What is the sum of the digits of all numbers from one to one billion?

Answer: 40500000001


Let \( ABC \) be a triangle and let \( D, E, \) and \( F \) be the bases of the heights of \( A, B, \) and \( C \) respectively. Let \( H \) be the orthocenter of triangle \( ABC \). The lines \( EF \) and \( AD \) intersect at \( G \). Let \( K \) be the point on the circumcircle of triangle \( ABC \) that lies diametrically opposite \( A \). The line \( AK \) intersects \( BC \) at \( M \). Show that the lines \( GM \) and \( HK \) are parallel.

Answer: GM\parallelHK


In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table. Each dragon said:

- 1st head: "The dragon to my left is green."
- 2nd head: "The dragon to my right is blue."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?

Answer: 176


Given the equation:

\[ A = \operatorname{ctg}\left(\frac{11 \pi}{4}+\frac{1}{2} \arccos \frac{2 b}{a}\right)+\operatorname{ctg}\left(\frac{11 \pi}{4}-\frac{1}{2} \arccos \frac{2 b}{a}\right) \]

Find the value of \( A \).

Answer: -\frac{}{b}


If $x$ is an odd number, then find the largest integer that always divides the expression\[(10x+2)(10x+6)(5x+5)\]

Answer: 960


For a geometric sequence $\{b_n\}$ with a common ratio of $4$, if $T_n$ is the product of the first $n$ terms of the sequence $\{b_n\}$, then $\frac{T_{20}}{T_{10}}$, $\frac{T_{30}}{T_{20}}$, $\frac{T_{40}}{T_{30}}$ also form a geometric sequence with a common ratio of $4^{100}$. Similarly, in an arithmetic sequence $\{a_n\}$ with a common difference of $3$, if $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, then \_\_\_\_\_\_ also form an arithmetic sequence with a common difference of \_\_\_\_\_\_.

Answer: 300


Given the equation \( \lg (a x+1)=\lg (x-1)+\lg (2-x) \) involving \( x \), determine the range of values for the real number \( a \) such that the equation has a unique real solution.

Answer: (-1,-\frac{1}{2}]\cup{3-2\sqrt{3}}


Triangle  $ABC$  is right angled such that  $\angle ACB=90^{\circ}$  and  $\frac {AC}{BC} = 2$ . Let the line parallel to side  $AC$  intersects line segments  $AB$  and  $BC$  in  $M$  and  $N$  such that  $\frac {CN}{BN} = 2$ . Let  $O$  be the intersection point of lines  $CM$  and  $AN$ . On segment  $ON$  lies point  $K$  such that  $OM+OK=KN$ . Let  $T$  be the intersection point of angle bisector of  $\angle ABC$  and line from  $K$  perpendicular to  $AN$ . Determine value of   $\angle MTB$ .

Answer: 90


While waiting for customers, a watermelon seller sequentially weighed 20 watermelons (with masses of 1 kg, 2 kg, 3 kg, ..., up to 20 kg), balancing a watermelon on one side of the scale with one or two weights on the other side (possibly identical weights). The seller recorded on paper the mass of the weights he used. What is the minimum number of different numbers that could appear in his records, given that the mass of each weight is an integer in kilograms?

Answer: 6


For what values of the parameter \( a \) does the equation

\[ \ln (x-2a) - 3(x-2a)^{2} + 2a = 0 \]

have a unique root?

Answer: \frac{\ln6+1}{4}


For how many integers \( n \) between 1 and 2005, inclusive, is \( 2 \cdot 6 \cdot 10 \cdots(4n - 2) \) divisible by \( n! \)?

Answer: 2005


At the class reunion, 45 people attended. It turned out that any two of them, who have the same number of acquaintances among the attendees, do not know each other. What is the largest number of pairs of acquaintances that could have been among the attendees?

Answer: 870


Given that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$, and $g(x) \neq 0$, $f''(x)g(x) < f(x)g''(x)$, $f(x)=a^{x}g(x)$, $\frac{f(1)}{g(1)}+ \frac{f(-1)}{g(-1)}= \frac{5}{2}$, determine the probability that the sum of the first $k$ terms of the sequence $\left\{ \frac{f(n)}{g(n)}\right\} (n=1,2,…,10)$ is greater than $\frac{15}{16}$.

Answer: \frac{3}{5}


In a dark room, a drawer contains 120 red socks, 100 green socks, 70 blue socks, and 50 black socks. A person selects socks one by one from the drawer without being able to see their color. What is the minimum number of socks that must be selected to ensure that at least 15 pairs of socks are selected, with no sock being counted in more than one pair?

Answer: 33


Jack has a 3-litre jug of water. What is the maximum number of 0.5-litre bottles that he can completely fill?
(A) 3
(B) 1.5
(C) 6
(D) 12
(E) 15

Answer: 6


Let \( AB \) be a chord of the parabola \( y^2 = 2px \) (with \( p > 0 \)) that passes through the focus and is not perpendicular to the x-axis. Let \( P \) be a point on the y-axis other than the origin \( O \). The y-coordinates of points \( A \), \( B \), and \( P \) are \( y_1 \), \( y_2 \), and \( y_0 \) respectively. If the four points \( P \), \( A \), \( B \), and \( O \) are concyclic, find the value of \( \frac{y_1 + y_2}{y_0} \).

Answer: 4


A circle of radius \( t \) is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius \( r = 1 + \sin \frac{\pi}{8} \). Find \( rt \).

Answer: \frac{2+\sqrt{2}}{4}


Find the smallest natural number \( n \) such that every convex 100-gon can be represented as the intersection of \( n \) triangles.

Answer: 50


Answer: \frac{8}{4+x}


All three-digit numbers from 100 to 999 are written consecutively without spaces. Kostya underlined $k$ consecutive digits in this sequence, and Andrey underlined another $k$ consecutive digits in this sequence. The $k$-digit numbers underlined by the boys are equal. What is the largest possible value of $k$ for which this could happen?

Answer: 5


At the intersection of perpendicular roads, a highway from Moscow to Kazan intersects with a road from Vladimir to Ryazan. Dima and Tolya are traveling at constant speeds from Moscow to Kazan and from Vladimir to Ryazan, respectively.

When Dima crossed the intersection, Tolya was 900 meters away from it. When Tolya crossed the intersection, Dima was 600 meters away from it. How many meters will be between the boys when Tolya travels 900 meters from the moment he crosses the intersection?

Answer: 1500


Given the ten digits 0, 1, 2, 3, …, 9 and the imaginary unit i, determine the total number of distinct imaginary numbers that can be formed.

Answer: 90


Let $\{a_n\}$ be a decreasing geometric sequence, where $q$ is the common ratio, and $S_n$ is the sum of the first $n$ terms. Given that $\{a_1, a_2, a_3\} \subseteq \{-4, -3, -2, 0, 1, 2, 3, 4\}$, find the value of $$\frac {S_{8}}{1-q^{4}}$$.

Answer: \frac {17}{2}


Consider the arithmetic sequence $\{a_n\}$ where each term is an integer and the first term $a_1 = 2019$. For any positive integer $n$, there always exists a positive integer $m$ such that $a_1 + a_2 + \cdots + a_n = a_m$. Determine the number of such sequences $\{a_n\}$.

Answer: 5


Let  $a$  and  $b$  be positive integers such that  $(2a+b)(2b+a)=4752$ . Find the value of  $ab$ .

*Proposed by James Lin*

Answer: 520


Answer: \frac{3-\sqrt{3}}{3}


A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let $S$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $S$?

Answer: 37


A coach decided to award 12 students who ran the distance in the best time. Each of them needs to be awarded with a "gold," "silver," or "bronze" medal. All three types of medals must be used (at least one of each), and a student who finished earlier cannot be awarded a less valuable medal than the one who finished later.

How many ways can the coach distribute the medals (assuming all runners have different times)?

Answer: 55


Given a positive integer $n$ that is not divisible by 2 or 3, and there do not exist non-negative integers $a$ and $b$ such that $\left|2^{a}-3^{b}\right|=n$, find the smallest value of $n$.

Answer: 35


Joey wrote a system of equations on a blackboard, where each of the equations was of the form \( a + b = c \) or \( a \cdot b = c \) for some variables or integers \( a, b, c \). Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads:

\[
\begin{array}{ll}
x & z = 15 \\
x & y = 12 \\
x & x = 36
\end{array}
\]

If \( x, y, z \) are integer solutions to the original system, find the sum of all possible values of \( 100x + 10y + z \).

Answer: 2037


Find the number of ordered tuples  $\left(C,A,M,B\right)$  of non-negative integers such that \[C!+C!+A!+M!=B!\]

*2019 CCA Math Bonanza Team Round #4*

Answer: 7


Find all real numbers \( a \) for which there exists a non-constant function \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying the following two equations for all \( x \in \mathbb{R} \):

1) \( f(a x) = a^{2} f(x) \)

2) \( f(f(x)) = a f(x) \)

Answer: 01


The perimeter of a semicircle with an area of ______ square meters is 15.42 meters.

Answer: 14.13


Find the largest positive integer  $n$  such that  $\sigma(n) = 28$ , where  $\sigma(n)$  is the sum of the divisors of  $n$ , including  $n$ .

Answer: 12


The sides of triangle $DEF$ are in the ratio of $3:4:5$. Segment $EG$ is the angle bisector drawn to the shortest side, dividing it into segments $DG$ and $GE$. What is the length, in inches, of the longer subsegment of side $DE$ if the length of side $DE$ is $12$ inches? Express your answer as a common fraction.

Answer: \frac{48}{7}


Given a convex quadrilateral \(ABCD\) with \(X\) as the midpoint of the diagonal \(AC\), it turns out that \(CD \parallel BX\). Find \(AD\) if it is known that \(BX = 3\), \(BC = 7\), and \(CD = 6\).

Answer: 14


1. Given that the line $l$ passing through the point $M(-3,0)$ is intercepted by the circle $x^{2}+(y+2)^{2}=25$ to form a chord of length $8$, what is the equation of line $l$?
2. Circles $C_{1}$: $x^{2}+y^{2}+2x+8y-8=0$ and $C_{2}$: $x^{2}+y^{2}-4x-4y-2=0$ intersect. What is the length of their common chord?
3. What is the equation of a circle passing through points $A(0,2)$ and $B(-2,2)$, with its center lying on the line $x-y-2=0$?
4. Given the circle with the equation $(x-2)^{2}+y^{2}=1$, a line passing through the external point $P(3,4)$ intersects the circle at points $A$ and $B$. What is the value of $\overrightarrow{PA}\cdot \overrightarrow{PB}$?

Answer: 16


The angle of inclination of the line $$\begin{cases} \left.\begin{matrix}x=3- \frac { \sqrt {2}}{2}t \\ y= \sqrt {5}- \frac { \sqrt {2}}{2}t\end{matrix}\right.\end{cases}$$ is ______.

Answer: \frac{\pi}{4}


Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where  $\lfloor x \rfloor$  denotes the greatest integer less than or equal to  $x$ .

(You may use the fact that   $\sin{n^\circ}$  is irrational for positive integers  $n$  not divisible by  $30$ .)

*Ray Li*

Answer: 178


Order the numbers $3$, $\frac{5}{2}$, and $\sqrt{10}$ from smallest to largest.

Answer: \frac{5}{2}, 3, \sqrt{10}


What is the least integer whose square is 75 more than its double?

Answer: -8


The school wants to distribute 90 story books to third-grade students. If each student receives 1 book, there will be some books left over; if the remaining books are then distributed so that every 2 students share 1 book, there will be no books left. How many third-grade students are there in total?

Answer: 60


Let $S$ be the sum of the interior angles of a polygon $P$ for which each interior angle is $7\frac{1}{2}$ times the
exterior angle at the same vertex. Then
$\textbf{(A)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may be regular}\qquad \\ \textbf{(B)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad \\ \textbf{(C)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is regular}\qquad \\ \textbf{(D)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad \\ \textbf{(E)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may or may not be regular}$

Answer: \textbf{(E)}\S=2700


Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$.  Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutation is guaranteed to intersect at least one of these $1006$
permutations.

Answer: 1006


A square sheet of paper has an area of $12 \text{ cm}^2$. The front is white and the back is black. When the paper is folded so that point $A$ rests on the diagonal and the visible black area is equal to the visible white area, how far is point A from its original position? Give your answer in simplest radical form.

Answer: 2\sqrt{6}


There are eight different symbols designed on  $n\geq 2$  different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any  $k$  symbols  $(1\leq k\leq 7)$  the number of shirts containing at least one of the  $k$  symbols is even. Determine the value of  $n$ .

Answer: 255


On the board, there are 101 numbers written: \(1^2, 2^2, \ldots, 101^2\). In one operation, it is allowed to erase any two numbers and write the absolute value of their difference instead.

What is the smallest number that can be obtained after 100 operations?

Answer: 1


Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat?
$\textbf{(A)}~\frac{1}{24}\qquad\textbf{(B)}~\frac{1}{12}\qquad\textbf{(C)}~\frac{1}{8}\qquad\textbf{(D)}~\frac{1}{6}\qquad\textbf{(E)}~\frac{1}{4}$

Answer: \textbf{(C)}~\frac{1}{8}


The circles \( S_{1} \) and \( S_{2} \) with centers \( O_{1} \) and \( O_{2} \) respectively intersect at points \( A \) and \( B \). Tangents to \( S_{1} \) and \( S_{2} \) at point \( A \) intersect line segments \( B O_{2} \) and \( B O_{1} \) at points \( K \) and \( L \) respectively. Prove that \( K L \| O_{1} O_{2} \).

Answer: KL\parallelO_1O_2


Given 25 vessels with capacities $1, 2,\ldots, 25$ liters, in how many ways can you choose ten of them such that any two selected vessels can be used to measure out exactly 1 liter of water? (An unlimited amount of water is available, and it is possible to transfer water from one vessel to another.)

Answer: 16


Using a permutation of the numbers $10, 20, 30, 40$ for $A, B, C, D$, maximize the value of the expression $\frac{1}{A-\frac{1}{B+\frac{1}{C-\frac{1}{D}}}}$. Then, find the value of $A+2B+3C+4D$.

Answer: 290


For which values of \(a\) and \(b\) do the equations

\[
19 x^{2} + 19 y^{2} + a x + b y + 98 = 0
\]

and

\[
98 x^{2} + 98 y^{2} + a x + b y + 19 = 0
\]

have a common solution?

Answer: ^2+b^2\ge13689


A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?

Answer: 115


A right triangle with an angle of \(30^{\circ}\). The segment connecting the midpoints of the bases is equal to 3. The angles at the larger base of the trapezoid are \(30^{\circ}\) and \(60^{\circ}\). Find the height.

Answer: \frac{3\sqrt{3}}{2}


There are $n$ identical-looking balls, one of which is lighter than the others (all other balls have the same weight). Using a balance scale without weights, and needing at least 5 weighings to identify the lighter ball, what is the maximum value of $n$?

Answer: 243


The graph of $y = \frac{p(x)}{q(x)}$ is shown below, where $p(x)$ and $q(x)$ are quadratic.  (Assume that the grid lines are at integers.)

[asy]
unitsize(0.6 cm);

real func (real x) {
  return (2*(x - 1)/(x + 2));
}

int i;

for (i = -8; i <= 8; ++i) {
  draw((i,-8)--(i,8),gray(0.7));
  draw((-8,i)--(8,i),gray(0.7));
}

draw((-8,0)--(8,0));
draw((0,-8)--(0,8));
draw((-2,-8)--(-2,8),dashed);
draw((-8,2)--(8,2),dashed);
draw(graph(func,-8,-2.1),red);
draw(graph(func,-1.9,8),red);
filldraw(Circle((5,func(5)),0.15),white,red);

limits((-8,-8),(8,8),Crop);
[/asy]

The horizontal asymptote is $y = 2,$ and the only vertical asymptote is $x = -2.$  Find $\frac{p(3)}{q(3)}.$

Answer: \frac{4}{5}


On a table there are  $100$  red and  $k$  white buckets for which all of them are initially empty. In each move, a red and a white bucket is selected and an equal amount of water is added to both of them. After some number of moves, there is no empty bucket and for every pair of buckets that are selected together at least once during the moves, the amount of water in these buckets is the same. Find all the possible values of  $k$ .

Answer: 100


The three-digit even numbers \( A \, , B \, , C \, , D \, , E \) satisfy \( A < B < C < D < E \). Given that \( A + B + C + D + E = 4306 \), find the smallest value of \( A \).

Answer: 326


Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart?
[asy] size((50)); dot((5,0)); dot((5,5)); dot((0,5)); dot((-5,5)); dot((-5,0)); dot((-5,-5)); dot((0,-5)); dot((5,-5)); [/asy]
$\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{2}{7}\qquad\textbf{(C)}\ \frac{4}{11}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{4}{7}$

Answer: \textbf{(B)}\\frac{2}{7}


Suppose we have 1,000,000 bus tickets numbered from 000000 to 999999. We call a ticket "lucky" if the sum of the first three digits of its number is equal to the sum of the last three digits. Let \( N \) be the number of lucky tickets. Prove the following equalities:

a) \(\left(1+x+\ldots+x^{9}\right)^{3}\left(1+x^{-1}+\ldots+x^{-9}\right)^{3}=x^{27}+\ldots+a_{1} x+N+a_{1} x+\ldots+x^{-27}\);

b) \(\left(1+x+\ldots+x^{9}\right)^{6}=1+\ldots+N x^{27}+\ldots+x^{54}\).

c) Find the number of lucky tickets.

Answer: 55252


Let  $n$  be a positive integer. Suppose that we have disks of radii  $1, 2, . . . , n.$  Of each size there are two disks: a transparent one and an opaque one. In every disk there is a small hole in the centre, with which we can stack the
disks using a vertical stick. We want to make stacks of disks that satisfy the following conditions:
  $i)$  Of each size exactly one disk lies in the stack.
  $ii)$  If we look at the stack from directly above, we can see the edges of all of the  $n$  disks in the stack. (So if there is an opaque disk in the stack,no smaller disks may lie beneath it.)
Determine the number of distinct stacks of disks satisfying these conditions.
(Two stacks are distinct if they do not use the same set of disks, or, if they do use the same set of disks and the orders in which the disks occur are different.)

Answer: (n+1)!


Let ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ have left and right foci $F_{1}$, $F_{2}$, and line $l$ passing through point $F_{1}$. If the point $P$, which is the symmetric point of $F_{2}$ with respect to line $l$, lies exactly on ellipse $C$, and $\overrightarrow{F_1P} \cdot \overrightarrow{F_1F_2} = \frac{1}{2}a^2$, find the eccentricity of $C$.

Answer: \frac{1}{2}


Each student inputs a natural number into a calculator, starting with 1 for the first student, 2 for the second student, and so on. The calculator's keys for numbers 4 and 8 are broken, so the next student can only input 5, followed by 6, 7, 9, 10, 11, 12, 13, 15, and so forth. Based on this input method, what natural number does Xiao Ming, the 100th student, input?

Answer: 155


In the addition problem, each digit has been replaced by a letter.  If different letters represent different digits then what is the value of $C$?

Answer: 1


Given a quadrilateral \( ABCD \) such that \( AB = BD \), \( \angle ABD = \angle DBC \), and \( \angle BCD = 90^\circ \). Point \( E \) is marked on segment \( BC \) such that \( AD = DE \). What is the length of segment \( BD \) if it is known that \( BE = 7 \) and \( EC = 5 \)?

Answer: 17


A fixed point of a function \( f \) is a value of \( x \) for which \( f(x) = x \). Let \( f \) be the quadratic function defined by \( f(x) = x^2 - c x + c \) where \( c \in \mathbb{R} \). Find, in interval notation, the set consisting of all values of \( c \) for which \( f \circ f \) has four distinct fixed points.

Answer: (-\infty,-1)\cup(3,\infty)


Given the function $f(x) = 3x^4 + 6$, find the value of $f^{-1}(150)$.

Answer: \sqrt[4]{48}


Given that \( n \in \mathbf{N}^{+} \), and subsets \( A_{1}, A_{2}, \cdots, A_{2n+1} \) of a set \( B \) satisfy:
1. \( \left|A_{i}\right| = 2n \) for \( i = 1, 2, \cdots, 2n+1 \);
2. \( \left|A_{i} \cap A_{j}\right| = 1 \) for \( 1 \leqslant i < j \leqslant 2n+1 \);
3. Each element in \( B \) belongs to at least two of the \( A_{i} \).

For which values of \( n \) is it possible to label the elements of \( B \) with 0 or 1 such that each \( A_{i} \) contains exactly \( n \) elements labeled 0?

Answer: n


A circle $\omega$ with center at point $I$ is inscribed in a convex quadrilateral $ABCD$ and touches side $AB$ at point $M$ and side $CD$ at point $N$, with the condition that $\angle BAD + \angle ADC < 180^\circ$. A point $K \neq M$ is chosen on line $MN$ such that $AK = AM$. In what ratio can line $DI$ divide segment $KN$? Provide all possible answers and prove that there are no others.

Answer: 1:1


Given a regular triangular pyramid $S-ABC$ with a base that is an equilateral triangle with side length 1 and lateral edges of length 2. If a section passing through the line $AB$ divides the volume of the pyramid into two equal parts, find the cosine of the dihedral angle between the section and the base.

Answer: \frac{2\sqrt{15}}{15}


There are \( n \) points in space \( P_{1}, \ldots, P_{n} \) such that there exists a point \( P \) such that \( P_{i} P < P_{i} P_{j} \) for all \( j \neq i \). Show that \( n < 15 \).

Answer: n<15


The volume of a hemispherical soup bowl is 8 liters. How much soup fills the bowl up to half its height?

Answer: 2.5


Any six points are taken inside or on a rectangle with dimensions $2 \times 1$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$.

Answer: \frac{\sqrt{5}}{2}


Using the same Rotokas alphabet, how many license plates of five letters are possible that begin with G, K, or P, end with T, cannot contain R, and have no letters that repeat?

Answer: 630


If $p$, $q$, $r$, $s$, $t$, and $u$ are integers such that $1728x^3 + 64 = (px^2 + qx + r)(sx^2 + tx + u)$ for all $x$, then what is $p^2+q^2+r^2+s^2+t^2+u^2$?

Answer: 23456


For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\sin x = \sin (n+1)x$ on the interval $[0, 2\pi]$. Calculate $\sum_{n=2}^{100} G(n)$.

Answer: 10296


In a lecture hall, there is a row of connected desks, each seating 6 people. Students can only leave the hall from either end of the row. The time at which each student finishes their exam varies. If a student seated inside the row wants to submit their exam first, they need to disturb others. Let the number of students who disturb others when submitting their exams be a random variable $X$. What is the mathematical expectation of $X$?

Answer: \frac{21}{10}


Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$?

Answer: 192


In the cube \(ABCD-A_1B_1C_1D_1\), point \(E\) is the midpoint of \(BC\) and point \(F\) is on \(AA_1\), with the ratio \(A_1F : FA = 1 : 2\). Find the dihedral angle between plane \(B_1EF\) and base \(A_1B_1C_1D_1\).

Answer: \arctan\frac{\sqrt{37}}{3}


Petya bought a microcalculator from the store "Turing Machines and Other Computing Devices," which can compute $x y + x + y + 1$ for any real numbers $x$ and $y$, and has no other operations. Petya wants to write a "program" to compute the polynomial $1 + x + x^{2} + \ldots + x^{1982}$. By a "program," he means a sequence of polynomials $f_{1}(x), \ldots, f_{n}(x)$ such that $f_{1}(x) = x$ and for any $i=2, \ldots, n$, $f_{i}(x)$ is either a constant or

$$f_{i}(x) = f_{j}(x) \cdot f_{k}(x) + f_{k}(x) + f_{j}(x) + 1,$$

where $j < i, k < i$, and $f_{n}(x) = 1 + x + \ldots + x^{1982}$.

a) Help Petya write the "program".

b) Is it possible to write the "program" if the calculator only has the operation $x y + x + y$?

Answer: No


Find an approximate value of $0.998^6$ such that the error is less than $0.001$.

Answer: 0.988


Given that $a_{1}, a_{2}, \cdots, a_{10}$ and $b_{1}, b_{2}, \cdots, b_{10}$ are permutations of $1, 2, \cdots, 10 $, specifically
$$
\begin{array}{l}
\left\{a_{1}, a_{2}, \cdots, a_{10}\right\} \\
=\left\{b_{1}, b_{2}, \cdots, b_{10}\right\} \\
=\{1, 2, \cdots, 10\}.
\end{array}
$$

Let $c_{k}=a_{k}^{2} + b_{k}^{2} (k=1,2, \cdots, 10)$. Suppose the maximum value among $c_{1}, c_{2}, \cdots, c_{10}$ is $M$, and the minimum value is $m$. Find:
(1) The minimum possible value of $M$;
(2) The maximum possible value of $m$.

Answer: 61


Given an inscribed quadrilateral $ABCD$ with perpendicular diagonals. On its circumscribing circle, there is a point $E$ diametrically opposite to $D$, such that segments $AB$ and $DE$ do not intersect. Find the ratio of the areas of triangle $BCD$ and quadrilateral $ABED$.

Answer: 1


Let \( X = \{1, 2, \ldots, 98\} \). Call a subset of \( X \) good if it satisfies the following conditions:
1. It has 10 elements.
2. If it is partitioned in any way into two subsets of 5 elements each, then one subset has an element coprime to each of the other 4, and the other subset has an element which is not coprime to any of the other 4.

Find the smallest \( n \) such that any subset of \( X \) of \( n \) elements has a good subset.

Answer: 50


Let triangle \(ABC\) be an acute triangle with circumcircle \(\Gamma\). Let \(X\) and \(Y\) be the midpoints of minor arcs \(\overparen{AB}\) and \(\overparen{AC}\) of \(\Gamma\), respectively. If line \(XY\) is tangent to the incircle of triangle \(ABC\) and the radius of \(\Gamma\) is \(R\), find, with proof, the value of \(XY\) in terms of \(R\).

Answer: \sqrt{3}R


In a group of people, there are 13 who like apples, 9 who like blueberries, 15 who like cantaloupe, and 6 who like dates. (A person can like more than 1 kind of fruit.) Each person who likes blueberries also likes exactly one of apples and cantaloupe. Each person who likes cantaloupe also likes exactly one of blueberries and dates. Find the minimum possible number of people in the group.

Answer: 22


We may say concerning the solution of $|x|^2 + |x| - 6 =0$ that:
$\textbf{(A)}\ \text{there is only one root}\qquad  \textbf{(B)}\ \text{the sum of the roots is }{+1}\qquad  \textbf{(C)}\ \text{the sum of the roots is }{0}\qquad \\ \textbf{(D)}\ \text{the product of the roots is }{+4}\qquad  \textbf{(E)}\ \text{the product of the roots is }{-6}$

Answer: \textbf{C}


A rectangular flag is divided into four triangles, labelled Left, Right, Top, and Bottom. Each triangle is to be colored one of red, white, blue, green, and purple such that no two triangles that share an edge are the same color. How many different flags can be made?

Answer: 260


Given that \( a_{1}=1, a_{2}=3, a_{n+2}=(n+3) a_{n+1}-(n+2) a_{n} \), find the smallest value of \( n \) such that \( a_{m} \) is divisible by 9 for all \( m \geq n \).

Answer: 5


Find all functions $f: R \to R$ such that, for any $x, y \in R$:
$f\left( f\left( x \right)-y \right)\cdot f\left( x+f\left( y \right) \right)={{x}^{2}}-{{y}^{2}}$

Answer: f(x) = x


Karl the old shoemaker made a pair of boots and sent his son Hans to the market to sell them for 25 talers. At the market, two people, one missing his left leg and the other missing his right leg, approached Hans and asked to buy one boot each. Hans agreed and sold each boot for 12.5 talers.

When Hans came home and told his father everything, Karl decided that he should have sold the boots cheaper to the disabled men, for 10 talers each. He gave Hans 5 talers and instructed him to return 2.5 talers to each person.

While Hans was looking for the individuals in the market, he saw sweets for sale, couldn't resist, and spent 3 talers on candies. He then found the men and gave them the remaining money – 1 taler each. On his way back home, Hans realized how bad his actions were. He confessed everything to his father and asked for forgiveness. The shoemaker was very angry and punished his son by locking him in a dark closet.

While sitting in the closet, Hans thought deeply. Since he returned 1 taler to each man, they effectively paid 11.5 talers for each boot: $12.5 - 1 = 11.5$. Therefore, the boots cost 23 talers: $2 \cdot 11.5 = 23$. And Hans had spent 3 talers on candies, resulting in a total of 26 talers: $23 + 3 = 26$. But there were initially only 25 talers! Where did the extra taler come from?

Answer: 25


Let  $k$  be a positive integer congruent to  $1$  modulo  $4$  which is not a perfect square and let  $a=\frac{1+\sqrt{k}}{2}$ .
Show that  $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$ .

Answer: {1,2,\ldots,\lfloor\rfloor}


If $e^{i \theta} = \frac{1 + i \sqrt{2}}{2}$, then find $\sin 3 \theta.$

Answer: \frac{\sqrt{2}}{8}


Somewhere in the universe, \(n\) students are taking a 10-question math competition. Their collective performance is called laughable if, for some pair of questions, there exist 57 students such that either all of them answered both questions correctly or none of them answered both questions correctly. Compute the smallest \(n\) such that the performance is necessarily laughable.

Answer: 253


Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is  $18$ . How many passcodes satisfy these conditions?

Answer: 36


Find the number of integer points \((x, y)\) that satisfy the equation \(\frac{1}{|x|} + \frac{1}{|y|} = \frac{1}{2017}\).

Answer: 12


The equation \( x^{2} + mx + 1 + 2i = 0 \) has real roots. Find the minimum value of the modulus of the complex number \( m \).

Answer: \sqrt{2 + 2\sqrt{5}}


Let \( c \) and \( d \) be integers. Prove: There exist infinitely many distinct integer pairs \(\left(x_n, y_n\right) (n = 1, 2, \ldots)\) such that \( x_n \) is a divisor of \( c y_n + d \) and \( y_n \) is a divisor of \( c x_n + d \) if and only if \( c \) is a divisor of \( d \).

Answer: \mid


In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?

Answer: \frac{63}{146}


Iveta sequentially wrote natural numbers consisting of the digits 1, 3, 5, and 7. She used no other digits, proceeded in ascending order from the smallest number, and did not omit any number. She wrote the numbers consecutively to form an exceptionally long number:

$$
1357111315173133 \ldots
$$

What digit is at the 1286th position in this number?

(E. Novotná)

Answer: 3


According to national regulations, only adults aged between 18 and 70 are eligible to apply for a motor vehicle driver's license. A sixth-grade student, Li Ming, says, "My dad has a driver's license. His age equals the product of the month and day of his birth, and that product is 2975." How old is Li Ming's father?

Answer: 35


The cafeteria is tiled with the same floor tiles, and it takes 630 tiles to cover an area with 18 square decimeters of tiles. How many tiles will it take if we switch to square tiles with a side length of 6 decimeters?

Answer: 315


Find the common ratio of the infinite geometric series: $$\frac{-4}{7}+\frac{14}{3}+\frac{-98}{9} + \dots$$

Answer: -\frac{49}{6}


Given the following conditions:①$asinB=bsin({A+\frac{π}{3}})$; ②$S=\frac{{\sqrt{3}}}{2}\overrightarrow{BA}•\overrightarrow{CA}$; ③$c\tan A=\left(2b-c\right)\tan C$. Choose one of the three conditions and fill in the blank below, then answer the following questions.
In $\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, the area of $\triangle ABC$ is $S$, and it satisfies _____.
$(1)$ Find the value of angle $A$;
$(2)$ Given that the area of $\triangle ABC$ is $2\sqrt{3}$, point $D$ lies on side $BC$ such that $BD=2DC$, find the minimum value of $AD$.

Answer: \frac{4\sqrt{3}}{3}


The distance between every two utility poles along the road is 50 meters. Xiao Wang travels at a constant speed in a car, and sees 41 utility poles in 2 minutes after seeing the first pole. How many meters does the car travel per hour?

Answer: 60000


Answer: \log_{}b


A large cube is made up of 27 small cubes. A plane is perpendicular to one of the diagonals of this large cube and bisects the diagonal. How many small cubes are intersected by this plane?

Answer: 19


12.1. Let the function $f:\left[\pi ; \frac{3 \pi}{2}\right] \rightarrow \mathbb{R}, f(x)=\frac{3 \cos (2 x)-4 \sin (2 x)+\cos x-3 \sin x}{(\sin x+3 \cos x)^{2022}}$. Determine the primitive $F:\left[\pi ; \frac{3 \pi}{2}\right] \rightarrow \mathbb{R}$ of the function $f$, for which $F\left(\frac{3 \pi}{2}\right)=\frac{1}{2021}$.

Answer: F(x) = -\frac{1}{2021 (\sin x + 3 \cos x)^{2021}} - \frac{1}{2020 (\sin x + 3 \cos x)^{2020}} + \frac{1}{2020}


## Task B-3.3.

The length of the side of the rhombus $A B C D$ is $6 \, \text{cm}$, and the measure of the angle at vertex $B$ is $120^{\circ}$. All points inside the rhombus that are closer to vertex $B$ than to the other vertices lie within the pentagon $P$. Determine its area.

Answer: 6\sqrt{3}


2. Given the line $l: y=x-a$ intersects the parabola $C$: $x^{2}=2 p y(p>0)$ at points $M$ and $N$, and the circle passing through $M$ and $N$ intersects the parabola $C$ at two other distinct points $E$ and $F$. Then the cosine of the inclination angle of the line $E F$ is $\qquad$.

Answer: -\dfrac{\sqrt{2}}{2}


Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for evert integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times $111$ appears on this sequence?

Answer: 14


36. (5 points)
Big Ears Toutu and his good friend Z Zhuang set off from $A$ and $B$ respectively at the same time, heading towards each other, originally planning to meet at point $C$, where $AC=\frac{4}{7} AB$. However, Toutu rested for 35 seconds along the way, resulting in Zhuang walking 60 meters more than originally planned before meeting Toutu. What is Toutu's speed in meters/second?

Answer: 4


Let \( a_{1}, \ldots, a_{25} \) be non-negative integers, and let \( k \) be the smallest among them. Prove that

\[
\left[\sqrt{a_{1}}\right]+\left[\sqrt{a_{2}}\right]+\ldots+\left[\sqrt{a_{25}}\right] \geq\left[\sqrt{a_{1}+\ldots+a_{25}+200 k}\right]
\]

(As usual, \([x]\) denotes the greatest integer that does not exceed \( x \).)

(S. Berlov, A. Khrabrov)

Answer: \left[\sqrt{a_{1}}\right]+\left[\sqrt{a_{2}}\right]+\ldots+\left[\sqrt{a_{25}}\right] \geq\left[\sqrt{a_{1}+\ldots+a_{25}+200 k}\right]


9. Is this possible? Can three people cover a distance of 60 km in 3 hours if they have a two-person motorcycle at their disposal? The motorcycle's speed is 50 km/h, and the pedestrian's speed is 5 km/h.

Answer: Yes


Ivan and Mirka were sharing pears from a plate. Ivan always took two pears, and Mirka took half of what was left on the plate. They proceeded in this manner: Ivan, Mirka, Ivan, Mirka, and finally Ivan, who took the last two pears.

Determine who ended up with more pears and by how many.

(M. Dillingerová)

Hint. How many pears did Mirka take the second time?

Answer: 2


One. (20 points) As shown in Figure 6, in the right triangle $\triangle ABC$, $\angle BAC=90^{\circ}$, $AD \perp BC$ at point $D$, $P$ is the midpoint of $AD$, the extension of $BP$ intersects $AC$ at point $E$, $EF \perp BC$ at point $F$, and the extension of $FE$ intersects the circumcircle $\odot O$ of $\triangle ABC$ at point $G$. If $AE=3$, $EC=12$, find the length of segment $EG$.

Answer: 6(\sqrt{2} - 1)


6. The number of triangles with different shapes that can be formed using the vertices of a regular tridecagon is $\qquad$ (Note: Congruent triangles are considered to have the same shape).

Answer: 14


Exercise 9. Let $\mathbb{R}_{+}^{*}$ be the set of strictly positive real numbers. Find all functions $f: \mathbb{R}_{+}^{*} \longrightarrow \mathbb{R}_{+}^{*}$ such that for all $x, y>0$, we have

$$
f\left(\frac{f(x)}{y f(x)+1}\right)=\frac{x}{x f(y)+1}
$$

Answer: f(x) = x


63. A, B, and C are playing a round-robin table tennis tournament, with the rule that the winner gets 2 points, the loser gets 0 points, and in case of a draw, each gets 1 point. After the tournament, there are $\qquad$ possible score situations for the three players.

Answer: 19


Answer: 4^8


## Task 1 - 090521

Six equally sized dice are stacked on a table as shown in the image. The number on the top face is 1.

Determine the sum of the numbers on the hidden faces of these dice!

Note that the sum of the numbers on opposite faces of each die is always 7.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0046.jpg?height=329&width=97&top_left_y=378&top_left_x=1619)

Answer: 41


10. In $\triangle A B C$, $A B=\sqrt{30}, A C=\sqrt{6}, B C$ $=\sqrt{15}$, there is a point $D$ such that $A D$ bisects $B C$ and $\angle A D B$ is a right angle, the ratio $\frac{S_{\triangle A D B}}{S_{\triangle A B C}}$ can be written as $\frac{m}{n}$. ($m, n$ are coprime positive integers). Then $m+n=$

Answer: 65


13. (3 points) As shown in the figure: In parallelogram $A B C D$, $O E=E F=F D$. The area of the parallelogram is 240 square centimeters, and the area of the shaded part is $\qquad$ square centimeters.

Answer: 20


1. $A, B, C, D, E$ five people participate in a chess tournament together and find that their average age is exactly 28 years old. One year later, $A, B, C, D, F$ participate together and find that their average age is exactly 30 years old. How many years older is $F$ than $E$?

Answer: 5


3. In a tournament with six teams, each team plays once against every other team. If a team wins a match, it receives 3 points and the loser gets 0 points. If a match ends in a draw, both teams receive 1 point.

Can the final scores of the teams be exactly six consecutive numbers $a, a+1, \ldots, a+5$? If so, determine all values of $a$ for which this is possible.

Answer: 4


Question 236, Oleg wrote a real number above each column square (outside the table) and to the left of each row square (outside the table) on a blank $50 \times 50$ grid. It is known that all 100 real numbers he wrote are distinct, and there are exactly 50 rational and 50 irrational numbers among them. Then, Oleg wrote the sum of the number to the left of the row and the number above the column in each cell (creating an addition table). How many rational numbers can appear in the grid at most?

Answer: 1250


5. Tina has been depositing 500 euros at the beginning of each year for five consecutive years into a bank that capitalizes annually at an interest rate of $5 \%$. How many years after the last deposit will she have 4500 euros?

## You have 120 minutes to solve the problems.

Solve each problem independently on the provided paper, with each problem on a separate page. Do not sign the sheets, write only your code. Write your answers in ink, clearly and neatly. Draw graphs of functions with a pencil. If you solve a problem in multiple ways, clearly indicate which solution the evaluator should grade. If you make a mistake, strike it through.

Each problem will be graded on a scale of 0 to 6 points.

The National Competition Committee wishes you great success.

## 7th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools

National Competition, April 21, 2007

## Solutions to Problems and Scoring Guide

## A contestant who arrives at the correct solution by any valid method (even if not anticipated by the scoring guide) receives all possible points.

A correct method is one that

- sensibly takes into account the problem statement,
- leads to the solution of the problem,
- is mathematically correct and complete.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the contestant generally receives all the points provided. However, if the solution is guessed (and cannot be arrived at without calculation), such a solution is scored with 0 points.

A contestant who has only partially solved a problem, with correct but incomplete steps, cannot receive more than half of the possible points.

## First Year

Answer: 10


Problem 8.5.1. In the cells of a $12 \times 12$ table, natural numbers are arranged such that the following condition is met: for any number in a non-corner cell, there is an adjacent cell (by side) that contains a smaller number. What is the smallest number of different numbers that can be in the table?

(Non-corner cells are those that are not in the corner of the table. There are exactly 140 of them.)

Answer: 11


4.76 Suppose the roads of a city form a $(n-1) \times(n-1)$ grid. The city plans to open several bus routes, with the requirement that each bus route turns at most once, and that from any of the $n^{2}$ intersections, one can reach any other intersection with at most one transfer. How many routes does the city need to set up at a minimum?

Answer: n


8. An $A M C$ exam consists of 30 questions, with the scoring rules specifying 5 points for each correct answer, 0 points for each wrong answer, and 2 points for each unanswered question.
The sum of the points a participant earns for each question is called the participant's total score. Assuming there are enough participants, the number of different possible total scores is $\qquad$

Answer: 145


8. Fill the circles with the numbers $0, 1, 2, 3, 4, 5, 6$ in the grid, with each number appearing exactly once, to form an integer equation consisting of single-digit and two-digit numbers. What number is filled in the grid?
$$
O \times O=\square=O \div O
$$

Answer: 12


# Task No. 8.2

## Condition:

Dr. Watson thinks that his watch is 5 minutes slow, but in reality, it is 13 minutes fast. He agreed to meet Sherlock Holmes at 16:00, and then called and said that he wouldn't make it on time and would be 20 minutes late. At what time will Watson actually arrive for the meeting?

Answer: 16:02


3. In Chernomorsk, there are $n$ residents. All of them formed a party. Then this party split into two non-intersecting factions, each of which declared itself a party. Each subsequent day, each party formed on the previous day, consisting of at least two members, split into two factions. Factions consisting of at least two people immediately declared themselves parties. No other parties were formed. When the process ended, each resident of Chernomorsk paid a membership fee of 1 ruble to each party they belonged to. What is the maximum amount the total fees could have been?

Answer: n(n+1)/2-1


18. (16th All-Russian Mathematics Competition) In the school football championship, each team must play a match against every other team. Each match awards 2 points to the winning team, 1 point to each team in the event of a draw, and 0 points to the losing team. It is known that one team has the highest score, but it has won fewer matches than any other team. How many teams must have participated at a minimum?

Answer: 6


4. The function $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} b x^{2}+c x+d$ has both a maximum and a minimum value in the interval $(0,2)$, then the range of $c^{2}+2 b c+4 c$ is $\qquad$ .

Answer: (0,1)


8. (10 points) Three households, A, B, and C, plan to subscribe to newspapers. There are 7 different newspapers available. It is known that each household subscribes to 3 different newspapers, and any two households have exactly one newspaper in common. Therefore, the three households have $\qquad$ different subscription methods.

Answer: 5670


28. Two brothers sold a flock of sheep that belonged to both of them, taking as many rubles for each sheep as there were sheep in the flock. The money received was divided as follows: first, the elder brother took ten rubles from the total amount, then the younger brother took ten rubles, after that the elder brother took another ten rubles, and so on. In the end, the younger brother did not have enough ten rubles; therefore, he took all the small change left after the division, and in addition, to make the division fair, the elder brother gave the younger brother his pocket knife. What was the pocket knife valued at?

Answer: 2


12.11. (CSSR, 80). The set $M$ is obtained from the plane by removing three distinct points $A, B$, and $C$. Find the smallest number of convex sets whose union is the set $M$.

Answer: 4


19. A seven-digit number, its hundred's digit is twice the million's digit, its ten's digit is twice the hundred thousand's digit, its unit's digit is twice the ten thousand's digit, its thousand's digit is 0, and it can definitely be divided by a five-digit number $a$, then $a$ is $\qquad$ .

Answer: 10002


Let $N \in \mathbb{N}$.

Discriminant the cat proposes the following game to you:

- You choose a finite sequence (of your choice and kept secret) of integers $\left(a_{0}, \ldots, a_{n}\right) \in \mathbb{N}^{n+1}$
- At each turn of the game, Discriminant gives you a real number $\beta \in \mathbb{N}$ and you respond with the value of $\sum_{k=0}^{n} a_{k} \beta^{k}$.
- The game continues until Discriminant has guessed the sequence of integers or until fewer than $N$ turns have taken place.
- If at the end of the game Discriminant has guessed the sequence of integers, he wins; otherwise, he loses.

If Discriminant wins, you owe him a can of cat pate.

For which values of $N$ would you accept to play this game?

Answer: 1


4. The bisectors of the exterior angles at vertices $A$ and $D$ of trapezoid $\mathrm{ABCD}(\mathrm{AB} \| \mathrm{CD}$) intersect at point $\mathrm{M}$, and the bisectors of the exterior angles at vertices B and C intersect at point $\mathrm{N}$ such that $\overline{\mathrm{MN}}=12 \mathrm{~cm}$. Calculate the perimeter of the trapezoid.

Answer: 24


Given a point $P$ and a circle $\mathcal{C}$ in the plane, the distance from $P$ to $\mathcal{C}$ is defined as the minimum length $PM$ between $P$ and a point $M$ on the circle $\mathcal{C}$. For example, if $P$ lies on the circle, then the distance from $P$ to $\mathcal{C}$ is zero, and if $P$ is the center of the circle, then the distance from $P$ to $\mathcal{C}$ is equal to the radius of $\mathcal{C}$.

Given four non-cyclic points $A, B, C, D$, what is the maximum number of circles that pass at equal distance from these four points?

Answer: 7


3. As shown in Figure 4, the parabola $y=a x^{2}+b x-3$ intersects the $x$-axis at points $A$ and $B$, and intersects the $y$-axis at point $C$, and $O B=O C=3 O A$. The line $y=-\frac{1}{3} x+1$ intersects the $y$-axis at point $D$, and $E$ is the vertex of the parabola. If $\angle D B C=\alpha, \angle C B E=\beta$, then $\alpha-\beta=$ . $\qquad$

Answer: 45


8. (5 points) The founder of a noble family received a plot of land. Each man in the family, upon dying, divided the land he inherited equally among his sons. If he had no sons, the land went to the state. No other members of the family gained or lost any land in any other way. In total, there were 150 people in the family. What is the smallest fraction of the original plot of land that any member of the family could have received?

Answer: \dfrac{1}{2 \times 3^{49}}


3. At a round table, raccoons, hedgehogs, and hamsters were sitting, a total of 134 animals. When asked: "Are there any animals of the same kind as you among your neighbors?", everyone answered "No." What is the maximum number of hedgehogs that could have been sitting at the table, given that hamsters and hedgehogs always tell the truth, raccoons almost always lie (except when a raccoon is sitting between two raccoons, in which case he tells the truth), and hamsters heard the question differently: "Are your neighbors animals of the same kind?"

Answer: 44


12.263. The height of a regular truncated triangular pyramid is $H$ and is the mean proportional between the sides of the bases. The lateral edge makes an angle $\alpha$ with the base. Find the volume of the pyramid.

Answer: \dfrac{\sqrt{3} H^3}{4 \sin^2 \alpha}


6・39 In a chess tournament, each pair of players plays exactly one game, with the winner getting 1 point, the loser getting 0 points, and each player getting $\frac{1}{2}$ point in the case of a draw. After the tournament, it was found that each player's score was exactly half from games against the 10 lowest-scoring players (the 10 lowest-scoring players got half of their points from games against each other). Find the total number of players in the tournament.

Answer: 25


12. The acute angle or right angle formed by the intersection of two lines is called the "angle" between the two lines. Now there are several lines on a plane, they intersect each other in pairs, and the "angle" can only be $30^{\circ}, 60^{\circ}$, or $90^{\circ}$. Question: What is the maximum number of lines?

Answer: 6


3 Suppose 2005 line segments are connected end-to-end, forming a closed polyline, and no two segments of the polyline lie on the same straight line. Then, what is the maximum number of intersection points where the polyline intersects itself?

Answer: 2007005


As shown in the figure, two squares $A B F E$ and $C D E F$ with side lengths of 6 are combined to form a rectangle $A B C D$. Point $G$ is the midpoint of $D E$. The line segment $B G$ intersects $E F$ at $H$. Find the area of the pentagon $C D G H F$.

Answer: 33


3.104. Find the angle between the generatrix and the base of a truncated cone, the total surface area of which is twice the surface area of a sphere inscribed in it.

Answer: \arctan(2)


The moments $\mathrm{E} X^{n}$ and absolute moments $\mathrm{E}|X|^{n}$ of the random variable $X$ can be expressed for integers $n \geq 1$ through the $n$-th order derivatives of the characteristic function $\varphi(t) = \mathrm{E} e^{i t X}$, $t \in \mathbb{R}$. For fractional $\alpha>0$, to obtain the corresponding results for $\mathrm{E} X^{\alpha}$ and $\mathrm{E}|X|^{\alpha}$, one has to resort to the concept of fractional derivatives, introduced as follows.

Let $n$ be a non-negative integer and $0<a<1$. The fractional derivative $\frac{d^{n+a} f(t)}{d t^{n+a}}$ of a function $f=f(t)$, $t \in \mathbb{R}$, is defined as

$$
\frac{a}{\Gamma(1-a)} \int_{-\infty}^{t} \frac{f^{(n)}(t)-f^{(n)}(s)}{(t-s)^{1+a}} d s
$$

(provided the integral is defined).

In particular, if $\varphi=\varphi(t)$ is the characteristic function of a random variable $X$, then

$$
\begin{aligned}
& \frac{d^{n+a} \varphi(0)}{d t^{n+a}}=\frac{a}{\Gamma(1-a)} \int_{\mathbb{R}_{+}} \frac{\varphi^{(n)}(0)-\varphi^{(n)}(-s)}{s^{1+a}} d s= \\
& =\frac{i^{n} a}{\Gamma(1-a)} \int_{\mathbb{R}_{+}}\left[\mathrm{E} X^{n}(1-\cos s X)+i \mathrm{E} X^{n} \sin s X\right] \frac{d s}{s^{1+a}}
\end{aligned}
$$

Show that for even $n$ the absolute moments $\mathrm{E}|X|^{n+a}$ are finite if and only if

(1) $\mathrm{E} X^{n}<\infty$;

(2) $\frac{d^{n+a} \text{Re} \varphi(0)}{d t^{n+a}}$ exists.

In this case

$$
\mathrm{E}|X|^{n+a}=\frac{(-1)^{n / 2}}{\cos \frac{a \pi}{2}} \frac{d^{n+a} \operatorname{Re} \varphi(0)}{d t^{n+a}}
$$

Answer: \mathrm{E}|X|^{n+a} = \frac{(-1)^{n / 2}}{\cos \frac{a \pi}{2}} \frac{d^{n+a} \operatorname{Re} \varphi(0)}{d t^{n+a}}


2. In $\triangle A B C$, $\angle C=100^{\circ}, \angle A=60^{\circ}$, take a point $P$ on the line $A C$ such that $\triangle P A B$ is an isosceles triangle, then the number of points $P$ that satisfy the condition is $\qquad$.

Answer: 2


92. Three consecutive vertices of a rhombus lie on the sides $A B, B C, C D$ of a given square with side 1. Find the area of the figure filled by the fourth vertices of such rhombi.

Answer: 1


5.1. It is known that in the past chess tournament, in each round all players were paired, the loser was eliminated (there were no draws). It is known that the winner played 6 games. How many participants in the tournament won at least 2 games more than they lost?

Answer: 8


Answer: 24


24. Preparing his son for school, the father bought him a new uniform for 24 r. How did he pay for the purchase, if he only had 5 r bills, and the cashier only had 3 r bills?

Answer: 6


"Pre-assembled" unit cube has a base and a top, as well as its sides. The top is rotated by an acute angle $\alpha$ around the line connecting the centers of the two faces, relative to the base. By how much does the top get closer to the base if the side edges remain the same length and the connection between the vertices also remains unchanged?

Answer: 1 - \sqrt{\cos \alpha}


1. Two boats start from places $A$ and $B$ heading towards each other. Each of them, when they reach one place, turns back to the other place. The first time the boats met was $5 \mathrm{~km}$ from $A$, and the second time was $3 \mathrm{~km}$ from $B$. Determine the distance from $A$ to $B$.

Answer: 12


2. In the Cartesian coordinate system, a line segment $A B$ of length 1 moves on the $x$-axis (point $A$ is to the left of $B$), point $P(0,1)$ is connected to $A$ by a line, and point $Q(1,2)$ is connected to $B$ by a line. Then the equation of the trajectory of the intersection point $R$ of lines $P A$ and $Q B$ is $\qquad$

Answer: xy - 2y + 2 = 0


Once, I decided to take a chairlift ride. At a certain moment, I noticed that the chairlift coming towards me had the number 95, and the next one had the number 0, followed by 1, 2, and so on. I looked at the number of my chair, which turned out to be 66. Have I traveled half the distance? When I encounter which chair will I have traveled half the distance?

Answer: 18


Given \( n \) points in a \( 10 \cdot 40 \) space (where \( n \geq 5 \)), no four points are coplanar, and they have the following property: if a sphere passes through any four of these points, then all the remaining points are located on or inside this sphere. Prove that all the given points lie on the same sphere.

Answer: \text{All given points lie on the same sphere.}


10. As shown in the figure, in square $\mathrm{ABCD}$, the area of isosceles right triangle $\mathrm{AEF}$ is $1 \mathrm{~cm}^2$, and the area of rectangle $\mathrm{EFGH}$ is $10 \mathrm{~cm}^2$. What is the area of square $\mathrm{ABCD}$ in $\mathrm{cm}^2$?

Answer: \dfrac{49}{2}


Let $ a$ be a positive real numbers. In the coordinate plane denote by $ S$ the area of the figure bounded by the curve $ y=\sin x\ (0\leq x\leq \pi)$ and the $ x$-axis and denote $ T$ by the area of the figure bounded by the curves $ y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\　y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right)$ and the $ x$-axis. Find the value of $ a$ such that $ S: T=3: 1$.

Answer: \dfrac{4}{3}


Through vertex \( A \) of rectangle \( ABCD \), a straight line \( \ell \) is drawn as shown in the figure. Perpendiculars \( BX \) and \( DY \) are dropped from points \( B \) and \( D \) onto line \( \ell \). Find the length of segment \( XY \), given that \( BX = 4 \), \( DY = 10 \), and \( BC = 2AB \).

Answer: 13


A fair coin is tossed 5 times. The probability of getting exactly one heads is not zero and is the same as the probability of getting exactly two heads.

Let the reduced fraction \(\frac{i}{j}\) represent the probability of getting exactly 3 heads when the coin is tossed 5 times. Find the value of \(i + j\).

Answer: 283


A lateral edge of a regular triangular pyramid forms an angle of $45^{\circ}$ with the plane of the base. Find the angle between the apothem of the pyramid and the plane of an adjacent lateral face.

Answer: \arcsin\frac{3}{5}


A natural number \( N \) has all of its divisors listed. The sum of the digits of each of these divisors is then calculated. It turns out that among these sums, all numbers from 1 to 9 are found. Find the smallest value of \( N \).

Answer: 288


In a 3 by 3 grid of unit squares, an up-right path is a path from the bottom left corner to the top right corner that travels only up and right in steps of 1 unit. For such a path \( p \), let \( A_{p} \) denote the number of unit squares under the path \( p \). Compute the sum of \( A_{p} \) over all up-right paths \( p \).

Answer: 90


Small and Big islands have a rectangular shape and are divided into rectangular counties. Each county has a road along one of the diagonals. On each island, these roads form a closed path that does not pass through any point more than once. Here's how the Small Island, consisting of 6 counties, is arranged (see Figure 1). Draw how the Big Island, which has an odd number of counties, could be arranged. How many counties did you get?

Answer: 9


In Mexico, ecologists have achieved the passage of a law whereby each car must not be driven at least one day a week (the owner informs the police of the car’s number and "day off" for that car). In a certain family, all adults wish to drive daily (each for their own errands!). How many cars (at minimum) does the family need if there are

a) 5 adults? b) 8 adults?

Answer: 10


In square \(ABCD\), point \(P\) lies on \(AB\), and \(DP\) bisects \(BC\) at point \(N\). Point \(Q\) lies on \(BA\), and \(CQ\) bisects \(AD\) at point \(M\). Line \(DP\) intersects line \(CQ\) at point \(O\). Given that the area of square \(ABCD\) is 192, find the area of \(\triangle POQ\).

Answer: 216


Five friends, one of whom had a monkey, bought a bag of nuts, which they planned to divide amongst themselves the next morning. However, during the night, one friend woke up and wanted some nuts. He divided all the nuts in the bag into five equal parts, with one nut left over, which he gave to the monkey, and took his fifth part. Following him, another friend woke up; unaware that someone had already taken nuts, he divided the remaining nuts in the bag again into five parts, with one nut left over, which he gave to the monkey, and took his fifth part. Subsequently, the remaining three friends woke up one after another, each performing the same operation: dividing the remaining nuts into five parts, taking their fifth part, with one nut left over each time, which was given to the monkey. Finally, in the morning, all five friends took out the bag, divided the remaining nuts into five parts, and the one nut left over was once again given to the monkey. Determine the smallest number of nuts in the bag such that this division is possible.

Answer: 15621


Let \( S \) be the set of points \(\left(x_{1}, x_{2}, \ldots, x_{2012}\right)\) in 2012-dimensional space such that \(\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{2012}\right| \leq 1\). Let \( T \) be the set of points in 2012-dimensional space such that \(\max _{i=1}^{2012}\left|x_{i}\right|=2\). Let \( p \) be a randomly chosen point on \( T \). What is the probability that the closest point in \( S \) to \( p \) is a vertex of \( S \)?

Answer: \frac{1}{2^{2011}}


Ivan, a full-time student, started a job on March 1st of the current year, a job he had long dreamed of. During the first two months of his probationary period, Ivan's salary was 20,000 rubles. After the probationary period ended, his salary increased to 25,000 rubles per month. In December, Ivan was rewarded with a 10,000 rubles bonus for exceeding his targets. Additionally, as a full-time budget-funded graduate student, he received a scholarship of 2,800 rubles per month throughout the year. What amount of personal income tax should Ivan pay into the budget? (Provide the answer as a whole number, without spaces or units of measurement)

Answer: 32500


Let \( f: \mathbf{Z} \rightarrow \mathbf{Z} \) be an even function such that \( f(1)=1 \) and \( f(2017) \neq 1 \). For any \( x, y \in \mathbf{Z} \), the following inequality holds:
$$
2 f(x+y)-f(x)-f(y) \leqslant|f(x)-f(y)|.
$$

Find all possible values of \( f(2018) \).

Answer: 1


A biologist watches a chameleon. The chameleon catches flies and rests after each catch. The biologist notices that:
(i) the first fly is caught after a resting period of one minute;
(ii) the resting period before catching the \(2m^{\text{th}}\) fly is the same as the resting period before catching the \(m^{\text{th}}\) fly and one minute shorter than the resting period before catching the \((2m+1)^{\text{th}}\) fly;
(iii) when the chameleon stops resting, he catches a fly instantly.

(a) How many flies were caught by the chameleon before his first resting period of 9 minutes?
(b) After how many minutes will the chameleon catch his 98th fly?
(c) How many flies were caught by the chameleon after 1999 minutes passed?

Answer: 462


Bob and Jane hold identical decks of twelve cards, three of each color: red, green, yellow, and blue. Bob and Jane shuffle their decks and then take turns dealing one card at a time onto a pile, with Jane going first. Find the probability that Jane deals all her red cards before Bob deals any of his red cards. Give your answer in the form of a fraction in lowest terms.

Answer: \frac{39}{1100}


Let \( A_{1}B_{1}C_{1}, A_{2}B_{2}C_{2} \), and \( A_{3}B_{3}C_{3} \) be three triangles in the plane. For \( 1 \leq i \leq 3 \), let \( D_{i}, E_{i} \), and \( F_{i} \) be the midpoints of \( B_{i}C_{i}, A_{i}C_{i} \), and \( A_{i}B_{i} \), respectively. Furthermore, for \( 1 \leq i \leq 3 \) let \( G_{i} \) be the centroid of \( A_{i}B_{i}C_{i} \). Suppose that the areas of the triangles \( A_{1}A_{2}A_{3}, B_{1}B_{2}B_{3}, C_{1}C_{2}C_{3}, D_{1}D_{2}D_{3}, E_{1}E_{2}E_{3} \), and \( F_{1}F_{2}F_{3} \) are 2, 3, 4, 20, 21, and 2020, respectively. Compute the largest possible area of \( G_{1}G_{2}G_{3} \).

Answer: 917


On the ceiling of a rectangular classroom measuring $10 \text{ m}$ in length, two lamps emitting conical light beams with an opening angle of $90^{\circ}$ are installed. The first lamp is located at the center of the ceiling and illuminates a circle on the floor with a diameter of $6 \text{ m}$. The second lamp is adjusted such that the illuminated area along the length of the classroom spans a $10 \text{ m}$ section, but the light does not reach the two opposite walls. How far apart are the two lamps?

Answer: 4


Below is the graph of \(y = a \sin (bx + c)\) for some positive constants \(a\), \(b\), and \(c\). The graph reaches its maximum at \(x = -\frac{\pi}{4}\) and has an amplitude of 3. Find the smallest possible value of \(c\).

Answer: \frac{3\pi}{4}


Alice took a mathematics test consisting of 90 questions: 15 arithmetic, 35 algebra, and 40 geometry problems. She answered $80\%$ of the arithmetic, $50\%$ of the algebra, and $70\%$ of the geometry problems correctly. However, she failed to pass the test as she got less than $65\%$ of the total problems right. How many more questions would she have needed to answer correctly to earn a $65\%$ passing grade?

Answer: 1


Triangle $PQR$ is similar to triangle $XYZ$. If $PQ = 8$, $QR = 16$, and $YZ = 32$ units, what are the lengths of segments $XY$ and $XZ$?

Answer: 32


In Mr. Greene's first-grade class, there are 36 students. If there are three times as many students with red hair as with green eyes, 12 students have red hair and green eyes, and 4 students have neither red hair nor green eyes, how many students have green eyes?

Answer: 11


The side length of a regular hexagon is 10 cm. Calculate the area of a trapezoidal region created by connecting opposite vertices of the hexagon. Express your answer in simplest radical form.

[asy]
size(120);
pair A,B,C,D,E,F;
A = dir(0); B = dir(60); C = dir(120); D = dir(180); E = dir(240); F = dir(300);

fill(A--C--E--cycle,heavycyan);

draw(A--B--C--D--E--F--A);
[/asy]

Answer: 100\sqrt{3} \text{ cm}^2


The Aquaflow system pumps water at a rate of 360 gallons per hour. How long will it take to pump a total of 540 gallons if after the first 30 minutes, the rate is increased to 480 gallons per hour?

Answer: 75 \text{ minutes}


Altitudes \( \overline{AD} \) and \( \overline{BE} \) of an acute triangle \( \triangle ABC \) intersect at point \( H \). If \( HD = 8 \) and \( HE = 3 \), compute \( (BD)(DC) - (AE)(EC) \).

Answer: 55


A circle with center \(A\) and radius 4 inches is tangent at \(C\) to a circle with center \(B\). Assume that point \(B\) is located on the smaller circle. What is the area of the shaded region formed by these circles? Express your answer in terms of \(\pi\).

Answer: 48\pi


A rectangular box has a volume of 5184 cubic inches and a surface area of 1944 square inches. The sum of the lengths of its 12 edges totals 216 inches. Calculate the volume of the box if its length, width, and height were each increased by two inches.

Answer: 7352 \text{ cubic inches}


For each positive integer $n$, define $h(n) = n^4 - 500n^2 + 625$. What is the sum of all values of $h(n)$ that are prime numbers?

Answer: 0


An equilateral triangle $ABC$ shares a side with square $BCDE$. Another equilateral triangle $AEF$ is positioned such that $AF$ is parallel to $BC$. Determine the number of degrees in the measure of angle $FAE$.

Answer: 60^\circ


At the mall's food court, Sarah wants to buy a meal consisting of one appetizer, one main course, one drink, and one dessert. The table below lists Sarah's options available in the food court, but she doesn't want to pick a soda if she chooses fries as an appetizer. How many distinct possible meals can she buy from these options?

\begin{tabular}{ |c | c | c | c|}
\hline \textbf{Appetizers} & \textbf{Main Courses} & \textbf{Drinks}& \textbf{Desserts} \\ \hline
Fries & Burger & Soda & Ice Cream \\ \hline
Salad & Chicken Wrap & Lemonade & Cake \\ \hline
Onion Rings & Pizza & Tea &  \\ \hline
\end{tabular}

Answer: 48


Trapezoid $ABCD$ (with bases $AB$ and $CD$) has an area of $48$ square units. Points $E$ and $F$ are the midpoints of the non-parallel sides $AD$ and $BC$, respectively. What is the area of triangle $BDF$?

Answer: 12


Mason is seeking to ascertain the Fermat point \(P\) of \(\triangle ABC\), where \(A\) is at the origin, \(B\) is at \( (12, 0) \), and \(C\) is at \( (4, 6) \). Thinking that \(P = (5,3)\) could be the Fermat point, he decides to compute the sum of the distances from \(P\) to the vertices. The final expression he achieves is in the form \(x\sqrt{a} + y\sqrt{b}\), where \(x\), \(y\), \(a\), and \(b\) are integers. Compute \(x + y\).

Answer: 3


Borisov L. The sage S. was told the sum of three natural numbers, while the sage P. was told their product. - If I had known, - said S., - that your number is greater than mine, I would have immediately named the three numbers. - My number is less than yours, - replied P., - and the numbers are ..., ... and ... . What numbers did P. name?

Answer: 1,1,4


10-1. At the base of the pyramid $S A B C D$ lies a trapezoid $A B C D$ with bases $B C$ and $A D$. Points $P_{1}$, $P_{2}, P_{3}$ belong to the side $B C$, such that $B P_{1}<B P_{2}<B P_{3}<B C$. Points $Q_{1}, Q_{2}, Q_{3}$ belong to the side $A D$, such that $A Q_{1}<A Q_{2}<A Q_{3}<A D$. Denote the points of intersection of $B Q_{1}$ with $A P_{1}$, $P_{2} Q_{1}$ with $P_{1} Q_{2}$, $P_{3} Q_{2}$ with $P_{2} Q_{3}$, and $C Q_{3}$ with $P_{3} D$ as $R_{1}, R_{2}, R_{3}$, and $R_{4}$, respectively. It is known that the sum of the volumes of the pyramids $S R_{1} P_{1} R_{2} Q_{1}$ and $S R_{3} P_{3} R_{4} Q_{3}$ is 78. Find the minimum value of the quantity

$$
V_{S A B R_{1}}^{2}+V_{S R_{2} P_{2} R_{3} Q_{2}}^{2}+V_{S C D R_{4}}^{2}
$$

In your answer, specify the integer closest to the found value.

Answer: 2028


Problem 5.8. In the large family of Ivanovs, there are no twins. A reporter came to the Ivanovs to interview them.

During the interview, each of the children said: "I have an older brother." After thinking for a moment, the reporter was very surprised. But the head of the family explained that some of the children were joking, and only six of them told the truth. How many children can there be in this family, given that there are 4 more boys than girls? List all possible options.

Answer: 8,10


3. Fill the numbers $1,2, \cdots, 36$ into a $6 \times 6$ grid, with each cell containing one number, such that the numbers in each row are in increasing order from left to right. The minimum value of the sum of the six numbers in the third column is $\qquad$
$(2015$, National Junior High School Mathematics League Competition)

Answer: 63


Problem 4. Five numbers $2,0,1,9,0$ are written in a circle on the board in the given order clockwise (the last zero is written next to the first two). In one move, the sum of each pair of adjacent numbers is written between them. For example, such an arrangement of numbers (on the right) will be after the first move:

![](https://cdn.mathpix.com/cropped/2024_05_06_edfa979e23ca6aeb44cfg-3.jpg?height=226&width=511&top_left_y=93&top_left_x=447)

After 5 moves, Polina calculated the sum of all numbers from the first zero (the one initially between 2 and 1) to the second zero (the one initially between 9 and 2) when moving around the circle clockwise, while Alina calculated the sum of all other numbers. What is the difference between Alina's and Polina's numbers?

Answer: 1944


Four. (50 points) On a plane, there are 100 different points and $n$ different lines $l_{1}, l_{2}, \cdots, l_{n}$, and the number of points that line $l_{k}$ passes through is denoted as $a_{k}$. If $a_{1}+a_{2}+\cdots+a_{n}=250$, find the minimum possible value of $n$.

Answer: 21


8. As shown in the figure, the perimeter of square $A B C D$ is $40 \mathrm{~m}$. Two people, Jia and Yi, start from $A$ and $B$ respectively at the same time and walk along the edges of the square. Jia walks $35 \mathrm{~m}$ per minute in a counterclockwise direction, and Yi walks $30 \mathrm{~m}$ per minute in a clockwise direction. If the notation $(a, b)$ represents that the two people have been walking for $a$ minutes and have met $b$ times, then when the two people first find themselves at two opposite vertices of the square after starting, the corresponding notation should be $(\quad$.

Answer: (6,10)


10. (10 points) As shown in the figure, there is a switch at point $A$ that can connect to $B$ or $C$. The circumference of the small loop track is 1.5 meters, and the circumference of the large loop track is 3 meters. Initially, $A$ is connected to $C$, and the train starts from point $A$, moving clockwise on the track. The switch changes the track connection every 1 minute. If the train's speed is 10 meters per minute, then the train will have used seconds when it returns to point $A$ for the 10th time.

Answer: 126


9. (10 points) At 8:10 AM, Feifei walked from home to school. 3 minutes later, the dog started running to catch up with her, and caught up with her 200 meters away from home; after catching up, it immediately ran back home, and then immediately turned around to catch Feifei again, catching up with her 400 meters away from home. After catching up, it ran back home again, and then went to catch Feifei, finally catching up with her at school. Feifei arrived at school at 8: $\qquad$ minutes.

Answer: 28


Problem 5.7. In a class, there are 31 students. Three of them have exactly three friends each, the next three have six each, the next three have nine each, ..., and the next three have thirty each. How many friends does the 31st student have? (Friendship between people is mutual.)

![](https://cdn.mathpix.com/cropped/2024_05_06_32e5791ee8081231aabeg-6.jpg?height=494&width=525&top_left_y=80&top_left_x=464)

Fig. 2: to the solution of problem 5.6

![](https://cdn.mathpix.com/cropped/2024_05_06_32e5791ee8081231aabeg-6.jpg?height=260&width=521&top_left_y=661&top_left_x=466)

Fig. 3: to the solution of problem 5.6

#

Answer: 15


[ [ divisibility rules for 3 and 9 ]

It is known that $35!=10333147966386144929 * 66651337523200000000$. Find the digit replaced by the asterisk.

The number 35! is divisible by 9.

#

Answer: 6


93.3. Find all solutions of the system of equations

$$
\left\{\begin{aligned}
s(x)+s(y) & =x \\
x+y+s(z) & =z \\
s(x)+s(y)+s(z) & =y-4
\end{aligned}\right.
$$

where $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digits in the decimal representations of $x, y$, and $z$, respectively.

Answer: (2,8,12)


Three. (50 points)
In a square with a side length of 4, 2002 points are placed. How many pairs of points are there with a distance less than 3 (find the largest specific number)?

Answer: 500000


7. (10 points) There are 37 people standing in a line to count off. The first person reports 1, and each subsequent person reports a number that is 3 more than the previous person's number. During the counting, one person made a mistake and reported a number that was 3 less than the previous person's number. The sum of the numbers reported by these 37 people is exactly 2011. Then, the person who made the mistake is the $\qquad$th person to report a number.

Answer: 34


24. A polynomial of degree 10 has three distinct roots. What is the maximum number of zero coefficients it can have?

(A. Khryabrov)

Answer: 9


7. Two identical resistors with resistance $R$ each are connected in series and connected to a source of constant voltage $U$. An ideal voltmeter is connected in parallel to one of the resistors. Its readings were $U_{v}=10 B$. After that, the voltmeter was replaced with an ideal ammeter. The ammeter readings were $-I_{A}=10 \mathrm{~A}$. Determine the value of $R$. (10 points)

Answer: 2


3. Two investors jointly invest in a business. Then one of them adds 1 million rubles, which increases his share in the business by 0.04. When he adds another 1 million rubles, his share increases by 0.02. Question: How much more money must he add to increase his share by another 0.04?

Answer: 8


14. A construction company

has contracted two projects, each to be constructed by two different teams. According to the progress of the projects, the construction company can adjust the number of people in the two teams at any time. If 70 people are transferred from Team A to Team B, then the number of people in Team B will be twice that of Team A; if a certain number of people are transferred from Team B to Team A, then the number of people in Team A will be three times that of Team B. How many people are there in Team A at least?

Answer: 130


9.5. At the summit, 2018 politicians gathered. Every two of them were supposed to hold private negotiations. At some point, it was found that among any four, there is one who has already talked to the other three. What is the maximum number of negotiations left to be held?

Answer: 3


Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A, B$ and the other at the points $C, D$ so that $|A B|=|B C|=|C D|=14 \mathrm{~cm}$. Another line intersects the circles at points $E, F$ and $G, H$ respectively, so that $|E F|=|F G|=|G H|=6 \mathrm{~cm}$. Find the radius $r$.

Answer: 13


7. Let $S$ be a subset of the set of numbers $\{1,2,3, \cdots, 1989\}$, and any two numbers in $S$ have a difference that is not equal to 4 or 7. How many numbers can $S$ contain at most?

Answer: 905


1. The arithmetic mean of several distinct prime numbers is 27. Determine the largest prime number among them.

Answer: 139


$\underline{\text { Akulich I.F. }}$

There are 40 identical gas cylinders, the pressure values in which are unknown to us and may be different. It is allowed to connect any cylinders to each other in a quantity not exceeding a given natural number $k$, and then disconnect them; in this case, the gas pressure in the connected cylinders is set equal to the arithmetic mean of the pressures in them before connection. For what smallest $k$ does there exist a way to equalize the pressures in all 40 cylinders regardless of the initial distribution of pressures in the cylinders?

#

Answer: 5


Ben "One Hunna Dolla" Franklin is flying a kite $KITE$ such that $IE$ is the perpendicular bisector of $KT$. Let $IE$ meet $KT$ at $R$. The midpoints of $KI, IT, TE, EK$ are $A, N, M, D,$ respectively. Given that $[MAKE] = 18$, $IT = 10$, $[RAIN] = 4$, find $[DIME]$.

Note: $[X]$ denotes the area of the figure $X$.

Answer: 16


Let $ABC$ be a non-equilateral triangle with integer sides. Let $D$ and $E$ be respectively the mid-points of $BC$ and $CA$; let $G$ be the centroid of $\Delta ABC$. Suppose, $D$, $C$, $E$, $G$ are concyclic. Find the least possible perimeter of $\Delta ABC$.

Answer: 37


Let $\Gamma_1$, $\Gamma_2$ be two circles, where $\Gamma_1$ has a smaller radius, intersecting at two points $A$ and $B$. Points $C$ and $D$ lie on $\Gamma_1$ and $\Gamma_2$ respectively such that point $A$ is the midpoint of the segment $CD$. Line $CB$ intersects the circle $\Gamma_2$ for the second time at point $F$, and line $DB$ intersects the circle $\Gamma_1$ for the second time at point $E$. The perpendicular bisectors of segments $CD$ and $EF$ intersect at point $P$. Knowing that $CA = 12$ and $PE = 5$, find $AP$.

Answer: 13


For positive integers $m$ and $n$, the decimal representation for the fraction $\frac{m}{n}$ begins with $0.711$ followed by other digits. Find the least possible value for $n$.

Answer: 45


Greta is completing an art project. She has twelve sheets of paper: four red, four white, and four blue. She also has twelve paper stars: four red, four white, and four blue. She randomly places one star on each sheet of paper. The probability that no star will be placed on a sheet of paper that is the same color as the star is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $n - 100m.$

Answer: 25


Let $ABC$ be a triangle with incenter $I$ such that $AB = 20$ and $AC = 19$. Point $P \neq A$ lies on line $AB$ and point $Q \neq A$ lies on line $AC$. Suppose that $IA = IP = IQ$ and that line $PQ$ passes through the midpoint of side $BC$. Suppose that $BC = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.

Answer: 3902


There are 25 ants on a number line; five at each of the coordinates $1$, $2$, $3$, $4$, and $5$. Each minute, one ant moves from its current position to a position one unit away. What is the minimum number of minutes that must pass before it is possible for no two ants to be on the same coordinate?

Answer: 126


Define a *small* prime to be a prime under $1$ billion. Find the sum of all *small* primes of the form $20^n + 1$, given that the answer is greater than $1000$.

Answer: 160402


Let $M$ be the maximum possible value of $x_1x_2 + x_2x_3 + \cdots + x_5x_1$ where $x_1, x_2, \cdots, x_5$ is a permutation of $(1, 2, 3, 4, 5)$. Let $N$ be the number of permutations for which this maximum is attained. Evaluate $M + N$.

Answer: 58


Determine the value of $f_{1991}(2^{1990})$ where for a given positive integer $k$, the function $f_1(k)$ is defined as the square of the sum of its digits. The sequence is recursively defined as $f_{n+1}(k) = f_1(f_n(k))$.

Answer: 256


For a positive integer $n$, let $f(n)$ be the number of (not necessarily distinct) primes in the prime factorization of $k$. For example, $f(1) = 0$, $f(2) = 1$, and $f(4) = f(6) = 2$. Let $g(n)$ be the number of positive integers $k \leq n$ such that $f(k) \geq f(j)$ for all $j \leq n$. Find $g(1) + g(2) + \ldots + g(100)$.

Answer: 136


Consider an odd prime $p$. For each $i$ in the set \( \{1, 2, \ldots, p-1\} \), let \( r_i \) be the remainder of \( i^p \) when it is divided by \( p^2 \). Find the sum:
\[
r_1 + r_2 + \ldots + r_{p-1}
\]

Answer: 0


Consider the points $A_1$ and $A_2$ on the side $AB$ of the square $ABCD$ such that $|AB| = 3 |AA_1|$ and $|AB| = 4 |A_2B|$. Similarly, consider points $B_1$ and $B_2$ on $BC$, $C_1$ and $C_2$ on $CD$, and $D_1$ and $D_2$ on $DA$.

The intersection point of lines $D_2A_1$ and $A_2B_1$ is $E$.
The intersection point of lines $A_2B_1$ and $B_2C_1$ is $F$.
The intersection point of lines $B_2C_1$ and $C_2D_1$ is $G$.
The intersection point of lines $C_2D_1$ and $D_2A_1$ is $H$.

Find the area of the square $EFGH$, given that the area of $ABCD$ is $1$.

Answer: 1


Consider a board with $n$ rows and $4$ columns. In the first line, there are $4$ zeros (one in each column). Each subsequent line is obtained from the previous line by performing the following operation: one of the columns, which you can choose, is maintained as in the previous line; the other three are changed:
- If in the previous line there was a $0$, then in the next line, a $1$ is placed.
- If in the previous line there was a $1$, then in the next line, a $2$ is placed.
- If in the previous line there was a $2$, then in the next line, a $0$ is placed.

Build the largest possible board with all its lines distinct and demonstrate that it is impossible to build a larger board.

Answer: 27


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Bob`, `Peter`
 - Each person has a unique favorite drink: `root beer`, `water`, `coffee`, `tea`, `milk`
 - Each person has a unique birthday month: `jan`, `april`, `sept`, `feb`, `mar`
 - Each person has an occupation: `teacher`, `lawyer`, `engineer`, `doctor`, `artist`
 - Each person prefers a unique type of vacation: `city`, `cruise`, `mountain`, `beach`, `camping`

## Clues:
1. The root beer lover is Bob.
2. The person who is an engineer is Bob.
3. The coffee drinker is not in the first house.
4. Peter is not in the second house.
5. The one who only drinks water is not in the fifth house.
6. The person who is a doctor is directly left of the person who prefers city breaks.
7. The person who likes going on cruises is somewhere to the left of the person whose birthday is in April.
8. The person whose birthday is in February is the person who loves beach vacations.
9. Eric is somewhere to the right of Arnold.
10. The tea drinker is the person who enjoys mountain retreats.
11. The person who enjoys mountain retreats is not in the second house.
12. The person whose birthday is in April is not in the second house.
13. The tea drinker is somewhere to the left of the person who enjoys camping trips.
14. The person who is an engineer is the person who enjoys camping trips.
15. The person who is an artist is the person whose birthday is in April.
16. The person whose birthday is in March is not in the first house.
17. Arnold is the coffee drinker.
18. The coffee drinker is the person whose birthday is in February.
19. The person who is a lawyer is somewhere to the right of the person who is a teacher.
20. The coffee drinker is not in the second house.
21. The person whose birthday is in April is the one who only drinks water.
22. The person whose birthday is in January is the root beer lover.
 What is the value of attribute Drink for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: water


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - People have unique heights: `average`, `tall`, `very short`, `short`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`, `dragonfruit`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`

## Clues:
1. Eric is the person who loves the rose bouquet.
2. The Dragonfruit smoothie lover is somewhere to the left of the person who loves the rose bouquet.
3. Arnold is the person who loves a carnations arrangement.
4. The Desert smoothie lover is in the third house.
5. The person who is very short is the Dragonfruit smoothie lover.
6. Alice is the person who likes Cherry smoothies.
7. The person who loves a bouquet of daffodils is the person who likes Cherry smoothies.
8. The person who loves a carnations arrangement is in the fourth house.
9. The person who is very short is directly left of the person who is tall.
10. The person who is short is the Desert smoothie lover.
 What is the value of attribute House for the person whose attribute Smoothie is watermelon? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `lilies`
 - People have unique heights: `very short`, `short`, `tall`, `average`
 - The mothers' names in different houses are unique: `Janelle`, `Kailyn`, `Holly`, `Aniya`
 - Each person has an occupation: `engineer`, `doctor`, `teacher`, `artist`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `soccer`

## Clues:
1. The person who loves swimming is the person who loves the rose bouquet.
2. The person who loves the rose bouquet is Eric.
3. Arnold is the person who is tall.
4. The person who loves a bouquet of daffodils is somewhere to the right of the person who is an engineer.
5. The person who loves soccer is the person who is short.
6. The person who is a teacher is in the first house.
7. The person whose mother's name is Janelle is the person who loves a carnations arrangement.
8. The person who loves basketball is the person who has an average height.
9. Arnold is not in the third house.
10. The person whose mother's name is Holly is somewhere to the right of the person who has an average height.
11. Peter is the person who is a doctor.
12. The person whose mother's name is Aniya is Alice.
13. Arnold is the person who loves the boquet of lilies.
 What is the value of attribute House for the person whose attribute Occupation is engineer? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Carol`, `Peter`, `Eric`, `Bob`, `Alice`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `modern`, `craftsman`, `mediterranean`, `victorian`
 - Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`, `stir fry`, `soup`
 - Each person prefers a unique type of vacation: `cultural`, `cruise`, `mountain`, `camping`, `city`, `beach`
 - People have unique heights: `average`, `very tall`, `very short`, `short`, `tall`, `super tall`
 - Everyone has a unique favorite cigar: `yellow monster`, `prince`, `dunhill`, `pall mall`, `blue master`, `blends`

## Clues:
1. Alice is in the fifth house.
2. The person who loves stir fry is the person living in a colonial-style house.
3. Alice is the person who loves the spaghetti eater.
4. Arnold is the person who loves the stew.
5. There is one house between the person who has an average height and Peter.
6. The person in a Craftsman-style house is not in the third house.
7. The person who has an average height is the person who loves stir fry.
8. The person who loves beach vacations is the person in a ranch-style home.
9. Eric is in the fourth house.
10. There is one house between the person living in a colonial-style house and the person who enjoys camping trips.
11. The person who enjoys mountain retreats is the person who smokes Yellow Monster.
12. The person who enjoys mountain retreats is the person who is very tall.
13. The person who enjoys mountain retreats and the Dunhill smoker are next to each other.
14. The person who loves the spaghetti eater is the person residing in a Victorian house.
15. The person who is tall is the person who loves beach vacations.
16. The person who is tall is somewhere to the left of the person residing in a Victorian house.
17. The person who loves stir fry is directly left of Bob.
18. The person in a modern-style house is somewhere to the left of Alice.
19. The person in a Craftsman-style house is somewhere to the left of the person who is short.
20. The person who loves stir fry is somewhere to the left of the Prince smoker.
21. There are two houses between the person who loves eating grilled cheese and the person who is super tall.
22. The person in a ranch-style home is the person who smokes Blue Master.
23. The person who smokes many unique blends is directly left of the person who smokes Blue Master.
24. The person who goes on cultural tours is the person who is a pizza lover.
25. The person who is a pizza lover is somewhere to the left of the person who likes going on cruises.
 What is the value of attribute Food for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: spaghetti


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Bob`, `Alice`
 - Each person lives in a unique style of house: `modern`, `craftsman`, `ranch`, `victorian`, `colonial`
 - The mothers' names in different houses are unique: `Penny`, `Kailyn`, `Holly`, `Janelle`, `Aniya`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `huawei p50`, `iphone 13`, `samsung galaxy s21`
 - Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`
 - The people keep unique animals: `fish`, `dog`, `horse`, `bird`, `cat`

## Clues:
1. The person who uses a Google Pixel 6 is not in the first house.
2. The one who only drinks water is Alice.
3. The person living in a colonial-style house is somewhere to the right of the person who uses a Huawei P50.
4. The person who keeps horses is the person who uses a OnePlus 9.
5. The person in a ranch-style home is The person whose mother's name is Kailyn.
6. The root beer lover is the cat lover.
7. The person living in a colonial-style house is not in the fourth house.
8. The bird keeper is in the fourth house.
9. The tea drinker is Bob.
10. The tea drinker is somewhere to the right of The person whose mother's name is Kailyn.
11. The root beer lover is somewhere to the left of The person whose mother's name is Kailyn.
12. The person who keeps horses is the person in a modern-style house.
13. The person who uses an iPhone 13 is the person who likes milk.
14. The dog owner is the person who likes milk.
15. The person who uses a Google Pixel 6 is the person in a Craftsman-style house.
16. Eric is not in the second house.
17. The tea drinker is in the fourth house.
18. The person who keeps horses is in the third house.
19. The person in a modern-style house is The person whose mother's name is Penny.
20. The root beer lover is Peter.
21. The person whose mother's name is Aniya is not in the fourth house.
22. The person whose mother's name is Janelle is the one who only drinks water.
 What is the value of attribute HouseStyle for the person whose attribute PhoneModel is iphone 13? Please reason step by step, and put your final answer within \boxed{}

Answer: ranch


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`, `Bob`, `Carol`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `lime`, `desert`, `blueberry`, `watermelon`
 - Each person has a unique type of pet: `rabbit`, `fish`, `bird`, `hamster`, `cat`, `dog`
 - Each person has a unique level of education: `master`, `bachelor`, `high school`, `trade school`, `associate`, `doctorate`
 - People have unique heights: `short`, `very tall`, `very short`, `tall`, `average`, `super tall`

## Clues:
1. The person with an aquarium of fish is in the first house.
2. The person with a doctorate is somewhere to the left of the person who owns a dog.
3. Peter and the person with a high school diploma are next to each other.
4. There is one house between the person with a pet hamster and the person who owns a rabbit.
5. The person who has a cat is Eric.
6. Bob and the person who owns a dog are next to each other.
7. The person who is tall is Carol.
8. The person with a master's degree is the person with a pet hamster.
9. Peter is the person who is very tall.
10. The person who likes Cherry smoothies is the person who is very tall.
11. The person who is super tall is somewhere to the right of Arnold.
12. The Watermelon smoothie lover is in the sixth house.
13. The person who is super tall is the Desert smoothie lover.
14. The person who is very short is the person with an associate's degree.
15. The person with an associate's degree and the Watermelon smoothie lover are next to each other.
16. The person who drinks Lime smoothies is the person with a pet hamster.
17. The person who is short is in the second house.
18. The person who drinks Blueberry smoothies is the person who attended trade school.
19. The person who is very short is somewhere to the right of Bob.
 What is the value of attribute House for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Carol`, `Arnold`, `Bob`, `Peter`
 - People have unique heights: `average`, `super tall`, `tall`, `short`, `very short`, `very tall`
 - The people are of nationalities: `chinese`, `brit`, `german`, `swede`, `norwegian`, `dane`
 - People have unique hair colors: `black`, `blonde`, `brown`, `auburn`, `gray`, `red`

## Clues:
1. Eric is the Norwegian.
2. There is one house between the person who is very short and the person who has auburn hair.
3. The person who is short is Arnold.
4. The Dane is the person who is super tall.
5. The British person is in the sixth house.
6. The person who is very short is somewhere to the right of the German.
7. The person who has gray hair is in the fifth house.
8. The person who has gray hair is Carol.
9. Bob is the person who has black hair.
10. The person who has red hair is in the second house.
11. The person who is super tall is directly left of Arnold.
12. There are two houses between Alice and the person who has an average height.
13. The Dane is not in the first house.
14. Bob is somewhere to the left of the Norwegian.
15. The person who is tall is the person who has gray hair.
16. The Chinese is in the fifth house.
17. There is one house between the person who is very tall and the person who has blonde hair.
 What is the value of attribute House for the person whose attribute Nationality is swede? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Peter`
 - Everyone has a unique favorite cigar: `dunhill`, `pall mall`, `prince`, `blue master`
 - Each person has a unique level of education: `associate`, `high school`, `master`, `bachelor`
 - They all have a unique favorite flower: `lilies`, `daffodils`, `carnations`, `roses`
 - Each person has a unique hobby: `cooking`, `photography`, `gardening`, `painting`

## Clues:
1. The person with a bachelor's degree is the person who enjoys gardening.
2. Arnold is somewhere to the left of the person who loves a carnations arrangement.
3. Eric is not in the second house.
4. The person who loves a bouquet of daffodils is the person who paints as a hobby.
5. The person who loves the rose bouquet is the person who smokes Blue Master.
6. The person partial to Pall Mall is Peter.
7. Arnold is in the third house.
8. Peter is directly left of the person who enjoys gardening.
9. Arnold is the person with a high school diploma.
10. The person who smokes Blue Master is not in the second house.
11. The Prince smoker is the photography enthusiast.
12. The person with a master's degree is somewhere to the left of the person who smokes Blue Master.
 What is the value of attribute Flower for the person whose attribute Cigar is dunhill? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies