Two isosceles triangles are placed side-by-side with their bases on the same line. If $\triangle ABC$ and $\triangle DEF$ are such triangles with $AB = AC$, $DE = DF$, $\angle BAC = 25^\circ$, and $\angle EDF = 40^\circ$, find the sum $\angle DAC + \angle ADE$.

Answer: 147.5^\circ


The planet Zeta follows an elliptical orbit with its sun at one focus. At its closest approach (perigee), it is 3 astronomical units (AU) from the sun, and at its farthest point (apogee), it is 15 AU away. Calculate the distance from the sun to Zeta when Zeta is exactly halfway along its orbit from perigee to apogee.

Answer: 9 \text{ AU}


A circular dartboard is divided into sectors. The probability of a dart randomly landing in one of the smaller sectors is $\frac{1}{8}$. There are 3 smaller sectors with equal probabilities and one larger sector. What is the central angle, in degrees, of one of the smaller sectors?

Answer: 45^\circ


A unit cube is cut once along a diagonal of one face and then along the diagonal of the opposite parallel face creating four triangular prisms. One of these prisms is cut again along a plane passing through one of its longest edges and the midpoint of its opposite edge. This results in six pieces in total. What is the volume of the triangular prism that includes vertex $W$ on the original cube?

Answer: \frac{1}{8}


8 Arrange fifteen students numbered $1,2,3, \ldots, 15$ in a circle facing inward, in numerical order. The first time, the student numbered 1 turns around. The second time, the students numbered 2 and 3 turn around. The third time, the students numbered $4,5,6$ turn around, .... The 15th time, all students turn around. When the 12th turn is completed, how many students are still facing outward?

Answer: 12


1. Given a set of data consisting of seven positive integers, the only mode is 6, and the median is 4. Then the minimum value of the sum of these seven positive integers is $\qquad$ .

Answer: 26


2.149. $\frac{6}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$.

Express the above text in English, keeping the original text's line breaks and format, and output the translation result directly.

2.149. $\frac{6}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$.

Answer: \dfrac{3\sqrt{2} + 2\sqrt{3} - \sqrt{30}}{2}


65. Suppose you have forgotten one digit of a phone number you need and dial it at random. What is the probability that you will have to make no more than two calls?

Answer: \dfrac{1}{5}


4. As shown in Figure 4, in $\triangle A B C$, $A B=A C$, $\angle B=40^{\circ}$, $B D$ is the angle bisector of $\angle B$, and $B D$ is extended to $E$ such that $D E=A D$. Then the degree measure of $\angle E C A$ is

Answer: 30


9. As shown in Figure 3, the center of the equilateral triangle $\triangle A_{1} B_{1} C_{1}$ with side length 1 is $O$. Rotate $\triangle A_{1} B_{1} C_{1}$ around the center $O$ to $\triangle A_{2} B_{2} C_{2}$ such that $A_{2} B_{2} \perp B_{1} C_{1}$. Then the area of the common part (i.e., the hexagon $A B C D E F$) of the two triangles is

Answer: \dfrac{3 - \sqrt{3}}{4}


Example 5 As shown in Figure 5, in $\triangle A B C$, $A C=7, B C=$ $4, D$ is the midpoint of $A B$, $E$ is a point on side $A C$, and $\angle A E D=$ $90^{\circ}+\frac{1}{2} \angle C$. Find the length of $C E$.

Answer: \dfrac{11}{2}


1.7 density, 400 gr weight liquid is mixed with 1.2 density $600 \mathrm{gr}$ solution. Measuring the density $D$ of the newly formed solution, we find that $D=1.4$. What is the contraction that occurs when mixing the two initial liquids?

Answer: 21


Amanda has two identical cubes. Each cube has one integer on each face so that the following statements are all true:

- Three adjacent faces of each cube have the numbers 15,9 and 4 as shown.
- The numbers on all pairs of opposite faces have the same sum $s$.
- When both cubes are rolled and the numbers on the top faces are added, the probability that the sum equals 24 is $\frac{1}{12}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_486fbaaa1172337e46fbg-2.jpg?height=320&width=287&top_left_y=1081&top_left_x=973)

Determine the sum of all possible values of $s$.

Answer: 64


3. In the city of hedgehogs, there are 1000 identical buildings, each with 1000 floors, and exactly one hedgehog lives on each floor. Each resident in the city wears a hat colored in one of 1000 possible colors, and any two residents in the same building have hats of different colors. We say that two hedgehogs are $\bar{u} p u j a \bar{u} e l i$ if they wear hats of the same color and live on consecutive floors (in different buildings). Determine the maximum possible number of (unordered) pairs of hedgehogs that are friends.

Answer: 250000000


6. (15 points) A capacitor with capacitance $C_{1}=20$ μF is charged to a voltage $U_{1}=20$ V. A second capacitor with capacitance $C_{2}=5$ μF is charged to a voltage $U_{2}=5$ V. The capacitors are connected with their oppositely charged plates. Determine the voltage that will establish across the plates.

Answer: 15


13.236. The master is giving a simultaneous chess exhibition on several boards. By the end of the first two hours, he won $10 \%$ of the total number of games played, while 8 opponents drew their games with the master. Over the next two hours, the master won $10 \%$ of the games with the remaining opponents, lost two games, and drew the remaining 7 games. On how many boards was the game being played?

Answer: 20


9. The military district canteen needs 1000 pounds of rice and 200 pounds of millet for dinner. The quartermaster goes to the rice store and finds that the store is having a promotion. “Rice is 1 yuan per pound, and for every 10 pounds purchased, 1 pound of millet is given as a gift (no gift for less than 10 pounds); millet is 2 yuan per pound, and for every 5 pounds purchased, 2 pounds of rice are given as a gift (no gift for less than 5 pounds).” What is the minimum amount of money the quartermaster needs to pay to buy enough rice and millet for dinner?

Answer: 1168


Example 6 Let $a_{1}, a_{2}, \cdots, a_{n}(n \geqslant 2)$ be $n$ distinct real numbers, $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=S$. Try to find $\max \min _{1 \leqslant i<j \leqslant n}\left(a_{i}-a_{j}\right)^{2}$.

Answer: \dfrac{12S}{n(n^2 -1)}


2. In 8 "Y" grade, there are quite a few underachievers, but Vovochka performs the worst of all. The pedagogical council decided that either Vovochka must correct his failing grades by the end of the quarter, or he will be expelled. If Vovochka corrects his grades, then 24% of the class will be underachievers, but if he is expelled, the percentage of underachievers will be 25%. What is the current percentage of underachievers in 8 "Y"?

#

Answer: 28


10. There is a sequence of numbers +1 and -1 of length $n$. It is known that the sum of every 10 neighbouring numbers in the sequence is 0 and that the sum of every 12 neighbouring numbers in the sequence is not zero.
What is the maximal value of $n$ ?

Answer: 15


Answer: 24


Find all solutions to the equation $\sqrt{3x+6}=x+2$.  If there are multiple solutions, order them from least to greatest, separated by comma(s).

Answer: 1


$A$, $B$ and $C$ are circular regions as shown. There are 7 items in circle $C$. There are exactly 20 items in $A$ and 10 of those items are not in $B$. How many items are in $B$, but not in $C$?

[asy]
draw(Circle((0,0),50));
label("$A$",(39,39));
draw(Circle((10,-10),32));
label("$B$",(-16,16));
draw(Circle((5,-5),20));
label("$C$",(22,-22));
[/asy]

Answer: 3


In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of $\triangle ABC$, as shown. [asy]
size(250);
pair A, B, C, P, Q, R, S;
R=(0,0);
Q=(-2,0);
S=(2,0);
P=(1,1.732);
B=(-5.73,-1);
C=(3.732,-1);
A=(1.366,3.098);
draw(A--B--C--A);
draw(circle(P, 1));
draw(circle(Q, 1));
draw(circle(R, 1));
draw(circle(S, 1));
label("A", A, N);
label("B", B, SW);
label("C", C, SE);
dot(P);
dot(Q);
dot(R);
dot(S);
label("P", P, N);
label("Q", Q, SW);
label("R", R, SW);
label("S", S, SE);
[/asy]

The radius of the circle with center $R$ is decreased so that

$\bullet$ the circle with center $R$ remains tangent to $BC$,

$\bullet$ the circle with center $R$ remains tangent to the other three circles, and

$\bullet$ the circle with center $P$ becomes tangent to the other three circles.

The radii and tangencies of the other three circles stay the same. This changes the size and shape of $\triangle ABC$. $r$ is the new radius of the circle with center $R$. $r$ is of the form $\frac{a+\sqrt{b}}{c}$. Find $a+b+c$.

Answer: 6


Points \( M \) and \( N \) are the midpoints of sides \( BC \) and \( AD \), respectively, of quadrilateral \( ABCD \). The segments \( AM \), \( DM \), \( BN \), and \( CN \) divide quadrilateral \( ABCD \) into seven parts (six of which are triangles, and the seventh is a quadrilateral). Prove that if the areas of these six parts are odd natural numbers, then all six parts are triangles.

Answer: \text{All six parts are triangles.}


The area of the large square $\mathrm{ABCD}$ is 18 square centimeters, and the side $\mathrm{MN}$ of the gray square $\mathrm{MNPQ}$ lies on the diagonal $\mathrm{BD}$, with vertex $\mathrm{P}$ on side $\mathrm{BC}$ and $\mathrm{Q}$ on side $\mathrm{CD}$. What is the area of the gray square $\mathrm{MNPQ}$ in square centimeters?

Answer: 4


Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$

Answer: 60^\circ


The line \( x + y = 1 \) intersects the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) at points \( A \) and \( B \). \( C \) is the midpoint of the line segment \( AB \). If \( |AB| = 2 \sqrt{2} \) and the slope of the line \( OC \) is \( k_C = \frac{1}{\sqrt{2}} \), find the equation of the ellipse.

Answer: \dfrac{x^2}{3} + \dfrac{2y^2}{3\sqrt{2}} = 1


B5. Simon has 2017 blue blocks numbered from 1 to 2017. He also has 2017 yellow blocks numbered from 1 to 2017. Simon wants to arrange his 4034 blocks in a row. He wants to do so in such a way that for every $k=1,2, \ldots, 2017$ the following conditions are met:

- to the left of the blue block with number $k$ there are $k$ or more yellow blocks;
- to the right of the yellow block with number $k$ there are $k$ or fewer blue blocks.

Determine all possible numbers of the 1000th block from the left in the row.

## C-problems

For the C-problems, not only the answer is important; your reasoning and the way of solving it must also be clearly written down. Solve each C-problem on a separate sheet of paper. Each correctly worked out C-problem is worth 10 points. Partial solutions can also earn points. Therefore, write everything clearly and submit your scratch paper (for each problem!) as well.

Answer: 500


Answer: -2 \ln 2


8. A three-digit number has 3 digits, none of which are 0, and its square is a six-digit number that has exactly 3 digits as 0. Write down one such three-digit number: $\qquad$

Answer: 448


There is a parabola and two of its tangents. Through the points of tangency, we draw chords parallel to the tangents. In what ratio does the intersection point divide the chords?

Answer: \dfrac{1}{3}


One. (40 points) Let $n(n>1)$ be a given positive integer, and $x_{1}, x_{2}, \cdots, x_{n}$ be any real numbers, satisfying $\sum_{i=1}^{n} x_{i}^{2}=1$. Find the maximum value of $\sum_{k=1}^{n} k x_{k}^{2}+\sum_{1 \leqslant i<j \leqslant n}(i+j) x_{i} x_{j}$.

Answer: \frac{n(n+1)}{4} + \frac{n}{2} \sqrt{ \frac{(n+1)(2n+1)}{6} }


5. A person is walking parallel to a railway track at a constant speed. A train also passes by him at a constant speed. The person noticed that depending on the direction of the train, it passes by him either in $t_{1}=1$ minute or in $t_{2}=2$ minutes. Determine how long it would take the person to walk from one end of the train to the other.

## $(15$ points)

Answer: 4


7. Let the set $T=\left\{(a, b, c) \mid a, b, c \in \mathbf{N}^{*}\right.$, and $a, b, c$ can form the side lengths of some triangle $\}$. Then the value of $\sum_{(a, b, c) \in T} \frac{2^{a}}{3^{b} \cdot 5^{c}}$ is $\qquad$.

Answer: \dfrac{17}{21}


Answer: 4


4. Let $O$ be the center of the base $\triangle ABC$ of the regular tetrahedron $P-ABC$. The dihedral angles between each pair of lateral edges are $\alpha$, and the angle between $PC$ and the plane $PAB$ is $\beta$. Denote the distance from point $O$ to each face as $d$. A moving plane through point $O$ intersects $PC$ at point $S$, and the extensions of $PA$ and $PB$ at points $Q$ and $R$, respectively. Then $\frac{1}{PQ}+\frac{1}{PR}+\frac{1}{PS}$ $\begin{array}{lllll}= & & & \end{array}$ $\qquad$ (the result should be expressed in terms of $\alpha, \beta, d$).

Answer: \dfrac{\sin \beta}{d}


Evie and Odette are playing a game. Three pebbles are placed on the number line; one at -2020 , one at 2020, and one at $n$, where $n$ is an integer between -2020 and 2020. They take it in turns moving either the leftmost or the rightmost pebble to an integer between the other two pebbles. The game ends when the pebbles occupy three consecutive integers.
Odette wins if their sum is odd; Evie wins if their sum is even. For how many values of $n$ can Evie guarantee victory if:
(a) Odette goes first;
(b) Evie goes first?

Answer: 2021


2. Given that $A$ is a two-digit number, the remainder when $A^{2}$ is divided by 15 is 1, then the number of $A$ that satisfies the condition is $($ )

The translation preserves the original text's line breaks and format.

Answer: 24


Five. (Total 20 points) Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the fixed point $A(1,0)$, and the foci are on the $x$-axis. The ellipse intersects the curve $|y|=x$ at points $B$ and $C$. There is a parabola with $A$ as the focus, passing through points $B$ and $C$, and opening to the left, with the vertex coordinates $M(m, 0)$. When the eccentricity $e$ of the ellipse satisfies $1>e^{2}>\frac{2}{3}$, find the range of the real number $m$.

---

The given ellipse is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ and it passes through the point $A(1,0)$. The foci of the ellipse are on the $x$-axis. The ellipse intersects the curve $|y|=x$ at points $B$ and $C$. A parabola with $A$ as the focus, passing through points $B$ and $C$, and opening to the left, has the vertex coordinates $M(m, 0)$. When the eccentricity $e$ of the ellipse satisfies $1>e^{2}>\frac{2}{3}$, find the range of the real number $m$.

Answer: \left(1, \dfrac{3 + \sqrt{2}}{4}\right)


2.3. Determine all functions $f: \mathbb{Z} \longrightarrow \mathbb{Z}$, where $\mathbb{Z}$ is the set of integers, such that
$$
f(m+f(f(n)))=-f(f(m+1))-n
$$
for all integers $m$ and $n$.

Answer: f(n) = -n - 1


Find all integers $n$ such that $20n+2$ divides $2003n+2002$.

The original text has been translated into English while preserving the original line breaks and format.

Answer: -42


1. As shown in Figure 1, it is known that rectangle $A B C D$ can be exactly divided into seven small rectangles of the same shape and size. If the area of the small rectangle is 3, then the perimeter of rectangle $A B C D$ is

Answer: 19


(5) A bag contains one black ball and one white ball. Each time a ball is randomly drawn from the bag, the drawn ball is not returned, but another black ball is added instead. The probability of drawing a black ball on the $n$-th draw is $a=$ $\qquad$ .

Answer: 1 - \dfrac{1}{2^n}


Let $a$ and $b$ be real numbers. Define $f_{a,b}\colon R^2\to R^2$ by $f_{a,b}(x;y)=(a-by-x^2;x)$. If $P=(x;y)\in R^2$, define $f^0_{a,b}(P) = P$ and $f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))$ for all nonnegative integers $k$.

The set $per(a;b)$ of the [i]periodic points[/i] of $f_{a,b}$ is the set of points $P\in R^2$ such that $f_{a,b}^n(P) = P$ for some positive integer $n$.

Fix $b$. Prove that the set $A_b=\{a\in R \mid per(a;b)\neq \emptyset\}$ admits a minimum. Find this minimum.

Answer: -\dfrac{(b + 1)^2}{4}


Ana baked 15 pasties. She placed them on a round plate in a circular
way: 7 with cabbage, 7 with meat and 1 with cherries in that exact order
and put the plate into a microwave. She doesn’t know how the plate has been rotated
in the microwave. She wants to eat a pasty with cherries. Is it possible for Ana, by trying no more than three pasties, to find exactly where the pasty with cherries is?

Answer: Yes


Subject 2. The arithmetic mean of the distinct natural numbers $a_{1}, a_{2}, a_{3}, \ldots, a_{2014}$, 2014, and 2016 is 2015.

a. Find the arithmetic mean of the numbers $a_{1}, a_{2}, a_{3}, \ldots, a_{2014}$.

b. If the numbers $a_{1}, a_{2}, a_{3}, \ldots, a_{2014}$ are even, show that $a_{1} \cdot a_{2} \cdot a_{3} \cdot \ldots \cdot a_{2014}=0$

Answer: 2015


Rubanov I.S.

Petya came up with 1004 reduced quadratic trinomials $f_{1}, \ldots, f_{1004}$, among the roots of which are all integers from 0 to 2007. Vasya considers all possible equations $f_{i}=f_{j}$ ( $i \neq j$ ), and for each root found, Petya pays Vasya one ruble. What is the smallest possible income for Vasya?

Answer: 0


12. (17 points) A rectangular wooden board, if 4 decimeters are cut off from both the longer and shorter sides, the area is reduced by 168 square decimeters. Please ask: What is the original perimeter of the rectangle in decimeters?

Answer: 92


## Task 15/66

For the calculation of the square root of a number $z=p^{2}+a$ with $0 \leq a \leq 2 p+1$, the approximation formula is

$$
\sqrt{z}=\sqrt{p^{2}+a} \approx p+\frac{a}{2 p+1}
$$

How large is the maximum error of this approximation in dependence on $a$? How does this change in dependence on $p$?

Answer: \dfrac{1}{8p + 4}


The natural numbers $1,2,3, \ldots, 10$ are written on a board. It is allowed to write the number $a^{2}$ if the number $a$ is already on the board, or write the least common multiple (LCM) of the numbers $a$ and $b$ if the numbers $a$ and $b$ are already on the board. Is it possible to obtain the number 1,000,000 using these operations?

Answer: \text{No}


Viswam walks half a mile to get to school each day. His route consists of $10$ city blocks of equal length and he takes $1$ minute to walk each block. Today, after walking $5$ blocks, Viswam discovers he has to make a detour, walking $3$ blocks of equal length instead of $1$ block to reach the next corner. From the time he starts his detour, at what speed, in mph, must he walk, in order to get to school at his usual time? If the answer is \(x\) mph, what is the value of \(10x\)?

Answer: 42


Sonya the frog chooses a point uniformly at random lying within the square \([0, 6] \times [0, 6]\) in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from \([0, 1]\) and a direction uniformly at random from \{north, south, east, west\}. All of her choices are independent. Find the probability that she lands outside the square. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 13


A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is \(3\sqrt{3}\) inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. The original answer is in the form \(k\sqrt{m} + n\pi\). Please give the value of k + m + n.

Answer: 10


Recall that in any row of Pascal's Triangle, the first and last elements of the row are $1$ and each other element in the row is the sum of the two elements above it from the previous row. With this in mind, define the *Pascal Squared Triangle* as follows:

- In the $n^{\text{th}}$ row, where $n \geq 1$, the first and last elements of the row equal $n^2$.
- Each other element is the sum of the two elements directly above it.

The first few rows of the Pascal Squared Triangle are shown below.

\[
\begin{array}{c@{\hspace{7em}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{3pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c}
\vspace{4pt}
\text{Row 1: } & & & & & & 1 & & & & & \\
\vspace{4pt}
\text{Row 2: } & & & & & 4 & & 4 & & & & \\
\vspace{4pt}
\text{Row 3: } & & & & 9 & & 8 & & 9 & & & \\
\vspace{4pt}
\text{Row 4: } & & &16& &17& &17& & 16& & \\
\vspace{4pt}
\text{Row 5: } & &25 & &33& &34 & &33 & &25 &
\end{array}
\]

Let $S_n$ denote the sum of the entries in the $n^{\text{th}}$ row. For how many integers $1 \leq n \leq 10^6$ is $S_n$ divisible by $13$?

Answer: 76926


Let $p$ and $q$ be prime numbers such that $(p-1)^{q-1}-1$ is a positive integer that divides $(2q)^{2p}-1$. Compute the sum of all possible values of $pq$.

Answer: 85


There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$, inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $100m + n$.

Answer: 200


The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 + \ldots + 10^{2013}$. If one writes down the number $S$, how often does the digit $5$ occur in the result?

Answer: 4022


Three friends play a game of marbles, agreeing that the loser of each round will double the number of marbles the other players have, meaning he gives the other two a number of marbles such that they end up with twice what they had at the beginning of the round. The first player lost the first round, the second player the second, the third player the third, and they all ended up with 64 marbles each. How many marbles did each of the three friends start with?

Answer: 104,56,32


8.3. In a cinema, five friends took seats numbered 1 to 5 (the leftmost seat is number 1). During the movie, Anya left to get popcorn. When she returned, she found that Varya had moved one seat to the right, Galia had moved three seats to the left, and Diana and Elia had swapped places, leaving the edge seat for Anya. Which seat did Anya occupy before she got up?

Answer: 3


17. What odd digit can the sum of two prime numbers end with, if it is not a single-digit number?

Answer: 1,3,5,9


## 53. Two Sisters

The age difference between two sisters is 4 years. If the cube of the age of the first one minus the cube of the age of the second one equals 988, how old is each sister?

Answer: 7,\quad11


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`, `Carol`
 - People have unique favorite sports: `volleyball`, `soccer`, `basketball`, `baseball`, `swimming`, `tennis`
 - Each person has a unique hobby: `gardening`, `cooking`, `painting`, `woodworking`, `knitting`, `photography`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Alice`, `Samantha`, `Fred`, `Timothy`
 - Each person has a unique birthday month: `feb`, `jan`, `mar`, `may`, `sept`, `april`
 - Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `pizza`, `spaghetti`, `stew`, `soup`

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

Answer: painting


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Bob`, `Alice`, `Eric`, `Carol`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Kailyn`, `Janelle`, `Penny`, `Sarah`
 - Everyone has something unique for lunch: `stir fry`, `stew`, `spaghetti`, `soup`, `pizza`, `grilled cheese`
 - The people keep unique animals: `dog`, `horse`, `cat`, `fish`, `rabbit`, `bird`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `lime`, `cherry`, `blueberry`, `dragonfruit`
 - Each person lives in a unique style of house: `mediterranean`, `craftsman`, `victorian`, `colonial`, `modern`, `ranch`

## Clues:
1. The person whose mother's name is Penny is the person residing in a Victorian house.
2. The person in a modern-style house is The person whose mother's name is Kailyn.
3. The person in a modern-style house is somewhere to the right of the person residing in a Victorian house.
4. The person who drinks Blueberry smoothies is not in the sixth house.
5. The person whose mother's name is Sarah is the person who drinks Blueberry smoothies.
6. The person who likes Cherry smoothies is not in the third house.
7. The person who likes Cherry smoothies is Eric.
8. The person whose mother's name is Janelle is not in the third house.
9. The person who loves the spaghetti eater is in the third house.
10. The fish enthusiast is The person whose mother's name is Holly.
11. The person who keeps horses is in the first house.
12. The person in a Mediterranean-style villa is the person who loves the stew.
13. The bird keeper is in the third house.
14. The person living in a colonial-style house is the person who drinks Blueberry smoothies.
15. Carol is the person who keeps horses.
16. There are two houses between the Desert smoothie lover and Peter.
17. The person who loves eating grilled cheese is the Dragonfruit smoothie lover.
18. The person who is a pizza lover is somewhere to the right of the Dragonfruit smoothie lover.
19. The person whose mother's name is Janelle is somewhere to the left of Arnold.
20. Carol is directly left of The person whose mother's name is Aniya.
21. The person in a ranch-style home is The person whose mother's name is Aniya.
22. Arnold is somewhere to the right of the person in a Craftsman-style house.
23. The cat lover is Bob.
24. There is one house between the Watermelon smoothie lover and the person who is a pizza lover.
25. There are two houses between the cat lover and the person who loves stir fry.
26. Arnold is the rabbit owner.
27. Peter is The person whose mother's name is Janelle.
 What is the value of attribute House for the person whose attribute Mother is Aniya? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Carol`, `Bob`, `Eric`, `Peter`
 - People use unique phone models: `iphone 13`, `samsung galaxy s21`, `xiaomi mi 11`, `oneplus 9`, `google pixel 6`, `huawei p50`
 - Each person has a unique birthday month: `april`, `mar`, `feb`, `may`, `jan`, `sept`
 - Each person has an occupation: `nurse`, `doctor`, `engineer`, `teacher`, `artist`, `lawyer`

## Clues:
1. The person who uses a Google Pixel 6 is directly left of the person whose birthday is in March.
2. The person who is a nurse is Eric.
3. Arnold is directly left of the person whose birthday is in April.
4. Alice and the person whose birthday is in January are next to each other.
5. The person who is an artist is the person whose birthday is in May.
6. The person whose birthday is in September is not in the sixth house.
7. There is one house between the person whose birthday is in March and the person who is a nurse.
8. The person whose birthday is in March is the person who is a lawyer.
9. The person who uses a Xiaomi Mi 11 is directly left of the person who is a nurse.
10. The person who is a teacher is not in the sixth house.
11. Eric is the person who uses an iPhone 13.
12. The person who is an engineer and Peter are next to each other.
13. The person whose birthday is in January is in the third house.
14. The person who uses a Samsung Galaxy S21 is Bob.
15. The person who is a lawyer and the person whose birthday is in April are next to each other.
16. The person who uses a OnePlus 9 is somewhere to the right of the person who is a doctor.
 What is the value of attribute PhoneModel for the person whose attribute House is 6? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Carol`, `Alice`, `Eric`, `Bob`
 - People have unique favorite music genres: `pop`, `hip hop`, `rock`, `jazz`, `classical`, `country`
 - The people keep unique animals: `fish`, `dog`, `cat`, `rabbit`, `horse`, `bird`
 - Each person lives in a unique style of house: `modern`, `colonial`, `mediterranean`, `ranch`, `victorian`, `craftsman`
 - Each person has a unique hobby: `gardening`, `woodworking`, `knitting`, `cooking`, `painting`, `photography`
 - The people are of nationalities: `brit`, `german`, `norwegian`, `swede`, `dane`, `chinese`

## Clues:
1. The cat lover is the person in a Mediterranean-style villa.
2. The person who loves country music is somewhere to the right of the rabbit owner.
3. The person who loves pop music is not in the second house.
4. Alice is somewhere to the right of Arnold.
5. The person who loves jazz music and the photography enthusiast are next to each other.
6. The person who loves rock music is the German.
7. The woodworking hobbyist is somewhere to the right of the person in a modern-style house.
8. The person who loves hip-hop music is the Norwegian.
9. The cat lover is somewhere to the left of the woodworking hobbyist.
10. Alice is the person who paints as a hobby.
11. The person in a Craftsman-style house is Carol.
12. Peter is the fish enthusiast.
13. The person who loves hip-hop music is not in the fifth house.
14. The person who loves pop music is not in the sixth house.
15. The person who loves cooking is somewhere to the left of the Swedish person.
16. The Dane is directly left of the person who keeps horses.
17. The rabbit owner is somewhere to the left of the British person.
18. There is one house between Alice and the person who loves country music.
19. The person who paints as a hobby is the person living in a colonial-style house.
20. The German is somewhere to the left of the photography enthusiast.
21. Eric is the Norwegian.
22. The person residing in a Victorian house is directly left of the person who enjoys gardening.
23. There are two houses between the person who loves cooking and the person who loves classical music.
24. The bird keeper is the person who loves cooking.
25. The cat lover is directly left of Bob.
26. Eric is not in the first house.
27. Carol is somewhere to the right of the German.
28. The rabbit owner is directly left of the person in a modern-style house.
 What is the value of attribute MusicGenre for the person whose attribute Hobby is gardening? Please reason step by step, and put your final answer within \boxed{}

Answer: hip hop


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`, `Arnold`, `Peter`, `Eric`, `Bob`, `Carol`
 - People have unique favorite sports: `baseball`, `tennis`, `swimming`, `basketball`, `soccer`, `volleyball`

## Clues:
1. The person who loves tennis is Eric.
2. The person who loves volleyball is in the fifth house.
3. Arnold is somewhere to the right of Carol.
4. Carol is not in the first house.
5. The person who loves baseball is Bob.
6. There is one house between the person who loves swimming and the person who loves basketball.
7. Alice is in the fourth house.
8. The person who loves swimming is Peter.
9. Alice and Arnold are next to each other.
10. Peter is somewhere to the right of Eric.
11. The person who loves swimming is not in the sixth house.
 What is the value of attribute House for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: 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: `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 Name for the person whose attribute House is 5? 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: `Eric`, `Alice`, `Arnold`, `Carol`, `Peter`, `Bob`
 - Each person has a unique level of education: `high school`, `trade school`, `bachelor`, `doctorate`, `master`, `associate`
 - Each person has a favorite color: `purple`, `blue`, `white`, `red`, `green`, `yellow`
 - Each person prefers a unique type of vacation: `cultural`, `cruise`, `beach`, `mountain`, `camping`, `city`
 - People have unique favorite book genres: `science fiction`, `biography`, `historical fiction`, `fantasy`, `romance`, `mystery`

## Clues:
1. Alice is the person with a master's degree.
2. There is one house between the person with an associate's degree and Peter.
3. The person who loves historical fiction books is somewhere to the left of the person who loves biography books.
4. The person who loves historical fiction books is the person with a high school diploma.
5. Bob is the person with a bachelor's degree.
6. Carol is somewhere to the right of Eric.
7. The person with a master's degree is somewhere to the right of the person who enjoys camping trips.
8. The person who loves science fiction books is somewhere to the left of the person who loves yellow.
9. The person who loves historical fiction books is somewhere to the left of the person whose favorite color is green.
10. The person with a high school diploma is not in the second house.
11. The person who loves beach vacations is the person with a doctorate.
12. The person who loves mystery books is somewhere to the left of the person with a bachelor's degree.
13. The person with a doctorate is in the first house.
14. The person with a high school diploma is directly left of the person who enjoys camping trips.
15. The person who loves romance books is not in the third house.
16. There is one house between the person who loves purple and the person with a bachelor's degree.
17. The person who loves biography books is the person whose favorite color is red.
18. There is one house between the person who loves fantasy books and the person who loves blue.
19. The person who goes on cultural tours is the person who loves blue.
20. Carol is not in the fifth house.
21. The person who loves purple is the person with a doctorate.
22. The person who enjoys mountain retreats is somewhere to the right of the person who loves science fiction books.
23. The person who prefers city breaks is the person whose favorite color is green.
 What is the value of attribute Color for the person whose attribute House is 4? 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: `Carol`, `Alice`, `Eric`, `Peter`, `Arnold`, `Bob`
 - The mothers' names in different houses are unique: `Penny`, `Janelle`, `Aniya`, `Sarah`, `Holly`, `Kailyn`
 - Everyone has a unique favorite cigar: `blends`, `dunhill`, `blue master`, `prince`, `yellow monster`, `pall mall`
 - Each person has a unique type of pet: `cat`, `dog`, `rabbit`, `hamster`, `bird`, `fish`
 - The people keep unique animals: `bird`, `horse`, `cat`, `rabbit`, `dog`, `fish`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `lime`, `cherry`, `blueberry`, `dragonfruit`

## Clues:
1. The dog owner is not in the third house.
2. The fish enthusiast is not in the fourth house.
3. The person who owns a dog is Eric.
4. The cat lover is not in the second house.
5. The person who smokes Blue Master is somewhere to the right of The person whose mother's name is Sarah.
6. The cat lover is not in the fourth house.
7. There are two houses between Carol and the person who owns a rabbit.
8. The person whose mother's name is Penny is the person with a pet hamster.
9. The person whose mother's name is Kailyn is somewhere to the right of the person with a pet hamster.
10. The dog owner and the person who drinks Lime smoothies are next to each other.
11. The Prince smoker is somewhere to the right of The person whose mother's name is Janelle.
12. There is one house between The person whose mother's name is Penny and Bob.
13. There are two houses between the person partial to Pall Mall and The person whose mother's name is Penny.
14. Peter is somewhere to the left of the person who drinks Lime smoothies.
15. The bird keeper is the Watermelon smoothie lover.
16. The person with a pet hamster is the Prince smoker.
17. The person who smokes Yellow Monster is the person who drinks Lime smoothies.
18. The person whose mother's name is Sarah is the person who smokes Yellow Monster.
19. The Dunhill smoker is somewhere to the left of the Watermelon smoothie lover.
20. The person whose mother's name is Holly is directly left of the person who owns a dog.
21. The bird keeper is The person whose mother's name is Kailyn.
22. The person whose mother's name is Sarah is not in the third house.
23. The person who smokes many unique blends is the rabbit owner.
24. The person who has a cat is somewhere to the right of the person with an aquarium of fish.
25. The person who smokes Yellow Monster is Eric.
26. The person who keeps a pet bird is the Desert smoothie lover.
27. Alice is The person whose mother's name is Kailyn.
28. The Dragonfruit smoothie lover is Carol.
29. The person who keeps horses is the person who likes Cherry smoothies.
 What is the value of attribute Mother for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: Sarah


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`, `Alice`, `Bob`, `Arnold`, `Eric`, `Carol`
 - People own unique car models: `tesla model 3`, `honda civic`, `toyota camry`, `chevrolet silverado`, `bmw 3 series`, `ford f150`

## Clues:
1. Alice is not in the fifth house.
2. Carol is somewhere to the left of the person who owns a Chevrolet Silverado.
3. The person who owns a Toyota Camry is Bob.
4. The person who owns a BMW 3 Series is not in the second house.
5. The person who owns a Honda Civic is somewhere to the right of Arnold.
6. There is one house between the person who owns a BMW 3 Series and the person who owns a Chevrolet Silverado.
7. There is one house between Arnold and Bob.
8. The person who owns a Tesla Model 3 is not in the sixth house.
9. The person who owns a Toyota Camry is not in the third house.
10. The person who owns a Tesla Model 3 is Peter.
11. The person who owns a Honda Civic and the person who owns a Toyota Camry are next to each other.
12. The person who owns a Toyota Camry is not in the fifth house.
13. The person who owns a Chevrolet Silverado is not in the second 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: 5


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`
 - People have unique heights: `short`, `very short`, `average`, `tall`
 - Each person has a favorite color: `yellow`, `white`, `red`, `green`
 - They all have a unique favorite flower: `lilies`, `roses`, `daffodils`, `carnations`
 - The people keep unique animals: `bird`, `fish`, `horse`, `cat`

## Clues:
1. The person who loves a carnations arrangement is somewhere to the left of the bird keeper.
2. Peter is not in the first house.
3. The person who loves the rose bouquet is directly left of Eric.
4. The person who loves a carnations arrangement and the person who loves the rose bouquet are next to each other.
5. Alice is directly left of the bird keeper.
6. The person who keeps horses is Peter.
7. The fish enthusiast is the person who loves a bouquet of daffodils.
8. The person whose favorite color is red is the person who loves a bouquet of daffodils.
9. The person who is tall is in the third house.
10. The person who has an average height is somewhere to the left of the person who loves white.
11. The person whose favorite color is green is not in the second house.
12. The person who is short is somewhere to the left of the person who is tall.
13. The person who is short is the person whose favorite color is green.
 What is the value of attribute House for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - People have unique favorite sports: `soccer`, `basketball`, `tennis`
 - The people keep unique animals: `cat`, `bird`, `horse`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`
 - The people are of nationalities: `swede`, `dane`, `brit`

## Clues:
1. The person who likes milk is not in the first house.
2. The person who keeps horses is somewhere to the left of Eric.
3. The person who loves basketball is Peter.
4. The Swedish person is somewhere to the left of the one who only drinks water.
5. The person who loves tennis is the tea drinker.
6. The person who loves soccer is the bird keeper.
7. The cat lover is the British person.
8. Peter is not in the third house.
9. The person who likes milk is the person who loves soccer.
 What is the value of attribute House for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Eric`
 - Each mother is accompanied by their child: `Bella`, `Meredith`, `Fred`
 - People use unique phone models: `samsung galaxy s21`, `iphone 13`, `google pixel 6`
 - People have unique favorite book genres: `romance`, `science fiction`, `mystery`
 - Everyone has a unique favorite cigar: `blue master`, `prince`, `pall mall`
 - People have unique heights: `short`, `very short`, `average`

## Clues:
1. Eric is directly left of the Prince smoker.
2. The person who has an average height is somewhere to the right of Peter.
3. The person who uses a Samsung Galaxy S21 is in the second house.
4. The person who smokes Blue Master is in the third house.
5. The person who uses a Google Pixel 6 is Eric.
6. Arnold is somewhere to the right of the person who loves science fiction books.
7. The person who uses an iPhone 13 is the person's child is named Bella.
8. The person who is short is the Prince smoker.
9. The person who loves mystery books is the person who uses a Google Pixel 6.
10. The person's child is named Meredith and the person who is short are next to each other.
 What is the value of attribute Children for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: Meredith


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - Each person has a unique birthday month: `sept`, `jan`, `april`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`
 - People have unique favorite book genres: `science fiction`, `romance`, `mystery`
 - The mothers' names in different houses are unique: `Janelle`, `Holly`, `Aniya`

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

Answer: spaghetti


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`
 - Everyone has something unique for lunch: `pizza`, `grilled cheese`, `spaghetti`
 - Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`
 - Each person has a unique type of pet: `fish`, `dog`, `cat`

## Clues:
1. The person who likes Cherry smoothies is the person with an aquarium of fish.
2. The person who loves the spaghetti eater is directly left of the person who has a cat.
3. The person who has a cat is Eric.
4. The Watermelon smoothie lover is Arnold.
5. The person who owns a dog is directly left of the person who is a pizza lover.
6. The Desert smoothie lover is not in the third house.
 What is the value of attribute Food for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


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`, `Arnold`, `Carol`, `Peter`, `Eric`, `Bob`
 - Each person has a unique type of pet: `cat`, `bird`, `fish`, `dog`, `hamster`, `rabbit`
 - People have unique heights: `super tall`, `average`, `tall`, `very short`, `very tall`, `short`
 - People have unique favorite sports: `baseball`, `tennis`, `soccer`, `swimming`, `basketball`, `volleyball`

## Clues:
1. There is one house between the person who has an average height and the person who is short.
2. The person with an aquarium of fish is in the fifth house.
3. The person who has a cat is the person who is tall.
4. The person who is very tall is the person who loves soccer.
5. Peter is in the first house.
6. The person who loves tennis is in the sixth house.
7. Carol is somewhere to the right of the person who is very tall.
8. The person who loves volleyball is directly left of the person who loves soccer.
9. The person who loves baseball is directly left of the person who is super tall.
10. Carol and Bob are next to each other.
11. The person who is super tall is not in the sixth house.
12. The person who is short is not in the fourth house.
13. There is one house between the person who loves swimming and Bob.
14. The person who loves basketball is somewhere to the left of the person who owns a dog.
15. The person who owns a rabbit is somewhere to the left of the person who is short.
16. The person who keeps a pet bird is the person who is short.
17. Arnold is not in the second house.
18. Bob is directly left of Eric.
19. The person who owns a rabbit and the person who is very tall are next to each other.
 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: 3


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`, `Bob`, `Alice`, `Eric`, `Peter`
 - People have unique heights: `very tall`, `average`, `tall`, `very short`, `short`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `spaghetti`, `pizza`, `stir fry`

## Clues:
1. Alice is the person who is short.
2. The person who is tall is in the third house.
3. The person who has an average height is not in the second house.
4. The person who has an average height is somewhere to the left of the person who loves the stew.
5. The person who loves stir fry is Arnold.
6. The person who is a pizza lover is the person who is tall.
7. Eric is the person who is tall.
8. Bob is somewhere to the right of Arnold.
9. The person who loves eating grilled cheese is somewhere to the right of Eric.
10. The person who is very short is somewhere to the left of Arnold.
 What is the value of attribute Food for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: grilled cheese


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`
 - People have unique favorite music genres: `pop`, `rock`, `classical`
 - People have unique favorite sports: `tennis`, `soccer`, `basketball`
 - Everyone has a unique favorite cigar: `blue master`, `pall mall`, `prince`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Janelle`

## Clues:
1. The person who loves basketball is somewhere to the right of the person partial to Pall Mall.
2. The Prince smoker is not in the third house.
3. Eric is The person whose mother's name is Holly.
4. The person whose mother's name is Aniya is the person who loves pop music.
5. Arnold is directly left of the Prince smoker.
6. The person who loves tennis is the person who smokes Blue Master.
7. The Prince smoker is somewhere to the right of the person who loves classical music.
8. The person whose mother's name is Aniya is not in the second house.
 What is the value of attribute House for the person whose attribute MusicGenre is pop? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Bob`, `Peter`, `Arnold`, `Carol`
 - People have unique heights: `very tall`, `tall`, `super tall`, `average`, `very short`, `short`
 - People use unique phone models: `oneplus 9`, `google pixel 6`, `samsung galaxy s21`, `iphone 13`, `huawei p50`, `xiaomi mi 11`

## Clues:
1. Bob is directly left of the person who is tall.
2. Peter is somewhere to the left of the person who uses an iPhone 13.
3. The person who is very short is somewhere to the right of the person who uses a Google Pixel 6.
4. Carol is the person who is very tall.
5. There is one house between the person who uses a Google Pixel 6 and the person who is short.
6. The person who uses a Samsung Galaxy S21 is not in the first house.
7. The person who uses a OnePlus 9 is directly left of the person who is short.
8. The person who is tall is Arnold.
9. The person who is super tall is in the first house.
10. The person who uses a Xiaomi Mi 11 is Carol.
11. The person who uses a Google Pixel 6 is somewhere to the right of Eric.
12. The person who is short is in the sixth house.
 What is the value of attribute House for the person whose attribute Height is super tall? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Alice`, `Peter`, `Bob`, `Arnold`
 - Each person has a unique type of pet: `hamster`, `fish`, `cat`, `dog`, `bird`
 - Each person has an occupation: `doctor`, `engineer`, `artist`, `lawyer`, `teacher`
 - Each person has a favorite color: `green`, `blue`, `yellow`, `red`, `white`
 - The people keep unique animals: `bird`, `cat`, `horse`, `fish`, `dog`

## Clues:
1. The person who owns a dog is Arnold.
2. The bird keeper is in the fourth house.
3. The person who keeps a pet bird is directly left of the dog owner.
4. The person who loves white is somewhere to the left of the person who is a lawyer.
5. The person who loves yellow is directly left of the person whose favorite color is green.
6. The cat lover is Bob.
7. The cat lover is somewhere to the left of Eric.
8. The person who keeps horses is in the fifth house.
9. The person who is a lawyer is directly left of the person who is a teacher.
10. The person who is a doctor is in the first house.
11. Alice is the person who loves yellow.
12. The person who loves blue is directly left of the person with an aquarium of fish.
13. The person who loves yellow is in the first house.
14. The person with a pet hamster is the person who is an artist.
15. Eric is the dog owner.
 What is the value of attribute House for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Peter`, `Eric`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `prince`, `dunhill`
 - People have unique favorite sports: `soccer`, `swimming`, `tennis`, `basketball`
 - Each person lives in a unique style of house: `ranch`, `colonial`, `craftsman`, `victorian`
 - Everyone has a favorite smoothie: `watermelon`, `desert`, `cherry`, `dragonfruit`

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

Answer: 3


On an $8 \times 8$ board, several dominoes (rectangles of $2 \times 1$) can be placed without overlapping. Let $N$ be the number of ways to place 32 dominoes in this manner, and $T$ be the number of ways to place 24 dominoes in this manner. Which is greater: $-N$ or $M$? Configurations that can be derived from one another by rotating or reflecting the board are considered distinct.

Answer: T


In a carriage, any group of $m \ (m \geq 3)$ passengers have only one common friend (where if A is a friend of B, then B is also a friend of A; no one is their own friend). What is the maximum number of friends that the person with the most friends in this carriage can have?

Answer: m


We intersected the unit cube $ABCDEFGH$ with a plane such that it intersects the edges $AB$ and $AD$ at the internal points $P$ and $Q$, respectively, both at a distance $x$ from $A$, and it intersects the edge $BF$ at the point $R$. What is the distance $BR$ if $\angle QPR = 120^\circ$?

Answer: 1-x


Given positive integers \( x, y, z \) and real numbers \( a, b, c, d \) such that \( x \leqslant y \leqslant z, x^a = y^b = z^c = 70^d \), and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d} \), determine the relationship between \( x + y \) and \( z \). Fill in the blank with “ \( > \)”, “ \( < \)” or “ \( = \)”.


Given natural numbers \( m \) and \( n \). There are two piles on the table: the first pile contains \( n \) stones, and the second pile contains \( m \) stones. Petya and Vasya play the following game. Petya starts. In one move, a player can break one of the available piles on the table into several smaller piles. The player who cannot make a move loses. For which values of \( m \) and \( n \) can Petya ensure victory regardless of Vasya's play?

Answer: m \neq n


There are three piles of stones. Each time A moves one stone from one pile to another, A earns a reward from B. The reward is equal to the difference in the number of stones between the pile to which A moves the stone and the pile from which A takes the stone. If this difference is negative, A must pay B that amount instead (if A doesn’t have enough money, A can owe it temporarily). At a certain moment, all the stones are in the piles where they initially started. Determine the maximum possible amount of money A can have by that moment.

Answer: 0


Several non-intersecting line segments are placed on a plane. Is it always possible to connect their endpoints with additional line segments so that a closed, non-self-intersecting polygonal chain is formed?

Answer: \text{No}


A three-digit natural number was written on the board. We wrote down all other three-digit numbers that can be obtained by rearranging its digits. Therefore, there were three new numbers in addition to the original number on the board. The sum of the smallest two of all four numbers is 1088. What digits does the original number contain?

Answer: 0, 5, 8


6. A point on the coordinate plane whose both horizontal and vertical coordinates are integers is called an integer point. The number of integer points in the region enclosed by the parabola $y=x^{2}+1$ and the line $2 x-y+81=0$ is $\qquad$ .

Answer: 988


Four, (Full marks 20 points) As shown in the figure, the side length of the equilateral $\triangle ABC$ is $1$, and points $R, P, Q$ on $AB, BC, CA$ satisfy $AR + BP + CQ = 1$ while moving. Let $BP = x, CQ = y$, $AR = z$, and the area of $\triangle PQR$ be $S$. Try to express $S$ in terms of $x, y, z$.

---

The translation retains the original text's line breaks and formatting.

Answer: S = \dfrac{\sqrt{3}}{4}(xy + yz + zx)


A person moves in the $x-y$ plane moving along points with integer co-ordinates $x$ and $y$ only. When she is at a point $(x,y)$, she takes a step based on the following rules:
(a) if $x+y$ is even she moves to either $(x+1,y)$ or $(x+1,y+1)$;
(b) if $x+y$ is odd she moves to either $(x,y+1)$ or $(x+1,y+1)$.
How many distinct paths can she take to go from $(0,0)$ to $(8,8)$ given that she took exactly three steps to the right $((x,y)$ to $(x+1,y))$?

Answer: 462


Let's determine the largest of the $A$ numbers for which the following statement holds. No matter how we write down the first 100 positive integers, there will always be ten consecutive numbers in the sequence whose sum is at least $A$.

Answer: 505


[ The sum of the angles of a triangle. The theorem about the exterior angle.] [ Criteria and properties of an isosceles triangle. ]

On the side $B C$ of the isosceles triangle $A B C(A B=B C)$, points $N$ and $M$ ( $N$ is closer to $B$ than $M$) are taken such that $N M=A M$ and $\angle M A C=\angle B A N$.

Find $\angle C A N$.

Answer: 60^\circ


Adrian has $2n$ cards numbered from $ 1$ to $2n$. He gets rid of $n$ cards that are consecutively numbered. The sum of the numbers of the remaining papers is $1615$. Find all the possible values of $n$.

Answer: 34


Show that \(X\) is an element of \(\mathcal{F}^{*}\) of order \(r\).

Also, introduce the notion of introspection as described by the authors of the AKS algorithm.

#### Definition 2.4.7.

An integer \(k\) is said to be introspective for a polynomial \(P(X) \in(\mathbb{Z} / p \mathbb{Z})[X]\) if \(P\left(X^{k}\right) \equiv P(X)^{k}\) \((\bmod H(X), p)\).

Answer: r


11.1. Solve the system $\left\{\begin{array}{c}x^{2}+4 \sin ^{2} y-4=0 \\ \cos x-2 \cos ^{2} y-1=0\end{array}\right.$.

Answer: (0, \frac{\pi}{2} + k\pi)


2. Solve the equation $\cos \left(\pi x^{2}\right)-\cos ^{2}\left(\frac{\pi x^{2}}{2}\right)+1+\cos \left(\pi x^{2}-4 \pi x\right)=\sin ^{2}\left(\frac{\pi x^{2}}{2}\right)$. In your answer, specify the third term of the increasing sequence of all positive roots of the equation.

Answer: \dfrac{3}{4}


For a positive integer $n$, there is a school with $2n$ people. For a set $X$ of students in this school, if any two students in $X$ know each other, we call $X$ [i]well-formed[/i]. If the maximum number of students in a well-formed set is no more than $n$, find the maximum number of well-formed set.

Here, an empty set and a set with one student is regarded as well-formed as well.

Answer: 3^n


$\mathrm{Az}$

$$
a_{n}=\frac{(n+3)^{2}+3}{n(n+1)(n+2)} \cdot \frac{1}{2^{n+1}}
$$

is a sequence defined by its general term. We form the sequence

$$
b_{n}=\sum_{k=1}^{n} a_{k}
$$

Determine the limit of the sequence $b_{n}$ as $n \rightarrow+\infty$.

Answer: 1


4. Let $H$ be the orthocenter of $\triangle A B C$ and let $A_{1}, B_{1}, C_{1}$ be the feet of the altitudes dropped from $A, B, C$ respectively. Let $\frac{\overline{A H}}{\overline{H A_{1}}}+\frac{\overline{B H}}{\overline{H B_{1}}}+\frac{\overline{C H}}{\overline{H C_{1}}}=2008$. Calculate the product $\frac{\overline{A H}}{\overline{H A_{1}}} \cdot \frac{\overline{B H}}{\overline{H B_{1}}} \cdot \frac{\overline{C H}}{\overline{H C_{1}}}$.

Answer: 2010


The curve \( L \) divides an equilateral triangle \( ABC \) with side length \( a \) into two parts of equal area. Prove that the length \( l \) of the curve \( L \) satisfies the inequality \( l \geqslant \frac{\sqrt{\pi}}{2 \sqrt[4]{3}} a \).

Answer: l \geqslant \frac{\sqrt{\pi}}{2 \sqrt[4]{3}} a


\section*{Task 4 - 121224}

In a city, a network of at least two bus lines is to be established. This network must satisfy the following conditions:

(1) Each line has exactly three stops.

(2) Each line has exactly one stop in common with every other line.

(3) It is possible to reach any stop from any other stop with one line, without having to change lines in between.

Determine all possibilities for the number of bus lines in such a network.

Answer: 7


The conductance of a $0.0015 \mathrm{M}$ aqueous solution of a weak monobasic acid was determined by using a conductivity cell consisting of platinized Pt electrodes. The distance between the electrodes is $120 \mathrm{~cm}$ with an area of cross section of $1 \mathrm{~cm}^{2}$. The conductance of this solution was found to be $5 \times 10^{-7} \mathrm{~S}$. The $\mathrm{pH}$ of the solution is 4 . The value of limiting molar conductivity $\left(\Lambda_{m}^{o}\right)$ of this weak monobasic acid in aqueous solution is $Z \times 10^{2} \mathrm{~S} \mathrm{~cm}^{-1} \mathrm{~mol}^{-1}$. What is the value of $Z$?
Please give the answer directly without any intermediate steps.

Answer: 6


13. From the consecutive natural numbers $1,2,3, \cdots, 2014$, select $n$ numbers such that these $n$ numbers satisfy: taking any two of them, one will not be 5 times the other. Try to find the maximum value of $n$, and explain the reason.

Answer: 1679


Murashkin M.V.

On each cell of a $10 \times 10$ board, there is a chip. It is allowed to choose a diagonal with an even number of chips and remove any chip from it.

What is the maximum number of chips that can be removed from the board using such operations?

Answer: 90


B3. We start with a square with side 1. In the first minute, squares with side $\frac{1}{3}$ grow in the middle of the vertical sides. In the next minute, squares with sides of $\frac{1}{3}$ of the length of those line segments grow in the middle of all vertical line segments of the new figure. See below for the situation after 0, 1, and 2 minutes.
![](https://cdn.mathpix.com/cropped/2024_04_17_fd6be77049c724ba1866g-1.jpg?height=178&width=880&top_left_y=1966&top_left_x=586)

This process continues. Every minute, a square with a side of $\frac{1}{3}$ of the length of each vertical line segment grows in the middle of each vertical line segment. After an hour, squares have grown 60 times.

What is the perimeter of the figure after an hour?

Answer: 84


1. Given the function $f(x)=[x[x[x]]]$, where $[x]$ denotes the greatest integer not exceeding $x$. If its domain is $[0,4]$, then its range is $\qquad$ .

Answer: \{0, 1, 8, 9, 12, 13, 14, 27, 28, 29, 33, 34, 35, 36, 40, 41, 42, 43, 64\}


Insert two digits into the number 2019 so that the resulting six-digit number

- starts with 2 and ends with 9,
- is composed of six different digits,
- is divisible by three,
- its first three-digit number is divisible by three,
- its first four-digit number is divisible by four,
- the sum of the inserted digits is odd.

Determine the difference between the largest and smallest six-digit number with the given properties.

(L. Růžičková)

Answer: 69180


Example 2. In space, there are 10 points, 4 of which lie on the same plane, and no other set of 4 points are coplanar; find the number of circular cones (not necessarily right circular cones) with one of the points as the vertex and a circle passing through 3 other points as the base.

Answer: 836


For any set $A=\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\}$ of four distinct positive integers with sum $s_{A}=a_{1}+a_{2}+a_{3}+a_{4}$, let $p_{A}$ denote the number of pairs $(i, j)$ with $1 \leq i<j \leq 4$ for which $a_{i}+a_{j}$ divides $s_{A}$. Among all sets of four distinct positive integers, determine those sets $A$ for which $p_{A}$ is maximal.  Answer. The sets $A$ for which $p_{A}$ is maximal are the sets the form $\{d, 5 d, 7 d, 11 d\}$ and $\{d, 11 d, 19 d, 29 d\}$, where $d$ is any positive integer. For all these sets $p_{A}$ is 4.

Answer: \{d, 5d, 7d, 11d\}


12th Chinese 1997 Problem B2 Let X be the set of residues mod 17. We regard two members of X as adjacent if they differ by 1, so 0 and 16 are adjacent. We say that a permutation of X is dispersive if it never takes two adjacent values to two adjacent values, and connective if it always takes two adjacent values to two adjacent values. What is the largest N for which we can find a permutation p on X such that p, p 2 , ... , p N-1 are all dispersive and p N is connective?

Answer: 8


Two cones have a common vertex, and the generatrix of the first cone is the height of the second. The angle at the vertex of the axial section of the first cone is $\arccos \frac{1}{3}$, and that of the second is $-120^{\circ}$. Find the angle between the generatrices at which the lateral surfaces of the cones intersect.

Answer: 60^\circ


4. Let the function be
$$
\begin{array}{l}
f(x)=\sqrt{10-6 \cos x}+\sqrt{\frac{17}{8}-\frac{3 \sqrt{2}}{2} \sin x}+ \\
\sqrt{19-2 \sqrt{2} \cos x-8 \sin x} \text {. } \\
\end{array}
$$

For all real numbers $x$, the minimum value of $f(x)$ is
$\qquad$ .

Answer: \dfrac{21\sqrt{2}}{4} - 1


12. (12 points) Person A and Person B start from points A and B respectively at the same time, moving towards each other at a constant speed. When A and B meet at point C, Person C starts from B, moving at a constant speed towards A. When A and C meet at point D, A immediately turns around and reduces their speed to 80% of the original speed. When A and C arrive at A at the same time, B is still 720 meters away from A. If the distance between C and D is 900 meters, then the distance between A and B is $\qquad$ meters.

Answer: 5265


133. Digits and Prime Numbers. Using each of the nine digits once and only once, form prime numbers (numbers that do not divide evenly by any integer except 1 and themselves) whose sum is the smallest.

For example, four prime numbers

61

+283

47

59

450

contain all nine digits once and only once, and their sum is 450, but it can be significantly reduced. This is a very simple puzzle.

Answer: 207


The non-parallel sides of a trapezoid are extended until they intersect, and a line parallel to the bases of the trapezoid is drawn through this point of intersection. Find the length of the segment of this line that is limited by the extensions of the diagonals if the lengths of the bases of the trapezoid are $a$ and $b$.

Answer: \dfrac{2ab}{|a - b|}


15. Given the function $f(x)=x^{2}+x-2$.
(1) Try to find the analytical expression of the function $g(x)=\frac{|f(x)|-f(x)}{2}$;
(2) If $a>0$, when the line $y=a x+b$ intersects the curve $y=g(x)$ at three different points, try to determine the relationship between $a$ and $b$, and draw a diagram to represent the region where the point $(a, b)$ lies with $a$ and $b$ as coordinates.

Answer: g(x) = \begin{cases} 0 & \text{if } x \leq -2 \text{ or } x \geq 1, \\ -x^2 - x + 2 & \text{if } -2 < x < 1. \end{cases}


5. (10 points) On a horizontal straight line, a square and two isosceles right triangles are placed. If the area of the square placed diagonally is 6 square centimeters, then the total area of the shaded parts is $\qquad$ square centimeters.

Answer: 3


There is a $2n\times 2n$ rectangular grid and a chair in each cell of the grid. Now, there are $2n^2$ pairs of couple are going to take seats. Define the distance of a pair of couple to be the sum of column difference and row difference between them. For example, if a pair of couple seating at $(3,3)$ and $(2,5)$ respectively, then the distance between them is $|3-2|+|3-5|=3$. Moreover, define the total distance to be the sum of the distance in each pair. Find the maximal total distance among all possibilities.

Answer: 4n^3


A variable tangent $t$ to the circle $C_1$, of radius $r_1$, intersects the circle $C_2$, of radius $r_2$ in $A$ and $B$. The tangents to $C_2$ through $A$ and $B$ intersect in $P$.
Find, as a function of $r_1$ and $r_2$, the distance between the centers of $C_1$ and $C_2$ such that the locus of $P$ when $t$ varies is contained in an equilateral hyperbola.

[b]Note[/b]: A hyperbola is said to be [i]equilateral[/i] if its asymptotes are perpendicular.

Answer: \sqrt{2} r_1


323. Find the area of the triangle formed by the intersection of a sphere of radius $R$ with a trihedral angle, the dihedral angles of which are equal to $\alpha, \beta$ and $\gamma$, and the vertex coincides with the center of the sphere.

Answer: (\alpha + \beta + \gamma - \pi) R^2


6. Five spherical surfaces can divide space into $\qquad$ parts.

Answer: 30


19. In base five, there is $(\overline{a b b})_{5} \times(\overline{a c d})_{5}=(\overline{b a e c a})_{5}$, where the same letters represent the same digits, and different letters represent different digits. Then, the product of this equation converted to a decimal number is . $\qquad$

Answer: 2021


6. A triangular piece of paper of area 1 is folded along a line parallel to one of the sides and pressed flat. What is the minimum possible area of the resulting figure?

Answer: \dfrac{2}{3}


Let us call a natural number useful if it does not contain any zeros or repeated digits in its decimal notation, and the product of all its digits is a multiple of their sum. Do there exist two consecutive three-digit useful numbers?

Answer: Yes


8. Hexagon $A B C D E F$ has a circumscribed circle and an inscribed circle. If $A B=9, B C=6, C D=2$, and $E F=4$. Find $\{D E, F A\}$.

Answer: \{1, 8\}


## Task A-2.2.

A 2021-digit number is written. Each two-digit number formed by two consecutive digits of this number (without changing the order) is divisible by 17 or 23. The unit digit of this number is 7. What is its first digit?

Answer: 4


1. Given circles $\odot O_{1}$ and $\odot O_{2}$ intersect at points $A$ and $B$, with radii $5$ and $7$ respectively, and $O_{1} O_{2}=6$. A line through point $A$ intersects $\odot O_{1}$ and $\odot O_{2}$ at points $C$ and $D$, respectively. Points $P$ and $O$ are the midpoints of segments $CD$ and $O_{1} O_{2}$, respectively. Find the length of $OP$.

Answer: 2\sqrt{7}


2.203. $\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2} \cdot\left(2 x+\sqrt{x^{2}-1}\right)}{\sqrt{(x+1)^{3}}-\sqrt{(x-1)^{3}}}$.

2.203. $\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2} \cdot\left(2 x+\sqrt{x^{2}-1}\right)}{\sqrt{(x+1)^{3}}-\sqrt{(x-1)^{3}}}$.

The expression is already in a mathematical form and does not require translation. However, if you need the problem statement in English, it would be:

2.203. $\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2} \cdot\left(2 x+\sqrt{x^{2}-1}\right)}{\sqrt{(x+1)^{3}}-\sqrt{(x-1)^{3}}}$.

Simplify the expression above.

Answer: (x^2 - 1)^{-1/4}


Let \( a_{1}, a_{2}, \ldots, a_{k} \) be fixed numbers and \( u_{n+k} = a_{1} u_{n+k-1} + \ldots + a_{k} u_{n} \). Suppose that \( x_{1}, \ldots, x_{k} \) are pairwise distinct roots of the equation \( x^{k} = a_{1} x^{k-1} + a_{2} x^{k-2} + \ldots + a_{k} \). Prove that \( u_{n} = c_{1} x_{1}^{n-1} + \ldots + c_{k} x_{k}^{n-1} \) for some fixed numbers \( c_{1}, \ldots, c_{k} \).

Answer: u_n = c_1 x_1^{n-1} + c_2 x_2^{n-1} + \ldots + c_k x_k^{n-1}


2. The sum of 16 consecutive natural numbers, including the number $n$, is fifteen times greater than the number $n$. Determine all numbers $n$ for which this is possible.

Answer: 24


Consider a sequence where the \( n \)-th term is the first digit of the number \( 2^n \). Prove that the number of different "words" of length 13—sets of 13 consecutive digits—is 57.

Answer: 57


A point on the hypotenuse, equidistant from both legs, divides the hypotenuse into segments of lengths 30 cm and 40 cm. Find the legs of the triangle.

Answer: 42\,


The highway from Lishan Town to the provincial capital is 189 kilometers long, passing through the county town. It is 54 kilometers from Lishan Town to the county town. At 8:30 AM, a bus left from Lishan Town to the county town and arrived at 9:15 AM. After a 15-minute stop, it continued to the provincial capital and arrived at 11:00 AM. Another bus left from the provincial capital to Lishan Town on the same morning at 8:50 AM, traveling at 60 kilometers per hour. What is the time when the two buses meet?

Answer: 10:08


In a scalene triangle, two of the medians are equal to two of the altitudes. Find the ratio of the third median to the third altitude.

Answer: 3.5


Inside a grid rectangle with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut along the boundaries of the cells (the hole does not contain boundary cells). If you cut this figure along all the horizontal lines of the grid, you get 20 strips, each 1 cell in width. How many strips will you get if, instead, you cut it along all the vertical lines of the grid? (Note: A $1 \times 1$ cell is also considered a strip!)

Answer: 21


A set of identical square tiles with side length 1 is placed on a (very large) floor. Every tile after the first shares an entire edge with at least one tile that has already been placed.
- What is the largest possible perimeter for a figure made of 10 tiles?
- What is the smallest possible perimeter for a figure made of 10 tiles?
- What is the largest possible perimeter for a figure made of 2011 tiles?
- What is the smallest possible perimeter for a figure made of 2011 tiles?

Answer: 180


Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function satisfying the following conditions:
(a) \( f(1)=1 \).
(b) \( f(a) \leq f(b) \) whenever \( a \) and \( b \) are positive integers with \( a \leq b \).
(c) \( f(2a)=f(a)+1 \) for all positive integers \( a \).

How many possible values can the 2014-tuple \((f(1), f(2), \ldots, f(2014))\) take?

Answer: 1007


Let $\omega_1$ and $\omega_2$ be two circles intersecting at points $P$ and $Q$. The tangent line closer to $Q$ touches $\omega_1$ and $\omega_2$ at $M$ and $N$ respectively. If $PQ = 3$, $QN = 2$, and $MN = PN$, what is $QM^2$?

Answer: 5


A paper cup has a base that is a circle with radius $r$, a top that is a circle with radius $2r$, and sides that connect the two circles with straight line segments. This cup has height $h$ and volume $V$. A second cup, exactly the same shape as the first, is held upright inside the first cup so that its base is a distance of $\frac{h}{2}$ from the base of the first cup. The volume of liquid that will fit inside the first cup and outside the second cup can be written as $\frac{m}{n} \cdot V$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 93


On $2020 \times 2021$ squares, cover the whole area with several L-Trominos and S-Tetrominos (Z-Tetrominos) along the square so that they do not overlap. The tiles can be rotated or flipped over. Find the minimum possible number of L-Trominos to be used.

Answer: 2020


Let $\triangle ABC$ be acute with $\angle BAC = 45^{\circ}$. Let $\overline{AD}$ be an altitude of $\triangle ABC$, let $E$ be the midpoint of $\overline{BC}$, and let $F$ be the midpoint of $\overline{AD}$. Let $O$ be the center of the circumcircle of $\triangle ABC$, let $K$ be the intersection of lines $DO$ and $EF$, and let $L$ be the foot of the perpendicular from $O$ to line $AK$. If $BL = 6$ and $CL = 8$, find $AL^2$.

Answer: 48


Evan, Larry, and Alex are drawing whales on the whiteboard. Evan draws 10 whales, Larry draws 15 whales, and Alex draws 20 whales. Michelle then starts randomly erasing whales one by one. The probability that she finishes erasing Larry's whales first can be expressed as $\frac{p}{q}$. Compute $p+q$.

Answer: 137


Five identical circles are placed in a line inside a larger circle as shown. If the shown chord has a length of $16$, find the radius of the large circle.

Answer: 10


A unit circle is centered at $(0, 0)$ on the $(x, y)$ plane. A regular hexagon passing through $(1, 0)$ is inscribed in the circle. Two points are randomly selected from the interior of the circle, and horizontal lines are drawn through them, dividing the hexagon into at most three pieces. The probability that each piece contains exactly two of the hexagon's original vertices can be written as \[ \frac{2\left(\frac{m\pi}{n}+\frac{\sqrt{p}}{q}\right)^2}{\pi^2} \] for positive integers $m$, $n$, $p$, and $q$ such that $m$ and $n$ are relatively prime and $p$ is squarefree. Find $m+n+p+q$.

Answer: 11


For a positive integer $n$, consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$.

For example, if $n = 117$, it has six divisors: $1, 3, 9, 13, 39, 117$. Two of these divisors, namely $3$ and $13$, have unit digits equal to $3$. Hence, for $n = 117$, $p = 33.33\ldots$.

Find the maximum possible value of $p$ for any positive integer $n$.

Answer: 50


Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\omega$. $X$, $Y$, and $Z$ are points on $\omega$ such that $X$ is on minor arc $AB$, $Y$ is on minor arc $CD$, and $Z$ is on minor arc $EF$, where $X$ may coincide with $A$ or $B$ (and similarly for $Y$ and $Z$). Compute the square of the smallest possible area of $\triangle XYZ$.

Answer: 7500


Tim has a working analog 12-hour clock with two hands that run continuously. He also has a clock that runs at half the correct rate. At noon one day, both clocks show the exact time. At any given instant, the hands on each clock form an angle between $0^\circ$ and $180^\circ$ inclusive. At how many times during that day are the angles on the two clocks equal?

Answer: 33


We color all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue. Each vertex is an endpoint of 20 red segments and 22 blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same color. Suppose there are 2022 blue monochromatic triangles. How many red monochromatic triangles are there?

Answer: 859


How many ways are there to express $1,000,000$ as a product of exactly three integers greater than $1$? For the purpose of this problem, $abc$ is not considered different from $bac$, etc.

Answer: 114


In how many rearrangements of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ do the numbers form a \textit{hill}? A \textit{hill} is defined as a sequence where the numbers increase to a peak and then decrease, such as in $129876543$ or $258976431$.

Answer: 254


In triangle $ABC$, points $D$, $E$, and $F$ are on sides $BC$, $CA$, and $AB$ respectively, such that $BF = BD = CD = CE = 5$ and $AE - AF = 3$. Let $I$ be the incenter of $\triangle ABC$. The circumcircles of $\triangle BFI$ and $\triangle CEI$ intersect at $X \neq I$. Find the length of $DX$.

Answer: 3


There were 35 yellow and white dandelions growing on the lawn. After eight white dandelions flew away, and two yellow ones turned white, the number of yellow dandelions became twice the number of white ones. How many white and how many yellow dandelions were there on the lawn at first?

#

Answer: 20


## Task 1 - 290721

Susi goes shopping. From the money her mother gave her, she spends $30 \%$ at the butcher's; at the dairy, she pays with a quarter of the amount her mother gave her. At the greengrocer's, she needs exactly four fifths of the amount she paid at the butcher's.

At the bakery, she finally spends twice as much money as the remaining amount she brings back home. Her mother gives her half of this remaining amount, which is exactly $1.05 \mathrm{M}$, so she can buy a soft ice cream.

Determine the amount of money Susi initially received from her mother!

Answer: 30


In a country with $n$ cities, all direct airlines are two-way. There are $r>2014$ routes between pairs of different cities that include no more than one intermediate stop (the direction of each route matters). Find the least possible $n$ and the least possible $r$ for that value of $n$.

Answer: 2016


2. Ivan Semenovich leaves for work at the same time every day, drives at the same speed, and arrives exactly at 9:00. One day he overslept and left 40 minutes later than usual. To avoid being late, Ivan Semenovich drove at a speed 60% faster than usual and arrived at 8:35. By what percentage should he have increased his usual speed to arrive exactly at 9:00?

Answer: 30


6. 15 cards, each card has 3 different Chinese characters, any 2 cards do not have exactly the same characters; among any 6 cards, there must be 2 cards that have a common character. Question: What is the maximum number of different Chinese characters on these 15 cards?

Answer: 35


Six. (20 points) As shown in Figure 3, let
$$
P_{1}(1, \sqrt{3}), P_{2}(4,2 \sqrt{3}), \cdots, P_{n}\left(x_{n}, y_{n}\right)
$$
$\left(0<y_{1}<y_{2}<\cdots<y_{n}\right)$ be $n$ points on the curve $C: y^{2}=3 x(y \geqslant 0)$. Points $A_{i}(i=1,2, \cdots, n)$ are on the positive $x$-axis, satisfying that $\triangle A_{i-1} A_{i} P_{i}$ is an equilateral triangle ($A_{0}$ is the origin).
(1) Find the expression for the $x$-coordinate $a_{n}$ of point $A_{n}\left(a_{n}, 0\right)$ in terms of $n$;
(2) Evaluate
$$
\left[\frac{4}{5}+\frac{1}{\sqrt[4]{a_{1}}}+\frac{1}{\sqrt[4]{a_{2}}}+\cdots+\frac{1}{\sqrt[4]{a_{2000}}}\right]
$$

where [a] denotes the integer part of the number. When $2005 < c < 2005.6$, take $\sqrt{c}=44.78$; when $2006 < c < 2006.5$, take $\sqrt{c}=44.785$. $\sqrt{2}=1.4$.

Answer: 88


19. Sesame Street has 100 households, and their house numbers are not consecutive, but are arranged according to the following pattern:
$$
1,2,4,5,7,9,10,12,14,16,17,19,21,23,25, \cdots \cdots
$$

Among these 100 house numbers, there are $\qquad$ odd numbers.

Answer: 49


2・-110 Let the sequence of positive real numbers $x_{0}, x_{1}, \cdots, x_{1995}$ satisfy the following two conditions:
(1) $x_{0}=x_{1995}$;
(2) $x_{i-1}+\frac{2}{x_{i-1}}=2 x_{i}+\frac{1}{x_{i}}, i=1,2, \cdots, 1995$.

Find the maximum value of $x_{0}$ for all sequences that satisfy the above conditions.

Answer: 2^{997}


8.4. The steamship "Raritet" after leaving the city moves at a constant speed for three hours, then drifts for an hour, moving with the current, then moves for three hours at the same speed, and so on. If the steamship starts its journey from city A and heads to city B, it takes 10 hours. If it starts from city B and heads to city A, it takes 15 hours. How long would it take to travel from city A to city B on a raft?

Answer: 60


8. A parallel beam of light falls normally on a thin lens with a focal length of $F=150$ cm. Behind the lens, there is a screen on which a circular spot of a certain diameter is visible. If the screen is moved 40 cm, a spot of the same diameter will again be visible on the screen. Determine the initial distance from the lens to the screen. (15 points)

Answer: 170


4. (10 points) Person A leaves location $A$ to find person B, and after walking 80 kilometers, arrives at location $B$. At this point, person B had left for location $C$ half an hour earlier. Person A has been away from location $A$ for 2 hours, so A continues to location $C$ at twice the original speed. After another 2 hours, both A and B arrive at location $C$ simultaneously. What is the speed of person B in kilometers per hour?

Answer: 64


10.251. Five circles are inscribed in an angle of $60^{\circ}$ such that each subsequent circle (starting from the second) touches the previous one. By what factor is the sum of the areas of all five corresponding circles greater than the area of the smallest circle?

Answer: 7381


Let \( f(n) \) satisfy \( f(0) = 0 \) and \( f(n) = n - f(f(n-1)) \) for \( n = 1, 2, 3, \cdots \). Determine all real-coefficient polynomials \( g(x) \) such that
\[ f(n) = \lfloor g(n) \rfloor \text{ for } n = 0, 1, 2, \cdots, \]
where \( \lfloor g(n) \rfloor \) denotes the greatest integer less than or equal to \( g(n) \).

Answer: \frac{\sqrt{5} - 1}{2} (x + 1)


3. In a convex quadrilateral $A B C D$, points $E$ and $F$ are the midpoints of sides $B C$ and $C D$ respectively. Segments $A E, A F$, and $E F$ divide the quadrilateral into 4 triangles, the areas of which are consecutive natural numbers. What is the maximum possible value of the area of triangle $A B D$?

Answer: 6


In the right triangle \( \triangle ABC \), with \( \angle ACB = 90^\circ \), \( BC = a \), and \( CA = b \) (where \( a > 0 \) and \( b > 0 \)), point \( P \) lies on side \( AB \). The right triangle \( \triangle ABC \) is folded along line \( PC \) to form the tetrahedron \( PACB \). Find the maximum possible volume of this tetrahedron.

Answer: \dfrac{a^2 b^2}{6 \left( a^{2/3} + b^{2/3} \right)^{3/2}}


Consider 2021 lines in the plane, no two of which are parallel and no three of which are concurrent. Let E be the set of their intersection points. We want to assign a color to each point in E such that any two points on the same line, and whose connecting segment contains no other point of E, are of different colors.

How many colors are needed at a minimum to achieve such a coloring?

Answer: 3


Consider two mirrors placed at a right angle to each other and two points A at $ (x,y)$ and B at $ (a,b)$. Suppose a person standing at point A shines a laser pointer so that it hits both mirrors and then hits a person standing at point B (as shown in the picture). What is the total distance that the light ray travels, in terms of $ a$, $ b$, $ x$, and $ y$? Assume that $ x$, $ y$, $ a$, and $ b$ are positive.

[asy]draw((0,4)--(0,0)--(4,0),linewidth(1));
draw((1,3)--(0,2),MidArrow);
draw((0,2)--(2,0),MidArrow);
draw((2,0)--(3,1),MidArrow);
dot((1,3));
dot((3,1));
label("$A (x,y)$", (1,3),NE);
label("$B (a,b)$", (3,1),NE);[/asy]

Answer: \sqrt{(x + a)^2 + (y + b)^2}


5. Let $P$ be the vertex of a cone, and $A, B, C$ be three points on the circumference of its base, satisfying $\angle A B C=90^{\circ}$, $M$ is the midpoint of $A P$. If $A B=1, A C=2, A P=\sqrt{2}$, then the size of the dihedral angle $M-B C-A$ is $\qquad$ .

Answer: \arctan\left(\dfrac{2}{3}\right)


Define the Fibonacci numbers via $F_0=0$, $f_1=1$, and $F_{n-1}+F_{n-2}$.

Olivia flips two fair coins at the same time, repeatedly, until she has flipped a tails on both, not necessarily on the same throw. She records the number of pairs of flips $c$ until this happens (not including the last pair, so if on the last flip both coins turned up tails $c$ would be $0$). What is the expected value $F_c$?

Answer: \dfrac{19}{11}


1. Line segment $A B$ divides a square into two polygons (points $A$ and $B$ are on the sides of the square), each polygon has an inscribed circle, one of which has a radius of 6, while the other has a radius greater than 6. What is the difference between the side length of the square and twice the length of line segment $A B$?

Answer: 12


Example 21 Given three lines $l_{1}, l_{2}, l_{3}$ on a plane that do not pass through the origin, their equations are:
$$
\begin{array}{l}
l_{1}: a_{1} x+b_{1} y+c_{1}=0, \\
\iota_{2}: a_{2} x+b_{2} y+c_{2}=0, \\
\iota_{3}: a_{3} x+b_{3} y+c_{3}=0 .
\end{array}
$$

A moving line $l$ passing through the origin intersects the lines $l_{1}, l_{2}, l_{3}$ at points $A\left(x_{1}, y_{1}\right), B\left(x_{2}\right.$, $\left.y_{2}\right), C\left(x_{3}, y_{3}\right)$. Let $\left|\overrightarrow{O A}_{i}\right|=\rho_{i}(i=1,2,3),$ and $P$ be a point on the line $l$, with $|\overrightarrow{O P}|=$ $\rho$, and satisfies: $\frac{1}{\rho_{1}}+\frac{1}{\rho_{2}}+\frac{1}{\rho_{3}}=\frac{1}{\rho}$. Try to find the equation of the trajectory of the moving point $P(x, y)$.

Answer: \left(\frac{a_{1}}{c_{1}}+\frac{a_{2}}{c_{2}}+\frac{a_{3}}{c_{3}}\right)x + \left(\frac{b_{1}}{c_{1}}+\frac{b_{2}}{c_{2}}+\frac{b_{3}}{c_{3}}\right)y + 1 = 0


11. Given a function $f(x)$ defined on $[0,1]$, $f(0)=0, f(1)=1$, and satisfies the following conditions:
(a) For any $x \in[0,1], f(x) \geq 0$;
(b) For any two numbers $x_{1} \geq 0, x_{2} \geq 0, x_{1}+x_{2} \leq 1$, it holds that $f\left(x_{1}+x_{2}\right) \geq f\left(x_{1}\right)+f\left(x_{2}\right)$.
Find the smallest positive number $c$, such that for any function $f(x)$ satisfying the above conditions and for any $x \in[0,1]$, we have $f(x) \leq c x$.

Answer: 2


9.6. Waiting for customers, a watermelon seller sequentially weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg), balancing the watermelon on one scale pan with one or two weights on the other pan (possibly identical). In the process, the seller wrote down on a piece of paper the weights he used. What is the smallest number of different numbers that could have appeared in his notes, if the weight of each weight is an integer number of kilograms?

Answer: 6


In an isosceles triangle, the angle at the apex is $36^{\circ}$, and the angle bisector of the base angle is equal to $\sqrt{20}$. Find the lengths of the sides of the triangle.

Answer: 2\sqrt{5}


13. (15 points) As shown in the figure, ten points are evenly marked on the circumference of a circle. Place the ten natural numbers $1 \sim 10$ on these ten points. Rotate a line passing through the center of the circle. When the line does not pass through any marked points, it divides the numbers 1 to 10 into two groups. For each arrangement, there are five ways to divide the numbers as the line rotates. For each division, there is a product of the sums of the two groups, and the smallest of these five products is denoted as $K$. What is the maximum value of $K$ among all possible arrangements?

Answer: 756


30 Given that the three sides of $\triangle A B C$ are all integers, $\angle A=2 \angle B, \angle C>90^{\circ}$. Find the minimum perimeter of $\triangle A B C$.

Answer: 77


13.150. In the manuscript of an arithmetic problem book, an example was written in which a given number had to be multiplied by 3 and 4 subtracted from the result. In the printing house, a misprint was made: instead of the multiplication sign, a division sign was placed, and instead of a minus sign, a plus sign. Nevertheless, the final result did not change. What example was intended to be placed in the problem book?

Answer: 3 \times 3 - 4


In the picture, you can see a pool with a long step on one of its walls. We started filling the empty pool with a constant flow and observed the water level. After

![](https://cdn.mathpix.com/cropped/2024_04_17_ac63ecbffe7e6b0bc0e7g-2.jpg?height=309&width=614&top_left_y=451&top_left_x=755)

8 min, the water level had risen to a height of $20 \mathrm{~cm}$ and had not yet reached the level of the step. After $23 \mathrm{~min}$ of filling, the water level was at $55 \mathrm{~cm}$ and the step had already been submerged for some time. After 35.5 min of filling, the pool was filled to a height of $80 \mathrm{~cm}$. At what height $h$ from the bottom of the pool is the step located?

(L. Šimünek)

Answer: 45


Let $n$ be a positive integer, and denote by $f(n)$ the smallest positive integer $k$ such that $\left[\frac{n^{2}}{k}\right]=\left[\frac{n^{2}}{k+1}\right]$. Determine the value of $\left[\frac{n^{2}}{f(n)}\right]+f(n)$!

Answer: 2n