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: `Alice`, `Peter`, `Eric`, `Arnold`, `Carol`, `Bob`
 - People have unique heights: `very short`, `very tall`, `short`, `super tall`, `tall`, `average`
 - Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `soup`, `stew`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Penny`, `Aniya`, `Kailyn`, `Sarah`
 - Each person has a unique favorite drink: `tea`, `water`, `milk`, `coffee`, `boba tea`, `root beer`
 - Each person has a unique type of pet: `rabbit`, `fish`, `bird`, `hamster`, `cat`, `dog`

## Clues:
1. Arnold is the person who loves the stew.
2. The person who owns a dog is directly left of the person who has a cat.
3. The person whose mother's name is Holly is in the sixth house.
4. The tea drinker is somewhere to the left of the boba tea drinker.
5. The tea drinker is in the second house.
6. The person who is very tall is Bob.
7. The person who has a cat is The person whose mother's name is Penny.
8. The person who is very short is directly left of Alice.
9. Carol is the person who is a pizza lover.
10. The person who has an average height is not in the sixth house.
11. The person who loves stir fry is Bob.
12. The person with a pet hamster is not in the third house.
13. The person whose mother's name is Janelle is somewhere to the left of the person with a pet hamster.
14. Eric is the person who loves the soup.
15. The person whose mother's name is Kailyn is directly left of the person who keeps a pet bird.
16. The one who only drinks water is somewhere to the right of the person who likes milk.
17. The person who owns a rabbit is the person who loves the stew.
18. The person who is tall is The person whose mother's name is Aniya.
19. The person who is very short is somewhere to the left of the person who likes milk.
20. The person who is tall is not in the fifth house.
21. The person who is very short is in the second house.
22. The person who loves the stew is the root beer lover.
23. The person whose mother's name is Kailyn is the boba tea drinker.
24. The person who is tall is not in the fourth house.
25. The person who is super tall is directly left of the person who loves eating grilled cheese.
26. Carol is directly left of the root beer lover.
 What is the value of attribute Mother for the person whose attribute Height is average? Please reason step by step, and put your final answer within \boxed{}

Answer: Sarah


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`, `Bob`, `Arnold`, `Eric`, `Peter`
 - People have unique favorite book genres: `romance`, `mystery`, `biography`, `fantasy`, `science fiction`
 - Each person has a unique birthday month: `sept`, `feb`, `jan`, `mar`, `april`
 - Each person prefers a unique type of vacation: `mountain`, `beach`, `city`, `camping`, `cruise`
 - The people keep unique animals: `horse`, `bird`, `fish`, `dog`, `cat`

## Clues:
1. The person whose birthday is in April is the person who loves mystery books.
2. The person whose birthday is in March is somewhere to the left of the person who loves mystery books.
3. The person who loves beach vacations is Arnold.
4. The person who loves beach vacations is the person whose birthday is in January.
5. The bird keeper is directly left of Peter.
6. The person who loves science fiction books is somewhere to the left of the person whose birthday is in February.
7. The fish enthusiast is somewhere to the right of Bob.
8. The fish enthusiast is Eric.
9. The fish enthusiast is the person who loves fantasy books.
10. The person who enjoys mountain retreats is directly left of the person whose birthday is in April.
11. The person whose birthday is in April is the person who enjoys camping trips.
12. The person whose birthday is in September is the person who loves science fiction books.
13. The cat lover is the person who prefers city breaks.
14. The person who loves romance books is somewhere to the right of the person who loves science fiction books.
15. The cat lover is the person whose birthday is in February.
16. The person who keeps horses is Peter.
 What is the value of attribute BookGenre for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


Julia Anna Orum's former student, now a speech therapist, is planning a series of therapy sessions for a group of children. She schedules 5 sessions per week, and each session can accommodate 3 children. Over the course of 4 weeks, how many individual therapy sessions will she conduct in total?

Answer: 60


Jamie owns a small bakery and is looking for an efficient and flexible financial service platform to manage her transactions. In one week, Jamie made 45 sales, with each sale averaging $12. She found a platform that charges a fixed monthly fee of $30 and an additional fee of $0.50 per transaction. If Jamie decides to use this platform, how much would she pay in fees for that week?

Answer: 30.00


A cloud hosting provider manages the infrastructure for an e-commerce platform. The platform needs to handle 300 transactions per hour during peak times. To ensure reliability, the cloud provider allocates 5 servers, each capable of processing 20 transactions per hour. If the cloud provider wants to increase the total capacity by 30% to accommodate future growth, how many additional servers, each with the same capacity, will be needed to meet the new demand?

Answer: 15


Alex is the trusted second-in-command at a company that specializes in strategic planning and internal operations. She is planning a team-building event for the company's 48 employees. Alex decides to divide the employees into 6 equal groups for various team activities. Each group will rotate through 5 different activities during the event. If each activity requires 4 sets of equipment and Alex can pack 3 sets of equipment into one box, how many boxes does she need to pack all the equipment for the event?

Answer: 7


Alex, a conspiracy theorist, believes that international cooperation is just a cover for secret meetings where countries plan to control the world. He has discovered that seven countries have been secretly meeting every 3 days. If these countries began their meetings on the first day of the year, how many meetings will they have had by the end of February in a non-leap year?

Answer: 20


Maria, a middle-aged woman, decided to learn more about the legal system by reading books. She started with a beginner's guide to law that had 240 pages. Maria plans to read 15 pages each day. However, she knows she will have to skip reading on Sundays to keep up with her family activities. If Maria starts reading on a Monday, how many days will it take her to finish the book, not counting Sundays?

Answer: 16


During the 17th century, a small village was conducting witch trials, and the historical novelist is researching the events that took place. She discovers that on the first day of the trials, 12 people were accused of witchcraft. On the second day, the number of accused individuals doubled. On the third day, the number of accused increased by 5 from the previous day. By the fourth day, the number of accused was halved, and then 7 more individuals were added to those accused. How many people in total were accused of witchcraft by the end of the fourth day?

Answer: 86


Oleksandr is a retired Ukrainian civil engineer who dedicated 40 years of his life to constructing bridges and roads that connected communities across Ukraine. He worked on three major projects in his career. The first project took 8 years to complete and involved building 50 kilometers of highway. The second project was a bridge that took 6 years and spanned 10 kilometers. His final project, which took 10 years, involved constructing a 60-kilometer railway line.

If Oleksandr worked an average of 200 days each year on these projects, how many kilometers of infrastructure did Oleksandr help build per working day over his entire career?

Answer: 0.025


In the global network of acoustic enthusiasts, there are 250 members who love sharing their knowledge. Each member decides to conduct an experiment with sound waves and shares their findings with 3 other members in the network. However, due to overlapping interests, each member receives shared insights from only 2 of the 3 members they shared their own experiment with. How many unique shared insight exchanges happen within the network?

Answer: 500


Mr. Edwards, a retired electronic repair technician, loves spending his afternoons debunking common myths about electronics. One day, he heard a myth that leaving a phone charger plugged in without a device wastes a lot of energy.

To test this, Mr. Edwards decided to measure the energy usage of his phone charger when it's plugged in without a phone. He found that the charger uses 0.1 watts per hour. Mr. Edwards knows that an average charger is left plugged in for about 10 hours a day when not in use.

Mr. Edwards also calculates that his electricity company charges him $0.12 per kilowatt-hour. He wants to find out how much money he spends in a year just from the energy wasted by leaving the charger plugged in without a phone.

How much does Mr. Edwards spend annually on this wasted energy?

Answer: 0.04


Professor Smith, an art history professor at the University of Edinburgh, is curating an exhibition that explores the connection between art and science. For the exhibition, she selects 12 paintings that each include a scientific element. She wants to display these paintings in the gallery in such a way that each painting is accompanied by 3 scientific artifacts. If Professor Smith has collected a total of 45 scientific artifacts for the exhibition, how many paintings will have to share the scientific artifacts with other paintings?

Answer: 0


Mrs. Thompson, a retired teacher from Ontario, receives a monthly pension of $2,500. She decided to invest 40% of her pension in a sustainable energy fund and the remaining 60% in a high-yield bond. After one year, the sustainable energy fund earned an annual return of 5%, and the high-yield bond earned an annual return of 3%. How much total money did Mrs. Thompson earn from her investments at the end of the year?

Answer: 1140


A social activist is organizing a community event to bring together people from different generations to watch a documentary. The activist has invited 40 children, 30 parents, and 20 grandparents. To foster unity, they plan a meal where each child will sit with two parents and one grandparent. Each of these groups of four people will be given a large tray of food to share, and 5 extra trays are ordered as a reserve. How many trays in total does the activist need to order for the event?

Answer: 20


A business owner, Alex, is reviewing a contract for a project deliverable. According to the contract, Alex needs to deliver 120 units of a product within a month. Alex's team can produce 8 units per day, and they work 5 days a week. However, due to a holiday, the team will not work for 2 days this month.

How many more units does the team need to produce each day to meet the contractual obligation of delivering 120 units by the end of the month?

Answer: 0


A young Yoruba woman named Amina loves running and goes for morning jogs every day. She runs a distance of 3 kilometers each time she jogs. This week, she invited you to join her for 5 of her morning jogs. If you both ran the same distance each time, how many kilometers did you run together in total this week?

Answer: 30


A writer is working on an article covering the history of music piracy and its impact on the music industry. They find that in the year 2000, there were approximately 10 million cases of music piracy worldwide. By the year 2005, the number of piracy cases increased by 25%. In 2010, due to new anti-piracy measures, the number of cases was reduced by 20% from the 2005 figure. How many cases of music piracy were there in 2010?

Answer: 10000000


At the rehabilitation center, the staff member is planning a week of activities for the patients. She organizes a daily exercise class, an arts and crafts session, and a relaxation hour each day from Monday to Friday. For each activity, she needs 3 volunteers to assist, and she has 10 volunteers available each day.

On Monday, she realizes she needs 5 extra volunteers for a special event. To manage, she decides to reduce the number of volunteers assisting in the arts and crafts session for that day.

How many volunteers will be assisting in the arts and crafts session on Monday after the adjustment?

Answer: 0


Alex runs a successful family bakery and is planning to expand the operation by opening two new branches. Each new branch requires an initial investment of $15,000. To fund the expansion, Alex decides to reinvest 40% of the bakery's monthly profit of $10,000 into the new branches. Additionally, Alex has saved $5,000 specifically for this expansion. After covering the initial investments for both branches, how much money will Alex have left from the reinvested profits and savings?

Answer: 0


A filmmaker releases a music video that is 4 minutes long. A dedicated viewer watches the video 5 times, pausing every 30 seconds to analyze each frame for hidden meanings and symbolism. If it takes the viewer an additional 2 minutes to analyze each paused frame, how many total minutes does the viewer spend watching and analyzing the video?

Answer: 100


Let $X = \{1, 2, \ldots, 2017\}$. Let $k$ be a positive integer. Given any $r$ such that $1 \le r \le k$, there exist $k$ subsets of $X$ such that the union of any $r$ of them is equal to $X$, but the union of any fewer than $r$ of them is not equal to $X$. Find, with proof, the greatest possible value for $k$.

Answer: 2017


In the diagram below, $\angle B = 43^\circ$ and $\angle D = 102^\circ$. Find $\angle A + \angle B + \angle C + \angle D + \angle E + \angle F$.

Answer: 720


A regular pentagon can have the line segments forming its boundary extended to lines, resulting in an arrangement of lines that intersect at ten points. How many ways are there to choose five points from these ten such that no three of the points are collinear?

Answer: 12


Let $\triangle ABC$ be a triangle with side lengths $AB = 9$, $BC = 10$, $CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC$, $E = BO \cap CA$, $F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{CF}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.

Answer: 43


In the diagram below, three squares are inscribed in right triangles. Their areas are $A$, $M$, and $N$, as indicated in the diagram. If $M = 5$ and $N = 12$, then $A$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$.

Answer: 36


Let $N_{15}$ be the answer to problem 15.

For a positive integer $x$ expressed in base ten, let $x'$ be the result of swapping its first and last digits. For example, if $x = 2024$, then $x' = 4022$. Let $C$ be the number of $N_{15}$-digit positive integers $x$ with a nonzero leading digit that satisfy the property that both $x$ and $x'$ are divisible by $11$. Note: $x'$ is allowed to have a leading digit of zero. Compute the sum of the digits of $C$ when $C$ is expressed in base ten.

Answer: 9


Each of the natural numbers from $1$ to $n$ is colored either red or blue, with each color being used at least once. It turns out that:
- Every red number is a sum of two distinct blue numbers; and
- Every blue number is a difference between two red numbers.

Determine the smallest possible value of $n$ for which such a coloring exists.

Answer: 9


8. (for 9th grade. 2 points). In triangle $A B C$, angle $A$ is equal to $40^{\circ}$. The triangle is randomly thrown onto a table. Find the probability that vertex $A$ will be to the east of the other two vertices.

Answer: \dfrac{7}{18}


# Task 5.

Once upon a time, Baba Yaga and Koschei the Deathless tried to divide a magical powder that turns everything into gold equally. Baba Yaga took out a scale and weighed all the powder. The scales showed 6 zolotniks. Then she started removing the powder until the scales showed 3 zolotniks. However, Koschei suspected that the scales were lying and weighed the portion that had been removed separately on the same scales (there were no others). The scales showed 2 zolotniks. Determine the exact weight of the two parts into which Baba Yaga divided the powder. Assume that if the scales are lying, they always lie by the same amount.

Answer: 3


5. Planet Beta has seven countries, each of which has exactly four friendly countries and two enemy countries. There are no three countries that are all enemies with each other. For such a planetary situation, the number of possible alliances of three countries that are all friendly with each other is $\qquad$.

Answer: 7


16. 7 (US MO 16) In the plane, there are three circles $C_{i}(i=1,2,3)$, where the diameter of $C_{1}$ is $A B=1$; $C_{2}$ is concentric with $C_{1}$, has a diameter of $k$, and satisfies $1<k<3$; $C_{3}$ has $A$ as its center and $2 k$ as its diameter ($k$ is a constant). Consider all line segments $X Y$, one end $X$ on $C_{2}$, the other end $Y$ on $C_{3}$, and $X Y$ contains point $B$. For what ratio $X B: B Y$ is the length of segment $X Y$ minimized?

Answer: 1


[b]p1.[/b] Let $S$ be the sum of the $99$ terms: $$(\sqrt1 + \sqrt2)^{-1},(\sqrt2 + \sqrt3)^{-1}, (\sqrt3 + \sqrt4)^{-1},..., (\sqrt{99} + \sqrt{100})^{-1}.$$ Prove that $S$ is an integer.

[b]p2.[/b] Determine all pairs of positive integers $x$ and $y$ for which $N=x^4+4y^4$ is a prime. (Your work should indicate why no other solutions are possible.)

[b]p3.[/b] Let $w,x,y,z$ be arbitrary positive real numbers. Prove each inequality:

(a) $xy \le \left(\frac{x+y}{2}\right)^2$
(b) $wxyz \le \left(\frac{w+x+y+z}{4}\right)^4$
(c) $xyz \le \left(\frac{x+y+z}{3}\right)^3$

[b]p4.[/b] Twelve points $P_1$,$P_2$, $...$,$P_{12}$ are equally spaaed on a circle, as shown. Prove: that the chords $\overline{P_1P_9}$, $\overline{P_4P_{12}}$ and $\overline{P_2P_{11}}$ have a point in common.
[img]https://cdn.artofproblemsolving.com/attachments/d/4/2eb343fd1f9238ebcc6137f7c84a5f621eb277.png[/img]

[b]p5.[/b] Two very busy men, $A$ and $B$, who wish to confer, agree to appear at a designated place on a certain day, but no earlier than noon and no later than $12:15$ p.m. If necessary, $A$ will wait $6$ minutes for $B$ to arrive, while $B$ will wait $9$ minutes for $A$ to arrive but neither can stay past $12:15$ p.m. Express as a percent their chance of meeting.

PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Answer: 9


5. As shown in Figure 1, given $\angle A C B=90^{\circ}, D A \perp$ plane $A B C, A E \perp D B$ at point $E, A F \perp D C$ at point $F$, and $A D=A B=2$. Then the maximum volume of the tetrahedron $D-A E F$ is . $\qquad$

Answer: \dfrac{\sqrt{2}}{6}


An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping?

Answer: \dfrac{9}{2}


$n$ is an odd integer greater than 1. $p$ is an odd prime number, and $x, y, k$ are natural numbers. If $x^{n} + y^{n} = p^{k}$, prove that $n$ is a power of $p$.

Answer: n \text{ is a power of } p


Variant 132. $[-5 ;-2018]$. Variant 133. $[4 ; 2016]$. Variant 134. $[-5 ;-2017]$.

Grading criteria: 20 points - correct (not necessarily the same as above) solution and correct answer; 15 points - mostly correct solution and correct answer, but there are defects (for example: the transition from (1) to (2) is made without justification or with incorrect justification); $\mathbf{1 0}$ points - it is not proven that the equation has no other solutions except $x \in(4 ; 5)$ (note that the proof must be clear and neat; arguments like "it is obvious that this function is better, and therefore there can be no intersections" do not qualify); 5 points - the transition to equation (2) is justified, but there is no further progress.

## Problem 6 (Variant 131)

There are a lot of homogeneous circular disks of the same thickness and density. These disks are placed on a horizontal board such that the planes of the disks are vertical. Two adjacent disks touch each other, and the radii of two adjacent disks are in the ratio $2: 1$. The disk on the left is always larger than the disk on the right (see figure). The radius of the largest disk is 2 m. Determine the distance from the center of the largest disk to the center of mass of the system.

Answer: \dfrac{6}{7}


[ Divisibility of numbers. General properties ] [ Mathematical logic (other) ]

From the statements "number $a$ is divisible by 2", "number $a$ is divisible by 4", "number $a$ is divisible by 12", and "number $a$ is divisible by $24$", three are true, and one is false. Which one

#

Answer: 24


The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: $P_{0}P_{1}P_{3}$, $P_{0}P_{3}P_{6}$, $P_{0}P_{6}P_{7}$, $P_{0}P_{7}P_{8}$, $P_{1}P_{2}P_{3}$, $P_{3}P_{4}P_{6}$, $P_{4}P_{5}P_{6}$. In how many ways can these triangles be labeled with the names $\triangle_{1}$, $\triangle_{2}$, $\triangle_{3}$, $\triangle_{4}$, $\triangle_{5}$, $\triangle_{6}$, $\triangle_{7}$ so that $P_{i}$ is a vertex of triangle $\triangle_{i}$ for $i = 1, 2, 3, 4, 5, 6, 7$? Justify your answer.

[img]6740[/img]

Answer: 1


1. When $a^{3}-a-1=0$, $a+\sqrt{2}$ is a root of some polynomial with integer coefficients. Find the polynomial of the lowest degree with leading coefficient 1 that satisfies the above condition.

Answer: x^6 - 8x^4 - 2x^3 + 13x^2 - 10x - 1


Segments \(AA'\), \(BB'\), and \(CC'\) with endpoints on the sides of an acute-angled triangle \(ABC\) intersect at a point \(P\) inside the triangle. On each of these segments, a circle is constructed with the segment as a diameter, and a chord passing through point \(P\) is drawn perpendicular to this diameter. It turns out that the three drawn chords have the same length. Prove that \(P\) is the orthocenter of triangle \(ABC\).

Answer: P \text{ is the orthocenter of triangle } ABC


3. Given that $A D$ is the angle bisector of $\triangle A B C$, $\angle C$ $=90^{\circ}, \frac{A B-A C}{B D-C D}=\sqrt{5}$. Then
$$
\frac{A B+A C}{A D+B C}=
$$
$-$ $\qquad$

Answer: \dfrac{5\sqrt{5}}{5 + 2\sqrt{6}}


Given two concentric circles \(\Omega\) and \(\omega\). Chord \(AD\) of circle \(\Omega\) is tangent to \(\omega\). Within the smaller segment \(AD\) of the circle bounded by \(\Omega\), an arbitrary point \(P\) is taken. Tangents from \(P\) to the circle \(\omega\) intersect the major arc \(AD\) of circle \(\Omega\) at points \(B\) and \(C\). Segments \(BD\) and \(AC\) intersect at point \(Q\). Prove that segment \(PQ\) divides segment \(AD\) into two equal parts.

Answer: PQ \text{ bisects } AD


One, (40 points) Let $x_{1}, x_{2}, \cdots, x_{n}$ be real numbers no less than -1, satisfying $\sum_{i=1}^{n} x_{i}^{3}=0$. Find the smallest constant $c$, such that for all positive integers $n$, we have $\sum_{i=1}^{n} x_{i}^{2} \leqslant c n$.

Answer: \dfrac{4}{3}


Martin decided to spend all his savings on sweets. He found out that he could buy three lollipops and $3 \mathrm{dl}$ of cola or $18 \mathrm{dkg}$ of yogurt-covered raisins or $12 \mathrm{dkg}$ of yogurt-covered raisins and half a liter of cola. In the end, he bought one lollipop and 6 dl of cola. How many grams of yogurt-covered raisins did he still have left?

(Bednářová)

Answer: 60


4. Given $P_{1}, P_{2}, \cdots, P_{100}$ as 100 points on a plane, satisfying that no three points are collinear. For any three of these points, if their indices are in increasing order and they form a clockwise orientation, then the triangle with these three points as vertices is called "clockwise". Question: Is it possible to have exactly 2017 clockwise triangles?

Answer: 2017


6. In tetrahedron $ABCD$, the opposite edges are equal, $E, F$ are the midpoints of $AB, CD$ respectively, and $EF \perp AB, EF \perp CD, EF=6, AC=10, BC=6\sqrt{2}$. Then the distance between the skew lines $AD$ and $BC$ is $\qquad$.

Answer: 8


[ Theorem of the length of a tangent and a secant; the product of the entire secant and its external part [ Mean proportionals in a right triangle

In rhombus $A B C D$, angle $B A D$ is acute. A circle inscribed in this rhombus touches sides $A B$ and $C D$ at points $M$ and $N$ respectively and intersects segment $C M$ at point $P$, and segment $B N$ at point $Q$. Find the ratio of $B Q$ to $Q N$, if $C P: P M=9: 16$.

#

Answer: \dfrac{1}{9}


Let  $M$  and  $N$  , respectively, be the points of the sides  $[AB]$  and  $[BC]$  of a rectangle  $ABCD$  . Let  $a,b,c$  be the area of the triangle  $AMD,MBN, NCD$  respectively. Express in terms of  $a,b$  and  $c$  , the area of the triangle  $DMN$  .

Answer: \sqrt{(a + b + c)^2 - 4ac}


NT 3. Find the largest integer $k(k \geq 2)$, for which there exists an integer $n(n \geq k)$ such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:

1. each chosen number is not divisible by 6 , by 7 and by 8 ;
2. the positive difference of any two different chosen numbers is not divisible by at least one of the numbers 6,7 or 8 .

Answer: 108


For each positive integer $m$, we define $L_m$ as the figure that is obtained by overlapping two $1 \times m$ and $m \times 1$ rectangles in such a way that they coincide at the $1 \times 1$ square at their ends, as shown in the figure.

[asy]
pair h = (1, 0), v = (0, 1), o = (0, 0);
for(int i = 1; i < 5; ++i)
{
  o = (i*i/2 + i, 0);
  draw(o -- o + i*v -- o + i*v + h -- o + h + v -- o + i*h + v -- o + i*h -- cycle);
  string s = "$L_" + (string)(i) + "$";
  label(s, o + ((i / 2), -1));
  for(int j = 1; j < i; ++j)
  {
    draw(o + j*v -- o + j*v + h);
    draw(o + j*h -- o + j*h + v);
  }
}
label("...", (18, 0.5));
[/asy]

Using some figures $L_{m_1}, L_{m_2}, \dots, L_{m_k}$, we cover an $n \times n$ board completely, in such a way that the edges of the figure coincide with lines in the board. Among all possible coverings of the board, find the minimal possible value of $m_1 + m_2 + \dots + m_k$.

Note: In covering the board, the figures may be rotated or reflected, and they may overlap or not be completely contained within the board.

Answer: \dfrac{n(n+1)}{2}


6. For a positive integer $n$, let the sum of its digits be denoted as $s(n)$, and the product of its digits be denoted as $p(n)$. If $s(n) +$ $p(n) = n$ holds, then $n$ is called a "coincidence number". The sum of all coincidence numbers is $\qquad$ .

Answer: 531


Calculate the sum of every pair of five distinct real numbers. If the smallest pair sums are $20$ and $200$, and the largest pair sums are $2014$ and $2000$, then what is the range of values for the smallest number $a$ among these five numbers?

Answer: (-793, 10)


The radii of two circles are 2 and 4. Their common internal tangents are perpendicular to each other. Find the length of each of them.

Answer: 6


3. How many ways are there to cut a $10 \times 10$ square into several rectangles along the grid lines such that the sum of their perimeters is 398? Ways that can be matched by rotation or flipping are considered different.

Answer: 180


A dedicated groupie attends every performance of their favorite band over a 10-day music festival. Each day, the band performs 3 songs. After each performance, the groupie reads 4 different critics' reviews and writes down 2 interesting opinions from each critic to discuss later. How many total opinions does the groupie write down over the entire 10-day festival?

Answer: 240


Putnam 1998 Problem A1 A cone has circular base radius 1, and vertex a height 3 directly above the center of the circle. A cube has four vertices in the base and four on the sloping sides. What length is a side of the cube? Solution

Answer: \dfrac{9\sqrt{2} - 6}{7}


4. Given a right triangle $\triangle A B C$ with the altitude from the right angle to the hypotenuse being $C D$, and $A D=\frac{1}{3} A B$. If $\triangle A C D$ is rotated around $C D$ to $\triangle A_{1} C D$, such that the dihedral angle $A_{1}-C D-B$ is $60^{\circ}$. Then the angle between the skew lines $A_{1} C$ and $A B$ is $\qquad$ (express the answer using inverse trigonometric functions).

Answer: \arccos \dfrac{\sqrt{3}}{6}


4. As shown in the figure, given the ellipse $\frac{x^{2}}{2} + y^{2}=1, P A \perp A B, C B \perp$ $A B$, and $D A=3 \sqrt{2}, C B$ $=\sqrt{2}$. Point $P$ moves on the arc $\overparen{A B}$. Then the minimum value of the area of $\triangle P C D$ is $\qquad$

Answer: 4 - \sqrt{6}


32. All students in the school are divided into two teams to participate in the school's radio gymnastics competition. They form two solid square formations, designated as Formation A and Formation B, where the number of people on each side of Formation A equals 8. If the two teams merge, they can form a hollow square Formation C, with the number of people on each side of the outermost layer of Formation C being 4 more than the number of people on each side of the outermost layer of Formation B. The number of people in Formation A is exactly enough to fill the hollow part of Formation C. The total number of students participating in the radio gymnastics competition is. $\qquad$

Answer: 260


Three, (50 points) Given a positive integer $n$. Find $\sum_{k=1}^{n}\left[\frac{n}{2^{k}}-\frac{1}{2}\right]$, where $[x]$ denotes the greatest integer not exceeding the real number $x$.

---

Please note that the format and line breaks have been preserved as requested.

Answer: 0


10.4. Let's call a snake in a convex $n$-gon an open, non-self-intersecting broken line with $n-1$ segments, the set of vertices of which coincides with the set of all vertices of the $n$-gon. Find the number of different snakes in an $n$-gon. (Snakes are equal if they coincide as geometric loci of points. For example, the number of snakes in a triangle is 3)

Answer: n \cdot 2^{n-3}


7. If non-negative integers $m, n$ add up with exactly one carry (in decimal), then the ordered pair $(m, n)$ is called "good". The number of all good ordered pairs whose sum is 2014 is $\qquad$ .

Answer: 195


## Task B-3.2.

Ana, Bruno, Cvita, Dino, and Ema are trying to arrange themselves in five seats in a row. In how many ways can they do this if Ana does not want to sit next to either Bruno or Cvita, and Dino does not want to sit next to Ema?

Answer: 28


Two mutually perpendicular rays, propagating in the same vertical plane, fall from the air onto a horizontal glass surface. The refractive index of the glass is $n = 1.5$. The angle of refraction for the first ray is $\beta = 25^{\circ}$. Determine the angle between the refracted rays.

Answer: 56^\circ


4. A train passes a building at a constant speed. For 30 seconds, the train is "visible from head to tail but not in full view", and for 10 seconds the train is "visible from head but not from tail". The train completely passes the building in $\qquad$ seconds.

Answer: 50


Example 1. Find $P_{n}=\prod_{k=1}^{n}\left(2 \cos 2^{k-1} \theta-1\right)$.

Answer: \dfrac{1 + 2 \cos 2^{n} \theta}{1 + 2 \cos \theta}


33 friends are collecting stickers for a 2011-sticker album. A distribution of stickers among the 33 friends is incomplete when there is a sticker that no friend has. Determine the least $m$ with the following property: every distribution of stickers among the 33 friends such that, for any two friends, there are at least $m$ stickers both don't have, is incomplete.

Answer: 1890


Traditional Tetris blocks are made of 4 squares pieced together and come in 7 different shapes (shapes that cannot coincide after rotating in the same plane are considered different). How many different shapes can be made by piecing 5 squares together for a new type of Tetris block?

Answer: 18


282. Cross of chips. Arrange 20 chips in the shape of a cross, as shown in the figure. How many different cases can you count where four chips form a perfect square?

For example, squares are formed by the chips at the ends of the cross, the chips located in the center, as well as the chips marked with the letters $A$ and $B$.

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-084.jpg?height=485&width=488&top_left_y=1431&top_left_x=790)

Which 6 chips should be removed so that no four of the remaining chips are located at the vertices of any square?

Answer: 6


19. As shown in the figure, $A, B, C, D$ are four consecutive vertices of a regular $n$-sided polygon, and $\triangle B C E$ is an equilateral triangle. If points $A, B, E$ are three consecutive vertices of another regular polygon, then the maximum value of $n$ is $\qquad$

Answer: 42


Answer: -\dfrac{30}{13} + \dfrac{72}{13}i


Example 5. Find the number of roots of the equation

$$
Q_{7}(z) \equiv z^{7}-2 z-5=0
$$

in the right half-plane.

Answer: 3


Let $N$ be the greatest positive integer that can be expressed using all seven Roman numerals $I$, $V$, $X$, $L$, $C$, $D$, and $M$ exactly once each, and let $n$ be the least positive integer that can be expressed using these numerals exactly once each. Find $N - n$. Note that the arrangement $CM$ is never used in a number along with the numeral $D$.

Answer: 222


The line \( y = kx + 1 \) intersects with the left branch of the hyperbola \( x^{2} - y^{2} = 1 \) at points \( A \) and \( B \). The line \( l \) passes through the point \( (-2, 0) \) and the midpoint of \( AB \). Find the range of values for the y-intercept \( b \) of the line \( l \).

Answer: (-\infty, -2 - \sqrt{2}) \cup (2, \infty)


54. The function $f: R \rightarrow R$ is continuous. For each real $x$, the equality $f(x) \cdot f(f(x))=1$ holds. It is known that $f(1000)=999$. Find $f(500)$.

Answer: \dfrac{1}{500}


3. In how many ways can a $3 \times 3$ table be filled with elements of the set $\{10,3,2018\}$ such that the sum of the numbers in each row, each column, and both diagonals is divisible by 3?

Answer: 27


5. In a regular tetrahedron $P-ABC$, $AB=1, AP=2$. A plane $\alpha$ passing through $AB$ divides its volume equally. Then the cosine of the angle formed by edge $PC$ and plane $\alpha$ is $\qquad$

Answer: \dfrac{3\sqrt{5}}{10}


IV OM - III - Task 3

Through each vertex of a tetrahedron of a given volume $ V $, a plane parallel to the opposite face of the tetrahedron has been drawn. Calculate the volume of the tetrahedron formed by these planes.

Answer: 27V


1. Given that the circumradius of $\triangle A B C$ is $R$, and
$$
2 R\left(\sin ^{2} A-\sin ^{2} C\right)=(\sqrt{2} a-b) \sin B,
$$

where $a$ and $b$ are the sides opposite to $\angle A$ and $\angle B$ respectively. Then the size of $\angle C$ is

Answer: 45^\circ


Given $\alpha$ and $\beta$ in the interval $\left(0, \frac{\pi}{2}\right)$, prove that $\alpha + \beta = \frac{\pi}{2}$ is a necessary and sufficient condition for $\frac{\sin^{4} \alpha}{\cos^{2} \beta} + \frac{\sin^{4} \beta}{\cos^{2} \alpha} = 1$. Necessity is evident.

Answer: \alpha + \beta = \frac{\pi}{2}


7. Let the elements in set $T$ be integers between 1 and $2^{30}$, and their binary representations contain exactly two 1s. If a number is randomly selected from set $T$, then the probability that this number is divisible by 9 is $\qquad$.

Answer: \dfrac{5}{29}


9. (12 points) As shown in the figure, $A$ and $B$ are the two endpoints of a diameter of a circular track. Three miniature robots, Jia, Yi, and Bing, start moving simultaneously on the circular track at a uniform speed. Jia and Yi start from $A$, and Bing starts from $B$; Yi moves clockwise, while Jia and Bing move counterclockwise. After 12 seconds, Jia reaches $B$, and after another 9 seconds, Jia catches up with Bing for the first time, which is also when Jia meets Yi for the first time; when Bing first reaches $A$, it will take another $\qquad$ seconds for Yi to reach $B$ for the first time.

Answer: 56


14. A four-digit number, if a comma is placed between the hundreds and tens place, can be written as two two-digit numbers $(3126 \rightarrow 31,62)$. If the two two-digit numbers have an integer multiple relationship, we call such a four-digit number a "clever number". If you select 4 numbers from $1, 2, 4, 6, 8$ to form a four-digit number, how many "clever numbers" are there? $\qquad$

Answer: 12


3. Given $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ are
points on the circle $\odot O: x^{2}+y^{2}=1$,
and $|A B|=1$. Then the maximum value of $\left|x_{1}+y_{1}-1\right|+$ $\left|x_{2}+y_{2}-1\right|$ is $\qquad$

Answer: 2 + \sqrt{6}


3. 3.1. Petya marked 5 points on a face of a cube, turned it and marked 6 points on an adjacent face, then turned it again and marked 7 points, and so on. He marked points on each face this way. What is the maximum number of points that can be on two opposite faces?

Answer: 18


Given the parallelogram $ABCD$. The circle $S_1$ passes through the vertex $C$ and touches the sides $BA$ and $AD$ at points $P_1$ and $Q_1$, respectively. The circle $S_2$ passes through the vertex $B$ and touches the side $DC$ at points $P_2$ and $Q_2$, respectively. Let $d_1$ and $d_2$ be the distances from $C$ and $B$ to the lines $P_1Q_1$ and $P_2Q_2$, respectively.
Find all possible values of the ratio $d_1:d_2$.

[i](I. Voronovich)[/i]

Answer: 1


## Task B-3.3.

An isosceles triangle $A B C$ with base $\overline{B C}, |B C|=20$, and legs of length 18 is divided by the line segment $\overline{D E}$ into two parts of equal perimeters and areas. Point $D$ is on the base, and point $E$ is on the leg $\overline{A C}$, and both are different from the vertices of the triangle. Determine the length of $|D C|$.

Answer: 18


The positive numbers  $a_1, a_2,...$  satisfy  $a_1 = 1$  and  $(m+n)a_{m+n }\le  a_m +a_n$  for all positive integers  $m$  and  $n$ . Show that  $\frac{1}{a_{200}} > 4  \cdot 10^7$  .
.

Answer: \frac{1}{a_{200}} > 4 \times 10^7


11. (20 points) Given an ellipse with eccentricity $\frac{1}{2}$, its left focus $F_{1}$ is the intersection of the directrix of the parabola $y^{2}=4 p x(p>0)$ with the $x$-axis, and its right focus $F_{2}$ is also the focus of the parabola. The intersection point of the ellipse and the parabola above the $x$-axis is $P$. Extend $P F_{1}$ to intersect the parabola at point $Q$. $M$ is a moving point on the parabola between points $P$ and $Q$. If the side lengths of $\triangle P F_{1} F_{2}$ are exactly three consecutive positive integers, find the maximum area of $\triangle M P Q$.

Answer: \dfrac{\sqrt{6}}{16}


Let $n > 3$ be a positive integer. Find all integers $k$ such that $1 \le k \le n$ and for
which the following property holds:
If $x_1, . . . , x_n$ are $n$ real numbers such that $x_i + x_{i + 1} + ... + x_{i + k - 1} = 0$ for all integers $i > 1$ (indexes are taken modulo $n$), then $x_1 = . . . = x_n = 0$.

Proposed by [i]Vincent Jugé and Théo Lenoir, France[/i]

Answer: k


1. Let the circumcenter, incenter, and orthocenter of non-isosceles $\triangle ABC$ be $O$, $I$, and $H$, respectively, with the circumradius being $1$ and $\angle A=60^{\circ}$. Then the circumradius of $\triangle OIH$ is $\qquad$.

Answer: \dfrac{1}{2}


8. A hotel has 90 vacant rooms, each with a unique key. 100 guests arrive, and keys need to be distributed so that any 90 of them can stay in the 90 rooms, with each person getting one room (assuming there is no limit to the number of keys that can be issued for each room or the number of keys each person can receive). What is the minimum number of keys that need to be issued? $\qquad$

Answer: 990


Let $A B C D A_{1} B_{1} C_{1} D_{1}$ be a unit cube. The midpoint of edge $B C$ is $F$, and the center of the square $D C C_{1} D_{1}$ is $O$. The plane of triangle $A F O$ divides the cube into two solids. Determine the ratio of the volumes of these two solids.

![](https://cdn.mathpix.com/cropped/2024_05_02_666a4c6d5a7717152565g-1.jpg?height=292&width=325&top_left_y=266&top_left_x=889)

Answer: \dfrac{7}{29}


Example 5 (2008 Chongqing College Entrance Examination Question) A person has 4 different colored light bulbs (with enough of each color), to be installed on the 6 points $A, B, C, A_{1}, B_{1}, C_{1}$ as shown in Figure 5, with the requirement that the light bulbs at the ends of the same line segment must be of different colors. The number of installation methods where each color of light bulb is used at least once is $\qquad$ (answer with a number).

Answer: 216


3.57. A line is drawn through the vertex of angle $\alpha$ at the base of an isosceles triangle, intersecting the opposite lateral side and forming an angle $\beta$ with the base. In what ratio does this line divide the area of the triangle?

Answer: \dfrac{\tan \alpha - \tan \beta}{2 \tan \beta}


11. Find the range of the function $y=x^{2}+x \sqrt{x^{2}-1}$.

Answer: (\frac{1}{2}, \infty)


33. Solve the equation $\frac{d y}{y+1}=\frac{d x}{x-1}$.

Answer: y = C(x - 1) - 1


## Task 6 - V00706

For the housing of a household scale, a rectangular metal strip measuring $390 \mathrm{~mm}$ in length and $85 \mathrm{~mm}$ in width was used at the VEB Thüringer Industriewerk Rauenstein. The thickness of the material was $2.5 \mathrm{~mm}$.

Through an improvement suggestion, it was possible to use $2 \mathrm{~mm}$ thick sheet metal.

Calculate the material savings in $\mathrm{t}$ for a production run of 100000 units!

(Density of iron $7.8 \mathrm{~g} \cdot \mathrm{cm}^{-3}$ )

Answer: 12.93


Let $n$ be a positive integer. In an $n\times n$ garden, a fountain is to be built with $1\times 1$ platforms covering the entire garden. Ana places all the platforms at a different height. Afterwards, Beto places water sources in some of the platforms. The water in each platform can flow to other platforms sharing a side only if they have a lower height. Beto wins if he fills all platforms with water.
Find the least number of water sources that Beto needs to win no matter how Ana places the platforms.

Answer: \left\lceil \dfrac{n^2}{2} \right\rceil


In Figure 1, \(ABCD\) is a parallelogram, \(BE \perp CD\), \(BF \perp AD\), \(CE = 2 \text{ cm}\), \(DF = 1 \text{ cm}\) and \(\angle EBF = 60^\circ\). If the area of the parallelogram \(ABCD\) is \(R \text{ cm}^2\), find the value of \(R\).

Answer: 12\sqrt{3}


The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq3$.

Let $m,n\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$, which satisfies $f(k)=F_{m+k}$ for all $k=0,1,...,n$.

Answer: n


Petr drew a regular hexagon whose vertices lay on a circle with a circumference of $16 \mathrm{~cm}$. Then, from each vertex of this hexagon, he drew a circle that passed through the two adjacent vertices. This resulted in a figure as shown in the following image.

Determine the perimeter of the marked flower.

(E. Novotná)

![](https://cdn.mathpix.com/cropped/2024_04_17_3ab73cf5e47d7567d598g-2.jpg?height=551&width=596&top_left_y=1518&top_left_x=730)

Hint. Look for (and justify) congruent parts of the image.

Answer: 32


3. In a football tournament, seven teams played: each team played once with each other. Teams that scored thirteen or more points advance to the next round. Three points are awarded for a win, one point for a draw, and zero points for a loss. What is the maximum number of teams that can advance to the next round?

Answer: 4


13. There are two equations:

Good + Good $=$ Wonderful, Wonderful $\times$ GoodGood $\times$ ReallyGood $=$ WonderfulProblemProblemWonderful, where each Chinese character represents a digit from $0 \sim 9$, the same character represents the same digit, and different characters represent different digits. Therefore, the number of all factors of the four-digit number "WonderfulProblemProblemWonderful" is $\qquad$

Answer: 16


Draw the altitudes in an acute-angled triangle \(ABC\) and denote the feet of the altitudes as \(A_1, B_1, C_1\), and the orthocenter as \(M\). Show that

\[ M A \times M A_{1} = 2 R r \]

where \(R\) is the circumradius of the given triangle and \(r\) is the inradius of the pedal triangle.

Answer: MA \times MA_{1} = 2 R r


11. As shown in Figure 1, the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ has left and right foci $F_{1}$ and $F_{2}$. Points $A$ and $B$ are on the upper half of the ellipse, and $F_{1} A \parallel F_{2} B$. Let $B F_{1}$ and $A F_{2}$ intersect at point $P$. Find the maximum value of the y-coordinate of point $P$.

Answer: \dfrac{3}{4}


B5. Given is the sequence of numbers $a_{0}, a_{1}, a_{2}, \ldots, a_{2020}$. It is given that $a_{0}=0$. Moreover, for every $k=1,2, \ldots, 2020$ it holds that

$$
a_{k}= \begin{cases}a_{k-1} \cdot k & \text { if } k \text { is divisible by } 8, \\ a_{k-1}+k & \text { if } k \text { is not divisible by } 8 .\end{cases}
$$

What are the last two digits of $a_{2020}$?

## C-Problems

For the C-Problems, not only the answer is important; a reasoning should also be provided that shows your answer is correct. Each correctly worked out C-Problem is worth 10 points. Partial solutions can also earn points. Therefore, write everything clearly and submit your scratch paper for each problem as well.

NOTE: Solve each C-Problem on a separate sheet of paper and also submit the corresponding scratch paper per problem.

Answer: 02


II. (Full marks 30 points) In the tetrahedron $A-BCD$, $\angle DAB + \angle BAC + \angle DAC = 90^{\circ}$, $\angle ADB = \angle BDC = \angle ADC = 90^{\circ}$.
It is also known that $DB = a$, $DC = b$.
Try to find the radius of the largest sphere that can fit inside the tetrahedron $A-BCD$.

Answer: \dfrac{ab}{2(a + b)}


6. (B. Frenkin) A rook has traversed an $n \times n$ board, visiting each cell exactly once, with each move being exactly one cell. The cells are numbered from 1 to $n^{2}$ in the order of the rook's traversal. Let $M$ be the maximum difference between the numbers of adjacent (by side) cells. What is the smallest possible value of $M$?

Answer: 2n - 1


## Task A-4.5.

Given are 100 points arranged in 10 rows and 10 columns. Each point is either red or blue. A point is in a corner if it is at the end of both a row and a column, and on the edge if it is at the end of a row or a column, but not both. Two points are adjacent if they are in the same row or column, with no other points between them. Each pair of adjacent points is connected by a line of red, blue, or green color, such that each red line connects two red points, each blue line connects two blue points, and each green line connects one red and one blue point. It is known that there are 52 red points in total, of which 16 are on the edge and two are in the corners. It is also known that there are a total of 98 green lines.

Determine the total number of blue lines.

Answer: 37


## Task 5 - 190735

Cathrin goes shopping. She has exactly 18 coins, and they are only two-mark and fifty-pfennig coins. She spends exactly half of the total amount of this money. After shopping, she realizes that she now has only two-mark and fifty-pfennig coins again, and she has as many two-mark coins as she had fifty-pfennig coins before, and as many fifty-pfennig coins as she had two-mark coins before. What amount of money did Cathrin have left after shopping?

Answer: 15


89. The triangle is defined by points $A(5 ; 2), B(-1 ;-4), C(-5$; -3). Form the equation of the line passing through point $B$ and parallel to $A C$.

Answer: x - 2y = 7


9-5. Two three-digit numbers are written on the board in a multiplication example. If the multiplication sign is replaced with 0, a seven-digit number is obtained, which is an integer multiple of the product. By what factor exactly

Answer: 73


6. There are 900 three-digit numbers $(100,101, \cdots, 999)$. These three-digit numbers are printed on cards, with ten three-digit numbers on each card. Some cards, when flipped, still show three-digit numbers, such as 198, which when flipped becomes 861 (1 is still considered 1 when flipped); some cards do not, such as 531, which when flipped shows I $E$. Therefore, some cards can be used in two ways. Thus, at most, _ fewer cards can be printed.

Answer: 34


Positive sequences $\{a_n\},\{b_n\}$ satisfy:$a_1=b_1=1,b_n=a_nb_{n-1}-\frac{1}{4}(n\geq 2)$.
Find the minimum value of $4\sqrt{b_1b_2\cdots b_m}+\sum_{k=1}^m\frac{1}{a_1a_2\cdots a_k}$,where $m$ is a given positive integer.

Answer: 5


Example 6. Father's age is 48 years old, son's age is 20 years old. How many years later will the father's age be 3 times the son's age?

Answer: 6


Four points $A, O, B, O^{\prime}$ are aligned in this order on a line. Let $C$ be the circle with center $O$ and radius 2015, and $C^{\prime}$ be the circle with center $O^{\prime}$ and radius 2016. Suppose that $A$ and $B$ are the intersections of two common tangents to the two circles. Calculate $A B$ given that $A B$ is an integer $<10^{7}$ and that $A O$ and $A O^{\prime}$ are integers.

## 3 Junior Eliminatory: Solutions

Translate the above text into English, retain the original text's line breaks and format, and output the translation result directly.

Answer: 8124480


7.20 A kind of single-player card game, its rules are as follows: Place 6 pairs of different cards into a backpack, the player randomly draws and returns cards each time, but when a pair is drawn, it is placed aside. If the player always draws three cards each time, and if the three cards drawn are all different (i.e., no pairs), the game ends; otherwise, drawing continues until the backpack is empty. Let the probability that the backpack is empty be $\frac{p}{q}$, where $p$ and $q$ are coprime positive integers, find $p+q$.

Answer: 394


The positive integers \( x_{1}, \ldots, x_{n}\), where \( n \geq 3 \), satisfy \( x_{1} < x_{2} < \cdots < x_{n} < 2 x_{1} \). Set \( P = x_{1} x_{2} \cdots x_{n} \). Prove that if \( p \) is a prime number, \( k \) a positive integer, and \( P \) is divisible by \( p^{k} \), then \( \frac{P}{P^{x}} \geq n! \).

Answer: \frac{P}{p^k} \geq n!


1. When $-1 \leqslant x \leqslant 2$, the function $y=(k-2)x + 2|k|-1$ is always positive for $x$. Then the range of values for $k$ is

Answer: (\dfrac{5}{4}, +\infty)


15. Given the sequence $\left\{a_{n}\right\}(n \geqslant 0)$ satisfies $a_{0}=0$, for all non-negative integers $n$, we have
$$
a_{n+1}=2 \sqrt{30 a_{n}\left(a_{n}+1\right)}+11 a_{n}+5 .
$$

Find the general term formula for $a_{n}$.

Answer: \dfrac{(11 + 2\sqrt{30})^n + (11 - 2\sqrt{30})^n - 2}{4}


7. Given $|\boldsymbol{a}|=1,|\boldsymbol{b}|=|c|=2, \boldsymbol{b} \cdot \boldsymbol{c}=0$, $\lambda \in(0,1)$. Then
$$
|a-b+\lambda(b-c)|+\left|\frac{1}{2} c+(1-\lambda)(b-c)\right|
$$

the minimum value is $\qquad$

Answer: \sqrt{5} - 1


267. Dubious Equality. The statement «$342=97$» can be made valid by inserting several algebraic signs between the digits, for example $(-3+4) \cdot 2=9-7$. Can the equality be given meaning without inserting any signs?

Answer: 342_5 = 97_{10}


Grandpa forgot the four-digit code of his mobile phone. He only remembered that the first digit was not zero, that in the middle were either two fours or two sevens or a four and a seven (in an unknown order), and that the number was divisible by 15. How many possibilities are there for the forgotten code? What digit could be on the first place?

(M. Volfová)

Hint. Start with the middle two-digit number, then consider the other conditions from the problem.

Answer: 24


1. Detective Podberezyakov is pursuing Maksim Detochkin (each driving their own car). At the beginning, both were driving on the highway at a speed of 60 km/h, with Podberezyakov lagging behind Detochkin by 2 km. Upon entering the city, each of them reduced their speed to 40 km/h, and upon exiting the city, finding themselves on a good highway, each increased their speed to 70 km/h. When the highway ended, at the border with a dirt road, each had to reduce their speed to 30 km/h again. What was the distance between them on the dirt road?

Answer: 1


In every row of a grid $100 \times n$ is written a permutation of the numbers $1,2 \ldots, 100$. In one move you can choose a row and swap two non-adjacent numbers with difference $1$. Find the largest possible $n$, such that at any moment, no matter the operations made, no two rows may have the same permutations.

Answer: 2^{99}


a) In the figure below, two overlapping squares are shown. The larger square has a side length of 4, and the smaller square has a side length of 3. What is the area of the gray region minus the area of the black region?

b) In the figure below, six squares are drawn, with side lengths (from left to right) of 6, 5, 4, 3, 2, and 1, respectively. What is the area of the gray region minus the area of the black region?

Answer: 7


The function \( f \) assigns a number \( f(\boldsymbol{v}) \) to each vector \( \boldsymbol{v} \) (with a common origin at point \( O \) ) in space, such that for any vectors \( \boldsymbol{u}, \boldsymbol{v} \) and any numbers \( \alpha, \beta \), the value \( f(\alpha \boldsymbol{u} + \beta \boldsymbol{v}) \) does not exceed at least one of the numbers \( f(\boldsymbol{u}) \) or \( f(\boldsymbol{v}) \). What is the maximum number of values that such a function can take?

Answer: 4


## Task 1

In the relay race (4 x 100 meters), 8-man teams compete in the final. How many runners are competing in total for the victory?

Answer: 32


89. The blackboard is written with a number $\underbrace{222 \cdots 2}_{9 \uparrow 2}$, and the following operation is performed: erase the last digit, multiply the number on the blackboard by 4, and then add the last digit that was just erased. Continue performing such operations until a single-digit number appears on the blackboard, then this number is $\qquad$.

Answer: 6


Triangle $ABC$ has $AB=40,AC=31,$ and $\sin{A}=\frac{1}{5}$ . This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$ . Find the maximum possible area of $AQRS$ .
Please give the answer directly without any intermediate steps.

Answer: 744


16. (15 points) Let $x, y, z \geqslant 0$, and at most one of them is 0. Find
$$
f(x, y, z)=\sum \sqrt{\frac{x^{2}+256 y z}{y^{2}+z^{2}}}
$$

the minimum value, where $\sum$ denotes the cyclic sum.

Answer: 12


Let $a_1, a_2, \dots, a_n$ be real numbers, and let $b_1, b_2, \dots, b_n$ be distinct positive integers. Suppose that there is a polynomial $f(x)$ satisfying the identity
\[
(1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}.
\]
Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \dots, b_n$ and $n$ (but independent of $a_1, a_2, \dots, a_n$).

Answer: \dfrac{b_1 b_2 \cdots b_n}{n!}


ABC is acute-angled. AD and BE are altitudes. The areas of triangles BDE, DEA, EAB, and ABD satisfy the inequality: area of BDE ≤ area of DEA ≤ area of EAB ≤ area of ABD. Show that the triangle ABC is isosceles.

Answer: \triangle ABC \text{ is isosceles}


Two robbers stole 300 gold coins. They decided to divide them as follows: the first robber puts some coins (possibly all) in a bag, and the second robber decides who will get the bag; then the process is repeated several more times. The division ends when

- either all the money has been used up,
- or one of them has received 11 bags - in which case all the remaining money immediately goes to the other robber.

What is the maximum number of coins that the first robber can be guaranteed to get?

Answer: 146


## Task 3

A number is called interesting if all its digits are mutually different, and its first digit is equal to the sum of all the other digits. For example, the numbers 321 and 80413 are interesting. How many interesting numbers are there?

Answer: 338


Problem 1. Vasya and Petya ran out from the starting point of a circular running track at the same time and ran in opposite directions. They met at some point on the track. Vasya ran a full lap and, continuing to run in the same direction, reached the place of their previous meeting at the moment when Petya had run a full lap. How many times faster did Vasya run than Petya?

Answer: \dfrac{1 + \sqrt{5}}{2}


Giuseppe has a plywood sheet measuring $22 \times 15$. He wants to cut out as many rectangular blanks of size $3 \times 5$ as possible from it. How to achieve this?

Answer: 22


In each cell of a $2019 \times 2019$ square, both diagonals are drawn. Is there a closed path consisting of these diagonals that does not pass through any diagonal more than once and visits all the cells of the square (i.e., contains at least one diagonal from each cell)?

Answer: \text{No}


10. 78 students stand in a row. Starting from the left, give a badminton shuttlecock to the first person, and then give a badminton shuttlecock every 2 people. Starting from the right, give a table tennis ball to the first person, and then give a table tennis ball every 3 people. The number of students who have both a badminton shuttlecock and a table tennis ball is $\qquad$ people.

Answer: 6


Let \( a, b > 1 \) be odd numbers such that \( a + b = 2^l \). Find \( k \in \mathbb{N}^{*} \) such that \( k^2 \mid a^k + b^k \).

Answer: 1


4. Let $p, \alpha, m \in \mathbf{Z}_{+}$, denote $p^{\alpha} \| m$ to mean $p^{\alpha} \mid m$, but $p^{\alpha+1} \nmid m$. For any positive integer $n$, the maximum value of the positive integer $\alpha$ satisfying $2^{\alpha} \|\left(3^{n}+1\right)$ is $\qquad$

Answer: 1


9. Find the maximum value of the volume of a regular quadrilateral frustum with the upper base edge and side edge both equal to 1 (accurate to 0.001).

Note: If the upper base area of the regular quadrilateral frustum is $S_{1}$, the lower base area is $S_{2}$, and the height is $h$, then its volume
$$
V=\frac{1}{3} h\left(S_{1}+S_{2}+\sqrt{S_{1} S_{2}}\right) .
$$

Answer: 1.674


10.1. It is known that the quadratic equations in $x$, $2017 x^{2} + p x + q = 0$ and $u p x^{2} + q x + 2017 = 0$ (where $p$ and $q$ are given real numbers) have one common root. Find all possible values of this common root and prove that there are no others.

Answer: 1


One, (20 points) In $\triangle A B C$, it is known that $\angle A>\angle B>$ $\angle C$, and $\angle A=2 \angle B$. If the three sides of the triangle are integers, and the area is also an integer, find the minimum value of the area of $\triangle A B C$.

Answer: 132


There is a stone on each square of $n\times n$ chessboard. We gather $n^2$ stones and distribute them to the squares (again each square contains one stone)  such that any two adjacent stones are again adjacent. Find all distributions such that at least one stone at the corners remains at its initial square. (Two squares are adjacent if they share a common edge.)

Answer: 3


Given a circle with radius $r$ and a segment $AB$ parallel to the plane of the circle, at a distance $r$ from the plane. The orthogonal projection of the segment $AB$ onto the plane of the circle is a diameter of the circle. From every interior point of the segment $AB$, we draw two half-lines perpendicular to $AB$, each of which intersects the circle. Determine the volume of the solid bounded by the circle and the half-lines.

The problem involves a circle with radius $r$ and a segment $AB$ that is parallel to the plane of the circle and is at a distance $r$ from the plane. The orthogonal projection of the segment $AB$ onto the plane of the circle is a diameter of the circle. From every interior point of the segment $AB$, we draw two half-lines perpendicular to $AB$, each of which intersects the circle. We need to determine the volume of the solid bounded by the circle and the half-lines.

Answer: \dfrac{\pi r^3}{2}


4. Inside triangle $ABC$, where $\angle C=70^{\circ}, \angle B=80^{\circ}$, a point $M$ is taken such that triangle $CMB$ is equilateral. Find the angles $MAB$ and $MAC$.

Answer: 20^{\circ}


1. How many numbers with at least four digits are divisible by 9 and can be formed from the digits $1,9,0,1,2,0,1,9$ (each digit can be used as many times as it is listed)?

Answer: 3


34. Consider an equilateral triangle $A B C$, where $A B=B C=C A=2011$. Let $P$ be a point inside $\triangle A B C$. Draw line segments passing through $P$ such that $D E\|B C, F G\| C A$ and $H I \| A B$. Suppose $D E: F G: H I=8: 7: 10$. Find $D E+F G+H I$.

Answer: 4022


Task B-4.5. If we draw a tangent $t_{1}$ to the ellipse $b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}$ at an angle of $45^{\circ}$ to the positive direction of the $x$-axis, its intercept on the $y$-axis is 4. If we draw a tangent $t_{2}$ at an angle of $60^{\circ}$, the intercept on the $y$-axis will increase by 2. Determine the area of the quadrilateral $F_{1} T_{1} F_{2} T_{2}$, where $F_{1}, F_{2}$ are the foci of the ellipse, and $T_{1}, T_{2}$ are the intersections of the tangents $t_{1}, t_{2}$ with the $y$-axis.

Answer: 4


9.6. First, Betya writes the numbers $1, 2$ in her notebook, and Nik writes $3, 4$ in his notebook. Then, at the beginning of each minute, Nik and Betya each write down a quadratic polynomial with real coefficients, whose roots are the two numbers in their notebooks, denoted as $f(x)$ and $g(x)$, respectively. If the equation $f(x)=g(x)$ has two distinct real roots $x_{1}$ and $x_{2}$, one of them immediately changes the numbers in their notebook to $x_{1}$ and $x_{2}$; if the equation $f(x)=g(x)$ does not have two distinct real roots $x_{1}$ and $x_{2}$, they do nothing and wait for the next minute to start. After some time, one of the numbers in Betya's notebook is 5. Find all possible values of the other number in her notebook.

Answer: \dfrac{14}{5}


Let $\lambda$ be a real number. Suppose that if \(ABCD\) is any convex cyclic quadrilateral such that \(AC=4\), \(BD=5\), and \(\overline{AB}\perp\overline{CD}\), then the area of $ABCD$ is at least $\lambda$. Then the greatest possible value of $\lambda$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.

[i]Proposed by Eric Shen[/i]

Answer: 9041


In a sequence of several zeros and ones, consider pairs of digits in this sequence (not necessarily adjacent) where the left digit is 1 and the right digit is 0. Let there be exactly $M$ pairs where there is an even number of digits between the 1 and the 0, and exactly $N$ pairs where there is an odd number of digits between the 1 and the 0.

Prove that $M \geq N$.

Answer: M \geq N


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`, `Bob`, `Arnold`, `Alice`, `Eric`, `Carol`
 - The people keep unique animals: `rabbit`, `bird`, `dog`, `horse`, `cat`, `fish`
 - Each person has an occupation: `doctor`, `teacher`, `lawyer`, `artist`, `engineer`, `nurse`

## Clues:
1. Arnold is not in the sixth house.
2. The person who is an engineer is directly left of Eric.
3. Carol and the rabbit owner are next to each other.
4. The person who is an artist is Peter.
5. The person who keeps horses is in the second house.
6. Eric is directly left of the person who is a lawyer.
7. The person who is an artist is somewhere to the left of the person who is a doctor.
8. The cat lover is not in the sixth house.
9. Peter is in the fifth house.
10. The dog owner is the person who is an artist.
11. The person who keeps horses is directly left of the rabbit owner.
12. The bird keeper is Alice.
13. The person who is a teacher and Alice are next to each other.
14. Alice is not in the first house.
 What is the value of attribute Name for the person whose attribute Animal is dog? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


There are 3 houses, numbered 1 to 3 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`, `Eric`, `Peter`
 - Each person has a unique level of education: `associate`, `bachelor`, `high school`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`

## Clues:
1. The person who likes milk is somewhere to the right of Eric.
2. The person who loves science fiction books is directly left of the person who owns a Toyota Camry.
3. The person who loves cooking is Peter.
4. The person who loves romance books is the person with a bachelor's degree.
5. The person who owns a Tesla Model 3 is Peter.
6. The person with an associate's degree is the person who loves mystery books.
7. The person who loves mystery books is the photography enthusiast.
8. The one who only drinks water is not in the second house.
9. The person who owns a Toyota Camry is not in the third house.
10. The person who owns a Ford F-150 is the person who loves mystery books.
 What is the value of attribute CarModel for the person whose attribute Education is bachelor? Please reason step by step, and put your final answer within \boxed{}

Answer: toyota camry


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`, `Bob`, `Alice`, `Eric`, `Peter`
 - People have unique favorite sports: `swimming`, `basketball`, `volleyball`, `soccer`, `tennis`, `baseball`
 - People have unique favorite music genres: `rock`, `pop`, `jazz`, `country`, `hip hop`, `classical`
 - Each person has a unique level of education: `doctorate`, `trade school`, `bachelor`, `high school`, `associate`, `master`

## Clues:
1. Alice is in the fourth house.
2. Alice is somewhere to the right of the person with a doctorate.
3. The person with a high school diploma is somewhere to the right of the person who loves basketball.
4. The person who loves volleyball is the person with a bachelor's degree.
5. Alice is directly left of the person who loves jazz music.
6. The person with a master's degree is the person who loves swimming.
7. There is one house between the person who loves rock music and the person with a bachelor's degree.
8. Carol is in the second house.
9. The person who loves rock music is somewhere to the left of the person who loves pop music.
10. The person who loves swimming is somewhere to the left of Carol.
11. Eric is directly left of the person who loves hip-hop music.
12. Alice is the person with an associate's degree.
13. Alice is the person who loves classical music.
14. Arnold is the person who loves soccer.
15. The person who loves basketball and the person who loves volleyball are next to each other.
16. Peter is the person who loves tennis.
17. The person with a doctorate is Peter.
 What is the value of attribute FavoriteSport for the person whose attribute MusicGenre is rock? Please reason step by step, and put your final answer within \boxed{}

Answer: tennis


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`, `Bob`, `Alice`, `Peter`, `Arnold`, `Carol`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `tulips`, `iris`, `lilies`
 - People have unique heights: `very tall`, `super tall`, `average`, `short`, `very short`, `tall`
 - Each person has a favorite color: `blue`, `yellow`, `purple`, `red`, `green`, `white`
 - Everyone has a unique favorite cigar: `yellow monster`, `pall mall`, `prince`, `blue master`, `blends`, `dunhill`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `xiaomi mi 11`, `samsung galaxy s21`, `huawei p50`, `oneplus 9`

## Clues:
1. The person who uses an iPhone 13 is the person who loves white.
2. Peter is the person whose favorite color is red.
3. The Prince smoker is directly left of the person partial to Pall Mall.
4. The person who smokes Blue Master and the Dunhill smoker are next to each other.
5. The person who loves blue is the person who uses a Xiaomi Mi 11.
6. The person who loves white is Alice.
7. The person who uses a OnePlus 9 is somewhere to the left of Arnold.
8. The person who is tall is the person who smokes Blue Master.
9. The person who uses a Samsung Galaxy S21 is the Dunhill smoker.
10. Bob is directly left of the person who is very short.
11. The person who loves a bouquet of daffodils is Eric.
12. The person who loves the boquet of iris is the person who uses a OnePlus 9.
13. The person who uses a Huawei P50 is the person who is very short.
14. The person who smokes many unique blends is directly left of Peter.
15. The Prince smoker is not in the first house.
16. The person who is very tall is the person who loves a bouquet of daffodils.
17. The person who uses a Google Pixel 6 is not in the fifth house.
18. The person who is tall is not in the third house.
19. The person who uses a Huawei P50 is somewhere to the left of the person who uses a Samsung Galaxy S21.
20. Arnold is somewhere to the left of the person who is short.
21. The person who loves the vase of tulips is not in the first house.
22. The person who loves the boquet of lilies is directly left of the person who loves yellow.
23. Arnold is the person who loves purple.
24. The person who uses a OnePlus 9 and the person who uses a Samsung Galaxy S21 are next to each other.
25. The person who has an average height is not in the first house.
26. The person who loves the rose bouquet is Alice.
 What is the value of attribute Color for the person whose attribute Cigar is prince? Please reason step by step, and put your final answer within \boxed{}

Answer: red


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`, `Eric`, `Bob`, `Peter`, `Arnold`
 - Each person has a unique birthday month: `mar`, `april`, `sept`, `feb`, `jan`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Kailyn`, `Penny`, `Aniya`
 - Each person has an occupation: `engineer`, `doctor`, `lawyer`, `artist`, `teacher`
 - People have unique hair colors: `red`, `blonde`, `black`, `gray`, `brown`

## Clues:
1. The person whose birthday is in March is in the fifth house.
2. The person whose birthday is in February is in the first house.
3. The person who is a doctor is Eric.
4. The person whose mother's name is Janelle is in the third house.
5. The person who is an artist is the person who has brown hair.
6. The person who is an artist is in the fourth house.
7. The person whose mother's name is Penny is somewhere to the left of the person who has black hair.
8. Peter is the person who has black hair.
9. The person who has gray hair is the person who is a teacher.
10. Alice is The person whose mother's name is Kailyn.
11. Arnold is somewhere to the right of the person whose birthday is in September.
12. The person who has brown hair is the person whose birthday is in January.
13. Arnold is the person who has blonde hair.
14. The person whose mother's name is Holly is the person who has black hair.
15. Peter is the person who is a lawyer.
16. The person whose birthday is in September is somewhere to the left of The person whose mother's name is Kailyn.
17. Alice is the person who has gray hair.
 What is the value of attribute Mother for the person whose attribute HairColor is gray? Please reason step by step, and put your final answer within \boxed{}

Answer: Kailyn


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`, `Peter`, `Bob`, `Eric`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Bella`, `Timothy`, `Fred`
 - Everyone has a favorite smoothie: `lime`, `dragonfruit`, `watermelon`, `desert`, `cherry`
 - Each person has a unique type of pet: `hamster`, `bird`, `fish`, `cat`, `dog`
 - Each person has a unique favorite drink: `tea`, `coffee`, `milk`, `root beer`, `water`
 - The mothers' names in different houses are unique: `Penny`, `Aniya`, `Holly`, `Janelle`, `Kailyn`

## Clues:
1. Alice is the person's child is named Samantha.
2. The person whose mother's name is Penny is somewhere to the right of the person's child is named Meredith.
3. Peter is The person whose mother's name is Penny.
4. The person who is the mother of Timothy is directly left of The person whose mother's name is Kailyn.
5. The person who likes Cherry smoothies is Eric.
6. There is one house between the person who drinks Lime smoothies and The person whose mother's name is Holly.
7. There is one house between the person who keeps a pet bird and the coffee drinker.
8. The person with a pet hamster is Arnold.
9. The tea drinker is in the fifth house.
10. There are two houses between the one who only drinks water and the tea drinker.
11. The person who likes Cherry smoothies is the person who keeps a pet bird.
12. The person whose mother's name is Holly is the person who owns a dog.
13. There is one house between the person who keeps a pet bird and the person's child is named Bella.
14. The person whose mother's name is Kailyn and The person whose mother's name is Penny are next to each other.
15. Peter is not in the fourth house.
16. The person with a pet hamster is not in the second house.
17. The person who likes Cherry smoothies is directly left of the person's child is named Meredith.
18. The Dragonfruit smoothie lover is the person who has a cat.
19. The root beer lover is in the first house.
20. The person's child is named Fred is not in the fourth house.
21. The Desert smoothie lover and The person whose mother's name is Aniya are next to each other.
 What is the value of attribute Name for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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: `Arnold`, `Peter`, `Alice`, `Bob`, `Eric`
 - People have unique heights: `very tall`, `average`, `tall`, `very short`, `short`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `cruise`, `city`, `camping`
 - Each person has a unique level of education: `master`, `doctorate`, `associate`, `high school`, `bachelor`

## Clues:
1. The person who is very tall is directly left of the person with an associate's degree.
2. The person who has an average height and the person who enjoys mountain retreats are next to each other.
3. Eric is the person who is short.
4. The person who prefers city breaks is in the fourth house.
5. The person who loves beach vacations is the person who is very tall.
6. The person with a master's degree is Bob.
7. The person who enjoys camping trips is not in the third house.
8. The person who has an average height is not in the second house.
9. The person with a doctorate is directly left of Alice.
10. The person with a master's degree is the person who enjoys camping trips.
11. The person who enjoys camping trips is not in the first house.
12. The person with a doctorate is the person who is tall.
13. The person with a high school diploma is somewhere to the right of the person who has an average height.
14. Arnold is the person with a doctorate.
 What is the value of attribute House for the person whose attribute Education is doctorate? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


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: `Alice`, `Carol`, `Bob`, `Peter`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `lilies`, `tulips`, `roses`, `daffodils`, `iris`, `carnations`
 - The people are of nationalities: `norwegian`, `dane`, `swede`, `german`, `brit`, `chinese`
 - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`, `dragonfruit`, `blueberry`, `lime`
 - Each person has a unique hobby: `photography`, `cooking`, `knitting`, `woodworking`, `painting`, `gardening`
 - Each person has a unique type of pet: `rabbit`, `hamster`, `fish`, `cat`, `bird`, `dog`

## Clues:
1. The Swedish person is the person who loves cooking.
2. The person who drinks Blueberry smoothies is Alice.
3. The person who loves the rose bouquet is not in the sixth house.
4. The photography enthusiast is somewhere to the right of the Dragonfruit smoothie lover.
5. The person who loves cooking is the person who loves a bouquet of daffodils.
6. The person who loves the boquet of lilies and the person who loves the boquet of iris are next to each other.
7. The person who paints as a hobby is Alice.
8. Arnold is somewhere to the right of the Watermelon smoothie lover.
9. The woodworking hobbyist is somewhere to the right of the German.
10. The woodworking hobbyist is somewhere to the left of the person who loves a bouquet of daffodils.
11. The Dane is the Watermelon smoothie lover.
12. The Dragonfruit smoothie lover is Peter.
13. The person who drinks Lime smoothies is the woodworking hobbyist.
14. There is one house between the Watermelon smoothie lover and the person who likes Cherry smoothies.
15. The person who keeps a pet bird is the person who drinks Blueberry smoothies.
16. The Desert smoothie lover is the person who enjoys gardening.
17. The person with a pet hamster is in the fifth house.
18. The person who owns a dog is the Norwegian.
19. The person who has a cat is somewhere to the right of Bob.
20. The British person is directly left of Carol.
21. The person who loves the boquet of iris is Arnold.
22. The German is somewhere to the right of the person who loves a carnations arrangement.
23. Bob is the photography enthusiast.
24. The person with an aquarium of fish is Arnold.
 What is the value of attribute Nationality for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: norwegian


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`, `Alice`, `Eric`, `Arnold`
 - Each person has a unique hobby: `cooking`, `painting`, `gardening`, `photography`
 - The people keep unique animals: `horse`, `fish`, `cat`, `bird`
 - People have unique favorite book genres: `fantasy`, `mystery`, `romance`, `science fiction`
 - Each person has a unique birthday month: `april`, `jan`, `sept`, `feb`
 - People have unique favorite music genres: `pop`, `rock`, `classical`, `jazz`

## Clues:
1. The person who loves cooking is the person who loves romance books.
2. The person whose birthday is in February is the person who loves pop music.
3. Eric is not in the second house.
4. The person who loves romance books is not in the fourth house.
5. The person whose birthday is in February is the fish enthusiast.
6. Alice is somewhere to the right of the person who loves fantasy books.
7. The person who keeps horses is the person who loves rock music.
8. The person who enjoys gardening is the person whose birthday is in April.
9. The person who loves jazz music is the person who loves cooking.
10. The person who loves rock music is the person who loves mystery books.
11. The person who paints as a hobby is directly left of the person who loves romance books.
12. Peter is the person who loves pop music.
13. The person who enjoys gardening is Arnold.
14. The person who loves rock music is directly left of the person whose birthday is in January.
15. The person who loves cooking is not in the third house.
16. The cat lover is somewhere to the right of the person who keeps horses.
 What is the value of attribute BookGenre for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: romance


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`, `Bob`, `Carol`, `Peter`, `Eric`, `Alice`
 - People have unique favorite music genres: `classical`, `hip hop`, `jazz`, `pop`, `rock`, `country`
 - Everyone has something unique for lunch: `soup`, `stir fry`, `grilled cheese`, `spaghetti`, `stew`, `pizza`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `mediterranean`, `modern`, `colonial`, `victorian`
 - Each person has a unique birthday month: `feb`, `sept`, `april`, `may`, `jan`, `mar`
 - People have unique hair colors: `red`, `black`, `blonde`, `brown`, `gray`, `auburn`

## Clues:
1. The person whose birthday is in April is Bob.
2. The person residing in a Victorian house is the person who loves stir fry.
3. There are two houses between the person who loves jazz music and the person who has auburn hair.
4. The person who loves the stew is in the second house.
5. The person who loves stir fry is the person whose birthday is in February.
6. The person who loves classical music is in the fifth house.
7. The person who loves jazz music is Alice.
8. The person living in a colonial-style house is in the fifth house.
9. The person who has gray hair is the person who loves the stew.
10. The person whose birthday is in March is the person who has red hair.
11. The person who loves country music is directly left of the person whose birthday is in September.
12. The person whose birthday is in May is somewhere to the left of the person who loves country music.
13. The person in a ranch-style home is the person who loves the soup.
14. The person whose birthday is in January is the person who loves the stew.
15. The person who has blonde hair is the person in a Mediterranean-style villa.
16. Arnold is somewhere to the right of the person whose birthday is in April.
17. The person whose birthday is in February is the person who loves hip-hop music.
18. The person who has blonde hair is somewhere to the left of Eric.
19. The person residing in a Victorian house is Carol.
20. The person who has black hair is the person in a ranch-style home.
21. The person in a modern-style house is not in the second house.
22. The person who loves the spaghetti eater is somewhere to the left of the person who loves country music.
23. The person who is a pizza lover is Bob.
24. The person who loves pop music is somewhere to the left of the person whose birthday is in September.
 What is the value of attribute Food for the person whose attribute HairColor is brown? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


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: `Alice`, `Peter`, `Arnold`, `Eric`
 - Each person has a unique type of pet: `cat`, `dog`, `bird`, `fish`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`, `Kailyn`
 - Each person lives in a unique style of house: `colonial`, `ranch`, `victorian`, `craftsman`
 - People have unique favorite sports: `soccer`, `basketball`, `swimming`, `tennis`

## Clues:
1. The person who loves soccer is Alice.
2. The person whose mother's name is Kailyn is somewhere to the left of the person in a Craftsman-style house.
3. The person with an aquarium of fish and The person whose mother's name is Janelle are next to each other.
4. Alice is the person who keeps a pet bird.
5. The person who loves swimming is the person in a ranch-style home.
6. Eric is in the first house.
7. Eric is the person residing in a Victorian house.
8. Arnold is the person in a ranch-style home.
9. Arnold is not in the second house.
10. The person whose mother's name is Holly is somewhere to the left of the person who owns a dog.
11. The person whose mother's name is Janelle is in the fourth house.
12. The person whose mother's name is Kailyn is the person who loves tennis.
13. Alice is The person whose mother's name is Janelle.
 What is the value of attribute House for the person whose attribute Mother is Holly? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 3 houses, numbered 1 to 3 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`
 - People have unique favorite music genres: `classical`, `rock`, `pop`
 - Each person has a unique type of pet: `cat`, `fish`, `dog`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`
 - The people keep unique animals: `cat`, `bird`, `horse`

## Clues:
1. The person whose mother's name is Aniya is not in the first house.
2. The person whose mother's name is Aniya is directly left of The person whose mother's name is Holly.
3. The person whose mother's name is Janelle is Peter.
4. The person who loves pop music is directly left of the person who owns a dog.
5. The person who keeps horses is in the first house.
6. Peter and the cat lover are next to each other.
7. Eric is the person who loves rock music.
8. The person whose mother's name is Janelle and the person who has a cat are next to each other.
 What is the value of attribute Mother for the person whose attribute MusicGenre is classical? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


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`, `Peter`, `Eric`, `Carol`, `Alice`, `Bob`
 - The people are of nationalities: `swede`, `brit`, `chinese`, `dane`, `german`, `norwegian`
 - People have unique favorite book genres: `science fiction`, `mystery`, `fantasy`, `biography`, `historical fiction`, `romance`
 - Everyone has a favorite smoothie: `blueberry`, `dragonfruit`, `desert`, `watermelon`, `cherry`, `lime`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `huawei p50`, `xiaomi mi 11`, `iphone 13`, `google pixel 6`

## Clues:
1. Alice is the British person.
2. There is one house between Eric and the Dragonfruit smoothie lover.
3. The Chinese is somewhere to the left of Alice.
4. The person who loves historical fiction books is directly left of the person who loves biography books.
5. The person who loves biography books is somewhere to the right of Carol.
6. The person who uses an iPhone 13 is somewhere to the right of the Desert smoothie lover.
7. The person who loves mystery books is the British person.
8. Bob is somewhere to the right of the person who uses a OnePlus 9.
9. Carol is the person who drinks Blueberry smoothies.
10. There are two houses between the person who loves romance books and the Watermelon smoothie lover.
11. The person who likes Cherry smoothies is the Swedish person.
12. The person who loves science fiction books is the person who uses a Google Pixel 6.
13. The person who drinks Lime smoothies is Eric.
14. Bob is in the fifth house.
15. Carol is in the second house.
16. The Norwegian and Alice are next to each other.
17. The Swedish person is not in the fifth house.
18. The German and the Norwegian are next to each other.
19. The person who loves fantasy books is the Dragonfruit smoothie lover.
20. The British person is somewhere to the left of Peter.
21. The person who loves romance books is the person who uses a Samsung Galaxy S21.
22. Alice is the person who uses a Xiaomi Mi 11.
23. The person who loves romance books is Eric.
 What is the value of attribute House for the person whose attribute PhoneModel is samsung galaxy s21? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 3 houses, numbered 1 to 3 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`, `Peter`, `Eric`
 - Each person has a unique birthday month: `sept`, `jan`, `april`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`
 - People have unique favorite book genres: `science fiction`, `romance`, `mystery`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Aniya`

## Clues:
1. The person whose birthday is in April is the person who enjoys gardening.
2. The person whose mother's name is Janelle is not in the third house.
3. The photography enthusiast is in the first house.
4. The person whose mother's name is Janelle is the person who loves mystery books.
5. The person who loves science fiction books and Eric are next to each other.
6. Arnold is not in the third house.
7. The person whose mother's name is Holly is the person whose birthday is in September.
8. The person whose birthday is in January is The person whose mother's name is Aniya.
9. The person whose mother's name is Holly is the person who loves science fiction books.
10. The person who loves eating grilled cheese is The person whose mother's name is Aniya.
11. The person who loves the spaghetti eater is Arnold.
 What is the value of attribute Name for the person whose attribute Food is grilled cheese? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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`, `Bob`, `Alice`, `Eric`, `Peter`, `Carol`
 - People have unique favorite sports: `volleyball`, `soccer`, `basketball`, `baseball`, `swimming`, `tennis`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `woodworking`, `knitting`, `photography`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Alice`, `Samantha`, `Fred`, `Timothy`
 - Each person has a unique birthday month: `feb`, `jan`, `mar`, `may`, `sept`, `april`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `pizza`, `spaghetti`, `stew`, `soup`

## Clues:
1. The person who loves soccer is somewhere to the right of Eric.
2. The person who paints as a hobby is somewhere to the left of the woodworking hobbyist.
3. The person whose birthday is in February is not in the third house.
4. The person who is the mother of Timothy is directly left of Arnold.
5. The person who loves cooking is somewhere to the left of the person's child is named Fred.
6. Alice is the person who paints as a hobby.
7. The person's child is named Alice is the person who loves cooking.
8. The person whose birthday is in March is the person's child is named Samantha.
9. The person who loves the soup is the person whose birthday is in May.
10. Carol is somewhere to the right of the person whose birthday is in May.
11. The person whose birthday is in March and the person who is a pizza lover are next to each other.
12. The person whose birthday is in February is not in the fourth house.
13. There are two houses between the person who loves baseball and the person who loves the stew.
14. The person whose birthday is in May is somewhere to the left of the person who loves volleyball.
15. The person whose birthday is in January is somewhere to the right of Arnold.
16. The person who loves cooking is not in the third house.
17. The person who enjoys knitting is the person whose birthday is in September.
18. The person who enjoys gardening is the person's child is named Meredith.
19. The person who is a pizza lover is the person who loves swimming.
20. There is one house between the person who loves basketball and the person whose birthday is in May.
21. The person who loves the soup is directly left of the person who loves the spaghetti eater.
22. The person who loves the stew is the person's child is named Bella.
23. The person who enjoys knitting is somewhere to the left of the person who loves the soup.
24. Bob is somewhere to the left of the person who loves the stew.
25. Arnold is the person who loves stir fry.
26. The woodworking hobbyist is directly left of the person who loves the soup.
 What is the value of attribute Food for the person whose attribute Hobby is gardening? Please reason step by step, and put your final answer within \boxed{}

Answer: stir fry


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: `Alice`, `Arnold`, `Peter`, `Eric`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `prince`, `dunhill`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `basketball`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `craftsman`, `victorian`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `cherry`, `dragonfruit`

## Clues:
1. The Dunhill smoker is in the first house.
2. The person who smokes Blue Master is the Watermelon smoothie lover.
3. The person in a ranch-style home is the person who loves basketball.
4. The person who loves soccer is in the second house.
5. The Dunhill smoker is the person who likes Cherry smoothies.
6. The person who loves swimming is somewhere to the right of the person who smokes Blue Master.
7. The Dragonfruit smoothie lover is Alice.
8. The Prince smoker is the Dragonfruit smoothie lover.
9. The Watermelon smoothie lover is not in the second house.
10. Arnold is the person who smokes Blue Master.
11. The person who loves swimming is the person living in a colonial-style house.
12. The person who loves swimming is Peter.
13. The person in a Craftsman-style house is directly left of the person in a ranch-style home.
 What is the value of attribute HouseStyle for the person whose attribute Cigar is dunhill? Please reason step by step, and put your final answer within \boxed{}

Answer: victorian


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: `Bob`, `Alice`, `Arnold`, `Peter`, `Eric`
 - Each person has a unique level of education: `doctorate`, `bachelor`, `associate`, `master`, `high school`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `google pixel 6`, `iphone 13`, `huawei p50`
 - People have unique favorite music genres: `rock`, `classical`, `jazz`, `pop`, `hip hop`
 - The people keep unique animals: `bird`, `dog`, `horse`, `cat`, `fish`
 - Each person has a unique hobby: `painting`, `cooking`, `knitting`, `gardening`, `photography`

## Clues:
1. The person who loves hip-hop music is in the second house.
2. The person who loves classical music is not in the fifth house.
3. The person who loves classical music is the fish enthusiast.
4. The person who uses an iPhone 13 is the person with a doctorate.
5. The person with a master's degree is not in the second house.
6. The person who enjoys knitting is not in the first house.
7. The person with a high school diploma is somewhere to the left of the photography enthusiast.
8. The person who paints as a hobby is somewhere to the right of the person who loves rock music.
9. Arnold is the bird keeper.
10. The person with a bachelor's degree is the person who uses a Google Pixel 6.
11. The person with a bachelor's degree is somewhere to the right of the person who paints as a hobby.
12. Bob is not in the fourth house.
13. The person who loves cooking is the person who loves classical music.
14. The person who loves cooking is the person who uses a Huawei P50.
15. The person who keeps horses is somewhere to the right of Arnold.
16. The person who paints as a hobby is the person who uses a OnePlus 9.
17. The person who uses a OnePlus 9 is somewhere to the left of the person with an associate's degree.
18. The person who uses an iPhone 13 is the dog owner.
19. The person with a high school diploma is Alice.
20. Peter is the person who enjoys gardening.
21. The person who paints as a hobby is the person who loves jazz music.
 What is the value of attribute Animal for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: fish