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3. Determine the primitive $F: \mathbb{R} \longrightarrow \mathbb{R}$ of the function $f: \mathbb{R} \longrightarrow \mathbb{R}$
$$
f(x)=\frac{\sin x \cdot \sin \left(x-\frac{\pi}{4}\right)}{e^{2 x}+\sin ^{2} x}
$$
for which $F(0)=0$.
|
\dfrac{\sqrt{2}}{4} \left( 2x - \ln\left(e^{2x} + \sin^2 x\right) \right)
|
1/8
|
In triangle $ABC$, let $AB = AC$. The angle bisectors of $\angle CAB$ and $\angle ABC$ intersect the sides $BC$ and $CA$ at points $D$ and $E$, respectively. Let $K$ be the incenter of triangle $ADC$. Suppose that $\angle BEK = 45^{\circ}$. Determine all possible values of $\angle CAB$.
---
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
|
90^\circ
|
3/8
|
If there are $2 k(k \geqslant 3)$ points on a plane, where no three points are collinear. Draw a line segment between any two points, and color each line segment red or blue. A triangle with three sides of the same color is called a monochromatic triangle, and the number of monochromatic triangles is denoted as $S$. For all possible coloring methods, find the minimum value of $S$.
|
2 \dbinom{k}{3}
|
0/8
|
Question 6 As shown in Figure 7, there is a semi-elliptical steel plate, with the length of the major semi-axis being $2 r$ and the length of the minor semi-axis being $r$. It is planned to cut this steel plate into the shape of an isosceles trapezoid, with the lower base $A B$ being the minor axis of the semi-ellipse, and the upper base $C D$ having endpoints on the ellipse, denoted as $C D=2 x$, and the area of the trapezoid as $S$.
(1) Find the function of the area $S$ with $x$ as the independent variable, and write down its domain;
(2) Find the maximum value of the area $S$.
|
\dfrac{3\sqrt{3}}{2} r^2
|
2/8
|
1. Find the smallest positive integer $n$, such that $n$ is divisible by 3 and the product of its digits is 882.
|
13677
|
3/8
|
Bakayev E.B.
A boy and a girl were sitting on a long bench. Twenty more children approached them one by one, and each of them sat between two of the already seated children. We will call a girl brave if she sat between two neighboring boys, and a boy brave if he sat between two neighboring girls. When everyone was seated, it turned out that the boys and girls were sitting on the bench, alternating. How many of them were brave?
|
10
|
4/8
|
5. A natural number, if the sum of its digits equals the product of its digits, is called a "coincidence number". Among five-digit numbers, there are $\qquad$ "coincidence numbers".
|
40
|
4/8
|
To [i]dissect [/i] a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals:
[img]https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc90412c3eada855ae52b45.png[/img]
An [i]integer-ratio[/i] right triangle is a right triangle whose side lengths are in an integer ratio. For example, a triangle with sides $3,4,5$ is an[i] integer-ratio[/i] right triangle, and so is a triangle with sides $\frac52 \sqrt3 ,6\sqrt3, \frac{13}{2} \sqrt3$. On the other hand, the right triangle with sides$ \sqrt2 ,\sqrt5, \sqrt7$ is not an [i]integer-ratio[/i] right triangle. Determine, with proof, all integers $n$ for which it is possible to completely [i]dissect [/i] a regular $n$-sided polygon into [i]integer-ratio[/i] right triangles.
|
4
|
5/8
|
B1. We call a sequence of consecutive positive integers balanced if the number of multiples of three in that sequence is equal to the number of multiples of five. For example, the sequence 30, 31, 32, 33, 34, 35, 36 is not balanced, because out of these 7 numbers, 3 are multiples of three (namely 30, 33, and 36) and only 2 are multiples of five (namely 30 and 35).
How many numbers can a balanced sequence of consecutive positive integers contain at most?
|
11
|
1/8
|
## Task B-3.5.
A trapez with mutually perpendicular diagonals has bases of length $a=12$ and $c=4$, and the extensions of the legs of the trapezoid intersect at an angle $\alpha$. If $\cos \alpha=\frac{4}{5}$, calculate the area of this trapezoid.
|
36
|
3/8
|
4. A cylinder is intersected by a plane, and the shape of the intersection is an ellipse. The eccentricity of this ellipse is $\frac{\sqrt{5}}{3}$. The acute angle formed between this intersection and the base of the cylinder is . $\qquad$
|
\arccos\left(\dfrac{2}{3}\right)
|
1/8
|
Let $n$ be a positive integer. Let $S$ be a set of ordered pairs $(x, y)$ such that $1\leq x \leq n$ and $0 \leq y \leq n$ in each pair, and there are no pairs $(a, b)$ and $(c, d)$ of different elements in $S$ such that $a^2+b^2$ divides both
$ac+bd$ and $ad - bc$. In terms of $n$, determine the size of the largest possible set $S$.
|
\dfrac{n(n +1)}{2}
|
0/8
|
The angle between the height of a regular triangular pyramid and a lateral edge is $\alpha (\alpha < \pi / 4)$. In what ratio does the center of the circumscribed sphere divide the height of the pyramid?
|
\cos 2\alpha : 1
|
2/8
|
Given a positive integer $n (n>2004)$, we put 1, 2, 3, …,$n^2$ into squares of an $n\times n$ chessboard with one number in a square. A square is called a “good square” if the square satisfies following conditions:
1) There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is less than the number within the square.
2) There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares is less than the number within the square.
Find the maximum value of the number of the “good square”.
|
n(n - 2004)
|
0/8
|
10. Centipedes and three-headed dragons. In a certain herd of centipedes and three-headed dragons, there are a total of 26 heads and 298 legs. Each centipede has one head. How many legs does a three-headed dragon have?
|
14
|
0/8
|
9.2. Six people - liars and knights - sat around a table. Liars always lie, while knights always tell the truth. Each of them was given a coin. Then each of them passed their coin to one of their two neighbors. After that, 3 people said: "I have one coin," while the other 3 said: "I have no coins." What is the maximum number of knights that could have been sitting at the table?
|
4
|
3/8
|
8. There are five table tennis balls, three of which are new, and two are old (i.e., used at least once). Each time a match is played, two balls are taken out and used, and then all are put back. Let the number of new balls taken in the second match be $\xi$, then the mathematical expectation $E \xi=$ $\qquad$ .
|
\dfrac{18}{25}
|
2/8
|
The vertices of the equilateral triangle $ABC$ are at distances of $2, 3, 5$ units from a point $D$ lying in the plane of the triangle. Calculate the length of the side of the triangle.
|
\sqrt{19}
|
4/8
|
In triangle $ABC$, let $I, O, H$ be the incenter, circumcenter and orthocenter, respectively. Suppose that $AI = 11$ and $AO = AH = 13$. Find $OH$.
[i]Proposed by Kevin You[/i]
|
10
|
4/8
|
7.1. (GDR, 74). What is greater: $\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}-\sqrt{2}$ or 0?
|
0
|
5/8
|
12. Six weights look the same, with masses of 101 grams, 102 grams, 103 grams, 104 grams, 105 grams, and 106 grams. Place these six weights on a balance, three on each side, and the result is that the right side of the balance is heavier. The probability that the 106-gram weight is on the right side of the balance is $\qquad$ $\%$.
|
80
|
5/8
|
2. In the final stage of a professional bowling tournament, the top five players compete as follows: First, the fifth-place player competes with the fourth-place player, the loser gets fifth place, the winner competes with the third-place player; the loser gets third place, the winner competes with the first-place player, the loser gets second place, the winner gets first place. How many different possible orders of finish are there?
(39th American High School Mathematics Examination)
|
16
|
2/8
|
## Task A-4.3.
Determine all natural numbers $m$ and $n$ for which $2^{n}+5 \cdot 3^{m}$ is a square of some natural number.
|
m=2
|
2/8
|
Find all integers $k$ for which, there is a function $f: N \to Z$ that satisfies:
(i) $f(1995) = 1996$
(ii) $f(xy) = f(x) + f(y) + kf(m_{xy})$ for all natural numbers $x, y$,where$ m_{xy}$ denotes the greatest common divisor of the numbers $x, y$.
Clarification: $N = \{1,2,3,...\}$ and $Z = \{...-2,-1,0,1,2,...\}$ .
|
-1
|
3/8
|
6. Klárka had a three-digit number written on a piece of paper. When she correctly multiplied it by nine, she got a four-digit number that started with the same digit as the original number, the middle two digits were the same, and the last digit was the sum of the digits of the original number. What four-digit number could Klárka have gotten?
|
2007
|
1/8
|
Example 4 Let real numbers $x_{1}, x_{2}, \cdots, x_{1999}$ satisfy the condition $\sum_{i=1}^{1990}\left|x_{i}-x_{i+1}\right|=1991$.
And $y_{k}=\frac{1}{k} \sum_{i=1}^{k} x_{i}(k=1,2, \cdots, 1991)$. Try to find the maximum value of $\sum_{i=1}^{1990}\left|y_{i}-y_{i+1}\right|$. ${ }^{[3]}$
|
1990
|
4/8
|
Problem 2. Let $\left\{a_{n}\right\}_{n=1}^{\infty}$ be a sequence of integer numbers such that
$$
(n-1) a_{n+1}=(n+1) a_{n}-2(n-1)
$$
for any $n \geq 1$. If 2000 divides $a_{1999}$, find the smallest $n \geq 2$ such that 2000 divides $a_{n}$.
Oleg Mushkarov, Nikolai Nikolov
|
249
|
3/8
|
8. A middle school has 35 lights on each floor. To save electricity while ensuring the lighting needs of the corridors, the following requirements must be met:
(1) Two adjacent lights cannot be on at the same time;
(2) Any three consecutive lights cannot be off at the same time.
If you were to design different lighting methods, what is the maximum number of different lighting methods you can design?
$\qquad$ kinds of different lighting methods.
|
31572
|
4/8
|
(3) Let $\left\{a_{n}\right\}$ be an arithmetic sequence with the sum of the first $n$ terms denoted as $S_{n}$. If $S_{15}>0, S_{16}<0$, then the largest among $\frac{S_{1}}{a_{1}}$,
$\frac{S_{2}}{a_{2}}, \cdots, \frac{S_{15}}{a_{15}}$ is $\qquad$.
|
\dfrac{S_{8}}{a_{8}}
|
1/8
|
There are three prisoners in a prison. A warden has 2 red and 3 green hats and he has decided to play the following game: He puts the prisoners in a row one behind the other and on the had of each prisoner he puts a hat. The first prisoner in the row can't see any of the hats, the second prisoner can see only the hat at the had of the first one, and the third prisoner can see the hats of the first two prisoners. If some of the prisoners tells the color of his own hat, he is free; but if he is wrong, the warden will kill him. If a prisoner remain silent for sufficiently long, he is returned to his cell. Of course, each of them would like to be rescued from the prison, but if he isn't sure about the color of his hat, he won't guess.
After noticing that second and third prisoner are silent for a long time, first prisoner (the one who doesn't see any hat) has concluded the color of his hat and told that to the warden. What is the color of the hat of the first prisoner? Explain your answer! (All prisoners know that there are 2 red and 3 green hats in total and all of them are good at mathematics.)
|
green
|
4/8
|
B3 A $3 \times 3$ grid of 9 dots labeled by $A, B, C, D, E, F, K, L$, and $M$ is shown in the figure. There is one path connecting every pair of adjacent dots, either orthogonal (i.e. horizontal or vertical) or diagonal. A turtle walks on this grid, alternating between orthogonal and diagonal moves. One could describe any sequence of paths in terms of the letters $A, \cdots, M$. For example, $A-B-F$ describes a sequence of two paths $A B$ and $B F$.
What is the maximum number of paths the turtle could traverse, given that it does not traverse any path more than once?
|
17
|
1/8
|
3. If $a>b>0$, then the minimum value of $a^{3}+\frac{1}{b(a-b)}$ is $\qquad$ .
|
\frac{20}{\sqrt[5]{1728}}
|
0/8
|
## Task B-2.7.
Let $A B C D$ be a parallelogram with side lengths $|A B|=a \mathrm{~cm}$ and $|B C|=b \mathrm{~cm} (a>b)$ and an acute angle $\alpha$. The area of the quadrilateral formed by the intersection of the angle bisectors of the internal angles of the parallelogram is $48 \mathrm{~cm}^{2}$, and $\sin \frac{\alpha}{2}=\frac{3}{5}$. Calculate the difference $a-b$.
|
10
|
2/8
|
10. Santa Claus has 36 identical gifts, which are placed in 8 bags. It is known that the number of gifts in the 8 bags is at least 1 and all different. Now, some bags are to be selected, and all the gifts in the selected bags are to be evenly distributed to 8 children, with exactly all gifts being distributed (each child gets at least one gift). Therefore, there are $\qquad$ different ways to choose.
|
31
|
4/8
|
II. (25 points) As shown in Figure 6, with the vertices of an equilateral triangle $\triangle ABC$ with side length 2 as centers and the side length as the radius, draw three equal circles, obtaining intersection points $D, E, F$. Connect $CF$, intersecting $\odot C$ at point $G$. With $E$ as the center and $EG$ as the radius, draw an arc intersecting sides $AB$ and $BC$ at points $M$ and $N$, respectively. Connect $MN$. Find the inradius $r$ of $\triangle BMN$.
|
\dfrac{3\sqrt{3} - \sqrt{15}}{6}
|
0/8
|
11. (20 points) Let the sequence of rational numbers $\left\{a_{n}\right\}$ be defined as follows:
$a_{k}=\frac{x_{k}}{y_{k}}$, where $x_{1}=y_{1}=1$, and
if $y_{k}=1$, then $x_{k+1}=1, y_{k+1}=x_{k}+1$;
if $y_{k} \neq 1$, then $x_{k+1}=x_{k}+1, y_{k+1}=y_{k}-1$.
How many terms in the first 2011 terms of this sequence are positive integers?
|
213
|
0/8
|
B2. The integer $N$ consists of 2009 nines written in sequence. A computer calculates $N^{3}=(99999 \ldots 99999)^{3}$. How many nines does the written-out number $N^{3}$ contain in total?
|
4017
|
2/8
|
4. Given that the circumradius of $\triangle A B C$ is 1, and $A B, B C, \frac{4}{3} C A$ form a geometric sequence in order, then the maximum value of $B C$ is $\qquad$ .
|
\dfrac{4\sqrt{2}}{3}
|
2/8
|
A graph has 6 points and 15 edges. Each edge is colored red or blue, so that there are no blue triangles. The number of red edges at each point is a, b, c, d, e, f. Show that (2a - 7)^2 + (2b - 7)^2 + (2c - 7)^2 + (2d - 7)^2 + (2e - 7)^2 + (2f - 7)^2 < 55.
|
(2a - 7)^2 + (2b - 7)^2 + (2c - 7)^2 + (2d - 7)^2 + (2e - 7)^2 + (2f - 7)^2 < 55
|
0/8
|
Example 5 Let $A B C D E F$ be a hexagon, and a frog starts at vertex $A$. It can jump to one of the two adjacent vertices at each step. If it reaches point $D$ within 5 jumps, it stops jumping; if it does not reach point $D$ within 5 jumps, it stops after 5 jumps. How many different jumping sequences are possible from the start to the stop?
(1997, National High School Mathematics Competition)
|
26
|
3/8
|
3. In trapezoid $A B C D$, $A B / / C D, A C 、 B D$ intersect at point $O$. If $A C=5, B D=12$, the midline length is $\frac{13}{2}$, the area of $\triangle A O B$ is $S_{1}$, and the area of $\triangle C O D$ is $S_{2}$, then $\sqrt{S_{1}}+\sqrt{S_{2}}$ $=$ . $\qquad$
|
\sqrt{30}
|
3/8
|
Suppose $m$ and $n$ are relatively prime positive integers. A regular $m$-gon and a regular
$n$-gon are inscribed in a circle. Let $d$ be the minimum distance in degrees (of the arc along
the circle) between a vertex of the $m$-gon and a vertex of the $n$-gon. What is the maximum
possible value of $d$?
|
\dfrac{180}{mn}
|
0/8
|
8,9
In trapezoid $ABCD$, side $AD$ is the larger base. It is known that $AD=CD=4 \frac{2}{3}$, $\angle BAD=90^{\circ}$, and $\angle BCD=150^{\circ}$. On base $AD$, triangle $AED$ is constructed, and points $B$ and $E$ lie on the same side of line $AD$ with $AE=DE$. The height of this triangle, drawn from vertex $E$, is $1 \frac{2}{5}$. Find the area of the common part of trapezoid $ABCD$ and triangle $AED$.
|
\dfrac{147\sqrt{3} - 245}{3}
|
0/8
|
6. Let $m>1$ be an integer. The sequence $a_{1}, a_{2}, \cdots$ is defined as follows:
$$
\begin{array}{l}
a_{1}=a_{2}=1, a_{3}=4, \\
a_{n}=m\left(a_{n-1}+a_{n-2}\right)-a_{n-3}(n \geqslant 4) .
\end{array}
$$
Determine all integers $m$ such that every term of the sequence is a perfect square.
|
2
|
4/8
|
Two three-digit numbers are given. The hundreds digit of each of them is equal to the units digit of the other. Find these numbers if their difference is $297$ and the sum of digits of the smaller number is $23$.
|
689
|
1/8
|
4. In quadrilateral $A B C D$, $\angle A=60^{\circ}, A B=1$, point $E$ is on side $A B$ such that $A E: E B=2: 1, P$ is a moving point on diagonal $A C$. Then the minimum value of $P E+P B$ is $\qquad$ .
|
\dfrac{\sqrt{7}}{3}
|
0/8
|
Three points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle's center?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
|
19
|
4/8
|
An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 5 minutes he is at the dot labeled $B$? [asy]
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
dot((0,0)); dot((1,0)); dot((2,0)); dot((-1,0)); dot((-2,0)); dot((0,1)); dot((0,2)); dot((0,-1)); dot((0,-2)); dot((1,1)); dot((1,-1)); dot((-1,-1)); dot((-1,1));
label("$A$",(0,0),SW);
label("$B$",(0,1),NE);
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
|
5
|
0/8
|
Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. What is the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$?The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
|
10
|
4/8
|
All of the triangles in the figure and the central hexagon are equilateral. Given that $\overline{AC}$ is 3 units long, how many square units, expressed in simplest radical form, are in the area of the entire star? [asy]
import olympiad; import geometry; import graph; size(150); defaultpen(linewidth(0.8));
pair[] vertices;
for(int i = 0; i < 6; ++i){
vertices[i] = dir(30 + 60*i);
}
draw(vertices[0]--vertices[2]--vertices[4]--cycle);
draw(vertices[1]--vertices[3]--vertices[5]--cycle);
label("$D$",vertices[0],NE); label("$C$",vertices[1],N); label("$B$",vertices[2],NW); label("$A$",vertices[3],SW);
label("$F$",vertices[4],S); label("$E$",vertices[5],SE);
[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
|
6
|
1/8
|
Cube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. Find the volume of the larger of the two solids.
[asy]
import cse5;
unitsize(8mm);
pathpen=black;
pair A = (0,0), B = (3.8,0), C = (5.876,1.564), D = (2.076,1.564), E = (0,3.8), F = (3.8,3.8), G = (5.876,5.364), H = (2.076,5.364), M = (1.9,0), N = (5.876,3.465);
pair[] dotted = {A,B,C,D,E,F,G,H,M,N};
D(A--B--C--G--H--E--A);
D(E--F--B);
D(F--G);
pathpen=dashed;
D(A--D--H);
D(D--C);
dot(dotted);
label("$A$",A,SW);
label("$B$",B,S);
label("$C$",C,SE);
label("$D$",D,NW);
label("$E$",E,W);
label("$F$",F,SE);
label("$G$",G,NE);
label("$H$",H,NW);
label("$M$",M,S);
label("$N$",N,NE);
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
|
89
|
5/8
|
A bag contains $8$ green candies and $4$ red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that you do not eat a green candy after you eat a red candy. Find $m+n$.
|
6
|
4/8
|
Each time you click a toggle switch, the switch either turns from [i]off[/i] to [i]on[/i] or from [i]on[/i] to [i]off[/i]. Suppose that you start with three toggle switches with one of them [i]on[/i] and two of them [i]off[/i]. On each move you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers so that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be [i]on[/i] and two of them will be [i]off[/i]. Find $m+n$.
|
142
|
4/8
|
Let $z$ be a complex number satisfying $(z+\tfrac{1}{z})(z+\tfrac{1}{z}+1)=1$. Evaluate $(3z^{100}+\tfrac{2}{z^{100}}+1)(z^{100}+\tfrac{2}{z^{100}}+3)$.
|
36
|
5/8
|
Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.
|
13
|
0/8
|
How many four-digit positive integers have no adjacent equal even digits? For example, count numbers such as $1164$ and $2035$ but not $6447$ or $5866$.
|
7801
|
4/8
|
One of the two Purple Comet! question writers is an adult whose age is the same as the last two digits of the year he was born. His birthday is in August. What is his age today?
|
56
|
0/8
|
Right triangle $ABC$ has a right angle at $C$. Point $D$ on side $\overline{AB}$ is the base of the altitude of $\triangle ABC$ from $C$. Point $E$ on side $\overline{BC}$ is the base of the altitude of $\triangle CBD$ from $D$. Given that $\triangle ACD$ has area $48$ and $\triangle CDE$ has area $40$, fi nd the area of $\triangle DBE$.
|
200
|
3/8
|
Consider the sequences of six positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ with the properties that $a_1=1$, and if for some $j > 1$, $a_j = m > 1$, then $m-1$ appears in the sequence $a_1,a_2,\dots,a_{j-1}$. Such sequences include $1,1,2,1,3,2$ and $1,2,3,1,4,1$ but not $1,2,2,4,3,2$. How many such sequences of six positive integers are there?
|
203
|
4/8
|
Larry and Diane start $100$ miles apart along a straight road. Starting at the same time, Larry and Diane
drive their cars toward each other. Diane drives at a constant rate of 30 miles per hour. To make it
interesting, at the beginning of each 10 mile stretch, if the two drivers have not met, Larry flips a fair coin.
If the coin comes up heads, Larry drives the next 10 miles at 20 miles per hour. If the coin comes up tails,
Larry drives the next 10 miles at 60 miles per hour. Larry and Diane stop driving when they meet. The expected number of times that Larry flips the coin is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n.$
|
413
|
0/8
|
Let $A = {1,2,3,4,5}$ and $B = {0,1,2}$. Find the number of pairs of functions ${{f,g}}$ where both f and g map the set A into the set B and there are exactly two elements $x \in A$ where $f(x) = g(x)$. For example, the function f that maps $1 \rightarrow 0,2 \rightarrow 1,3 \rightarrow 0,4 \rightarrow 2,5 \rightarrow 1$ and the constant function g which maps each element of A to 0 form such a pair of functions.
|
9720
|
1/8
|
In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three
5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as
necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile.
Find the number of different positive weights of chemicals that Gerome could measure.
|
129
|
1/8
|
In the figure below $\angle$LAM = $\angle$LBM = $\angle$LCM = $\angle$LDM, and $\angle$AEB = $\angle$BFC = $\angle$CGD = 34 degrees. Given that $\angle$KLM = $\angle$KML, find the degree measure of $\angle$AEF. This is #8 on the 2015 Purple comet High School. For diagram go to http://www.purplecomet.org/welcome/practice
|
107
|
0/8
|
The graph of the function \( y = x^2 + ax + b \) is shown in the figure. It is known that line \( AB \) is perpendicular to the line \( y = x \).
Find the length of the line segment \( OC \).
|
1
|
2/8
|
The set \( S \) contains 6 distinct positive integers. The average of the two smallest numbers is 5 and the average of the two largest is 22. What is the greatest possible average of all the numbers in the set \( S \)?
|
15.5
|
4/8
|
Today is January 30th, and we write down 130. The rule for writing the subsequent numbers is as follows: If the last number written is even, divide it by 2 and add 2 to get the next number. If the last number written is odd, multiply it by 2 and subtract 2 to get the next number. Starting with 130, we get: 130, 67, 132, 68, etc. What is the 2016th number in this sequence?
|
6
|
5/8
|
Let \( n \) be a positive integer. Claudio has \( n \) cards, each labeled with a different number from 1 to \( n \). He takes a subset of these cards, and multiplies together the numbers on the cards. He remarks that, given any positive integer \( m \), it is possible to select some subset of the cards so that the difference between their product and \( m \) is divisible by 100. Compute the smallest possible value of \( n \).
|
17
|
0/8
|
Your national football coach brought a squad of 18 players to the 2010 World Cup, consisting of 3 goalkeepers, 5 defenders, 5 midfielders, and 5 strikers. Midfielders are versatile enough to play as both defenders and midfielders, while the other players can only play in their designated positions. How many possible teams of 1 goalkeeper, 4 defenders, 4 midfielders, and 2 strikers can the coach field?
|
2250
|
0/8
|
The number 770 is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either 40 or 41 from the number on the board. She performs moves until the number is not positive, and then she stops. Let \( N \) be the number of sequences of moves that Melody could perform. Suppose \( N = a \cdot 2^b \) where \( a \) is an odd positive integer and \( b \) is a nonnegative integer. Compute \( 100a + b \).
|
318
|
0/8
|
In an inscribed quadrilateral $ABCD$, the degree measures of the angles are in the ratio $\angle A : \angle B : \angle C = 2 : 3 : 4$. Find the length of $AC$ if $CD = 15$ and $BC = 18\sqrt{3} - 7.5$.
|
39
|
0/8
|
Triangle $ABC$ with $\measuredangle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\tfrac{BE}{EF} = \tfrac{2}{3}$, then $\tfrac{CF}{FE} = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Lewis Chen
[/i]
|
7
|
3/8
|
I2.1 If $x, y$ and $z$ are positive real numbers such that $\frac{x+y-z}{z}=\frac{x-y+z}{y}=\frac{-x+y+z}{x}$ and $a=\frac{(x+y) \cdot(y+z) \cdot(z+x)}{x y z}$, find the value of $a$.
I2.2 Let $u$ and $t$ be positive integers such that $u+t+u t=4 a+2$. If $b=u+t$, find the value of $b$.
|
8
|
0/8
|
Chebotarev A.S.
On a plane, there is a circle. What is the minimum number of lines that need to be drawn so that, by symmetrically reflecting the given circle relative to these lines (in any order a finite number of times), it can cover any given point on the plane?
|
3
|
4/8
|
Initially 243 Hexagon $A B C D E F$ is inscribed in $\odot O$, $A F // D C, F E // C B, E D // B A, A B+B C=$ $2 C D$. Constructing squares on each of the six sides, the sum of the areas of these six squares is 2008. Find the perimeter of hexagon $A B C D E F$.
|
108
|
1/8
|
$A$ and $B$ are playing a game on a $25 \times 25$ grid. Initially, $A$ can mark some squares. After marking, they take turns placing coins on the grid, with $B$ placing the first coin. The rules for placing the coins are as follows: (1) Coins cannot be placed in marked squares; (2) Once a coin is placed, no coins can be placed in the same row or column as that coin; (3) The game ends when no more coins can be placed on the grid. The player who places the last coin wins. How many squares does $A$ need to mark at the beginning to ensure they win the game, assuming both $A$ and $B$ play optimally?
|
25
|
5/8
|
B-4. $p(x)$ is a non-zero polynomial of degree less than 1992, and $p(x)$ has no common factor with $x^{3}-x$. Let
$$
\frac{d^{1992}}{d x^{1992}}\left(\frac{p(x)}{x^{3}-x}\right)=\frac{f(x)}{g(x)},
$$
where $f(x), g(x)$ are polynomials. Find the smallest possible degree of $f(x)$.
|
3984
|
2/8
|
Three. (35 points) The real number sequence $a_{1}, a_{2} \cdots, a_{1997}$ satisfies:
$$
\left|a_{1}-a_{2}\right|+\left|a_{2}-a_{3}\right|+\cdots+\left|a_{1996}-a_{1997}\right|=
$$
1997. If the sequence $\left\{b_{n}\right\}$ satisfies:
$$
b_{k}=\frac{a_{1}+a_{2}+\cdots+a_{k}}{k}(k=1,2, \cdots, 1997),
$$
find the maximum possible value of $\left|b_{1}-b_{2}\right|+\left|b_{2}-b_{3}\right|+\cdots+\left|b_{1996}-b_{1997}\right|$.
|
1996
|
5/8
|
4. Find two numbers whose sum is 2017, and the sum of the numbers written with the same digits but in reverse order is 8947.
|
1408
|
1/8
|
Each vertex of a convex quadrilateral with area \( S \) is reflected symmetrically with respect to the diagonal that does not contain this vertex. Let the area of the resulting quadrilateral be denoted by \( S' \). Prove that \( \frac{S'}{S} < 3 \).
|
\frac{S'}{S} < 3
|
0/8
|
8.3. Let $p(n, k)$ denote the number of divisors of the positive integer $n$ that are not less than $k$. Try to find
$$
\begin{array}{l}
p(1001,1)+p(1002,2)+\cdots+ \\
p(2000,1000) .
\end{array}
$$
|
2000
|
1/8
|
6. On the porch of the house, a boy and a girl are sitting next to each other. Sasha says: "I am a boy." Zhena says: "I am a girl." If at least one of the children is lying, then who is the boy and who is the girl?
|
Sasha \text{ is a girl and Zhena is a boy}
|
5/8
|
8. (5 points) Person A and Person B stand facing each other 30 meters apart. They play "Rock-Paper-Scissors," with the winner moving forward 8 meters, the loser moving back 5 meters, and in the case of a tie, both moving forward 1 meter. After 10 rounds, they are 7 meters apart. How many rounds did they tie?
|
7
|
2/8
|
356. Find the sum of all products of the form $1 \cdot 2, 1 \cdot 3, \ldots$ $n(n-1)$, that can be formed from the numbers $1, 2, 3, \ldots n$.
|
\dfrac{n(n+1)(n-1)(3n+2)}{24}
|
1/8
|
8. In the cyclic quadrilateral $A B C D$, $\angle A C B=15^{\circ}, \angle C A D=45^{\circ}, B C=\sqrt{6}$, $A C=2 \sqrt{3}, A D=$ $\qquad$ .
|
2\sqrt{6}
|
5/8
|
# Problem 3. (3 points)
$4^{27000}-82$ is divisible by $3^n$. What is the greatest natural value that $n$ can take?
|
5
|
0/8
|
11. Let's take a positive integer $n$, sum its digits, and then add the digits of this sum again to get an integer $S$. What is the smallest $n$ that allows us to obtain $S \geq 10?$
|
199
|
5/8
|
47. As shown in the figure, $A B / / C D / / E F / / G H, A E / / D G$, point $C$ is on $A E$, point $F$ is on $D G$. Let the number of angles equal to $\angle \alpha$ be $m$ (excluding $\angle \alpha$ itself), and the number of angles supplementary to $\angle \beta$ be $n$. If $\alpha \neq \beta$, then the value of $m+n$ is $\qquad$.
|
11
|
0/8
|
1.1. If the 200th day of some year is Sunday and the 100th day of the following year is also Sunday, then what day of the week was the 300th day of the previous year? Enter the number of this day of the week (if Monday, then 1, if Tuesday, then 2, etc.).
|
1
|
3/8
|
Example 5. Integrate the differential equation $y^{\prime}=\frac{x \sqrt{1+x^{2}}+1}{\sqrt{1+x^{2}}\left(y+e^{y}\right)}$ and also find the particular solution that satisfies the Cauchy condition: $x_{0}=2, y_{0}=1$.
|
\frac{1}{2}y^2 + e^y = \frac{1}{2}x^2 + \ln\left(x + \sqrt{1 + x^2}\right) + e - \frac{3}{2} - \ln\left(2 + \sqrt{5}\right)
|
4/8
|
* $a, b, c, d$ are pairwise distinct positive integers, and $a+b=c d, a b=c+d$. Find all quadruples $(a, b, c, d)$ that satisfy the above requirements.
|
(1, 5, 2, 3)
|
0/8
|
1. In a $13 \times 13$ grid, numbers are arranged such that the numbers in each row and each column form an arithmetic progression in the order they are written. The grid is colored in two colors in a checkerboard pattern. The numbers on the corner white cells of the grid are $1, 2, 3,$ and $6$. Find the sum of the numbers on the black cells of the grid.
|
252
|
2/8
|
3. Inside the circle $\omega$ are located intersecting at points $K$ and $L$ circles $\omega_{1}$ and $\omega_{2}$, touching the circle $\omega$ at points $M$ and $N$. It turned out that points $K, M$, and $N$ lie on the same line. Find the radius of the circle $\omega$, if the radii of the circles $\omega_{1}$ and $\omega_{2}$ are 3 and 5, respectively.
|
8
|
1/8
|
4. As shown in Figure 4, the diameter $AB = d$ of semicircle $O$. $M$ and $N$ are the midpoints of $OA$ and $OB$, respectively, and point $P$ is on the semicircle. Draw $MC \parallel PN$ and $ND \parallel PM$, with points $C$ and $D$ both on the semicircle. Then $PC^2 + PD^2 + CM^2 + DN^2 =$ $\qquad$
|
\dfrac{5d^2}{8}
|
1/8
|
In the following figure, $ABC$ is an isosceles triangle with $BA = BC$. Point $D$ is inside it such that $\angle ABD = 13^\circ$, $\angle ADB = 150^\circ$, and $\angle ACD = 30^\circ$. Additionally, $ADE$ is an equilateral triangle. Determine the value of the angle $\angle DBC$.
|
39
|
2/8
|
7. Let the incircle of equilateral triangle $ABC$ have a radius of 2, with the center at $I$. If point $P$ satisfies $PI=1$, then the maximum value of the ratio of the areas of $\triangle APB$ to $\triangle APC$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
|
\dfrac{3 + \sqrt{5}}{2}
|
2/8
|
10.5. Find the number of all possible arrangements of chips in some cells of an 8 by 8 chessboard such that the number of chips in each row is different and the number of chips in each column is different.
|
2 \times (8!)^2
|
0/8
|
6. Solve the equation $\sqrt{3 x^{2}+5 x+7}-\sqrt{3 x^{2}+5 x+2}=1$.
|
-2
|
0/8
|
Let $\alpha$ and $\beta$ be positive integers such that $\frac{43}{197} < \frac{\alpha}{\beta} < \frac{17}{77}$. Find the minimum possible value of $\beta$.
|
32
|
5/8
|
Consider the following three lines in the Cartesian plane:
\[
\begin{cases}
\ell_1: & 2x - y = 7\\
\ell_2: & 5x + y = 42\\
\ell_3: & x + y = 14
\end{cases}
\]
Let \( f_i(P) \) correspond to the reflection of the point \( P \) across \( \ell_i \). Suppose \( X \) and \( Y \) are points on the \( x \) and \( y \) axes, respectively, such that \( f_1(f_2(f_3(X)))= Y \). Let \( t \) be the length of segment \( XY \); what is the sum of all possible values of \( t^2 \)?
|
260
|
4/8
|
Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$.
|
665
|
3/8
|
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.