\right)$ is . $\qquad$
|
[8,\frac{32}{3})
|
2. If $\mathrm{i}$ is the imaginary unit, and the complex number $z=\frac{\sqrt{3}}{2}+\frac{1}{2} \mathrm{i}$, then the value of $z^{2016}$ is
A. -1
B. $-\mathrm{i}$
C. i
D. 1
|
1
|
Problem 11.6. Oleg wrote down several composite natural numbers less than 1500 on the board. It turned out that the greatest common divisor of any two of them is 1. What is the maximum number of numbers that Oleg could have written down?
|
12
|
2. (7 points) Find all natural solutions to the equation $2 n-\frac{1}{n^{5}}=3-\frac{2}{n}$.
#
|
1
|
In an acute triangle $ ABC$, let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[AB\right]$ such that $ \angle ADB\equal{}\angle AEC\equal{}90{}^\circ$. If perimeter of triangle $ AED$ is 9, circumradius of $ AED$ is $ \frac{9}{5}$ and perimeter of triangle $ ABC$ is 15, then $ \left|BC\right|$ is
$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ \frac{24}{5} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \frac{48}{5}$
|
\frac{24}{5}
|
$12.40 f(x)=\frac{2^{2 x}}{\sqrt{2-2^{2 x}}} ; f^{\prime}(0)=?$
|
3\ln2
|
A positive integer $n$ is called "strong" if there exists a positive integer $x$ such that $x^{nx} + 1$ is divisible by $2^n$.
a. Prove that $2013$ is strong.
b. If $m$ is strong, determine the smallest $y$ (in terms of $m$) such that $y^{my} + 1$ is divisible by $2^m$.
|
2^m - 1
|
[ Extremal Properties (other) ] [ Examples and Counterexamples. Constructions ]
The number of edges of a convex polyhedron is 99. What is the maximum number of edges that a plane, not passing through its vertices, can intersect?
|
66
|
2. Given $x, y \in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right], a \in R$, and $x^{3}+\sin x-2 a=0,4 y^{3}+\sin y \cos y+a=0$. Then $\cos (x+2 y)=$ $\qquad$ .
|
1
|
2.3. Find all values of $a$ for which the quadratic function $f(x)=a x^{2}+2 a x-1$ takes values, the modulus of which does not exceed 3, at all points of the interval $[-2 ; 0]$. In your answer, specify the total length of the intervals to which the found values of $a$ belong.
|
6
|
8-2. On a distant planet in the mangrove forests, there lives a population of frogs. The number of frogs born each year is one more than the difference between the number of frogs born in the previous two years (the larger number minus the smaller number).
For example, if 5 frogs were born last year and 2 in the year before, then 4 frogs will be born this year; if 3 frogs were born last year and 10 in the year before, then 8 frogs will be born this year.
In 2020, 2 frogs were born, and in 2021, 9. In which nearest year will only 1 frog be born for the first time?
|
2033
|
3. In how many different ways can four friends Ankica, Gorana, Ivana, and Melita divide 11 identical fries among themselves, so that each gets at least two? The fries cannot be cut into smaller pieces.
|
20
|
2. Find the pattern and fill in the number:
$$
100,81,64,49 \text {, }
$$
$\qquad$ $, 25,16,9,4,1$ .
|
36
|
4. In $\triangle A B C$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are denoted as $a$, $b$, and $c$ respectively. If $b^{2}=a^{2}+c^{2}-a c$, and $c-a$ equals the height $h$ from vertex $A$ to side $A C$, then $\sin \frac{C-A}{2}=$ $\qquad$
|
\frac{1}{2}
|
Second Part (8 points per question, total 40 points)
6. A box contains 10 black balls, 9 white balls, and 8 red balls. If you close your eyes and draw balls from the box, how many balls must you draw to ensure that you have at least one red ball and one white ball? $\qquad$
|
20
|
5 Let $0<2a<1, M=1-a^{2}, N=1+a^{2}, P=\frac{1}{1-a}, Q=\frac{1}{1+a}$, then
A. $Q<P<M<N$
B. $M<N<Q<P$
C. $Q<M<N<P$
D. $M<Q<P<N$
|
C
|
1. Given real numbers $a>0, b>0$, satisfying $a+\sqrt{a}=2008, b^{2}+b=2008$. Then the value of $a+b$ is $\qquad$
|
2008
|
Example 9 Let $k \geqslant 2$ be a fixed positive integer, and $k$ positive integers $a_{1}, a_{2}, \cdots, a_{k}$ such that $n=\frac{\left(a_{1}+a_{2}+\cdots+a_{k}\right)^{2}}{a_{1} a_{2} \cdots a_{k}}$ is a positive integer. Find the maximum value of $n$.
|
k^{2}
|
37. Find the remainder when $1^{2017}+2^{2017}+3^{2017}+4^{2017}+5^{2017}$ is divided by 5. (where $a^{2017}$ denotes 2017 instances of $a$ multiplied together)
|
0
|
13.335. If a two-digit number is divided by the product of its digits, the quotient is 3 and the remainder is 8. If the number, formed by the same digits but in reverse order, is divided by the product of the digits, the quotient is 2 and the remainder is 5. Find this number.
|
53
|
214. A dog is chasing a rabbit that is 150 feet ahead of it. It makes a leap of 9 feet every time the rabbit jumps 7 feet. How many jumps must the dog make to catch the rabbit?
## Herbert's Problem.
|
75
|
## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{x^{3}-6 x^{2}+11 x-10}{(x+2)(x-2)^{3}} d x
$$
|
\ln|x+2|+\frac{1}{2(x-2)^{2}}+C
|
8. Recently, an article on the internet has gone viral. It originates from a common question, as shown in Figure 1. This seemingly easy-to-solve problem actually contains profound wisdom.
Let $a, b \in\{2,3, \cdots, 8\}$. Then the maximum value of $\frac{a}{10 b+a}+\frac{b}{10 a+b}$ is $\qquad$
|
\frac{89}{287}
|
## Problem Statement
Find the $n$-th order derivative.
$y=\frac{4 x+7}{2 x+3}$
|
y^{(n)}=\frac{(-1)^{n}\cdot2^{n}\cdotn!}{(2x+3)^{n+1}}
|
Example 2. If $\lg ^{2} x \lg 10 x<0$, find the value of $\frac{1}{\lg 10 x} \sqrt{\lg ^{2} x+\lg 10 x^{2}}$.
|
-1
|
13.024. A tractor team can plow 5/6 of a plot of land in 4 hours and 15 minutes. Before the lunch break, the team worked for 4.5 hours, after which 8 hectares remained unplowed. How large was the plot?
|
68
|
15. Let $0<\theta<\pi$, find the maximum value of $\sin \frac{\theta}{2}(1+\cos \theta)$.
|
\frac{4\sqrt{3}}{9}
|
Let $S$ be a real number. It is known that however we choose several numbers from the interval $(0, 1]$ with sum equal to $S$, these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5.
Find the greatest possible value of $S$.
|
5.5
|
20. Let $a_{1}, a_{2}, \ldots$ be a sequence satisfying the condition that $a_{1}=1$ and $a_{n}=10 a_{n-1}-1$ for all $n \geq 2$. Find the minimum $n$ such that $a_{n}>10^{100}$.
|
102
|
8,9 |
The center of a circle with a radius of 5, circumscribed around an isosceles trapezoid, lies on the larger base, and the smaller base is equal to 6. Find the area of the trapezoid.
|
32
|
2.85 Given that $a, b, c, d, e$ are real numbers satisfying
$$\begin{array}{l}
a+b+c+d+e=8 \\
a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16
\end{array}$$
determine the maximum value of $e$.
|
\frac{16}{5}
|
## Task 3 - 180613
Fred, Gerd, Hans, and Ingo are students in classes 6a, 6b, 7a, and 7b, and each of these classes has one of the four students.
In a conversation involving only Fred and the two students from 7th grade, Hans notes that three of the four students read only one of the magazines "alpha" and "technikus," namely Fred, Gerd, and the student from 6a.
The student from 7b, however, reads both "technikus" and the magazine "alpha."
To which class does each of the four students belong according to these statements, and which student reads both magazines "alpha" and "technikus"?
|
Hans
|
Let $ABC$ be an equilateral triangle whose angle bisectors of $B$ and $C$ intersect at $D$. Perpendicular bisectors of $BD$ and $CD$ intersect $BC$ at points $E$ and $Z$ respectively.
a) Prove that $BE=EZ=ZC$.
b) Find the ratio of the areas of the triangles $BDE$ to $ABC$
|
\frac{1}{9}
|
## Task Condition
Find the derivative of the specified order.
$$
y=\left(1-x-x^{2}\right) e^{\frac{x-1}{2}}, y^{IV}=?
$$
|
-\frac{1}{16}\cdot(55+17x+x^{2})e^{\frac{x-1}{2}}
|
153 Let the function $f(x)=1-|1-2 x|, g(x)=x^{2}-2 x+1, x \in[0,1]$, and define
$$
F(x)=\left\{\begin{array}{ll}
f(x) & f(x) \geqslant g(x), \\
g(x) & f(x)<g(x) .
\end{array}\right.
$$
Then the number of real roots of the equation $F(x) \cdot 2^{x}=1$ is
|
3
|
9. Ancient Greek mathematicians associated natural numbers with polygons in the following way, defining polygonal numbers: Triangular numbers: 1, 3, 6, 10, 15...
Square numbers: $1, 4, 9, 16, 25, \ldots$
Pentagonal numbers: $1, 5, 12, 22, 35, \ldots$
Hexagonal numbers: $1, 6, 15, 28, 45, \ldots$
$\qquad$
Then, according to the above sequence, the 8th hexagonal number is ( )
|
120
|
10.4 Let $n$ - be a natural number greater than 10. What digit can stand immediately after the decimal point in the decimal representation of the number $\sqrt{n^{2}+n}$? Provide all possible answers and prove that there are no others.
|
4
|
An $n \times n$ complex matrix $A$ is called \emph{t-normal} if
$AA^t = A^t A$ where $A^t$ is the transpose of $A$. For each $n$,
determine the maximum dimension of a linear space of complex $n
\times n$ matrices consisting of t-normal matrices.
Proposed by Shachar Carmeli, Weizmann Institute of Science
|
\frac{n(n+1)}{2}
|
3. Inside triangle $ABC$, a point $P$ is chosen such that $AP=BP$ and $CP=AC$. Find $\angle CBP$, given that $\angle BAC = 2 \angle ABC$.
---
Here is the translation of the provided text, maintaining the original formatting and structure.
|
30
|
Let $C$ and $D$ be points on the semicircle with center $O$ and diameter $AB$ such that $ABCD$ is a convex quadrilateral. Let $Q$ be the intersection of the diagonals $[AC]$ and $[BD]$, and $P$ be the intersection of the lines tangent to the semicircle at $C$ and $D$. If $m(\widehat{AQB})=2m(\widehat{COD})$ and $|AB|=2$, then what is $|PO|$?
$
\textbf{(A)}\ \sqrt 2
\qquad\textbf{(B)}\ \sqrt 3
\qquad\textbf{(C)}\ \frac{1+\sqrt 3} 2
\qquad\textbf{(D)}\ \frac{1+\sqrt 3}{2\sqrt 2}
\qquad\textbf{(E)}\ \frac{2\sqrt 3} 3
$
|
\frac{2\sqrt{3}}{3}
|
6. (i) Consider two positive integers $a$ and $b$ which are such that $\boldsymbol{a}^{\boldsymbol{a}} \boldsymbol{b}^{\boldsymbol{b}}$ is divisible by 2000 . What is the least possible value of the product $a b$ ?
(ii) Consider two positive integers $a$ and $b$ which are such that $\boldsymbol{a}^{\boldsymbol{b}} \boldsymbol{b}^{\boldsymbol{a}}$ is divisible by 2000 . What is the least possible value of the product $a b$ ?
|
20
|
Find the continuous function $ f(x)$ such that $ xf(x)\minus{}\int_0^x f(t)\ dt\equal{}x\plus{}\ln (\sqrt{x^2\plus{}1}\minus{}x)$ with $ f(0)\equal{}\ln 2$.
|
f(x) = \ln (1 + \sqrt{x^2 + 1})
|
## Problem A1
Find all positive integers $n<1000$ such that the cube of the sum of the digits of $n$ equals $n^{2}$.
|
1,27
|
2. How many positive values can the expression
$$
a_{0}+3 a_{1}+3^{2} a_{2}+3^{3} a_{3}+3^{4} a_{4}
$$
take if the numbers $a_{0}, a_{1}, a_{2}, a_{3}$ and $a_{4}$ are from the set $\{-1,0,1\}$?
|
121
|
Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, and $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$, where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$.
|
108
|
4. Given an acute triangle $\triangle A B C$ with three interior angles satisfying $\angle A>\angle B>\angle C$, let $\alpha$ represent the minimum of $\angle A-\angle B$, $\angle B-\angle C$, and $90^{\circ}-\angle A$. Then the maximum value of $\alpha$ is $\qquad$
|
15^{\circ}
|
## Task 1 - 050611
Two cyclists start simultaneously from Leipzig and Dresden (distance $119 \mathrm{~km}$). The cyclist from Leipzig is heading to Dresden, and the one from Dresden is heading to Leipzig. One of them travels at $15 \mathrm{~km}$ per hour, the other at $20 \mathrm{~km}$ per hour.
a) What is the distance between the two cyclists after $2 \frac{1}{2}$ hours?
b) How far are they from both cities when they meet?
|
31.5
|
10.147. In a parallelogram with a perimeter of 32 cm, the diagonals are drawn. The difference between the perimeters of two adjacent triangles is 8 cm. Find the lengths of the sides of the parallelogram.
|
12
|
Suppose a continuous function $f:[0,1]\to\mathbb{R}$ is differentiable on $(0,1)$ and $f(0)=1$, $f(1)=0$. Then, there exists $0<x_0<1$ such that
$$|f'(x_0)| \geq 2018 f(x_0)^{2018}$$
|
|f'(x_0)| \geq 2018 f(x_0)^{2018}
|
The numbers which contain only even digits in their decimal representations are written in ascending order such that \[2,4,6,8,20,22,24,26,28,40,42,\dots\] What is the $2014^{\text{th}}$ number in that sequence?
$
\textbf{(A)}\ 66480
\qquad\textbf{(B)}\ 64096
\qquad\textbf{(C)}\ 62048
\qquad\textbf{(D)}\ 60288
\qquad\textbf{(E)}\ \text{None of the preceding}
$
|
62048
|
6. A circle can be circumscribed around quadrilateral $A B C D$. The lengths of sides $A B$ and $A D$ are equal.
On side $C D$ is point $Q$ such that $D Q=1$, and on side $B C$ is point $P$ such that $B P=5$.
Furthermore, $\measuredangle D A B=2 \measuredangle Q A P$. Find the length of segment $P Q$.
## Solutions and answers
## Variant 1
Problem 1 Answer: $7-\frac{\sqrt{2}}{2}$
|
7-\frac{\sqrt{2}}{2}
|
3. Given a function $f(n)$ defined on the set of positive integers satisfies the conditions:
(1) $f(m+n)=f(m)+f(n)+m n\left(m, n \in \mathbf{N}_{+}\right)$;
(2) $f(3)=6$.
Then $f(2011)=$ . $\qquad$
|
2023066
|
# Task 3.
Maximum 15 points
In an equilateral triangle with area $S_{1}$, a circle is inscribed, and in this circle, an equilateral triangle with area $S_{2}$ is inscribed. In the resulting new triangle, another circle is inscribed, and in this circle, another equilateral triangle with area $S_{3}$ is inscribed. The procedure is repeated $n$ times. The area of the triangle $S_{n}$ turned out to be 1. Find the function $S_{1}(n)$.
|
S_{1}(n)=4^{n-1}
|
Positive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $ax^2-bx+c=0$ has exactly one real solution $r$, and this value $r$ is less than $100$. Compute the maximum possible value of $r$.
|
64
|
86. Chen Gang and Zhang Qiang started from their respective homes at the same time and walked towards each other. They first met at a point 5 kilometers away from Chen Gang's home. After meeting, they continued to walk at their original speeds. Chen Gang reached Zhang Qiang's home, and Zhang Qiang reached Chen Gang's home, and both immediately returned. As a result, they met again at a point 3 kilometers away from Zhang Qiang's home. Therefore, the distance between Chen Gang's home and Zhang Qiang's home is $\qquad$ kilometers.
|
12
|
Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:
(a) If $16$, $15$, or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and
(b) There are three integers $0 < x < y < z < 14$ such that when $x$, $y$, or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.
Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.
|
148
|
13.4. 19 ** In $\triangle A B C$ with a fixed perimeter, it is known that $|A B|=6$, and when vertex $C$ is at a fixed point $P$, $\cos C$ has a minimum value of $\frac{7}{25}$.
（1）Establish an appropriate coordinate system and find the equation of the locus of vertex $C$;
(2) Draw a line through point $A$ that intersects the curve from (1) at points $M$ and $N$, and find the minimum value of $|\overrightarrow{B M}| \cdot|\overrightarrow{B N}|$.
|
16
|
17. Given a triangle $\triangle A B C, \angle A B C=80^{\circ}, \angle A C B=70^{\circ}$ and $B C=2$. A perpendicular line is drawn from $A$ to $B C$, another perpendicular line drawn from $B$ to $A C$. The two perpendicular lines meet at $H$. Find the length of AH.
(2 marks)
17. 在 $\triangle A B C$ 中, $\angle A B C=80^{\circ}, \angle A C B=70^{\circ}$, 且 $B C=2$ 。由 $A$ 作線垂直於 $B C$, 由 $B$ 作線垂直於 $A C$,兩垂直線相交於 $H$ 。求 $A H$ 的長度。
|
2\sqrt{3}
|
2. Calculate $\sin ^{2} 20^{\circ}+\cos ^{2} 50^{\circ}+\sin 20^{\circ} \cos 50^{\circ}=$
|
\frac{3}{4}
|
Find the functions from $\mathbb{R}$ to $\mathbb{R}$ that satisfy:
$$
f(x+y)+f(x) f(y)=f(x y)+2 x y+1
$$
|
f(x)=x^2-1
|
1B. Does there exist a real number $y$ such that the numbers $\sqrt{y^{2}+2 y+1}, \frac{y^{2}+3 y-1}{3}$, $y-1$ in the given order, are three consecutive terms of an arithmetic progression? Explain your answer.
|
y\in{-2,-1,1}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.