There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`, `Bob`
 - Each person has a favorite color: `red`, `green`, `white`, `blue`, `yellow`
 - Each mother is accompanied by their child: `Bella`, `Timothy`, `Samantha`, `Meredith`, `Fred`
 - They all have a unique favorite flower: `tulips`, `daffodils`, `carnations`, `lilies`, `roses`

## Clues:
1. Alice is not in the fifth house.
2. Eric is the person who is the mother of Timothy.
3. Alice and Bob are next to each other.
4. The person who loves blue is not in the fourth house.
5. The person's child is named Samantha is the person who loves the vase of tulips.
6. The person who loves the vase of tulips is Alice.
7. The person's child is named Fred is the person whose favorite color is red.
8. The person's child is named Bella is in the first house.
9. There is one house between the person who loves blue and the person who loves yellow.
10. The person who loves yellow is the person who loves the rose bouquet.
11. The person's child is named Meredith is somewhere to the left of Bob.
12. The person who loves white is not in the second house.
13. The person who loves a carnations arrangement is not in the second house.
14. The person who loves white is Arnold.
15. The person who loves the boquet of lilies is in the first house.
 What is the value of attribute Color for the person whose attribute Children is Timothy? Please reason step by step, and put your final answer within \boxed{}

Answer: yellow


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`, `Eric`, `Peter`, `Carol`, `Bob`, `Arnold`
 - Everyone has a favorite smoothie: `watermelon`, `blueberry`, `desert`, `cherry`, `dragonfruit`, `lime`
 - Everyone has something unique for lunch: `stew`, `pizza`, `grilled cheese`, `stir fry`, `soup`, `spaghetti`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `xiaomi mi 11`, `huawei p50`, `samsung galaxy s21`, `oneplus 9`
 - People own unique car models: `tesla model 3`, `honda civic`, `toyota camry`, `ford f150`, `chevrolet silverado`, `bmw 3 series`
 - Each person lives in a unique style of house: `craftsman`, `ranch`, `modern`, `victorian`, `mediterranean`, `colonial`

## Clues:
1. Bob is the person who uses a Xiaomi Mi 11.
2. The person who loves the soup is in the fourth house.
3. The Dragonfruit smoothie lover is somewhere to the left of the person in a ranch-style home.
4. There is one house between the person who owns a Chevrolet Silverado and the person residing in a Victorian house.
5. The person in a Mediterranean-style villa is the person who drinks Lime smoothies.
6. Eric is in the sixth house.
7. The Desert smoothie lover is the person who is a pizza lover.
8. The person living in a colonial-style house is the person who drinks Blueberry smoothies.
9. The Dragonfruit smoothie lover and the person who uses a Google Pixel 6 are next to each other.
10. The person who loves the soup is Peter.
11. Alice is somewhere to the right of the person who owns a BMW 3 Series.
12. The person who loves stir fry is the person in a ranch-style home.
13. The person who owns a Ford F-150 is the person living in a colonial-style house.
14. The person in a Craftsman-style house is somewhere to the right of the person in a modern-style house.
15. The person who loves the stew is directly left of the person in a ranch-style home.
16. The person who owns a Tesla Model 3 is directly left of the person who loves stir fry.
17. The person who loves eating grilled cheese is the person who owns a Honda Civic.
18. The person in a Mediterranean-style villa is the person who uses a Google Pixel 6.
19. The person in a Craftsman-style house is the Watermelon smoothie lover.
20. The person in a modern-style house is Carol.
21. The person who uses an iPhone 13 is in the first house.
22. The person who owns a Honda Civic is somewhere to the left of the person who owns a BMW 3 Series.
23. The person who uses a Xiaomi Mi 11 is the person who is a pizza lover.
24. The person who uses a Samsung Galaxy S21 is the person who loves stir fry.
25. The person residing in a Victorian house is directly left of the person who uses a OnePlus 9.
 What is the value of attribute Food for the person whose attribute PhoneModel is huawei p50? Please reason step by step, and put your final answer within \boxed{}

Answer: stew


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 House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Alice`, `Eric`
 - People have unique heights: `tall`, `short`, `very short`, `average`
 - Each person has a unique birthday month: `april`, `jan`, `sept`, `feb`
 - People have unique hair colors: `red`, `black`, `brown`, `blonde`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`

## Clues:
1. Eric is the person who has blonde hair.
2. The person who is short is somewhere to the left of the person who has brown hair.
3. Eric is the person who is tall.
4. Arnold is the person who loves fantasy books.
5. The person who loves science fiction books is Eric.
6. The person who is very short is the person who has black hair.
7. The person whose birthday is in April is the person who has brown hair.
8. The person who has blonde hair and the person who has black hair are next to each other.
9. The person who has black hair is the person who loves fantasy books.
10. The person who has red hair is in the first house.
11. The person whose birthday is in September is in the second house.
12. The person whose birthday is in January is the person who has black hair.
13. The person who loves romance books is somewhere to the right of Alice.
 What is the value of attribute Height for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: tall


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

## Clues:
1. The Dragonfruit smoothie lover is in the first house.
2. The person who is a doctor is somewhere to the right of Eric.
3. Eric is the person who likes Cherry smoothies.
4. Peter is directly left of the person who is an artist.
5. The Desert smoothie lover is the person who loves a carnations arrangement.
6. The person who is a teacher is directly left of the Desert smoothie lover.
7. The person who loves the boquet of lilies is directly left of the person who likes Cherry smoothies.
8. The person who is a teacher is the person who loves the rose bouquet.
9. The person who is an engineer is Alice.
 What is the value of attribute Smoothie for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: watermelon


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`, `Arnold`, `Peter`
 - The people keep unique animals: `horse`, `bird`, `dog`, `cat`, `fish`
 - Each person has an occupation: `lawyer`, `teacher`, `doctor`, `engineer`, `artist`

## Clues:
1. The dog owner is not in the first house.
2. The bird keeper is somewhere to the right of the person who is an engineer.
3. Arnold is the fish enthusiast.
4. The bird keeper is directly left of the cat lover.
5. The person who is a teacher is not in the second house.
6. The fish enthusiast is the person who is a lawyer.
7. The person who is an engineer is Alice.
8. The fish enthusiast is in the fourth house.
9. The person who is a doctor is Peter.
10. Bob is the person who is a teacher.
11. Peter is somewhere to the right of Arnold.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


Example 1 Why does the equation $\left|x^{2}-2 x-3\right|=k$ have four distinct real roots when $k$ is a real number?

When $k$ is a real number, the equation $\left|x^{2}-2 x-3\right|=k$ can have four distinct real roots because the expression inside the absolute value, $x^{2}-2 x-3$, is a quadratic equation that can take both positive and negative values. The quadratic equation $x^{2}-2 x-3$ can be factored as $(x-3)(x+1)$, which means it has roots at $x=3$ and $x=-1$. The graph of this quadratic equation is a parabola that opens upwards, with its vertex at the point where $x=1$ (the midpoint of the roots), and the value of the quadratic at this point is $-4$.

When $k>0$, the equation $\left|x^{2}-2 x-3\right|=k$ can be split into two cases:
1. $x^{2}-2 x-3=k$
2. $x^{2}-2 x-3=-k$

For $k>4$, both equations will have two distinct real roots, leading to a total of four distinct real roots. For $0<k<4$, the first equation will have two distinct real roots, and the second equation will also have two distinct real roots, again leading to a total of four distinct real roots. When $k=4$, the equation $x^{2}-2 x-3=-4$ has a double root at $x=1$, and the equation $x^{2}-2 x-3=4$ has two distinct real roots, leading to a total of three distinct real roots. For $k=0$, the equation $\left|x^{2}-2 x-3\right|=0$ has exactly two distinct real roots, which are the roots of the quadratic equation $x^{2}-2 x-3=0$. For $k<0$, there are no real roots because the absolute value of a real number cannot be negative.

Answer: 0 < k < 4


Problem 3. Excellent students Alyosha and Vasya are listing four-digit numbers. Alyosha lists numbers where the first digit is equal to the product of the other three, while Vasya lists numbers where the last digit is equal to the product of the other three. Who will list more numbers and by how many?

Answer: 171


B. As shown in Figure $5, \odot O$ has a radius of $20, A$ is a point on $\odot O$. A rectangle $O B A C$ is constructed with $O A$ as the diagonal, and $O C=$ 12. Extend $B C$ to intersect $\odot O$ at points $D$ and $E$, then $C E-B D$ $=$ . $\qquad$

Answer: \dfrac{28}{5}


1. In one year, January 1 and April 1 fell on a Thursday. How many months in that year had 5 Fridays?

Answer: 5


7. Given a tetrahedron $ABCD$ with the lengths of its 6 edges being $4, 7, 20, 22, 28, x$, then the minimum possible value of $[x]$ is $\qquad$ (here, $[x]$ denotes the greatest integer less than or equal to $x$).

Answer: 8


11. (20 points) It is known that a box contains 100 red and 100 blue cards, each color of cards containing one card labeled with each of the numbers $1, 3, 3^2, \cdots, 3^{99}$. The total sum of the numbers on the cards of both colors is denoted as $s$.

For a given positive integer $n$, if it is possible to pick several cards from the box such that the sum of their labels is exactly $n$, then it is called a "scheme for $n$". The number of different schemes for $n$ is denoted as $f(n)$. Try to find the value of $f(1) + f(2) + \cdots + f(s)$.

Answer: 4^{100} - 1


9. (5 points) A cylindrical container with a depth of 30 cm, an outer diameter of 22 cm, and a wall thickness of 1 cm, is already filled with water to a depth of 27.5 cm. Now, a conical iron block with a base diameter of 10 cm and a height of 30 cm is placed inside. Then, $\qquad$ cubic centimeters of water will overflow.

Answer: 0


A natural number $n \geq 2$ is given. Consider all colorings of the cells of an $n \times n$ board in $k$ colors, such that each cell is painted in exactly one color and all $k$ colors are used. What is the smallest $k$ for which there will always be four cells painted in four different colors, situated at the intersections of two rows and two columns in any such coloring?

Answer: 2n


(7) After the math competition, Xiao Ming, Xiao Le, and Xiao Qiang each won a medal, with one getting the gold, one the silver, and one the bronze. The teacher guessed: "Xiao Ming got the gold, Xiao Le did not get the gold, Xiao Qiang did not get the bronze." It turned out that the teacher was only right about one of these guesses. From this, we can deduce: The ones who got the gold, silver, and bronze medals are, in order,

Answer: Xiao Le, Xiao Qiang, Xiao Ming


In the figure below, parallelograms $A B C D$ and $B F E C$ have areas $1234 \mathrm{~cm}^{2}$ and $2804 \mathrm{~cm}^{2}$, respectively. Points $M$ and $N$ are chosen on sides $A D$ and $F E$, respectively, so that segment $M N$ passes through $B$. Find the area of $\triangle M N C$.

![](https://cdn.mathpix.com/cropped/2024_04_17_3170889cf6798dffe43cg-2.jpg?height=721&width=653&top_left_y=775&top_left_x=736)

#

Answer: 2019


14. Now there are 4 containers weighing 3 tons each, 5 containers weighing 2.5 tons each, 14 containers weighing 1.5 tons each, and 7 containers weighing 1 ton each. How many trucks with a load capacity of 4.5 tons are needed at minimum to transport all the containers in one trip?

Answer: 12


$$
\begin{array}{l}
\text { 2 For a row consisting of } n \text { A's, } n \text { B's, and } n \text { C's, define a new row below it (one letter shorter) such that if the two letters above it are different, write the third letter, and if they are the same, write that letter. Repeat this operation on the newly obtained row until only one letter remains. The table below gives an example for } n=2. \\
\text { B A A A C } \\
\text { C A A B } \\
\text { B A C } \\
\text { C B } \\
\text { A } \\
\end{array}
$$

Find all positive integers \( n \) such that for any initial arrangement, after the above operation, the three vertices of the resulting large triangle are either all the same or all different.

Answer: 1


1. There were 10 chatterboxes sitting in a circle. At first, one of them told one joke, the next one clockwise told two jokes, the next one told three, and so on around the circle, until one of them told 100 jokes at once. At this point, the chatterboxes got tired, and the next one clockwise told 99 jokes, the next one told 98, and so on around the circle, until one of them told just one joke, and everyone dispersed. How many jokes did each of these 10 chatterboxes tell in total?

Answer: 1000


## 21. Rue Sainte-Catherine

I was walking with my mother-in-law. We were slowly, at a speed of 3 km/h, walking down Rue Sainte-Catherine in Bordeaux, which, as everyone knows (or maybe doesn't know), is straight. Suddenly, I remembered that I needed to drop a letter in a mailbox located a bit further down the street. Leaving my mother-in-law, who continued her walk calmly, I hurried ahead, and after dropping the letter, I walked back to meet her at the same speed (5 km/h). I clearly remember that I left her alone for only 3 minutes.

How far from the mailbox were we when we parted?

Answer: 0.2


Task 2. Let $n \geq 4$ be an integer. Consider a regular $2 n$-gon where each vertex is assigned an integer, which we call the value of that vertex. If four distinct vertices of this $2 n$-gon form a rectangle, then the sum of the values of these vertices is called a rectangle sum.

Determine for which integers (not necessarily positive) $m$ we can assign the numbers $m+1, m+2$, ..., $m+2 n$ as values to the vertices (in some order) so that every rectangle sum is a prime number. (Prime numbers are by definition positive.)

Answer: -n


Schön D.…

On the island of knights and liars, a traveler came to visit his acquaintance, a knight, and saw him at a round table with five guests.

- I wonder, how many of you are knights? - he asked.
- Why don't you ask each of us a question and find out yourself, - one of the guests suggested.
- Alright. Tell me each: who are your neighbors? - the traveler asked.

To this question, everyone answered the same.

- Not enough data! - said the traveler.
- But today is my birthday, don't forget about that, - said one of the guests.
- Yes, today is his birthday! - said his neighbor.

And the traveler was able to figure out how many knights were at the table. Indeed, how many of them are there?

Answer: 2


On a smooth table, there are three wooden balls with radii of $1, 2, 4$ respectively, all externally tangent to each other. Additionally, there is another small wooden ball on the table with a radius less than 1, which is tangent to the table and externally tangent to all three balls. Find the radius of this small wooden ball.

Answer: \dfrac{14 - 2\sqrt{21}}{7}


Problem 5. Mom baked pies - three with rice, three with cabbage, and one with cherries - and laid them out in a circle on a plate (see fig.). Then she put the plate in the microwave to warm them up. On the outside, all the pies look the same.

![](https://cdn.mathpix.com/cropped/2024_05_06_03d5e2dee9da9ea28fdcg-04.jpg?height=307&width=311&top_left_y=1516&top_left_x=951)
Masha knows how they were arranged, but she doesn't know how the plate was rotated. She wants to eat the cherry pie, considering the others to be unappetizing. How can Masha definitely achieve this, biting into as few unappetizing pies as possible

[7 points] (A. V. Khachaturyan)

![](https://cdn.mathpix.com/cropped/2024_05_06_03d5e2dee9da9ea28fdcg-05.jpg?height=736&width=917&top_left_y=395&top_left_x=266)

Answer: 2


8. Find the integer pairs $(x, y)$ that satisfy the condition $x^{4}+x^{3}+x^{2}+x=y^{2}+y$.

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

Answer: (-1, -1)


\section*{Exercise 1 - 041211}

From a four-digit table, we obtain the following approximate values:

\[
\sqrt[3]{636000} \approx 86.00 \text { and } \sqrt[3]{389000} \approx 73.00
\]

Therefore, \(z=\sqrt[3]{636000}-\sqrt[3]{389000} \approx 13\).

Without using any further table, it should be decided whether \(z\) is greater, smaller, or equal to 13.

Answer: z < 13


8. [30] Consider the following two-player game. Player 1 starts with a number, $N$. He then subtracts a proper divisor of $N$ from $N$ and gives the result to player 2 (a proper divisor of $N$ is a positive divisor of $N$ that is not equal to 1 or $N$ ). Player 2 does the same thing with the number she gets from player 1 , and gives the result back to player 1. The two players continue until a player is given a prime number, at which point that player loses. For how many values of $N$ between 2 and 100 inclusive does player 1 have a winning strategy?

Answer: 47


Example 2. Approximately calculate the increase in the volume of a cylinder with height $H=40 \mathrm{~cm}$ and base radius $R=30 \mathrm{~cm}$ when the radius is increased by $0.5 \mathrm{~cm}$.

Answer: 1200\pi


Let $r$ and $s$ be positive real numbers, and let $A=(0,0)$, $B=(1,0)$, $C=(r,s)$, and $D=(r+1,s)$ be points on the coordinate plane. Find a point $P$, whose coordinates are expressed in terms of $r$ and $s$, with the following property: if $E$ is any point on the interior of line segment $AB$, and $F$ is the unique point other than $E$ that lies on the circumcircles of triangles $BCE$ and $ADE$, then $P$ lies on line $\overleftrightarrow{EF}$.

Answer: \left( \dfrac{3r + 1}{2}, \dfrac{s}{2} \right)


## 9. Rhombi

Rhombus $A_{0} B_{0} C_{0} D_{0}$ is located inside rhombus $A B C D$ such that their sides are parallel, and their centers are at the same point. The segments $\overline{A A_{0}}, \overline{B B_{0}}, \overline{C C_{0}}$, and $\overline{D D_{0}}$ are divided into five congruent parts. The resulting points are connected by segments parallel to the sides of the given rhombi, thus forming four new rhombi.

The area of rhombus $A B C D$ is $2250 \mathrm{~cm}^{2}$, and the area of rhombus $A_{0} B_{0} C_{0} D_{0}$ is $810 \mathrm{~cm}^{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_30_12924d306592e3a90c60g-08.jpg?height=505&width=733&top_left_y=701&top_left_x=1164)

What is the total colored area in the figure?

## Result: $\quad 864$

Answer: 864


5. On the side $\overline{A B}$ of the rectangle $A B C D$, several points have been chosen, on the sides $\overline{B C}$ and $\overline{D A}$ four points each, and on the side $\overline{C D}$ five points. None of the chosen points is a vertex of the given rectangle. Using the chosen points as vertices, exactly 4040 quadrilaterals can be drawn such that at most one vertex lies on the segment $\overline{A B}$. How many points were chosen on the side $\overline{A B}$?

Answer: 13


9. The number on a certificate of honor is a ten-digit number, with some of the digits written in the boxes below. It is known that the product of every three consecutive digits is 24. What is this ten-digit number?
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline & & & & $\mathbf{2}$ & & & & $\mathbf{3}$ & \\
\hline
\end{tabular}

Answer: 4234234234


【Question 14】Select 5 digits from $1 \sim 9$, to form a five-digit number, such that this five-digit number can be evenly divided by any of the 5 selected digits, but cannot be evenly divided by any of the 4 unselected digits. Then, the smallest value of this five-digit number is $\qquad$.

---

Note: The blank at the end of the question is intended for the answer to be filled in.

Answer: 14728


Several merchants contributed to a common business venture an amount of rubles that was 100 times the number of merchants. They sent a trustee to Venice who received from each hundred rubles a number of rubles that was twice the number of merchants. The question is: how many merchants were there if the trustee received 2662 rubles?

Answer: 11


7. There are a total of 10040 numbers written on the blackboard, including 2006 ones, 2007 twos, 2008 threes, 2009 fours, and 2010 fives. Each operation involves erasing 4 different numbers and writing the fifth type of number (for example, erasing 1, 2, 3, and 4 each once and writing one 5; or erasing 2, 3, 4, and 5 each once and writing one 1, etc.). If after a finite number of operations, there are exactly two numbers left on the blackboard, what is the product of these two numbers?

Answer: 8


7. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right]\left(n \in \mathbf{N}^{*}\right)$, where $[x]$ denotes the greatest integer not exceeding the real number $x$. Let $C$ be a real number, and for any positive integer $n$, we have $\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C$, then the minimum value of $C$ is $\qquad$

Answer: \dfrac{1}{288}


3. Option 1.

In the Ivanov family, both the mother and the father, and their three children, were born on April 1st. When the first child was born, the parents' combined age was 45 years. The third child in the family was born a year ago, when the sum of the ages of all family members was 70 years. How old is the middle child now, if the sum of the ages of the children is 14 years?

Answer: 5


Let $ABC$ be a triangle with $\angle BAC = 60^\circ, BA = 2$, and $CA = 3$. A point $M$ is located inside $ABC$ such that $MB = 1$ and $MC = 2$. A semicircle tangent to $MB$ and $MC$ has its center $O$ on $BC$. Let $P$ be the intersection of the angle bisector of $\angle BAC$ and the perpendicular bisector of $AC$. If the ratio $OP/MO$ is $a/b$, where $a$ and $b$ are positive integers and $\gcd(a, b) = 1$, find $a + b$.

Answer: 3


$5$ people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that $\frac{m}{n}$ is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

Answer: 137


Three circles, with radii of $1, 1$, and $2$, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to all three circles can be written as $a + b\sqrt{c}$ where $c$ is not divisible by any perfect square larger than $1$. Find $a + b + c$

Answer: 22


Cyclic quadrilateral $ABCD$ has side lengths $AB = 2, BC = 3, CD = 5, AD = 4$.
Find $\sin A \sin B(\cot A/2 + \cot B/2 + \cot C/2 + \cot D/2)^2$.
Your answer can be written in simplest form as $a/b$. Find $a + b$.

Answer: 5101


Two white pentagonal pyramids, with side lengths all the same, are glued to each other at their regular pentagon bases. Some of the resulting $10$ faces are colored black. How many rotationally distinguishable colorings may result?

Answer: 120


Let f be a polynomial of degree $3$ with integer coefficients such that $f(0) = 3$ and $f(1) = 11$.
If f has exactly $2$ integer roots, how many such polynomials $f$ exist?

Answer: 0


Six ants are placed on the vertices of a regular hexagon with an area of $12$. At each point in time, each ant looks at the next ant in the hexagon (in counterclockwise order), and measures the distance, $s$, to the next ant. Each ant then proceeds towards the next ant at a speed of $\frac{s}{100}$ units per year. After T years, the ants’ new positions are the vertices of a new hexagon with an area of $4$. T is of the form $a \ln b$, where $b$ is square-free. Find $a + b$.

Answer: 103


Let $\vartriangle ABC$ be a triangle with $\angle BAC = 45^o, \angle BCA = 30^o$, and $AB = 1$. Point $D$ lies on segment $AC$ such that $AB = BD$. Find the square of the length of the common external tangent to the circumcircles of triangles $\vartriangle BDC$ and $\vartriangle ABC$.

Answer: 2


Let $p_1 = 2012$ and $p_n = 2012^{p_{n-1}}$ for $n > 1$. Find the largest integer $k$ such that $p_{2012}- p_{2011}$ is divisible by $2011^k$.

Answer: 2012


Let an be a sequence such that $a_0 = 0$ and:
$a_{3n+1} = a_{3n} + 1 = a_n + 1$
$a_{3n+2} = a_{3n} + 2 = a_n + 2$
for all natural numbers $n$. How many $n$ less than $2012$ have the property that $a_n = 7$?

Answer: 379


A square is inscribed in an ellipse such that two sides of the square respectively pass through the two foci of the ellipse. The square has a side length of $4$. The square of the length of the minor axis of the ellipse can be written in the form $a + b\sqrt{c}$ where $a, b$, and $c$ are integers, and $c$ is not divisible by the square of any prime. Find the sum $a + b + c$.

Answer: 21


Proctors Andy and Kristin have a PUMaC team of eight students labelled $s_1, s_2, ... , s_8$ (the PUMaC staff being awful with names). The following occurs:
$1$. Andy tells the students to arrange themselves in a line in arbitrary order.
$2$. Kristin tells each student $s_i$ to move to the current spot of student $s_j$ , where $j \equiv 3i + 1$ (mod $8$).
$3$. Andy tells each student $s_i$ to move to the current spot of the student who was in the $i$th position of the line after step $1$.
How many possible orders can the students be in now?

Answer: 1260


A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$.  Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$.  Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges.  What is the minimal number of cycles possible in $G$?

Answer: 6006


(Following question 1) Now instead consider an infinite strip of squares, labeled with the integers $0, 1, 2, \ldots$ in that order. You start at the square labeled $0$. You want to end up at the square labeled $3$. In how many ways can this be done in exactly $15$ moves?

Answer: 2002


Let $k$ be a positive integer with the following property: For every subset $A$ of $\{1,2,\ldots, 25\}$ with $|A|=k$, we can find distinct elements $x$ and $y$ of $A$ such that $\tfrac23\leq\tfrac xy\leq\tfrac 32$.  Find the smallest possible value of $k$.

Answer: 7


In the figure, segment \(DE\) divides square sheet \(ABCD\) into \(\triangle ADE\) and quadrilateral \(EDCB\). Given that the area ratio \(S_{\triangle ADE} : S_{EDCB} = 5 : 19\), determine the ratio of the perimeter of \(\triangle ADE\) to the perimeter of quadrilateral \(EDCB\).

Answer: \frac{15}{22}


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 HouseStyle for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: ranch


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Arnold`, `Alice`, `Bob`, `Eric`, `Peter`
 - Each person has an occupation: `artist`, `doctor`, `engineer`, `teacher`, `nurse`, `lawyer`
 - People have unique favorite book genres: `biography`, `mystery`, `romance`, `fantasy`, `science fiction`, `historical fiction`

## Clues:
1. The person who is an artist is somewhere to the left of the person who loves science fiction books.
2. The person who is an artist is somewhere to the left of Eric.
3. The person who is an artist is in the fourth house.
4. Eric is not in the sixth house.
5. Bob is the person who is a doctor.
6. The person who is an engineer is the person who loves biography books.
7. Peter is somewhere to the right of the person who is a nurse.
8. The person who loves science fiction books is Bob.
9. Arnold is not in the second house.
10. The person who is a teacher is Carol.
11. The person who is an artist is somewhere to the right of the person who loves fantasy books.
12. There are two houses between the person who loves romance books and Eric.
13. Eric is the person who loves historical fiction books.
14. There is one house between the person who is a teacher and Alice.
 What is the value of attribute BookGenre for the person whose attribute Occupation is doctor? Please reason step by step, and put your final answer within \boxed{}

Answer: science fiction


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`, `Arnold`, `Peter`, `Alice`, `Eric`
 - Each person has a unique favorite drink: `milk`, `root beer`, `coffee`, `water`, `tea`
 - Each person has a unique type of pet: `dog`, `bird`, `hamster`, `cat`, `fish`
 - Each mother is accompanied by their child: `Timothy`, `Meredith`, `Fred`, `Bella`, `Samantha`
 - The mothers' names in different houses are unique: `Kailyn`, `Penny`, `Aniya`, `Janelle`, `Holly`
 - People have unique favorite sports: `tennis`, `soccer`, `swimming`, `baseball`, `basketball`

## Clues:
1. The tea drinker is the person who loves tennis.
2. The person whose mother's name is Aniya is Arnold.
3. The person whose mother's name is Janelle is somewhere to the right of the person who loves swimming.
4. The person whose mother's name is Janelle is the person's child is named Bella.
5. The person's child is named Samantha is the person who likes milk.
6. The coffee drinker is not in the fifth house.
7. Bob is the person's child is named Bella.
8. The person who loves swimming and the person who loves basketball are next to each other.
9. The person who is the mother of Timothy is somewhere to the left of the person who owns a dog.
10. The person whose mother's name is Kailyn is in the fifth house.
11. The root beer lover is Alice.
12. The person whose mother's name is Holly is directly left of the root beer lover.
13. The person who likes milk is not in the second house.
14. There is one house between the person who has a cat and the person who loves baseball.
15. The person whose mother's name is Penny is somewhere to the right of the person who loves tennis.
16. The person's child is named Fred is the person who keeps a pet bird.
17. The person who likes milk is directly left of the person with a pet hamster.
18. The person who keeps a pet bird is Alice.
19. There are two houses between Eric and The person whose mother's name is Aniya.
 What is the value of attribute Children for the person whose attribute Mother is Holly? Please reason step by step, and put your final answer within \boxed{}

Answer: Timothy


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`, `Eric`, `Alice`, `Bob`, `Arnold`, `Carol`
 - Everyone has a unique favorite cigar: `yellow monster`, `blends`, `prince`, `dunhill`, `pall mall`, `blue master`
 - People use unique phone models: `xiaomi mi 11`, `iphone 13`, `google pixel 6`, `huawei p50`, `samsung galaxy s21`, `oneplus 9`
 - People own unique car models: `bmw 3 series`, `honda civic`, `tesla model 3`, `ford f150`, `chevrolet silverado`, `toyota camry`
 - The people are of nationalities: `dane`, `brit`, `chinese`, `swede`, `norwegian`, `german`

## Clues:
1. The person who uses a Xiaomi Mi 11 is somewhere to the left of the person who uses a Google Pixel 6.
2. Carol is somewhere to the left of the person who owns a Ford F-150.
3. Peter is not in the second house.
4. There is one house between the Dane and the person who uses a Google Pixel 6.
5. The Chinese is the person who uses a Google Pixel 6.
6. The person who smokes many unique blends is somewhere to the right of the person who owns a Honda Civic.
7. The person who owns a Tesla Model 3 is in the fourth house.
8. The person who uses a Samsung Galaxy S21 is Alice.
9. There is one house between the person who uses a Xiaomi Mi 11 and the person who smokes Blue Master.
10. The person partial to Pall Mall is directly left of the person who uses a Samsung Galaxy S21.
11. The person who smokes many unique blends is the German.
12. The person who owns a Chevrolet Silverado is the Prince smoker.
13. The person who owns a BMW 3 Series is Eric.
14. The person who owns a Ford F-150 is directly left of Bob.
15. The person who smokes Yellow Monster is not in the second house.
16. The Swedish person is somewhere to the right of the person who smokes Yellow Monster.
17. The person who owns a Chevrolet Silverado is directly left of the person who uses a Huawei P50.
18. The person who smokes Blue Master is in the fifth house.
19. The person who smokes Blue Master is directly left of the person who uses a OnePlus 9.
20. The Norwegian is the person who owns a Toyota Camry.
 What is the value of attribute Name for the person whose attribute CarModel is tesla model 3? 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: `Bob`, `Peter`, `Arnold`, `Eric`, `Alice`, `Carol`
 - People have unique hair colors: `red`, `gray`, `auburn`, `blonde`, `brown`, `black`
 - People have unique favorite book genres: `historical fiction`, `biography`, `romance`, `fantasy`, `mystery`, `science fiction`
 - People have unique heights: `very short`, `very tall`, `tall`, `short`, `super tall`, `average`
 - Each person has a unique level of education: `associate`, `doctorate`, `high school`, `trade school`, `bachelor`, `master`
 - Each person has a unique favorite drink: `boba tea`, `milk`, `water`, `root beer`, `coffee`, `tea`

## Clues:
1. The person with a doctorate and Bob are next to each other.
2. There is one house between Carol and the person who has black hair.
3. Peter is the person with a doctorate.
4. The person who loves science fiction books is Eric.
5. The one who only drinks water is the person who has an average height.
6. Alice is the person who loves mystery books.
7. The person who loves romance books is Bob.
8. The person who has brown hair is in the second house.
9. The person who has an average height is in the fourth house.
10. The boba tea drinker is the person who is very tall.
11. The person with a high school diploma is Bob.
12. The person who is short is somewhere to the left of the person who is super tall.
13. The person who loves historical fiction books and the person with a bachelor's degree are next to each other.
14. The person who attended trade school is the person who has red hair.
15. Carol is the person who has auburn hair.
16. The person who is very short is somewhere to the right of the person who has gray hair.
17. There are two houses between Eric and the person who loves fantasy books.
18. There are two houses between the tea drinker and the person who is very short.
19. Eric is somewhere to the left of the person with a bachelor's degree.
20. The root beer lover is Alice.
21. The person with an associate's degree is Arnold.
22. The person who is tall is not in the second house.
23. The person who loves science fiction books is the boba tea drinker.
24. The person who has black hair is not in the fourth house.
25. The person who is short is the coffee drinker.
26. The coffee drinker and the person with a high school diploma are next to each other.
27. The coffee drinker is not in the second house.
 What is the value of attribute Drink for the person whose attribute HairColor is auburn? Please reason step by step, and put your final answer within \boxed{}

Answer: water


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: `Bob`, `Eric`, `Arnold`, `Carol`, `Alice`, `Peter`
 - Each person has a unique hobby: `cooking`, `painting`, `knitting`, `photography`, `woodworking`, `gardening`
 - People have unique hair colors: `brown`, `auburn`, `black`, `red`, `gray`, `blonde`
 - People have unique favorite sports: `swimming`, `volleyball`, `basketball`, `baseball`, `tennis`, `soccer`
 - The mothers' names in different houses are unique: `Sarah`, `Holly`, `Penny`, `Janelle`, `Kailyn`, `Aniya`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `pall mall`, `prince`, `yellow monster`, `blue master`

## Clues:
1. The person who has gray hair is directly left of The person whose mother's name is Janelle.
2. The person who loves soccer is in the sixth house.
3. Arnold is the person who has blonde hair.
4. The photography enthusiast is the person who smokes Yellow Monster.
5. There is one house between the Dunhill smoker and the person who loves cooking.
6. The person who paints as a hobby is the person who has blonde hair.
7. The Dunhill smoker is directly left of Alice.
8. The person who paints as a hobby is directly left of the person who loves volleyball.
9. The person who loves soccer is The person whose mother's name is Aniya.
10. The person partial to Pall Mall is somewhere to the left of Carol.
11. The person who enjoys knitting and the person who loves basketball are next to each other.
12. The person partial to Pall Mall is directly left of The person whose mother's name is Holly.
13. The person who loves soccer is the Prince smoker.
14. The person who loves swimming is The person whose mother's name is Penny.
15. The person whose mother's name is Kailyn is the person who loves volleyball.
16. The person who loves basketball is The person whose mother's name is Janelle.
17. The person who enjoys knitting is the person who loves volleyball.
18. The person who enjoys knitting is not in the third house.
19. The person who has auburn hair is somewhere to the left of the photography enthusiast.
20. The person whose mother's name is Sarah is not in the fifth house.
21. Bob is directly left of the person who has red hair.
22. The Dunhill smoker is the person who enjoys gardening.
23. The person who loves baseball is the person who has auburn hair.
24. Eric is the person who smokes many unique blends.
25. The person who has brown hair is somewhere to the right of The person whose mother's name is Penny.
 What is the value of attribute House for the person whose attribute FavoriteSport is swimming? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Arnold`, `Carol`, `Eric`, `Peter`
 - People use unique phone models: `xiaomi mi 11`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`, `huawei p50`, `iphone 13`
 - People have unique favorite book genres: `romance`, `historical fiction`, `fantasy`, `mystery`, `science fiction`, `biography`
 - They all have a unique favorite flower: `tulips`, `roses`, `carnations`, `daffodils`, `iris`, `lilies`
 - The people keep unique animals: `dog`, `cat`, `bird`, `fish`, `rabbit`, `horse`
 - Each person has a unique level of education: `trade school`, `high school`, `master`, `associate`, `bachelor`, `doctorate`

## Clues:
1. The bird keeper is the person who uses a Google Pixel 6.
2. The person who loves a bouquet of daffodils is the person with an associate's degree.
3. The person who uses a OnePlus 9 is not in the fourth house.
4. There are two houses between the person with a master's degree and the rabbit owner.
5. The person who loves a carnations arrangement is Arnold.
6. The person who loves the boquet of lilies is not in the sixth house.
7. The person who loves the vase of tulips is Carol.
8. The person with a high school diploma is Bob.
9. The person who attended trade school is somewhere to the left of the person who keeps horses.
10. The person who loves mystery books is Eric.
11. The fish enthusiast is directly left of Carol.
12. Bob is the fish enthusiast.
13. Eric is the dog owner.
14. The person who loves a carnations arrangement is the person who uses a Google Pixel 6.
15. The person who loves fantasy books is in the second house.
16. The person who uses a Samsung Galaxy S21 is the person who loves the boquet of iris.
17. Alice is the person with an associate's degree.
18. The person who uses a Google Pixel 6 is not in the second house.
19. The dog owner is the person with a doctorate.
20. The fish enthusiast is the person who uses a Samsung Galaxy S21.
21. The person who uses an iPhone 13 is somewhere to the right of the person who loves fantasy books.
22. The person who loves mystery books is somewhere to the right of the person who loves the boquet of iris.
23. Alice is the person who uses an iPhone 13.
24. The person who attended trade school is not in the fourth house.
25. The person who loves biography books is directly left of Eric.
26. There is one house between the rabbit owner and the person who loves historical fiction books.
27. The person who loves the boquet of iris is somewhere to the right of the person who uses a Xiaomi Mi 11.
28. The person who loves science fiction books is the person who uses a Samsung Galaxy S21.
29. The person who loves the boquet of lilies is not in the second house.
 What is the value of attribute PhoneModel for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: google pixel 6


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`, `Eric`, `Arnold`, `Bob`, `Alice`, `Carol`
 - The people keep unique animals: `rabbit`, `dog`, `fish`, `bird`, `cat`, `horse`
 - Each person has a unique hobby: `cooking`, `knitting`, `painting`, `woodworking`, `photography`, `gardening`
 - People have unique favorite sports: `basketball`, `soccer`, `swimming`, `tennis`, `baseball`, `volleyball`
 - Each person prefers a unique type of vacation: `cruise`, `cultural`, `city`, `camping`, `mountain`, `beach`
 - Each person has a unique favorite drink: `water`, `root beer`, `boba tea`, `tea`, `milk`, `coffee`

## Clues:
1. The person who enjoys camping trips is the person who loves cooking.
2. The person who loves baseball is the photography enthusiast.
3. The person who loves volleyball is somewhere to the left of the person who prefers city breaks.
4. Carol is the person who loves tennis.
5. The dog owner is the person who loves swimming.
6. The photography enthusiast is the tea drinker.
7. The person who loves baseball is directly left of the person who goes on cultural tours.
8. Peter is the person who enjoys camping trips.
9. The person who enjoys knitting is the person who keeps horses.
10. The coffee drinker is directly left of the person who loves soccer.
11. The person who enjoys mountain retreats is the person who loves soccer.
12. The boba tea drinker is not in the fourth house.
13. The cat lover is somewhere to the left of the dog owner.
14. The person who loves beach vacations is Bob.
15. The coffee drinker is somewhere to the right of Alice.
16. The tea drinker is the fish enthusiast.
17. The person who likes milk is Carol.
18. There is one house between the rabbit owner and the woodworking hobbyist.
19. The person who loves cooking is not in the third house.
20. The person who loves tennis is the rabbit owner.
21. The dog owner is directly left of the person who likes going on cruises.
22. The woodworking hobbyist is the dog owner.
23. There are two houses between the person who paints as a hobby and the person who keeps horses.
24. Arnold is the root beer lover.
 What is the value of attribute Drink for the person whose attribute FavoriteSport is soccer? Please reason step by step, and put your final answer within \boxed{}

Answer: root beer


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`, `Arnold`, `Carol`, `Bob`, `Alice`, `Peter`
 - Each person prefers a unique type of vacation: `mountain`, `camping`, `beach`, `cultural`, `city`, `cruise`
 - Everyone has something unique for lunch: `spaghetti`, `soup`, `stir fry`, `stew`, `grilled cheese`, `pizza`
 - Each person has a unique favorite drink: `boba tea`, `root beer`, `water`, `milk`, `coffee`, `tea`
 - People have unique heights: `tall`, `average`, `very tall`, `short`, `very short`, `super tall`

## Clues:
1. The person who enjoys mountain retreats and the person who prefers city breaks are next to each other.
2. Alice is in the fourth house.
3. The coffee drinker is the person who is a pizza lover.
4. The person who is very tall is directly left of the person who enjoys mountain retreats.
5. Peter is not in the first house.
6. The boba tea drinker is in the fourth house.
7. The person who loves stir fry is somewhere to the left of Eric.
8. The person who likes milk is Bob.
9. The person who loves the soup is not in the second house.
10. The person who loves the spaghetti eater is not in the second house.
11. The person who loves the stew is Eric.
12. The root beer lover is the person who is short.
13. The person who loves beach vacations is in the first house.
14. The person who loves stir fry is the person who enjoys mountain retreats.
15. The person who loves the soup is somewhere to the left of the person who loves the spaghetti eater.
16. The boba tea drinker is somewhere to the left of the person who is a pizza lover.
17. Arnold is the person who likes going on cruises.
18. The person who is very short is somewhere to the right of the person who has an average height.
19. The person who is tall is not in the fourth house.
20. The person who prefers city breaks is the coffee drinker.
21. Bob and the one who only drinks water are next to each other.
22. The person who loves the stew is the person who enjoys camping trips.
23. Arnold is the person who is very short.
24. Carol is in the fifth house.
 What is the value of attribute Height for the person whose attribute Drink is root beer? Please reason step by step, and put your final answer within \boxed{}

Answer: short


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: `Bob`, `Carol`, `Peter`, `Alice`, `Eric`, `Arnold`
 - They all have a unique favorite flower: `daffodils`, `iris`, `lilies`, `roses`, `tulips`, `carnations`
 - People have unique hair colors: `gray`, `red`, `blonde`, `auburn`, `brown`, `black`
 - Each person has a favorite color: `white`, `blue`, `green`, `purple`, `red`, `yellow`
 - People own unique car models: `tesla model 3`, `ford f150`, `honda civic`, `toyota camry`, `chevrolet silverado`, `bmw 3 series`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Kailyn`, `Sarah`, `Penny`, `Aniya`

## Clues:
1. There is one house between Arnold and the person who loves blue.
2. The person whose mother's name is Janelle is directly left of the person who has auburn hair.
3. The person who owns a Chevrolet Silverado and the person who owns a BMW 3 Series are next to each other.
4. Carol is the person who loves a carnations arrangement.
5. The person who loves the boquet of iris is the person who has gray hair.
6. The person who loves the rose bouquet is The person whose mother's name is Kailyn.
7. Bob is the person who loves the boquet of lilies.
8. The person who loves a bouquet of daffodils is somewhere to the left of the person who owns a Chevrolet Silverado.
9. Bob is in the third house.
10. The person who owns a Honda Civic is in the sixth house.
11. The person who loves the vase of tulips is somewhere to the left of The person whose mother's name is Penny.
12. The person who has brown hair is in the fourth house.
13. The person who has brown hair is somewhere to the left of the person who has blonde hair.
14. The person whose mother's name is Holly is Bob.
15. The person who owns a Ford F-150 is in the third house.
16. Eric is directly left of the person who has auburn hair.
17. The person who owns a Tesla Model 3 is the person who loves yellow.
18. There are two houses between the person who loves purple and The person whose mother's name is Aniya.
19. The person who loves purple is in the first house.
20. The person whose favorite color is green is not in the fourth house.
21. The person who has gray hair is the person who owns a BMW 3 Series.
22. The person who has gray hair is somewhere to the left of Alice.
23. The person whose mother's name is Holly is somewhere to the left of the person who loves the vase of tulips.
24. The person who has red hair is the person who loves white.
 What is the value of attribute House for the person whose attribute Flower is lilies? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


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`
 - People own unique car models: `honda civic`, `tesla model 3`, `ford f150`, `toyota camry`
 - Each person has a unique type of pet: `bird`, `cat`, `fish`, `dog`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `google pixel 6`, `samsung galaxy s21`

## Clues:
1. The person who uses an iPhone 13 is the person who owns a dog.
2. Alice is not in the second house.
3. The person who uses an iPhone 13 is the person who owns a Toyota Camry.
4. The person who uses a Google Pixel 6 is not in the fourth house.
5. The person who owns a Ford F-150 is not in the second house.
6. The person with an aquarium of fish is directly left of the person who keeps a pet bird.
7. Peter is the person who uses a Google Pixel 6.
8. The person who owns a Tesla Model 3 is directly left of the person who uses an iPhone 13.
9. Eric is not in the second house.
10. Eric is not in the first house.
11. The person who uses a Google Pixel 6 and the person who owns a Ford F-150 are next to each other.
12. The person who uses a OnePlus 9 is somewhere to the left of the person who owns a Ford F-150.
 What is the value of attribute CarModel for the person whose attribute Pet is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: ford f150


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: `Bob`, `Carol`, `Alice`, `Peter`, `Arnold`, `Eric`
 - People have unique heights: `average`, `super tall`, `very short`, `short`, `very tall`, `tall`
 - Each person has a unique favorite drink: `coffee`, `milk`, `boba tea`, `root beer`, `tea`, `water`
 - Each person has a favorite color: `blue`, `yellow`, `red`, `green`, `purple`, `white`
 - Each person lives in a unique style of house: `mediterranean`, `colonial`, `modern`, `ranch`, `victorian`, `craftsman`
 - They all have a unique favorite flower: `lilies`, `roses`, `iris`, `daffodils`, `carnations`, `tulips`

## Clues:
1. Eric is not in the second house.
2. The coffee drinker is somewhere to the right of the person residing in a Victorian house.
3. The person who is very tall is directly left of the person in a Craftsman-style house.
4. The person living in a colonial-style house is the person who likes milk.
5. The root beer lover is directly left of the person whose favorite color is green.
6. The person who loves yellow is the person who loves the boquet of lilies.
7. Arnold is the person whose favorite color is red.
8. The person who loves the rose bouquet is Bob.
9. The person who is very tall is the root beer lover.
10. Alice is somewhere to the left of the person who loves a bouquet of daffodils.
11. The boba tea drinker is Carol.
12. There is one house between the person in a ranch-style home and the person who loves the boquet of iris.
13. The person who loves a carnations arrangement is the tea drinker.
14. The boba tea drinker is the person in a Mediterranean-style villa.
15. The person who loves a bouquet of daffodils is in the second house.
16. The person who is tall and the person who loves the vase of tulips are next to each other.
17. Carol is somewhere to the left of the person who loves the boquet of iris.
18. The person who has an average height is directly left of Arnold.
19. The person who is super tall and Eric are next to each other.
20. The person who loves the vase of tulips is somewhere to the right of the person who loves the boquet of lilies.
21. The person who is super tall is the tea drinker.
22. The person who is short is Carol.
23. The coffee drinker is the person who loves purple.
24. The person who loves blue is somewhere to the right of the person who loves a carnations arrangement.
 What is the value of attribute HouseStyle for the person whose attribute Height is very tall? Please reason step by step, and put your final answer within \boxed{}

Answer: victorian


There are 2 houses, numbered 1 to 2 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`
 - Each person has an occupation: `engineer`, `doctor`
 - Each person has a unique birthday month: `april`, `sept`
 - Each person lives in a unique style of house: `victorian`, `colonial`
 - People have unique heights: `very short`, `short`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`

## Clues:
1. The person who is an engineer is in the first house.
2. The person whose birthday is in April and the person who is a doctor are next to each other.
3. The person living in a colonial-style house is the person who is an engineer.
4. The person who is very short is the person who is an engineer.
5. The person who is short is the person partial to Pall Mall.
6. The person who is an engineer is Eric.
 What is the value of attribute Birthday for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


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: `Arnold`, `Alice`, `Peter`, `Eric`
 - Everyone has something unique for lunch: `grilled cheese`, `stew`, `pizza`, `spaghetti`
 - People have unique hair colors: `blonde`, `black`, `red`, `brown`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Janelle`, `Kailyn`

## Clues:
1. Alice is the person whose favorite color is red.
2. The person who has black hair is the person who loves the stew.
3. The person who loves white is Arnold.
4. The person whose favorite color is red and The person whose mother's name is Kailyn are next to each other.
5. The person who loves yellow and the person who has black hair are next to each other.
6. The person who is a pizza lover is somewhere to the left of the person whose favorite color is green.
7. Peter is The person whose mother's name is Janelle.
8. The person who has blonde hair is the person who loves white.
9. The person who has red hair is Alice.
10. The person who loves white is somewhere to the right of the person who loves eating grilled cheese.
11. The person who loves white is not in the second house.
12. The person whose mother's name is Kailyn is the person who is a pizza lover.
13. The person whose mother's name is Holly is the person who has red hair.
 What is the value of attribute Color for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: white


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`, `Alice`, `Bob`, `Peter`, `Eric`
 - People have unique hair colors: `blonde`, `black`, `gray`, `brown`, `red`
 - They all have a unique favorite flower: `tulips`, `carnations`, `daffodils`, `lilies`, `roses`
 - Each person has a favorite color: `white`, `red`, `green`, `blue`, `yellow`

## Clues:
1. The person who loves the rose bouquet and the person who loves yellow are next to each other.
2. The person who loves a carnations arrangement is Peter.
3. The person who loves the rose bouquet is not in the third house.
4. The person whose favorite color is green is not in the fifth house.
5. The person who loves a bouquet of daffodils is Alice.
6. The person who loves yellow is Eric.
7. The person who has blonde hair is in the fifth house.
8. The person who has black hair is directly left of Arnold.
9. The person who loves yellow is the person who has brown hair.
10. Alice is the person whose favorite color is green.
11. The person who has red hair is the person who loves blue.
12. The person whose favorite color is red is somewhere to the right of the person who loves blue.
13. The person who has black hair is the person who loves the vase of tulips.
 What is the value of attribute Flower for the person whose attribute Color is yellow? Please reason step by step, and put your final answer within \boxed{}

Answer: lilies


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: `Eric`, `Arnold`, `Peter`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `google pixel 6`
 - Each person has a unique birthday month: `april`, `jan`, `sept`

## Clues:
1. The person who uses an iPhone 13 is Peter.
2. The tea drinker is somewhere to the right of Eric.
3. The person whose birthday is in April is not in the third house.
4. The person whose birthday is in January is directly left of the person who likes milk.
5. The person who uses a Samsung Galaxy S21 and the person who uses an iPhone 13 are next to each other.
6. The person whose birthday is in September is Peter.
7. Eric is directly left of the person who uses an iPhone 13.
 What is the value of attribute Drink for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: milk


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`, `Bob`, `Carol`, `Arnold`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `blue master`, `yellow monster`, `pall mall`, `prince`
 - People have unique favorite book genres: `fantasy`, `romance`, `science fiction`, `mystery`, `historical fiction`, `biography`

## Clues:
1. The person partial to Pall Mall is Alice.
2. The Prince smoker is not in the sixth house.
3. Carol is directly left of the person who loves fantasy books.
4. The Prince smoker is not in the fifth house.
5. Peter is in the first house.
6. The person partial to Pall Mall is directly left of the Dunhill smoker.
7. The person partial to Pall Mall is in the second house.
8. Carol is the person who loves mystery books.
9. Bob is not in the fourth house.
10. The person who smokes Yellow Monster is not in the sixth house.
11. The person who loves biography books is not in the third house.
12. The person who loves historical fiction books is in the first house.
13. The person who smokes many unique blends is not in the fifth house.
14. The person who loves romance books is somewhere to the right of Eric.
15. There is one house between Arnold and the person who smokes Yellow Monster.
16. There is one house between the person who smokes many unique blends and Bob.
 What is the value of attribute Cigar for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: pall mall


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`, `Eric`, `Arnold`
 - Each person prefers a unique type of vacation: `mountain`, `beach`, `city`
 - Each person has a unique hobby: `gardening`, `cooking`, `photography`
 - People have unique favorite sports: `tennis`, `soccer`, `basketball`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`

## Clues:
1. The person who loves soccer is the person who loves a bouquet of daffodils.
2. Arnold is the person who loves the boquet of lilies.
3. The person who loves beach vacations is directly left of Eric.
4. The Desert smoothie lover is the person who loves basketball.
5. The person who loves the boquet of lilies is directly left of the person who loves a carnations arrangement.
6. Eric is directly left of Peter.
7. The person who loves basketball and the person who loves a bouquet of daffodils are next to each other.
8. The person who loves a bouquet of daffodils is the person who likes Cherry smoothies.
9. The person who enjoys mountain retreats is the person who loves cooking.
10. The photography enthusiast is the person who loves basketball.
 What is the value of attribute FavoriteSport for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: tennis


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`, `Alice`, `Eric`, `Bob`, `Peter`
 - People have unique hair colors: `gray`, `blonde`, `red`, `brown`, `black`
 - Each person has an occupation: `engineer`, `artist`, `lawyer`, `teacher`, `doctor`

## Clues:
1. The person who has black hair is the person who is a teacher.
2. The person who is a doctor is somewhere to the right of the person who is an artist.
3. There is one house between Eric and Alice.
4. Eric is somewhere to the left of Alice.
5. The person who has brown hair and Alice are next to each other.
6. Alice is the person who has red hair.
7. There are two houses between the person who has brown hair and the person who has blonde hair.
8. Alice is the person who is an engineer.
9. Arnold is the person who has blonde hair.
10. The person who is a doctor is Peter.
11. The person who has brown hair is somewhere to the left of the person who is a teacher.
 What is the value of attribute House for the person whose attribute Name is Eric? 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: `Eric`, `Peter`, `Arnold`
 - Each person has a unique favorite drink: `water`, `milk`, `tea`
 - They all have a unique favorite flower: `carnations`, `lilies`, `daffodils`
 - Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`

## Clues:
1. The person who smokes Blue Master is the person who loves a bouquet of daffodils.
2. Arnold is somewhere to the right of the person who loves a carnations arrangement.
3. Eric is somewhere to the right of the Prince smoker.
4. The one who only drinks water is the person partial to Pall Mall.
5. The person who loves the boquet of lilies is Peter.
6. The tea drinker is not in the third house.
 What is the value of attribute Cigar for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: prince


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`, `Eric`, `Carol`, `Alice`, `Arnold`
 - The people keep unique animals: `rabbit`, `dog`, `horse`, `fish`, `bird`, `cat`
 - People have unique favorite sports: `basketball`, `swimming`, `tennis`, `soccer`, `baseball`, `volleyball`
 - People have unique hair colors: `blonde`, `gray`, `auburn`, `red`, `black`, `brown`
 - People have unique heights: `average`, `very short`, `very tall`, `tall`, `super tall`, `short`
 - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `blue master`, `prince`, `dunhill`, `blends`

## Clues:
1. The person who is very tall and the dog owner are next to each other.
2. There are two houses between the person who loves tennis and the person who has an average height.
3. The cat lover is the person who loves baseball.
4. The person who is very short is not in the fifth house.
5. There is one house between the fish enthusiast and the person who has red hair.
6. The cat lover is somewhere to the left of the Dunhill smoker.
7. The rabbit owner is somewhere to the left of the person who is super tall.
8. The Dunhill smoker is Alice.
9. The person who has red hair is the Dunhill smoker.
10. The person who is super tall is not in the fifth house.
11. The Prince smoker is somewhere to the left of the person who smokes Yellow Monster.
12. The person who loves volleyball is somewhere to the right of the person who smokes Yellow Monster.
13. The person who smokes Blue Master is directly left of the person who has black hair.
14. There is one house between the person who loves basketball and the person who smokes Blue Master.
15. The person who smokes Yellow Monster is the person who is tall.
16. Bob and the fish enthusiast are next to each other.
17. The Dunhill smoker is the person who loves soccer.
18. The person who loves tennis is not in the third house.
19. Arnold is the person who smokes many unique blends.
20. The Prince smoker is Peter.
21. There is one house between the person who has blonde hair and the person who is tall.
22. The person who loves swimming is directly left of Alice.
23. The person who is super tall is Eric.
24. The person who is short is the rabbit owner.
25. The person who keeps horses is not in the fifth house.
26. The cat lover and the person who has brown hair are next to each other.
27. The person who is short and the person who has auburn hair are next to each other.
 What is the value of attribute FavoriteSport for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}

Answer: soccer


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`, `Eric`, `Arnold`
 - People have unique favorite sports: `soccer`, `basketball`, `tennis`
 - Each person lives in a unique style of house: `victorian`, `colonial`, `ranch`
 - Each person has a unique type of pet: `fish`, `dog`, `cat`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - The people are of nationalities: `swede`, `dane`, `brit`

## Clues:
1. The person who loves soccer is the person residing in a Victorian house.
2. The person who owns a dog is the person whose birthday is in April.
3. The person who owns a dog is the person residing in a Victorian house.
4. The person living in a colonial-style house is Arnold.
5. The person living in a colonial-style house is the person whose birthday is in January.
6. Arnold is directly left of the person whose birthday is in April.
7. The British person is the person who loves basketball.
8. The Swedish person is the person whose birthday is in April.
9. The Dane is Eric.
10. The person who has a cat is the person whose birthday is in January.
11. The person who loves soccer is in the third house.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


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`, `Arnold`, `Carol`, `Bob`, `Peter`, `Alice`
 - Each person lives in a unique style of house: `ranch`, `victorian`, `modern`, `colonial`, `craftsman`, `mediterranean`
 - Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`, `boba tea`
 - Each person has an occupation: `engineer`, `nurse`, `doctor`, `artist`, `lawyer`, `teacher`
 - The people keep unique animals: `horse`, `fish`, `dog`, `cat`, `rabbit`, `bird`
 - Each person has a favorite color: `red`, `purple`, `white`, `green`, `yellow`, `blue`

## Clues:
1. The fish enthusiast is in the first house.
2. Carol is somewhere to the left of the person in a Mediterranean-style villa.
3. The bird keeper is the person who loves purple.
4. The person in a ranch-style home is the one who only drinks water.
5. The coffee drinker is not in the fourth house.
6. The person living in a colonial-style house and the boba tea drinker are next to each other.
7. The cat lover is the person who is a teacher.
8. The person in a modern-style house is in the third house.
9. The person who is an engineer and Peter are next to each other.
10. Arnold is the person who is an artist.
11. The person whose favorite color is red is Arnold.
12. The coffee drinker is somewhere to the right of the person who is a teacher.
13. The bird keeper is the person in a ranch-style home.
14. The rabbit owner is in the sixth house.
15. The boba tea drinker is Peter.
16. Eric is the cat lover.
17. The person who is a nurse is directly left of the person in a modern-style house.
18. The bird keeper is Carol.
19. There are two houses between the cat lover and the boba tea drinker.
20. Bob is the tea drinker.
21. The person residing in a Victorian house is the dog owner.
22. The person whose favorite color is green is in the fourth house.
23. The person who loves blue is the root beer lover.
24. The person who loves white is not in the sixth house.
25. Peter is the person who is a doctor.
 What is the value of attribute House for the person whose attribute Drink is boba tea? Please reason step by step, and put your final answer within \boxed{}

Answer: 6


Colour a $20000 \times 20000$ square grid using 2000 different colours, with one colour per square. Two squares are considered neighbours if they share a vertex. Define a path as a sequence of squares where each pair of successive squares are neighbours.

Mark $k$ squares on the grid. For each unmarked square $x$, there is exactly one marked square $y$ of the same colour such that $x$ and $y$ are connected by a path consisting of squares of the same colour.

Furthermore, for any two marked squares of the same colour, any path connecting them must pass through squares of all the different colours.

Find the maximum value of $k$.

Answer: 2000


In a game of ping-pong, the score is $4-10$. Six points later, the score is $10-10$. You remark that it was impressive that I won the previous $6$ points in a row, but I remark back that you have won $n$ points in a row. What is the largest value of $n$ such that this statement is true regardless of the order in which the points were distributed?

Answer: 2


You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2 \times 1$ and $3 \times 1$. The playing field should be completely covered with these dominoes without any overlaps, and no domino should protrude beyond the playing field. Furthermore, all dominoes must be aligned in the same direction, meaning their longer sides must be parallel to each other. How many such coverings are possible?

Answer: 257


A square grid $100 \times 100$ is tiled in two ways: only with dominoes and only with $2 \times 2$ squares. What is the least number of dominoes that are entirely inside some $2 \times 2$ square?

Answer: 100


In a chess tournament, there are $100$ players. Each player plays one game with every other player. For a win, $1$ point is awarded; for a loss, $0$ points are given; and for a draw, both players receive $0.5$ points.

Ion scored more points than any other player. Mihai lost only one game but accumulated fewer points than any other player. Determine all possible values of the difference between the points accumulated by Ion and the points accumulated by Mihai.

Answer: 50


Let $k$ be an integer. If the equation $(x-1)\left|x+1\right|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there?

Answer: 4544


Let $a_1, a_2, a_3, \ldots$ be a non-decreasing sequence of positive integers. For $m \ge 1$, define $b_m = \min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n \ge m$. If $a_{19} = 85$, determine the maximum value of $a_1 + a_2 + \cdots + a_{19} + b_1 + b_2 + \cdots + b_{85}$.

Answer: 1700


Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$.

Answer: 84


The Fibonacci numbers are defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$. It is well-known that the sum of any 10 consecutive Fibonacci numbers is divisible by 11. Determine the smallest integer $N$ such that the sum of any $N$ consecutive Fibonacci numbers is divisible by 12.

Answer: 24


Let $n$ be a positive integer and let $d_1, d_2, \ldots, d_k$ be its divisors, such that $1 = d_1 < d_2 < \ldots < d_k = n$. Find all values of $n$ for which $k \geq 4$ and $n = d_1^2 + d_2^2 + d_3^2 + d_4^2$.

Answer: 130


A mahogany bookshelf has four identical-looking books which are $200$, $400$, $600$, and $800$ pages long. Velma picks a random book off the shelf, flips to a random page to read, and puts the book back on the shelf. Later, Daphne also picks a random book off the shelf and flips to a random page to read. Given that Velma read page $122$ of her book and Daphne read page $304$ of her book, the probability that they chose the same book is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.

Answer: 6425


8. A light ray $l$ emitted from point $A(-3,3)$ is reflected by the $x$-axis. The straight line of the reflected ray is tangent to the circle $x^{2}+y^{2}-4 x-4 y+7=0$. Then the equation of the straight line $l$ is $\qquad$ .

Answer: 3x +4y -3 =0


Task B-1.4. Points $M$ and $N$ belong to sides $\overline{A C}$ and $\overline{B C}$ of triangle $A B C$, respectively. Point $O$ is the intersection of segments $\overline{A N}$ and $\overline{B M}$. Determine the area of triangle $C M N$ if the areas of triangles $O M A, O A B$, and $O B N$ are 2, 3, and 4, respectively.

Answer: \dfrac{280}{3}


22. Let $C$ be a circle with radius 2006 . Suppose $n$ points are placed inside the circle and the distance between any two points exceed 2006. What is the largest possible $n$ ?

Answer: 5


Sergei and Lena have several chocolate bars, each weighing no more than 100 grams. No matter how they divide these chocolate bars, the total weight of the chocolate bars for one of them will not exceed 100 grams. What is the greatest possible total weight of all the chocolate bars?

Answer: 300


39th BMO 2003 Problem 4 Let X be the set of non-negative integers and f : X → X a map such that ( f(2n+1) ) 2 - ( f(2n) ) 2 = 6 f(n) + 1 and f(2n) >= f(n) for all n in X. How many numbers in f(X) are less than 2003?

Answer: 128


34. [?] The Game. Eric and Greg are watching their new favorite TV show, The Price is Right. Bob Barker recently raised the intellectual level of his program, and he begins the latest installment with bidding on following question: How many Carmichael numbers are there less than 100,000 ?
Each team is to list one nonnegative integer not greater than 100,000. Let $X$ denote the answer to Bob's question. The teams listing $N$, a maximal bid (of those submitted) not greater than $X$, will receive $N$ points, and all other teams will neither receive nor lose points. (A Carmichael number is an odd composite integer $n$ such that $n$ divides $a^{n-1}-1$ for all integers $a$ relatively prime to $n$ with $1<a<n$.)

Answer: 16


The sum of a set is the sum of all its elements. Let \( S \) be a set consisting of positive integers not exceeding 15. The sum of any two disjoint subsets of \( S \) must be different. Among all sets having this property, find the maximum possible sum of \( S \).

Answer: 61


7. There are less than 100 cells in a row, and at the beginning, only the cells at both ends contain a chess piece each. Several students take turns placing chess pieces in the cells. Each person places one piece at a time, and it must be placed in the middle of two adjacent cells that already contain pieces (for example, if the 3rd and 7th cells from left to right contain pieces, and the 4th, 5th, and 6th cells do not, a piece can be placed in the 5th cell; but if the 4th and 7th cells contain pieces, and the 5th and 6th cells do not, neither the 5th nor the 6th cell can be used). Each of these students placed 9 pieces, resulting in each cell containing exactly one piece. How many students are there? $\qquad$

Answer: 7


(given to Jean-Alix David and William Fujiwara). Find all natural numbers $x, y$ and $z$ satisfying $3^{x}+4^{y}=5^{z}$.

Answer: x = 2


G5. In triangle $A B C$, let $J$ be the centre of the excircle tangent to side $B C$ at $A_{1}$ and to the extensions of sides $A C$ and $A B$ at $B_{1}$ and $C_{1}$, respectively. Suppose that the lines $A_{1} B_{1}$ and $A B$ are perpendicular and intersect at $D$. Let $E$ be the foot of the perpendicular from $C_{1}$ to line $D J$. Determine the angles $\angle B E A_{1}$ and $\angle A E B_{1}$.

Answer: 90^\circ


The base of an isosceles triangle is $a$, its leg is $b$, and its perimeter is $k$. Prove that

$$
\frac{k}{2}<a+b<\frac{3 k}{4}
$$

How does the statement change if the triangle degenerates (becomes a line segment counted twice)?

Answer: \frac{k}{2} < a + b < \frac{3k}{4}


Example 5. Given that $\theta_{1}, \theta_{2}, \cdots, \theta_{n}$ are all non-negative, and $\theta_{1}+\theta_{2}+\cdots+\theta_{n}=\pi$. Find the maximum value of $\sin ^{2} \theta_{1}+\sin ^{2} \theta_{2}+\cdots+\sin ^{2} \theta_{n}$.
(1985, IMO Candidate Problem 12)

Answer: \dfrac{9}{4}


Five, (15 points) A school assigns numbers to the contestants participating in a math competition, with the smallest number being 0001 and the largest number being 2014. No matter which contestant steps forward to calculate the average of the numbers of all other contestants in the school, the average is always an integer. How many contestants can the school have at most?

Will the above text be translated into English, preserving the original text's line breaks and format? Please output the translation directly.

Answer: 34


6. Given the set
$$
A=\left\{n \in \mathbf{Z} \left\lvert\,\left(1+\frac{1}{n}\right)^{n+1}=\left(1+\frac{1}{2020}\right)^{2000}\right.\right\} \text {. }
$$

Then the number of elements in set $A$ is $\qquad$ .

Answer: 1


5. There is a ruler 10 cm long without divisions. What is the smallest number of intermediate divisions that need to be made on the ruler so that segments of length 1 cm, 2 cm, 3 cm, ..., 10 cm can be laid off, applying the ruler in each case only once.

Answer: 4


11. Given the function $f(x)=\ln x+\frac{a}{x}(a \in \mathbf{R})$.
(1) Determine the monotonicity of $f(x)$ within its domain;
(2) If $f(x)<2 x^{2}$ holds for all $x$ in $\left(\frac{1}{2},+\infty\right)$, find the range of values for $a$.

Answer: (-\infty, \frac{1}{4} + \frac{\ln 2}{2}]


4. Decipher the message RTOLK TOIK, which was enciphered using the affine transformation $C \equiv 3 P+24(\bmod 26)$.

Answer: PHONE HOME


4. Given $P(2,1)$, draw a line $l$ through point $P$ that intersects the positive $x$-axis and $y$-axis at points $A$ and $B$ respectively. Then the equation of the line $l$ that minimizes the perimeter of $\triangle A O B$ (where $O$ is the origin) is $\qquad$

Answer: 3x + 4y = 10


6. Find the minimum value of the expression $\left[\frac{7(a+b)}{c}\right]+\left[\frac{7(a+c)}{b}\right]+\left[\frac{7(b+c)}{a}\right]$, where $a, b$ and $c$ are arbitrary natural numbers.

Answer: 40


At 9 a.m., ships "Anin" and "Vanin" departed from port O to port E. At the same moment, the ship "Sanin" set off from port E to port O. All three vessels are traveling along the same course (with "Sanin" heading towards "Anin" and "Vanin") at constant but different speeds. At 10 a.m., "Vanin" was equidistant from both "Anin" and "Sanin." At 10:30 a.m., "Sanin" was equidistant from both "Anin" and "Vanin." At what moment will "Anin" be exactly in the middle between "Vanin" and "Sanin"?

Answer: 12:00


G4.1 In figure $1, A B C$ and $E B C$ are two right-angled triangles, $\angle B A C=\angle B E C=90^{\circ}, A B=A C$ and $E D B$ is the angle bisector of $\angle A B C$. Find the value of $\frac{B D}{C E}$.

Answer: 2


$[$ Theory of algorithms (miscellaneous) $]$ $[$ Extreme principle (miscellaneous) ]

![](https://cdn.mathpix.com/cropped/2024_05_06_b41405b02b6e888d68cag-05.jpg?height=43&width=243&top_left_y=1825&top_left_x=16)

Vasya solved problems for 10 days - at least one problem each day. Each day (except the first), if the weather was cloudy, he solved one more problem than the previous day, and if it was sunny, one less problem. In the first 9 days, Vasya solved 13 problems. What was the weather like on the tenth day?

#

Answer: \text{cloudy}


## Problem 3

Three vertices of a regular $2 n+1$ sided polygon are chosen at random. Find the probability that the center of the polygon lies inside the resulting triangle.

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


The set $ S$ consists of $ n > 2$ points in the plane. The set $ P$ consists of $ m$ lines in the plane such that every line in $ P$ is an axis of symmetry for $ S$. Prove that $ m\leq n$, and determine when equality holds.

Answer: m \leq n


## Task B-3.4.

How many nine-digit numbers divisible by 75 are there, all of whose digits are different, and the hundreds digit is 7?

Answer: 4680


$16 、  A$ nine-digit number contains the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once, and any two adjacent digits in this nine-digit number can form a two-digit number that can be expressed as the product of two single-digit numbers. This nine-digit number is $\qquad$ -.

Answer: 728163549


On weekdays, the Absent-Minded Scientist travels to work on the ring line of the Moscow Metro from "Taganskaya" station to "Kievskaya" station, and in the evening - back (see diagram).

![](https://cdn.mathpix.com/cropped/2024_05_06_c1015ea0b2d2805d8876g-25.jpg?height=672&width=692&top_left_y=1499&top_left_x=683)

Upon entering the station, the Scientist boards the first arriving train. It is known that trains run in both directions with approximately equal intervals, and that the train traveling on the northern route (via "Belorusskaya") takes 17 minutes to travel from "Kievskaya" to "Taganskaya" or vice versa, while on the southern route (via "Paveletskaya") it takes 11 minutes. Out of habit, the Scientist always calculates everything. Once, he calculated that over many years of observation:

- the train traveling counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after the train traveling clockwise arrives. The same is true for "Taganskaya".
- the trip from home to work takes the Scientist on average 1 minute less than the trip from work to home. Find the expected interval between trains traveling in the same direction.

Answer: 3


187. For which integer values of $x$ will the number $\frac{5 x+2}{17}$ be an integer?

Answer: x = 17k + 3


Two points $A$ and $B$ are selected independently and uniformly at random along the perimeter of a unit square with vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The probability that the $y$-coordinate of $A$ is strictly greater than the $y$-coordinate of $B$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.

Answer: 716


4. Gavrila placed 7 smaller boxes into a large box. After that, Glafira placed 7 small boxes into some of these seven boxes, and left others empty. Then Gavrila placed 7 boxes into some of the empty boxes, and left others empty. Glafira repeated this operation and so on. At some point, there were 34 non-empty boxes. How many empty boxes were there at this moment?

Answer: 205


Let \( \mathrm{p} \) be an odd prime number. Consider the set \(\{ 1, 2, \ldots, 2\mathrm{p} \}\) and find the number of subsets \( A \) that satisfy the following conditions:
1. \( A \) contains exactly \( \mathrm{p} \) elements.
2. The sum of all elements in \( A \) is divisible by \( \mathrm{p} \).

Answer: \dfrac{1}{p} \left( \dbinom{2p}{p} + 2(p - 1) \right)


1. The domain of the function $f(x)$ is $D$. If
(1) $f(x)$ is a monotonic function in $D$,
(2) There exists an interval $[a, b] \subseteq D$ such that the range of $f(x)$ on the interval $[a, b]$ is $[a, b]$,
then $y=f(x)$ is called a closed function.
If $f(x)=\sqrt{x+2}+k$ is a closed function, then the range of values for $k$ is

Answer: (-\dfrac{9}{4}, -2]


Task 1 - 130911 Two points $A$ and $B$ at the same height above the ground are at the same distance and on the same side of a straight, high wall. The segment $A B$ is $51 \mathrm{~m}$ long. A sound generated at $A$ reaches $B$ directly exactly $\frac{1}{10}$ s earlier than via the reflection off the wall.

Determine the distance of each of the two points $A$ and $B$ from the wall, assuming that the sound travels exactly $340 \mathrm{~m}$ per second.

Answer: 34


In triangle \(ABC\), an incircle \(\omega\) with radius \(r\) is inscribed, touching side \(AB\) at point \(X\). A point \(Y\), diametrically opposite to \(X\), is marked on the circle. Line \(CY\) intersects side \(AB\) at point \(Z\). Find the area of triangle \(ABC\), given that \(CA + AZ = 1\).

Answer: r


Example 3. Using the Kellogg method, calculate the smallest characteristic number of the kernel $K(x, t)=x^{2} t^{2}, 0 \leqslant x, t \leqslant 1$.

Answer: 5


5. As shown in Figure 2, $\triangle A E F$ is an inscribed triangle in the square $A B C D$ with side length $x$. If $\angle A E F=90^{\circ}$, $A E=a, E F=b(a>b)$, then $x=$

Answer: \dfrac{a^2}{\sqrt{2a^2 - 2ab + b^2}}


The diagram below shows two parallel rows with seven points in the upper row and nine points in the lower row. The points in each row are spaced one unit apart, and the two rows are two units apart. How many trapezoids which are not parallelograms have vertices in this set of $16$ points and have area of at least six square units?
[asy]
import graph; size(7cm);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
dot((-2,4),linewidth(6pt) + dotstyle);
dot((-1,4),linewidth(6pt) + dotstyle);
dot((0,4),linewidth(6pt) + dotstyle);
dot((1,4),linewidth(6pt) + dotstyle);
dot((2,4),linewidth(6pt) + dotstyle);
dot((3,4),linewidth(6pt) + dotstyle);
dot((4,4),linewidth(6pt) + dotstyle);
dot((-3,2),linewidth(6pt) + dotstyle);
dot((-2,2),linewidth(6pt) + dotstyle);
dot((-1,2),linewidth(6pt) + dotstyle);
dot((0,2),linewidth(6pt) + dotstyle);
dot((1,2),linewidth(6pt) + dotstyle);
dot((2,2),linewidth(6pt) + dotstyle);
dot((3,2),linewidth(6pt) + dotstyle);
dot((4,2),linewidth(6pt) + dotstyle);
dot((5,2),linewidth(6pt) + dotstyle); [/asy]

Answer: 361


6. (10 points) A deck of playing cards, excluding the joker, has 4 suits totaling 52 cards, with each suit having 13 cards, numbered from 1 to 13. Feifei draws 2 hearts, 3 spades, 4 diamonds, and 5 clubs. If the sum of the face values of these 14 cards Feifei drew is exactly 35, then how many of them are 1?

$\qquad$ cards are 1.

Answer: 4


4. As shown in Figure 5, positive real numbers $a_{1}, a_{2}, \cdots, a_{8}$ are marked at the corresponding vertices of a cube, such that the number at each vertex is the average of the three numbers marked at the adjacent vertices. Let
$$
\begin{array}{l}
M=\left(a_{1}+2 a_{2}+3 a_{3}+4 a_{4}\right) . \\
\quad\left(5 a_{5}+6 a_{6}+7 a_{7}+8 a_{8}\right), \\
N=a_{1}^{2}+a_{2}^{2}+\cdots+a_{8}^{2} . \\
\text { Then } \frac{M}{N}=
\end{array}
$$

Answer: \dfrac{65}{2}


The distance between points $A$ and $B$ is 40 km. A pedestrian left $A$ at 4 o'clock. When he had walked half the distance, he was caught up by a cyclist who had left $A$ at 7:20. An hour after this, the pedestrian met another cyclist who had left $B$ at 8:30. The speeds of the cyclists are the same. Determine the speed of the pedestrian.

Answer: 5


6. (10 points) Doudou's family has 4 people. This year, Doudou's brother is 3 years older than Doudou, and Doudou's mother is 2 years younger than Doudou's father. 5 years ago, the total age of the family was 59 years, and 5 years later, the total age of the family will be 97 years. Doudou's mother is $\qquad$ years old this year.

Answer: 33


A parallelogram's interior angle bisectors define a quadrilateral whose area is one-third of the parallelogram's area. Determine the ratio of the adjacent sides of the parallelogram.

Answer: \dfrac{4 + \sqrt{7}}{3}


A ticket consists of six digits $a, b, c, d, e, f$. This ticket is called "happy" if $a+b+c=$ $d+e+f$. How many happy tickets are there (including the ticket 000000)?

## Graphs

We will not present the results on graphs as they are covered in the graph theory poly on the Animath website.

Answer: 55252


A bag contains exactly 15 marbles of which 3 are red, 5 are blue, and 7 are green. The marbles are chosen at random and removed one at a time from the bag until all of the marbles are removed. One colour of marble is the first to have 0 remaining in the bag. What is the probability that this colour is red?

Answer: \dfrac{21}{40}


Question 1
In $\triangle ABC$, $D$ is a point on side $BC$. The incenters of $\triangle ABD$ and $\triangle ACD$ are $I_{1}$ and $I_{2}$, respectively. The circumcenters of $\triangle AI_{1}D$ and $\triangle AI_{2}D$ are $O_{1}$ and $O_{2}$, respectively. Line $I_{1}O_{2}$ intersects $I_{2}O_{1}$ at point $P$. Prove that $PD \perp BC$.

Question 2
In the quadrilateral $AI_{1}DI_{2}$, $\angle I_{1}DI_{2}$ is a right angle, and $\angle DI_{1}A$, $\angle DI_{2}A$ are obtuse angles. $O_{1}$ and $O_{2}$ are the circumcenters of $\triangle AI_{1}D$ and $\triangle AI_{2}D$, respectively. Point $P$ is the intersection of the rays $I_{1}O_{2}$ and $I_{2}O_{1}$. Prove that $\angle I_{1}DP = \angle PDI_{2}$.

Question 3
In the rectangle $DEKF$, $I_{1}$ and $I_{2}$ are points on $DE$ and $DF$, respectively. $X$ and $Y$ are the midpoints of $DI_{1}$ and $DI_{2}$, respectively. Through $X$ and $Y$, lines parallel to the sides $DF$ and $DE$ of the rectangle are drawn, intersecting $EF$ at points $O_{1}$ and $O_{2}$, respectively. Prove that $I_{1}O_{2}$, $I_{2}O_{1}$, and $DK$ are concurrent.

Answer: PD \perp BC


Around a large round table, 60 people are sitting, each of whom is either a knight or a liar. Each of them said: “Of the five people sitting in a row to my right, at least two are liars.” How many knights can sit at this table?

Answer: 40


## Zadatak A-1.3.

Biljarski stol ima oblik pravokutnika $A B C D$ i dimenzije $|A B|=2 \mathrm{~m}$ i $|B C|=1 \mathrm{~m}$. Biljarska kugla giba se po stolu pravocrtno dok ne dođe do ruba pravokutnika, a tada se odbija tako da putanja kugle prije i poslije odbijanja zatvara s rubom sukladne kutove. Ako biljarska kugla započne gibanje u točki $A$ te nakon odbijanja od stranica $\overline{C D}, \overline{B C}$ i $\overline{A B}$ redom završi gibanje u točki $D$, odredi ukupnu udaljenost koju

![](https://cdn.mathpix.com/cropped/2024_05_30_bdf789747fbcdaaac460g-03.jpg?height=323&width=399&top_left_y=1455&top_left_x=1271)
je kugla prešla. Kuglu promatramo kao materijalnu točku.

Answer: 5


Quadrilateral \(ABCD\) has sides \(AB\) and \(CD\) intersecting at point \(O\). A line \(l\) through \(O\) intersects \(AC\) at \(P\) and \(BD\) at \(Q\). A conic section passing through \(A\), \(B\), \(C\), and \(D\) intersects \(l\) at \(R\) and \(S\). Prove: \(\frac{1}{OP} + \frac{1}{OQ} = \frac{1}{OR} + \frac{1}{OS}\).

Answer: \frac{1}{OP} + \frac{1}{OQ} = \frac{1}{OR} + \frac{1}{OS}