Let $S$ be the number of bijective functions $f:\{0,1,\dots,288\}\rightarrow\{0,1,\dots,288\}$ such that $f((m+n)\pmod{17})$ is divisible by $17$ if and only if $f(m)+f(n)$ is divisible by $17$. Compute the largest positive integer $n$ such that $2^n$ divides $S$.

Answer: 270


The teacher wrote a four-digit number on a piece of paper for Xiaowei to guess. They had four rounds of questions and answers.
Xiaowei: "Is it 8765?"
Teacher: "You guessed two digits correctly, but both are in the wrong positions."
Xiaowei: "Is it 1023?"
Teacher: "You guessed two digits correctly, but both are in the wrong positions."
Xiaowei: "Is it 8642?"
Teacher: "You guessed two digits correctly, and both are in the correct positions."
Xiaowei: "Is it 5430?"
Teacher: "None of the digits are correct."
What is this four-digit number $\qquad$?

Answer: 7612


Exercise 13. A set of $n$ non-zero and distinct real numbers is said to be regular if, when these numbers are written on the board in ascending order, the difference between two adjacent numbers is always the same, regardless of the two adjacent numbers chosen. For example, the set $\{4,18,-3,11\}$ is regular because if the numbers are written in ascending order, $18-11=11-4=4-(-3)$. A set $A$ of non-zero and distinct real numbers is said to be super-regular if it is regular and the set formed by the inverses of the numbers in set $A$ is also regular.

What is the largest integer $n$ such that there exists a super-regular set of $n$ non-zero and distinct real numbers?

Answer: 4


8. Given a hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ with an eccentricity of $\frac{\sqrt{6}}{2}$, the left and right foci are $F_{1}, F_{2}$, and $P$ is a point on the hyperbola. Let $R, r$ be the circumradius and inradius of $\triangle P F_{1} F_{2}$. If $\angle F_{1} P F_{2}=60^{\circ}$, then $\frac{R}{r}=$ $\qquad$ .

Answer: 2(1+\sqrt{2})


Find all positive prime numbers $p$ and $q$ such that $p^{2}-q^{3}=1$.

Remark. $p$ is prime if it has only two divisors: 1 and itself. The numbers $2,3,5,7,11,13$ are prime, but $1,4,6,8,9$ are not.

Answer: p=3


6. As shown in Figure 1, given that line $l \perp$ plane $\alpha$, with the foot of the perpendicular being $O$, in the right triangle $\triangle ABC$, $BC=1, AC=2, AB=\sqrt{5}$. The right triangle moves freely in space under the following conditions: (1) $A \in l$, (2) $C \in \alpha$. Then the maximum distance between points $B$ and $O$ is

Answer: 1 + \sqrt{2}


Let \( a \) be a positive number. If \( a \) is not an integer, denote the largest integer less than \( a \) by \( [a] \) (read: the integer part of \( a \)). If \( a \) is an integer, then \( [a]=a \). Show that if \( a \) is a positive number and \( n \) is an integer, then:

\[ \left[\frac{[a]}{n}\right]=\left[\frac{a}{n}\right] \]

Answer: \left\lfloor \frac{\lfloor a \rfloor}{n} \right\rfloor = \left\lfloor \frac{a}{n} \right\rfloor


3. 15 children are playing hide-and-seek. Xiao Zhi is searching, and the other children are hiding. Now Xiao Zhi has already found 8 children, he still has $\qquad$ children left to find.

Answer: 6


Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$, are placed inside an angle whose vertex is $O$. $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the rays of the angle at points $A$ and $B$, with $AO=BO$. Find the distance of point $A$ to the line $OB$.

Answer: 2


196. Composite Squares. Can you find two three-digit squares (with no zeros) that, when written consecutively, form a six-digit number which itself is a square? For example, from 324 and $900\left(18^{2}\right.$ and $\left.30^{2}\right)$ we get $324900\left(570^{2}\right)$, but the number 900 contains two zeros, which is prohibited by the condition.

The problem has only one solution.

Answer: 225


At a school, a table tennis tournament was held, consisting of several rounds. In each round, each participant played exactly one match, and each match was refereed by one of the players not participating in it.

After several rounds, it turned out that each participant played exactly once with each of the others. Can it happen that all participants of the tournament refereed the same number of matches?

Answer: \text{No}


5. In a given hemisphere of radius $R$, three spheres of equal radius are inscribed such that they touch each other and also touch the given hemisphere. Calculate the radius of the inscribed spheres.

Answer: \left( \dfrac{\sqrt{21} - 3}{4} \right) R


In the right-angled triangle $ABC$, the lengths of the legs $AC=3$ and $BC=4$. Point $A$ was moved to point $A_{1}$ parallel to $BC$, then point $B$ was moved to point $B_{1}$ parallel to $A_{1}C$, and finally, point $C$ was moved to point $C_{1}$ parallel to $A_{1}B_{1}$, such that the resulting triangle $A_{1}B_{1}C_{1}$ is right-angled at $B_{1}$, and the length of the leg $A_{1}B_{1}$ is 1 unit. What is the length of the leg $B_{1}C_{1}$?

Answer: 12


Let \( n > 2 \) be an integer. A deck contains \( \frac{n(n-1)}{2} \) cards, numbered
\[ 1, 2, 3, \ldots, \frac{n(n-1)}{2} \]

Two cards form a magic pair if their numbers are consecutive, or if their numbers are \( 1 \) and \( \frac{n(n-1)}{2} \).

For which \( n \) is it possible to distribute the cards into \( n \) stacks in such a manner that, among the cards in any two stacks, there is exactly one magic pair?

Answer: n \text{ is odd}


4. Given that the circumradius of $\triangle A B C$ is $1, \angle A$, $\angle B$, and $\angle C$'s angle bisectors intersect the circumcircle of $\triangle A B C$ at points $A_{1}$, $B_{1}$, and $C_{1}$, respectively. Then
$$
\frac{A A_{1} \cos \frac{A}{2}+B B_{1} \cos \frac{B}{2}+C C_{1} \cos \frac{C}{2}}{\sin A+\sin B+\sin C}=
$$
$\qquad$

Answer: 2


A bag of coconuts is to be shared among five children and a monkey. One of the five children, without the others knowing, divided the coconuts in the bag into five equal parts. This way, he got one coconut as a remainder, which he gave to the monkey, and then left with one-fifth of the coconuts. The second child also divided the remaining coconuts into five equal parts, and he too got one as a remainder, which he gave to the monkey, and left with one-fifth of the coconuts. The other three children did the same. Finally, the five children came together and divided the remaining coconuts into five equal parts, and once again got one coconut as a remainder, which they gave to the monkey.

What is the minimum number of coconuts that were originally in the bag?

Answer: 15621


1. [3] Tim the Beaver can make three different types of geometrical figures: squares, regular hexagons, and regular octagons. Tim makes a random sequence $F_{0}, F_{1}, F_{2}, F_{3}, \ldots$ of figures as follows:
- $F_{0}$ is a square.
- For every positive integer $i, F_{i}$ is randomly chosen to be one of the 2 figures distinct from $F_{i-1}$ (each chosen with equal probability $\frac{1}{2}$ ).
- Tim takes 4 seconds to make squares, 6 to make hexagons, and 8 to make octagons. He makes one figure after another, with no breaks in between.
Suppose that exactly 17 seconds after he starts making $F_{0}$, Tim is making a figure with $n$ sides. What is the expected value of $n$ ?

Answer: 7


37. A mechanical beetle crawls along a square spiral on the ground. It starts from the lower left corner and crawls in the direction of the arrows, sequentially for $1 \mathrm{~cm}, 1 \mathrm{~cm}, 1 \mathrm{~cm}, 1 \mathrm{~cm}, 2 \mathrm{~cm}, 2 \mathrm{~cm}, 1 \mathrm{~cm}, 3 \mathrm{~cm}, 3 \mathrm{~cm}$, $\qquad$. It crawls $1 \mathrm{~cm}$ per second. How many seconds after departure does it just finish crawling the 100th line segment?

Answer: 1156


44. Given 5 points, no three of which lie on the same line. Find the probability that, by randomly selecting two points, a student will get the required line.

Answer: \dfrac{1}{10}


A rabbit initially stands at the position $0$, and repeatedly jumps on the real line. In each jump, the rabbit can jump to any position corresponds to an integer but it cannot stand still. Let $N(a)$ be the number of ways to jump with a total distance of $2019$ and stop at the position $a$. Determine all integers $a$ such that $N(a)$ is odd.

Answer: 1


Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions:
[list]
[*]For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square.
[*]There exists infinitely many positive integers $n$ such that $a_n=4\times 2022-3$.
[/list]
Prove that there exists a positive integer $N$ such that $\sum_{k=n}^{n+2021}a_k$ is constant for every integer $n \geq N$.
And determine the value of $\sum_{k=N}^{N+2021}a_k$.

Answer: (2*2022 -1)^2


9.5. A football team coach loves to experiment with the lineup. During training sessions, he divides 20 available field players into two teams of 10 players each, adds goalkeepers, and arranges a game between the teams. He wants any two field players to end up on different teams at some training session. What is the minimum number of training sessions that must take place for this to happen?

Answer: 5


$m$ and $n$ are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows:

Starting from one of the hexagons, the Donkey moves $m$ cells in one of the $6$ directions, then it turns $60$ degrees clockwise and after that moves $n$ cells in this new direction until it reaches it's final cell.

At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey?

[i]Proposed by Shayan Dashmiz[/i]

Answer: m^2 + mn + n^2


$10 \cdot 2$ On a plane, given $n(n \geqslant 3)$ points $A_{1}, A_{2}, \cdots, A_{n}$, where no three points are collinear. Let $\alpha$ denote the minimum value of all angles $\angle A_{i} A_{j} A_{k}$, where $A_{i}, A_{j}, A_{k}$ are three different given points. For each $n$, find the maximum value of $\alpha$, and determine when these points are distributed to achieve the maximum value.

Answer: \dfrac{180^\circ}{n}


7. In a race with six runners, $A$ finished between $B$ and $C, B$ finished between $C$ and $D$, and $D$ finished between $E$ and $F$. If each sequence of winners in the race is equally likely to occur, what is the probability that $F$ placed last?

Answer: \dfrac{5}{16}


5. Given two circles $\Gamma_{1}$ and $\Gamma_{2}$ are externally tangent at point $A$, circle $\Gamma$ is externally tangent to $\Gamma_{1}$ and $\Gamma_{2}$ at points $B$ and $C$ respectively. Extend the chord $B A$ of circle $\Gamma_{1}$ to intersect circle $\Gamma_{2}$ at point $D$, extend the chord $C A$ of circle $\Gamma_{2}$ to intersect circle $\Gamma_{1}$ at point $E$, and extend the chords $E B$ and $D C$ to intersect circle $\Gamma$ at points $F$ and $G$ respectively: If $B C=5, B F=12$, find the length of $B G$.

Answer: 13


The sum of positive numbers \(a, b, c,\) and \(d\) is equal to 3. Prove the inequality \( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} + \frac{1}{d^3} \leq \frac{1}{a^3 b^3 c^3 d^3} \).

Answer: \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} + \frac{1}{d^3} \leq \frac{1}{a^3 b^3 c^3 d^3}


Find the number of integers $c$ such that the equation \[\left||20|x|-x^2|-c\right|=21\] has $12$ distinct real solutions.
Please give the answer directly without any intermediate steps.

Answer: 57


15th Chinese 2000 Problem A2 The sequence a n is defined by a 1 = 0, a 2 = 1, a n = (n a n-1 + n(n-1) a n-2 + (-1) n-1 n)/2 + (-1) n . Find a n + 2 nC1 a n-1 + 3 nC2 a n-2 + ... + n nC(n-1) a 1 , where nCm is the binomial coefficient n!/(m! (n-m)! ). Solution

Answer: 2n! - n - 1


There is one piece of paper with the digit 5, two pieces of paper with the digit 3, and one hundred pieces of paper with the digit 2.

a) In how many ways can a ten-digit number be formed such that the product of its digits ends in 0?

b) All such numbers were written in ascending order. What number is in the 455th position?

Answer: 5322222322


A convex $n$-gon $P$, where $n>3$, is divided into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is described?

Answer: 4


Given \(a, b, c \in \mathbb{R}\), with \(a \neq 0\), consider the graph of the function \(P(x) = a x^{2} + b x + c\). Show that by using a dilation and a translation, one can obtain the graph of the function \(Q(x) = x^{2}\).

Answer: Q(x) = x^2


If the product of three consecutive integers is divided by their sum, the quotient is 5. What are these numbers?

In general: when will the quotient obtained in the aforementioned manner be an integer? For instance, which group of numbers will directly follow the group identified in the previous part in magnitude?

Answer: 4, 5, 6


In the figure, the graph of the function \( y = x^2 + ax + b \) is shown. It is known that the line \( AB \) is perpendicular to the line \( y = x \). Find the length of the segment \( OC \).

Answer: 1


On a rectangular sheet of graph paper of size \( m \times n \) cells, several squares are placed such that their sides align with the vertical and horizontal lines of the paper. It is known that no two squares coincide, and no square contains another square within itself. What is the maximum number of such squares?

Answer: m \times n


Let \( k \) be a given positive integer, and let \( P \) be a point on plane \( \alpha \). In plane \( \alpha \), if it is possible to draw \( n \) lines that do not pass through point \( P \) such that any ray starting from \( P \) and lying in this plane intersects with at least \( k \) of these \( n \) lines, find the minimum value of \( n \).

Answer: 2k + 1


In the trapezoid \(ABCD (BC \parallel AD)\), the diagonals intersect at point \(M\). Given \(BC = b\) and \(AD = a\), find the ratio of the area of triangle \(ABM\) to the area of trapezoid \(ABCD\).

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


Let \( I \) be the universal set, and let non-empty sets \( P \) and \( Q \) satisfy \( P \subsetneq Q \subsetneq I \). If there exists a set operation expression involving \( P \) and \( Q \) such that the result of the operation is the empty set \( \varnothing \), then one such expression could be _____ (just write one expression).

Answer: P \cap Q^c


Given a regular 200-sided polygon \( A_{1} A_{2} \cdots A_{200} \), connect the diagonals \( A_{i} A_{i+9} \) for \( i=1, 2, \cdots, 200 \), where \( A_{i+200}=A_{i} \) for \( i=1,2, \cdots, 9 \). How many different intersection points are there inside the regular 200-sided polygon formed by these 200 diagonals?

Answer: 800


Petya showed Vasya 37 identical-looking cards laid out in a row. He said that the hidden sides of the cards contain all the numbers from 1 to 37 (each exactly once) in such a way that the number on any card starting from the second one is a divisor of the sum of the numbers written on all the preceding cards. Then Petya revealed to Vasya that the number 37 is written on the first card and the number 1 is written on the second card. Vasya said that he then knows what number is written on the third card. What number is written on the third card?

Answer: 2


9. (HUN 3) The incenter of the triangle $A B C$ is $K$. The midpoint of $A B$ is $C_{1}$ and that of $A C$ is $B_{1}$. The lines $C_{1} K$ and $A C$ meet at $B_{2}$, the lines $B_{1} K$ and $A B$ at $C_{2}$. If the areas of the triangles $A B_{2} C_{2}$ and $A B C$ are equal, what is the measure of angle $\angle C A B$?

Answer: 60^\circ


5.

the driver's marital status

Correct answers: bonus-malus coefficient, engine power, driver's age

Question 14

Score: 7.00

Maria Ivanovna has retired. She did not have a funded pension, only a social insurance pension, and her individual pension coefficient amount is 120. In addition, Maria Ivanovna has a bank deposit of 600,000 rubles, placed at an annual interest rate of $4 \%$ with monthly interest payments, which she does not reinvest but withdraws from the account monthly. How much money does Maria Ivanovna have available each month if the fixed part of her pension is 6050 rubles per month, and the cost of the pension coefficient is 99 rubles?

Answer: 19930


9. (16 points) Given the function
$$
f(x)=10 x^{2}+m x+n(m, n \in \mathbf{Z})
$$

has two distinct real roots in the interval $(1,3)$. Find the maximum possible value of $f(1) f(3)$.

Answer: 99


A quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). Two circles \(\Omega_1\) and \(\Omega_2\) of equal radii with centers \(O_1\) and \(O_2\) are inscribed in the angles \(BAD\) and \(BCD\) respectively. The first circle is tangent to side \(AB\) at point \(L\), and the second circle is tangent to side \(BC\) at point \(F\).

a) Find the radius of the circle \(\Omega_2\) if \(AL=\sqrt{2}\) and \(CF=2\sqrt{2}\).

b) Additionally, given that point \(O_2\) is the center of the circle circumscribed around triangle \(BOC\), find the angle \(BDC\).

Answer: 2


9. As shown in Figure 2, in $\triangle A B C$, $A B=B C=5, A C$ $=7, \triangle A B C$ has an inscribed circle $\odot O$ that is tangent to side $A C$ at point $M$. A line $M N$ parallel to side $B C$ is drawn through point $M$ and intersects $\odot O$ at point $N$. A tangent to $\odot O$ is drawn through point $N$ and intersects $A C$ at point $P$. Then $M N-N P=$ $\qquad$ .

Answer: \dfrac{3}{5}


4. Consider a 20 -sided convex polygon $K$, with vertices $A_{1}, A_{2}, \ldots, A_{20}$ in that order. Find the number of ways in which three sides of $K$ can be chosen so that every pair among them has at least two sides of $K$ between them. (For example $\left(A_{1} A_{2}, A_{4} A_{5}, A_{11} A_{12}\right)$ is an admissible triple while $\left(A_{1} A_{2}, A_{4} A_{5}, A_{19} A_{20}\right)$ is not.)

Answer: 520


95. Four people, A, B, C, and D, made predictions about the rankings before the competition. The table below shows their predictions:
\begin{tabular}{|c|c|c|c|c|}
\hline & A's prediction & B's prediction & C's prediction & D's prediction \\
\hline 1st place & B & A & C & B \\
\hline 2nd place & C & B & D & C \\
\hline 3rd place & A & D & A & D \\
\hline 4th place & D & C & B & A \\
\hline
\end{tabular}

After the competition, when comparing the predictions with the actual results, it was found that B and D correctly predicted the rankings of two people, while A and C only correctly predicted the ranking of one person. If A, B, C, and D placed in positions $A$, $B$, $C$, and $D$ respectively, then the four-digit number $\overline{A B C D}=$ . $\qquad$

Answer: 1423


2. The chord $AB$ divides a circle with radius $r$ into two arcs in the ratio $1:2$. In the larger segment, a square is inscribed such that one of its sides lies on this chord. Express the length of the side of the square in terms of the radius $r$.

Answer: \dfrac{(2 + \sqrt{19})r}{5}


19. Next year's 'leap day', 29th February 2024 , may be written in 'ddmmyy' format as the number 290224. What is the sum of all the two-digit factors of 290224 ?

Answer: 397


## Zadatak B-3.7.

Iz skupa $S=\{(x, y): x, y \in \mathbb{R},-1 \leqslant x \leqslant 4,0 \leqslant y \leqslant 5\}$ na slučajan se način bira uređeni par $(x, y)$. Kolika je vjerojatnost da za članove $x$ i $y$ toga uređenog para vrijedi $x+y>2$ i $|x-y|<2$ ?

Answer: \dfrac{23}{50}


10. Given the ellipse $C: \frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\frac{1}{2}$, the upper and lower foci are $F_{1}, F_{2}$, and the right vertex is $D$. A line perpendicular to $D F_{2}$ through $F_{1}$ intersects the ellipse $C$ at points $A, B$, and $|B D|-\left|A F_{1}\right|=\frac{8 \sqrt{3}}{39}$.
(1) Find the value of $|A D|+|B D|$.
(2) Construct the tangents to the ellipse $C$ at points $A, B$ which intersect at point $E$. If $F_{1} E$ intersects the $x$-axis at $P$ and $F_{2} E$ intersects the $x$-axis at $Q$, find the value of $|P Q|$.

Answer: \dfrac{112\sqrt{3}}{39}


For a positive integer \( n (n \geq 2) \), define \( f(n) \) as the minimum number of elements in a set \( S \) such that the smallest element is \( 1 \in S \), the largest element is \( n \in S \), and each element other than 1 is the sum of two (possibly identical) elements in \( S \). Prove the following:
(1) \( f(n) \geq \left\lfloor \log_{2} n \right\rfloor + 1 \);
(2) There are infinitely many \( n \) such that \( f(n) = f(n+1) \).

Answer: f(n) \geq \left\lfloor \log_{2} n \right\rfloor + 1


In the image, three identical, overlapping equilateral triangles are illustrated. Determine the area of each triangle, given the following conditions:

$\triangleright$ The intersection of triangle $T_{1}$ and triangle $T_{2}$ is an equilateral triangle with an area of $\sqrt{3} \mathrm{~cm}^{2}$.

$\triangleright$ The intersection of triangle $T_{2}$ and triangle $T_{3}$ is an equilateral triangle with an area of $\frac{9}{4} \sqrt{3} \mathrm{~cm}^{2}$.

$\triangleright$ The intersection of triangle $T_{1}$ and triangle $T_{3}$ is an equilateral triangle with an area of $\frac{1}{4} \sqrt{3} \mathrm{~cm}^{2}$.

Answer: 4\sqrt{3}


5. On the edge $B B_{1}$ of a regular triangular prism $A B C A_{1} B_{1} C_{1}$, a point $T$ is taken such that $B T: B_{1} T=2: 5$. The point $T$ is the vertex of a right circular cone such that three vertices of the prism lie on the circumference of its base.

a) Find the ratio of the height of the prism to the edge of its base.

b) Suppose it is additionally known that $C C_{1}=7$. Find the volume of the cone.

Answer: \dfrac{\sqrt{21}}{3}


Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral. The original answer is in the format $1-\frac{m}{n\pi^2}$, please provide the value of m + n.

Answer: 47


Let $n$ be a positive integer. Find the number of pairs $P,Q$ of polynomials with real coefficients such that $$(P(X))^2+(Q(X))^2=X^{2n}+1$$ and $\deg P>\deg Q.$ Provide the value when $n=3$. The answer should be an integer.

Answer: 16


Determine all rational numbers $a$ for which the matrix
$$
\left(\begin{array}{cccc}
a & -a & -1 & 0 \\
a & -a & 0 & -1 \\
1 & 0 & a & -a \\
0 & 1 & a & -a
\end{array}\right)
$$
is the square of a matrix with all rational entries. Provide the integer value of $a$.

Answer: 0


Determine all pairs ( $a, p$ ) of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all values of $a$ and $p$ for the pairs that satisfy the condition.

Answer: 30


Let $a_{1}, a_{2}, \\ldots, a_{n}, b_{1}, b_{2}, \\ldots, b_{n}$ be $2 n$ positive integers such that the $n+1$ products $$\begin{gathered} a_{1} a_{2} a_{3} \\cdots a_{n} \\ b_{1} a_{2} a_{3} \\cdots a_{n} \\ b_{1} b_{2} a_{3} \\cdots a_{n} \\ \vdots \\ b_{1} b_{2} b_{3} \\cdots b_{n} \end{gathered}$$ form a strictly increasing arithmetic progression in that order. Determine the smallest positive integer that could be the common difference of such an arithmetic progression.

Answer: 1


A $2\times3$ rectangle has vertices at $(0,0)$, $(2,0)$, $(0,3)$, and $(2,3)$. It rotates $90^{\circ}$ clockwise about the point $(2,0)$. It then rotates $90^{\circ}$ clockwise about the point $(5,0)$, then $90^{\circ}$ clockwise about the point $(7,0)$, and finally, $90^{\circ}$ clockwise about the point $(10,0)$. (The side originally on the x-axis is now back on the x-axis.) Find the area of the region above the x-axis and below the curve traced out by the point whose initial position is $(1,1)$. The original answer is in the format $\frac{m\pi}{n} + k$. Please give the value of m + n + k.

Answer: 15


For each positive integer $n$, let $a(n)$ be the number of zeros in the base 3 representation of $n$. For which positive real numbers $x$ does the series $$ \sum_{n=1}^{\infty} \frac{x^{a(n)}}{n^3} $$ converge? Please provide the sum of the endpoints of the interval for which the series converges.

Answer: 25


Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points? (It is understood that each point is independently chosen relative to a uniform distribution on the sphere.) The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 9


Let $a_0=5/2$ and $a_k=a^2_{k-1}-2$ for $k\geq1.$ Compute $$\prod\limits_{k=0}^{\infty}\left(1-\frac{1}{a_k}\right)$$ in closed form. The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 10


Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that $$f(x)+f\left(1-\dfrac{1}{x}\right)=\arctan x$$ for all real $x\neq0.$ (As usual, y=arctan $x$ means $-\pi/2<$ $y<\pi/2$ and tan$y=x.)$ Find $$\int_0^1f(x)\:dx.$$ The original answer is in the format \(\frac{m \pi}{n}\). Please give the value of \(m + n\).

Answer: 11


Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ is called $k$-good if $\gcd(f(m)+n,f(n)+m)\leqslant k$ for all $m\neq n.$ Find the smallest positive integer $k$ such that there exists a $k$-good function.

Answer: 2


We have drawn the inscribed and excircles of a triangle, as well as the six other lines, each of which touches two of these circles, but not the sides of the triangle. In how many ways can we select from these six lines

a) the side lines of a trapezoid,

b) the side lines of a parallelogram, if no two sides of the original triangle are equal in length?

Answer: 12


Consider a given cube's 3 mutually skew edges, and all triangles whose vertices lie on these edges respectively. Which one has the smallest perimeter?

Answer: \dfrac{3\sqrt{6}}{2}


Point \( C \) is the midpoint of one of the arcs \(k\) corresponding to the chord \( AB \) of a given circle. Point \( P \) is an arbitrary point on the arc \( k \), and point \( M \) is a point on the segment \( PC \) such that \( PM = \frac{1}{n} |PA - PB| \). Prove that as point \( P \) varies along the arc \( k \), all points \( M \) lie on the same circle.

Answer: \text{All points } M \text{ lie on a fixed circle.}


[ Processes and Operations ]

When organizing an expedition to Everest, the participants set up four high camps (not counting the base camp), each a day's journey apart from each other, after which everyone descended. After recounting the supplies, the leader decided that one more oxygen cylinder needed to be brought to the fourth camp, and then everyone would go down again.

1) can carry no more than three cylinders uphill,
2) consumes exactly one cylinder of oxygen per day.

What is the minimum number of cylinders that will need to be taken from the camp to achieve the set goal?

(Leaving cylinders is only allowed in the camps.)

Answer: 81


$\left[\begin{array}{l}{\left[\begin{array}{l}\text { Tangent circles } \\ {\left[\text { Right triangle with an angle of } 30^{\circ}\right.}\end{array}\right]}\end{array}\right]$

In an equilateral triangle, a circle is inscribed. Three smaller circles are tangent to this circle and the sides of the triangle. Find the side of the triangle if the radius of the smaller circle is $r$.

Answer: 6\sqrt{3}r


9.4. In the castle, there are 9 identical square rooms, forming a $3 \times 3$ square. Nine people, consisting of liars and knights (liars always lie, knights always tell the truth), each occupied one of these rooms. Each of these 9 people said: "At least one of the neighboring rooms to mine is occupied by a liar." What is the maximum number of knights that could be among these 9 people? Rooms are considered neighboring if they share a common wall.

Answer: 6


8. There are 10 young men, each with a different weight and height; for any two young men $\mathbf{A}$ and $\mathbf{B}$, if $\mathbf{A}$ is heavier than $\mathbf{B}$, or $\mathbf{A}$ is taller than $\mathbf{B}$, then we say “$\mathrm{A}$ is not worse than B”; if a young man is not worse than the other 9 people, he is called a “great guy”. Then, how many “great guys” can there be at most among these 10 people.

Answer: 10


2. In the acute $\triangle A B C$, take a point $P$ such that
$$
\angle P A C=60^{\circ}, \angle P C A=\angle P B A=30^{\circ} \text {, }
$$
$M 、 N$ are the midpoints of sides $A C 、 B C$ respectively. Then the degree of $\angle P N M$ is $\qquad$

Answer: 90


Let \( ABCD \) be a convex quadrilateral. Assume that the incircle of triangle \( ABD \) is tangent to \( \overline{AB} \), \( \overline{AD} \), \( \overline{BD} \) at points \( W, Z, K \). Also assume that the incircle of triangle \( CBD \) is tangent to \( \overline{CB} \), \( \overline{CD} \), \( \overline{BD} \) at points \( X, Y, K \). Prove that quadrilateral \( WXYZ \) is cyclic.

Answer: WXYZ \text{ is cyclic}


[Equilateral (Equiangular) Triangle]

[Properties and Characteristics of a Parallelogram]

[Auxiliary Congruent Triangles]

[Rotation Helps Solve the Problem.

On the sides $AB, BC$, and $CA$ of an equilateral triangle $ABC$, points $D, E$, and $F$ are chosen respectively such that $DE \parallel AC$ and $DF \parallel BC$.

Find the angle between the lines $AE$ and $BF$.

#

Answer: 60^\circ


Let $n $ be a positive integer. Let $f $ be a function from nonnegative integers to themselves. Let $f (0,i)=f (i,0)=0$, $f (1, 1)=n $, and $ f(i, j)= [\frac {f(i-1,j)}{2}]+ [\frac {f(i, j-1)}{2}] $ for positive integers $i, j$ such that $i*j>1$. Find the number of pairs $(i,j) $ such that $f (i, j) $ is an odd number.( $[x]$ is the floor function).

Answer: n


4. Let $H$ be the foot of the altitude from vertex $C$ in triangle $\triangle A B C$. Let $R$ and $S$ be the points where the incircles of triangles $\triangle A H C$ and $\triangle B C H$ touch $\overline{C H}$, respectively. If $|A B|=2018$, $|A C|=2017$, and $|B C|=2016$, calculate $|R S|$.

Answer: \dfrac{2015}{4036}


Suppose the function $\psi$ satisfies $\psi(1) = \sqrt{2 + \sqrt{2 + \sqrt{2}}}$ and
\[\psi(3x) + 3 \psi(x) = \psi^3(x)\]for all real $x.$  Determine $\prod_{n = 1}^{100} \psi(3^n).$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Answer: 4098


In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\overline{AC}$ and $\overline{AB},$ respectively, so that $AP = PQ = QB = BC.$  Find the ratio of $\angle ACB$ to $\angle APQ.$  Enter your answer as a fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 11


A sequence $(a_n)$ is defined as follows: $a_1 = 1,$ $a_2 = \frac{1}{2},$ and
\[a_n = \frac{1 - a_{n - 1}}{2a_{n - 2}}\]for all $n \ge 3.$  Find $a_{120}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Answer: 61


There is a triangle \(ABC\). On the ray \(BA\), take a point \(A_1\) such that the segment \(BA_1\) is equal to \(BC\). On the ray \(CA\), take a point \(A_2\) such that the segment \(CA_2\) is equal to \(BC\). Similarly, construct points \(B_1, B_2\) and \(C_1, C_2\). Prove that the lines \(A_1A_2\), \(B_1B_2\), and \(C_1C_2\) are parallel.

Answer: \text{The lines } A_1A_2, B_1B_2, \text{ and } C_1C_2 \text{ are parallel.}


In triangle $ABC$, circles with centers $O_1$ and $O_2$ of equal radius are inscribed respectively in vertices $C$ and $A$. Point $O$ is the center of the circle inscribed in triangle $ABC$. These circles touch side $AC$ at points $K_1$, $K_2$, and $K$ respectively, with $CK_1 = 6$, $AK_2 = 8$, and $AC = 21$.

a) Find the length of segment $CK$.

b) Let the circle with center $O_1$ touch side $BC$ at point $K_3$. Find the angle $BCA$ if it is known that $O_1$ is the center of the circle circumscribed around triangle $OK_1K_3$.

Answer: 9


6. In space, there are 2017 points. The midpoints of the line segments connecting each pair of points are colored red. The minimum number of red points is

Answer: 4031


20. Given that $a, b, x, y$ are non-negative real numbers, and $a+b=27$. Try to find the maximum value of $\lambda$ such that the inequality $\left(a x^{2}+b y^{2}+ 4 x y\right)^{3} \geqslant \lambda\left(a x^{2} y+b x y^{2}\right)^{2}$ always holds, and find the conditions for equality.

Answer: 4


Problem 2. The sum of 15 consecutive natural numbers is a number with different digits, including the digits $0,1,2$ and 4. What is the smallest possible number among the 15 numbers?

Answer: 676


Problem 2.4. Consider an alphabet of $n$ letters, with which we will form words. We will say that a word contains a palindrome if a segment of that word, of more than one letter, reads the same forwards as backwards. For example, the word OLIMPIADAS contains the palindrome ADA. Given an integer $k$ greater than 2, determine how many words of length $k$ can be formed, with our alphabet of $n$ letters, that do not contain any palindrome of odd length.

Answer: n^2 (n - 1)^{k - 2}


## Task 22/90

Determine all four-digit natural numbers $n$ in the decimal system with the following properties:

1. All digits $a_{i}$ and the cross sum $Q$ of $n$ are prime numbers.
2. It holds that $n=Q \cdot P+2$, where $P$ is the cross product of $n$.

Answer: 3572


Let $S(n)$ be the sum of the digits of an integer $n$. For example, $S(327)=3+2+7=12$. Find the value of

$$
A=S(1)-S(2)+S(3)-S(4)+\ldots-S(2016)+S(2017)
$$

Answer: 1009


34th Eötvös 1930 Problem 2 A straight line is drawn on an 8 x 8 chessboard. What is the largest possible number of the unit squares with interior points on the line?

Answer: 15


10. (6 points) As shown in the figure, two people, A and B, start from point $A$ at the same time and walk along the sides of the square $A B C D$ in the direction of the arrow. The side length of the square $A B C D$ is 100 meters. A's speed is 1.5 times B's speed. The two people meet for the first time at point $E$. Then, the area of triangle $A D E$ is larger than the area of triangle $B C E$ by square meters.

Answer: 1000


In triangle $A B C$, point $P$ lies on the segment $A B$ at one-third of its length closer to point $A$, point $R$ is at one-third of the segment $P B$ closer to point $P$, and point $Q$ lies on the segment $B C$ such that angles $P C B$ and $R Q B$ are equal.

Determine the ratio of the areas of triangles $A B C$ and $P Q C$.

(L. Růžičková)

Hint. In the described arrangement of points, multiple triangles can be found whose areas can be compared.

Answer: \dfrac{9}{2}


2. The lines drawn to the legs of a right triangle form an angle $\varphi$. If it is known that $\operatorname{tg} \varphi=\frac{3}{4}$, find the angles of the triangle.

Answer: 45^\circ


Two mirror walls are placed to form an angle of measure $\alpha$. There is a candle inside the angle. How many reflections of the candle can an observer see?

Answer: \left\lfloor \dfrac{360^\circ}{\alpha} \right\rfloor


4. The range of the function $y=\arcsin [\sin x]+\arccos [\cos x](x \in[0,2 \pi),[x]$ represents the greatest integer not exceeding the real number $x$) is $\qquad$ .

Answer: \left\{0, \dfrac{\pi}{2}, \pi\right\}


14. In space, there are five points, no four of which are coplanar. If several line segments are drawn such that no tetrahedron exists in the graph, then the maximum number of triangles in the graph is $\qquad$.

Answer: 4


We start with 5000 forints in our pocket to buy gifts. We enter three stores. In each store, we like a gift item, which we will buy if we can afford it. The prices of the items in each store, independently of each other, are 1000, 1500, or 2000 Ft with a probability of $\frac{1}{3}$. What is the probability that we will manage to buy three gift items and still have money left?

Answer: \dfrac{17}{27}


Two players $A$ and $B$ play a game with a ball and $n$ boxes placed onto the vertices of a regular $n$-gon where $n$ is a positive integer. Initially, the ball is hidden in a box by player $A$. At each step, $B$ chooses a box, then player $A$ says the distance of the ball to the selected box to player $B$ and moves the ball to an adjacent box. If $B$ finds the ball, then $B$ wins. Find the least number of steps for which $B$ can guarantee to win.

Answer: n


Let $m$ be the largest real solution to the equation \[\dfrac{3}{x-3} + \dfrac{5}{x-5} + \dfrac{17}{x-17} + \dfrac{19}{x-19} = x^2 - 11x - 4\] There are positive integers $a, b,$ and $c$ such that $m = a + \sqrt{b + \sqrt{c}}$ . Find $a+b+c$ .
Please give the answer directly without any intermediate steps.

Answer: 263


Answer: (-221, 224)


BC is a diameter of a circle with center O. A is any point on the circle such that the angle AOC > 60°. EF is the chord that is the perpendicular bisector of AO. D is the midpoint of the minor arc AB. The line through O parallel to AD meets AC at J. Show that J is the incenter of triangle CEF.

Answer: J


Let \( a \) be a real number. The equation \(\left(z^{2}-2z+5\right)\left(z^{2}+2az+1\right)=0\) has 4 distinct roots. These roots correspond to 4 points that lie on a circle in the complex plane. Determine the range of values for the real number \( a \).

Answer: (-1, 1) \cup \{-3\}


Construct a curve of constant width \( h \), composed of five, seven, or any odd number of arcs of circles with radius \( h \). What is the length of each of these curves?

Answer: \pi h


Several boxes with a total weight of 10 tons are unloaded from a cargo ship. The weight of each box does not exceed 1 ton. To ensure that all these boxes can be transported in one trip, what is the minimum number of trucks, each with a capacity of 3 tons, required?

Answer: 5


A natural number \( n \) is said to be lucky if \( 2^{-n} \) and \( 2^{-n-1} \) have the same number of significant figures when written in decimal notation. For example, 3 is lucky since \( 2^{-3} = 0.125 \) and \( 2^{-4} = 0.0625 \) have the same number of significant figures. On the other hand, \( 2^{-5} = 0.03125 \) has one more significant figure than 0.0625, so 4 is not lucky. Given that \( \log 2 = 0.301 \) (correct to 3 significant figures), how many lucky numbers are less than 2004?

Answer: 603


Professor Smith is an English literature professor who loves analyzing and interpreting literary elements in comics. In his literature class, he assigns his students the task of analyzing 5 different comic books. Each comic book has exactly 32 pages. On average, there are 6 panels on each page. Professor Smith wants his students to focus on one specific panel per page to discuss the narrative and artistic techniques used.

If each student in the class is assigned to analyze 1 comic book, how many total panels will all the students have analyzed if there are 15 students in the class?

Answer: 480


Alex is an entry-level accountant learning the basics of audit principles and financial management. As part of a training exercise, Alex is reviewing the company's petty cash fund. At the beginning of the month, the petty cash fund had $150.00. During the month, the following transactions occurred:

- A purchase of office supplies for $25.50
- A reimbursement for travel expenses totaling $30.75
- A payment for a small client lunch costing $40.00
- A refund of $10.00 from a previous overcharge

How much money should be left in the petty cash fund at the end of the month after these transactions?

Answer: 63.75


Alex is an avid Salesforce user and enjoys blogging about tips, tricks, and tutorials on Salesforce development. She has decided to write a series of blog posts, each focusing on different aspects of Salesforce. She plans to write 3 blog posts per week for 4 weeks.

In each blog post, Alex includes 5 tips, 3 tricks, and 2 tutorials.

Calculate the total number of tips, tricks, and tutorials Alex will share in her blog series by the end of the 4 weeks.

Answer: 120


A $3 \times 3 \times 3$ cube composed of $27$ unit cubes rests on a horizontal plane. Determine the number of ways to select two distinct unit cubes from a $3 \times 3 \times 1$ block (the order is irrelevant) such that the line joining the centers of the two cubes forms a $45^\circ$ angle with the horizontal plane.

Answer: 60


A round robin tournament is held with $2016$ participants. Each player plays each other player once, and no games result in ties. We say a pair of players $A$ and $B$ is a dominant pair if all other players either defeat both $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number of dominant pairs.

Answer: 2015


Let $a_n$ be the product of the complex roots of $x^{2n} = 1$ that are in the first quadrant of the complex plane. That is, roots of the form $a + bi$ where $a, b > 0$. Let $r = a_1 \cdot a_2 \cdot \ldots \cdot a_{10}$. Find the smallest integer $k$ such that $r$ is a root of $x^k = 1$.

Answer: 1260


For a positive integer $n$, let $f(n)$ be equal to $n$ if there exists an integer $x$ such that $x^2 - n$ is divisible by $2^{12}$, and let $f(n)$ be $0$ otherwise. Determine the remainder when \( \sum_{n=0}^{2^{12}-1} f(n) \) is divided by $2^{12}$.

Answer: 0


John is flipping his favorite bottle, which currently contains $10$ ounces of water. However, his bottle is broken from excessive flipping, so after he performs a flip, one ounce of water leaks out of his bottle. When his bottle contains $k$ ounces of water, he has a $\frac{1}{k+1}$ probability of landing it on its bottom. What is the expected number of flips it takes for John’s bottle to land on its bottom?

Answer: 6


The teacher secretly thought of a three-digit number $S$. Students $A$, $B$, $C$, and $D$ tried to guess the number by suggesting $541$, $837$, $291$, and $846$ respectively. The teacher informed them that each of them got exactly one digit correct, and it was in the correct position. What is the number $S$?

Answer: 247


How many distinct positive integers can be expressed in the form $ABCD - DCBA$, where $ABCD$ and $DCBA$ are 4-digit positive integers? Here, $A$, $B$, $C$, and $D$ are digits, possibly equal.

**Clarification:** $A$ and $D$ cannot be zero because otherwise $ABCD$ or $DCBA$ wouldn't be a true 4-digit integer.

Answer: 161


Let $p=101$. The sum \[\sum_{k=1}^{10}\frac{1}{\binom{p}{k}}\] can be written as a fraction of the form $\dfrac{a}{p!}$, where $a$ is a positive integer. Compute $a \pmod{p}$.

Answer: 5


Evin and Jerry are playing a game with a pile of marbles. On each player's turn, they can remove $2$, $3$, $7$, or $8$ marbles. If they can't make a move because there's $0$ or $1$ marble left, they lose the game. Given that Evin goes first and both players play optimally, for how many values of $n$ from $1$ to $1434$ does Evin lose the game?

Answer: 573


Let $a_0, a_1, \ldots$ be a sequence of positive integers such that $a_0 = 1$, and for all positive integers $n$, $a_n$ is the smallest composite number relatively prime to all of $a_0, a_1, \ldots, a_{n-1}$. Compute $a_{10}$.

Answer: 841


Let $f(x)$ be a real function with smallest period $5$, and $g(x)$ be defined as $f(2^x)$. Given that $g(x)$ has $25$ zeros in the range $[0,5)$, find the number of zeros that $g(x)$ has in the range $[1,4)$.

Answer: 11


In a basketball tournament, every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$, and there are no draws. A single team wins the tournament with $26$ points, and exactly two teams share the last position with $20$ points. How many teams participated in the tournament?

Answer: 12


In each cell of a chessboard with $2$ rows and $2019$ columns, a real number is written such that:
1. There are no two numbers written in the first row that are equal to each other.
2. The numbers written in the second row coincide with (in some other order) the numbers written in the first row.
3. The two numbers written in each column are different, and they add up to a rational number.

Determine the maximum quantity of irrational numbers that can be in the chessboard.

Answer: 4032


In the convex quadrilateral $ABCD$, we have the following side lengths: $AB = 15$, $BC = 16$, $CD = 12$, $DA = 25$, and the diagonal $BD = 20$. Let $M$ be the circumcenter and $\gamma$ the circumcircle of $\triangle ABD$. Line $CB$ intersects $\gamma$ again at point $F$. Line $AF$ intersects $MC$ at point $G$, and line $GD$ intersects $\gamma$ again at point $E$. Determine the area of the pentagon $ABCDE$.

Answer: 396


Jamie, an outgoing traveling salesperson, loves attending 90's revival concerts whenever possible. Last month, Jamie attended three concerts in different cities. The first concert was 120 miles away, the second was 80 miles away, and the third was 150 miles away from home. Jamie's car gets 25 miles per gallon, and gas costs $3.50 per gallon. If Jamie wants to know the total cost of gas for attending all three concerts, how much did Jamie spend on gas last month?

Answer: 98


Marko, a Serbian professional footballer, has been researching up-and-coming players in Europe. He decided to track the performance of three promising players each from different European leagues: the English Premier League, the Spanish La Liga, and the Italian Serie A.

In the past month, the player from the English Premier League scored 5 goals, the player from the Spanish La Liga scored 7 goals, and the player from the Italian Serie A scored 4 goals. Marko is curious about their total goal contributions.

Additionally, he noticed that the Premier League player provided 3 assists, the La Liga player provided 2 assists, and the Serie A player provided 6 assists.

Marko wants to calculate the total number of goals and assists contributed by these three players over the past month.

How many goals and assists did the three players contribute in total?

Answer: 27


Professor Thompson, a Medieval Studies scholar, is preparing a syllabus for her upcoming course. She needs to gather a set of resources for her class, which includes 12 books, 8 journal articles, and 5 manuscripts. Each book costs $25, each journal article costs $10, and each manuscript costs $15. Additionally, she plans to make 3 copies of each resource for her students. How much will Professor Thompson spend in total for all the resources, including the copies?

Answer: 1820


Mr. Johnson is a retired teacher who actively attends union meetings to support his fellow educators. Last month, he attended 5 different union meetings. At each meeting, he brought along 3 retired friends to also show their support. If each meeting lasted 2 hours, and they spent an additional 30 minutes in discussions after each meeting, how many total hours did Mr. Johnson and his friends spend attending and discussing union meetings last month?

Answer: 50


Jamie, a passionate history major, is conducting interviews for a research project on civil rights activists. Jamie plans to interview 5 activists each week. If each interview takes 1 hour, and Jamie schedules 2 hours every day from Monday to Friday for interviews, how many weeks will it take Jamie to complete interviews with a total of 60 civil rights activists?

Answer: 6


In her heyday, a retired runway model with striking features and a great fan following in the 80's and 90's used to walk in 5 fashion shows per week. If each fashion show had an audience of 500 people, and each person in the audience told 3 friends about the model's performance, how many people knew about her performance from a single week of shows?

Answer: 10000


In Tokyo, Mr. Tanaka, an extreme right-wing political supporter, decides to organize a rally to express his views. He plans to distribute flyers to gather supporters. He starts with printing 1,200 flyers. After distributing some flyers in the morning, he has 750 flyers left. In the afternoon, he prints an additional 300 flyers. By the end of the day, he has distributed a total of 1,050 flyers. How many flyers does Mr. Tanaka have left at the end of the day?

Answer: 450


The master magician, known for their secretive nature, has 5 different magic tricks that they perform at special shows. Each trick requires a unique set of magic cards, and the magician has a total of 60 magic cards in their collection. To prevent anyone from learning their secrets, the magician ensures that no two tricks share more than 2 cards in common.

If each magic trick uses the same number of magic cards, how many cards does each trick use?

Answer: 16


Mr. Thompson, a father who enjoys long drives, decided to take a memorial trip in honor of his son, who sadly passed away in a motorway accident. He plans to visit 5 different places that his son loved. Each place is 45 miles apart from the next. If he starts his journey from home and returns home after visiting all 5 places, how many miles will Mr. Thompson drive in total?

Answer: 270


Jamie is a jazz theory expert who offers personalized feedback and resources through virtual lessons. This week, Jamie has scheduled 5 virtual lessons, each lasting 45 minutes. During each lesson, Jamie spends 15 minutes discussing jazz theory concepts, and the remaining time is dedicated to providing personalized feedback. After completing all 5 lessons, Jamie spends an additional 30 minutes preparing resources for each student. How many minutes does Jamie spend in total on lessons and preparation this week?

Answer: 375


Sarah is an alumna who recently started working at the International Monetary Fund (IMF). To help future graduates, she hosts a workshop where she shares insights on how they can also land a job at the IMF. During the workshop, she explains that she applied to 5 different positions at the IMF. Each application required a $50 processing fee, which her university reimbursed. Before joining the IMF, Sarah attended 3 conferences related to international finance, each costing $120, which she paid out of pocket. During her first month at the IMF, Sarah earned a monthly salary of $3,500. How much more money did Sarah earn in her first month at the IMF compared to the total amount she spent on applications and conferences?

Answer: 3140


Alex, a seasoned budget traveler, is planning a trip to three different cities. He has found the following deals for flights:

- A flight to City A costs $120, but he has a 15% discount code he can use.
- A flight to City B is advertised as "Buy one, get one 50% off." The regular price for the flight is $200.
- A flight to City C is $150, with an additional $30 baggage fee if you don't book through a special promotion link that offers free baggage.

Alex, being skeptical of too-good-to-be-true offers, wants to calculate the total cost for traveling to all three cities, assuming he books the cheapest possible options. What is the total cost for Alex's flights, including any discounts and fees?

Answer: 452


A unit cube has vertices \(P_1,P_2,P_3,P_4,P_1',P_2',P_3',\) and \(P_4'\). Vertices \(P_2\), \(P_3\), and \(P_4\) are adjacent to \(P_1\), and for \(1\le i\le 4,\) vertices \(P_i\) and \(P_i'\) are opposite to each other. A regular octahedron has one vertex in each of the segments \(P_1P_2\), \(P_1P_3\), \(P_1P_4\), \(P_1'P_2'\), \(P_1'P_3'\), and \(P_1'P_4'\). The original answer is in the form \(\frac{k\sqrt{n}}{m}\). Please find the value of k + m + n.

Answer: 9


Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?

Answer: 204


Let \(A_{1}, A_{2}, A_{3}, A_{4}, A_{5}\), and \(A_{6}\) be six points on a circle in this order such that \(\overline{A_{1} A_{2}}=\overline{A_{2} A_{3}}\), \(\widehat{A_{3} A_{4}}=\widehat{A_{4} A_{5}}\), and \(\widehat{A_{5} A_{6}}=\widehat{A_{6} A_{1}}\), where \(\widehat{A_{1} A_{2}}\) denotes the arc length of the arc \(A_{1} A_{2}\) etc. It is also known that \(\angle A_{1} A_{3} A_{5}=72^{\circ}\). Find the size of \(\angle A_{4} A_{6} A_{2}\) in degrees.

Answer: 54


A tournament with 100 competitors, each with different skill levels, will take place. The most skilled competitor always wins against the less skilled one. Each participant plays exactly twice, with two randomly chosen opponents (once with each). A competitor who wins both matches receives a medal. Determine the minimum number of medals that can be distributed in the tournament.

Answer: 1


In the diagram, point \( P \) is inside quadrilateral \( ABCD \). Also, \( DA = DP = DC \) and \( AP = AB \). If \(\angle ADP = \angle CDP = 2x^\circ\), \(\angle BAP = (x+5)^\circ\), and \(\angle BPC = (10x-5)^\circ\), what is the value of \( x \)?

Answer: 13


Point $M$ is the midpoint of side $AB$ of triangle $ABC$. Circle $\omega_{1}$ passes through points $A$ and $M$ and is tangent to line $AC$, while circle $\omega_{2}$ passes through points $B$ and $M$ and is tangent to line $BC$. The circles $\omega_{1}$ and $\omega_{2}$ intersect again at point $D$. Point $E$ lies inside triangle $ABC$ and is symmetric to point $D$ with respect to line $AB$. Find the angle $CEM$.

Answer: 180


As shown in the diagram, a plane section of the cube $ABCDEFGH$ passes through vertices $A$ and $C$ and a point $K$ on the edge $EF$. This section divides the cube into two parts with a volume ratio of $3:1$. What is the value of $\frac{EK}{KF}$?

Answer: \sqrt{3}


Fill the eight numbers $2, 3, 4, 5, 6, 7, 8, 9$ into the eight boxes below (without repetition) to form many different subtraction equations. To make the calculation result the smallest natural number, what is the calculation result?

Answer: 247


Carl is on a vertex of a regular pentagon. Every minute, he randomly selects an adjacent vertex (each with probability \(\frac{1}{2}\)) and walks along the edge to it. What is the probability that after 10 minutes, he ends up where he started?

Answer: \frac{127}{512}


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

## Clues:
1. The person who loves tennis is somewhere to the right of Arnold.
2. Alice is the person who loves baseball.
3. Eric is somewhere to the left of the person who loves basketball.
4. The person who loves baseball is somewhere to the left of the person who loves swimming.
5. Eric is directly left of the person who loves baseball.
6. The person who loves basketball is somewhere to the left of Arnold.
7. Bob is not in the fifth house.
 What is the value of attribute FavoriteSport for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: swimming


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`, `Alice`
 - Each person has a unique birthday month: `may`, `sept`, `jan`, `mar`, `feb`, `april`
 - People have unique hair colors: `brown`, `blonde`, `red`, `auburn`, `black`, `gray`
 - Each mother is accompanied by their child: `Alice`, `Samantha`, `Fred`, `Meredith`, `Timothy`, `Bella`
 - Everyone has a favorite smoothie: `blueberry`, `cherry`, `watermelon`, `lime`, `dragonfruit`, `desert`
 - The mothers' names in different houses are unique: `Sarah`, `Aniya`, `Janelle`, `Holly`, `Kailyn`, `Penny`

## Clues:
1. The person whose birthday is in September and the person's child is named Alice are next to each other.
2. The person whose birthday is in April is the person who is the mother of Timothy.
3. The person who likes Cherry smoothies is the person who has gray hair.
4. Alice is somewhere to the right of Arnold.
5. The person's child is named Alice is somewhere to the right of The person whose mother's name is Sarah.
6. Peter is the person who drinks Blueberry smoothies.
7. The person whose mother's name is Janelle is the person whose birthday is in May.
8. The person whose birthday is in January is somewhere to the left of The person whose mother's name is Sarah.
9. The Watermelon smoothie lover is the person who has brown hair.
10. The person who has red hair is somewhere to the right of the person who has gray hair.
11. The person who drinks Lime smoothies is in the fifth house.
12. The person who has black hair and the person's child is named Meredith are next to each other.
13. The person whose birthday is in March is somewhere to the right of the person who has red hair.
14. The person whose mother's name is Sarah is somewhere to the left of the person who has auburn hair.
15. The person who has black hair is directly left of The person whose mother's name is Penny.
16. The person's child is named Samantha is The person whose mother's name is Holly.
17. The person who has blonde hair and Bob are next to each other.
18. The person who has gray hair is the person whose birthday is in January.
19. The person who has black hair is in the first house.
20. The person whose mother's name is Sarah is the person who has blonde hair.
21. The person whose mother's name is Kailyn is Arnold.
22. The person whose birthday is in January is The person whose mother's name is Kailyn.
23. The person whose birthday is in September is Carol.
24. The person who drinks Lime smoothies is somewhere to the right of the Desert smoothie lover.
25. The person's child is named Fred is not in the third house.
 What is the value of attribute Mother for the person whose attribute HairColor is brown? Please reason step by step, and put your final answer within \boxed{}

Answer: Penny


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`, `Alice`
 - Each person has a unique birthday month: `may`, `sept`, `jan`, `mar`, `feb`, `april`
 - People have unique hair colors: `brown`, `blonde`, `red`, `auburn`, `black`, `gray`
 - Each mother is accompanied by their child: `Alice`, `Samantha`, `Fred`, `Meredith`, `Timothy`, `Bella`
 - Everyone has a favorite smoothie: `blueberry`, `cherry`, `watermelon`, `lime`, `dragonfruit`, `desert`
 - The mothers' names in different houses are unique: `Sarah`, `Aniya`, `Janelle`, `Holly`, `Kailyn`, `Penny`

## Clues:
1. The person whose birthday is in September and the person's child is named Alice are next to each other.
2. The person whose birthday is in April is the person who is the mother of Timothy.
3. The person who likes Cherry smoothies is the person who has gray hair.
4. Alice is somewhere to the right of Arnold.
5. The person's child is named Alice is somewhere to the right of The person whose mother's name is Sarah.
6. Peter is the person who drinks Blueberry smoothies.
7. The person whose mother's name is Janelle is the person whose birthday is in May.
8. The person whose birthday is in January is somewhere to the left of The person whose mother's name is Sarah.
9. The Watermelon smoothie lover is the person who has brown hair.
10. The person who has red hair is somewhere to the right of the person who has gray hair.
11. The person who drinks Lime smoothies is in the fifth house.
12. The person who has black hair and the person's child is named Meredith are next to each other.
13. The person whose birthday is in March is somewhere to the right of the person who has red hair.
14. The person whose mother's name is Sarah is somewhere to the left of the person who has auburn hair.
15. The person who has black hair is directly left of The person whose mother's name is Penny.
16. The person's child is named Samantha is The person whose mother's name is Holly.
17. The person who has blonde hair and Bob are next to each other.
18. The person who has gray hair is the person whose birthday is in January.
19. The person who has black hair is in the first house.
20. The person whose mother's name is Sarah is the person who has blonde hair.
21. The person whose mother's name is Kailyn is Arnold.
22. The person whose birthday is in January is The person whose mother's name is Kailyn.
23. The person whose birthday is in September is Carol.
24. The person who drinks Lime smoothies is somewhere to the right of the Desert smoothie lover.
25. The person's child is named Fred is not in the third house.
 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: Penny


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Peter`, `Eric`, `Carol`, `Arnold`
 - People own unique car models: `toyota camry`, `ford f150`, `bmw 3 series`, `tesla model 3`, `chevrolet silverado`, `honda civic`
 - Each person has a unique hobby: `knitting`, `gardening`, `woodworking`, `painting`, `photography`, `cooking`
 - Each person has a unique favorite drink: `coffee`, `root beer`, `milk`, `water`, `tea`, `boba tea`

## Clues:
1. There is one house between the coffee drinker and the woodworking hobbyist.
2. The person who enjoys knitting is in the first house.
3. There are two houses between the person who loves cooking and the person who likes milk.
4. The person who paints as a hobby is the person who owns a Toyota Camry.
5. Arnold is the tea drinker.
6. There is one house between the person who owns a Chevrolet Silverado and the person who owns a BMW 3 Series.
7. The woodworking hobbyist is somewhere to the right of Carol.
8. The person who paints as a hobby is not in the fourth house.
9. The boba tea drinker is in the third house.
10. Bob is somewhere to the left of the person who likes milk.
11. The person who owns a Chevrolet Silverado is not in the sixth house.
12. The person who owns a Toyota Camry is the root beer lover.
13. The person who owns a Chevrolet Silverado is Alice.
14. Peter is the person who owns a Toyota Camry.
15. The root beer lover is not in the sixth house.
16. The person who owns a Tesla Model 3 is not in the first house.
17. The person who enjoys gardening is Eric.
18. The tea drinker is somewhere to the left of the one who only drinks water.
19. The person who owns a Honda Civic and the boba tea drinker are next to each other.
20. Bob is in the third house.
 What is the value of attribute CarModel for the person whose attribute Drink is root beer? Please reason step by step, and put your final answer within \boxed{}

Answer: toyota camry


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`, `Arnold`, `Alice`, `Bob`, `Peter`
 - Each person lives in a unique style of house: `colonial`, `modern`, `ranch`, `victorian`, `craftsman`
 - People have unique favorite sports: `tennis`, `basketball`, `soccer`, `baseball`, `swimming`
 - People have unique favorite music genres: `pop`, `classical`, `jazz`, `rock`, `hip hop`

## Clues:
1. The person in a Craftsman-style house and the person who loves basketball are next to each other.
2. Alice is the person living in a colonial-style house.
3. Peter is somewhere to the right of the person who loves swimming.
4. The person living in a colonial-style house is the person who loves soccer.
5. The person in a ranch-style home is the person who loves baseball.
6. The person who loves pop music is somewhere to the left of the person who loves swimming.
7. The person who loves baseball is not in the second house.
8. The person in a modern-style house is in the fourth house.
9. The person who loves basketball and Peter are next to each other.
10. The person in a Craftsman-style house is somewhere to the left of Arnold.
11. The person who loves jazz music is not in the fourth house.
12. There are two houses between the person who loves tennis and the person in a ranch-style home.
13. Eric is not in the third house.
14. The person who loves hip-hop music is the person in a ranch-style home.
15. The person who loves tennis is the person who loves classical music.
 What is the value of attribute House for the person whose attribute MusicGenre is hip hop? 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: `Peter`, `Arnold`, `Alice`, `Eric`
 - People have unique heights: `tall`, `short`, `very short`, `average`
 - Each person has a unique birthday month: `april`, `jan`, `sept`, `feb`
 - People have unique hair colors: `red`, `black`, `brown`, `blonde`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`

## Clues:
1. Eric is the person who has blonde hair.
2. The person who is short is somewhere to the left of the person who has brown hair.
3. Eric is the person who is tall.
4. Arnold is the person who loves fantasy books.
5. The person who loves science fiction books is Eric.
6. The person who is very short is the person who has black hair.
7. The person whose birthday is in April is the person who has brown hair.
8. The person who has blonde hair and the person who has black hair are next to each other.
9. The person who has black hair is the person who loves fantasy books.
10. The person who has red hair is in the first house.
11. The person whose birthday is in September is in the second house.
12. The person whose birthday is in January is the person who has black hair.
13. The person who loves romance books is somewhere to the right of Alice.
 What is the value of attribute House for the person whose attribute Birthday is feb? 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: `Arnold`, `Peter`, `Bob`, `Eric`, `Alice`
 - People have unique hair colors: `gray`, `brown`, `black`, `red`, `blonde`
 - Each person prefers a unique type of vacation: `city`, `mountain`, `camping`, `cruise`, `beach`
 - They all have a unique favorite flower: `roses`, `tulips`, `carnations`, `lilies`, `daffodils`
 - Everyone has something unique for lunch: `stew`, `stir fry`, `spaghetti`, `pizza`, `grilled cheese`
 - Each person has a unique hobby: `painting`, `photography`, `gardening`, `knitting`, `cooking`

## Clues:
1. The person who loves the stew is directly left of the person who loves the vase of tulips.
2. The person who enjoys camping trips is the person who loves eating grilled cheese.
3. The person who has black hair is somewhere to the left of Bob.
4. The photography enthusiast is the person who has gray hair.
5. The person who is a pizza lover is the person who likes going on cruises.
6. Alice is the person who prefers city breaks.
7. The person who loves beach vacations is directly left of the person who paints as a hobby.
8. The person who loves the spaghetti eater is Arnold.
9. Bob is in the fifth house.
10. The person who loves the boquet of lilies is in the third house.
11. The person who enjoys knitting is in the fourth house.
12. The person who has brown hair is directly left of the person who paints as a hobby.
13. There are two houses between the person who likes going on cruises and the person who enjoys knitting.
14. The person who has red hair is the person who loves the spaghetti eater.
15. The person who loves a carnations arrangement is not in the second house.
16. Eric is somewhere to the left of the person who enjoys gardening.
17. The photography enthusiast is the person who loves the stew.
18. Eric is the person who loves a bouquet of daffodils.
 What is the value of attribute HairColor for the person whose attribute Food is spaghetti? Please reason step by step, and put your final answer within \boxed{}

Answer: red


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`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`
 - People have unique favorite music genres: `classical`, `pop`, `rock`
 - They all have a unique favorite flower: `carnations`, `daffodils`, `lilies`
 - People have unique heights: `average`, `very short`, `short`
 - Everyone has a unique favorite cigar: `prince`, `pall mall`, `blue master`

## Clues:
1. Peter is the person who has an average height.
2. The person who smokes Blue Master is the person who is short.
3. The person who loves a bouquet of daffodils is directly left of the person partial to Pall Mall.
4. The person who loves pop music is the person who loves a bouquet of daffodils.
5. The person whose mother's name is Janelle is the person who loves the boquet of lilies.
6. The person whose mother's name is Aniya is in the first house.
7. The person who has an average height is in the second house.
8. The person who loves pop music and the person who loves classical music are next to each other.
9. Eric is in the third house.
10. The person whose mother's name is Holly is the person who is very short.
 What is the value of attribute Height for the person whose attribute Mother is Aniya? Please reason step by step, and put your final answer within \boxed{}

Answer: short


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

## Clues:
1. The person's child is named Meredith is the person who loves a carnations arrangement.
2. The person's child is named Bella is directly left of the person who loves the vase of tulips.
3. The person's child is named Bella is the person who loves the rose bouquet.
4. The person who loves the boquet of lilies is Eric.
5. There are two houses between Bob and Peter.
6. Alice is the person's child is named Meredith.
7. Peter is the person who loves the rose bouquet.
8. The person's child is named Fred is in the third house.
9. The person's child is named Samantha is in the fifth house.
10. The person's child is named Meredith is in the second 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: 1


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Arnold`, `Peter`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blue master`, `pall mall`
 - Each person lives in a unique style of house: `victorian`, `colonial`, `ranch`, `craftsman`
 - Each person has a unique birthday month: `april`, `sept`, `feb`, `jan`
 - Each person has an occupation: `teacher`, `artist`, `doctor`, `engineer`
 - Each mother is accompanied by their child: `Samantha`, `Bella`, `Meredith`, `Fred`

## Clues:
1. The person who is a doctor is the person's child is named Fred.
2. The person's child is named Samantha and the person whose birthday is in January are next to each other.
3. Arnold is the person whose birthday is in September.
4. The person in a Craftsman-style house is the person who is an engineer.
5. The Dunhill smoker is the person's child is named Samantha.
6. Alice is directly left of Arnold.
7. The person whose birthday is in April and the Prince smoker are next to each other.
8. Arnold is the person who is a teacher.
9. The person living in a colonial-style house is not in the first house.
10. The person who smokes Blue Master is somewhere to the left of Eric.
11. The person who smokes Blue Master is somewhere to the right of the person's child is named Fred.
12. The person whose birthday is in April is somewhere to the left of the person who is an engineer.
13. Eric is not in the fourth house.
14. The person's child is named Bella is somewhere to the left of the person whose birthday is in April.
15. The person in a ranch-style home is Arnold.
 What is the value of attribute Children for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: Samantha


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Arnold`
 - People own unique car models: `honda civic`, `toyota camry`, `ford f150`, `tesla model 3`
 - Each person has a unique birthday month: `feb`, `april`, `sept`, `jan`
 - Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`
 - People have unique heights: `average`, `tall`, `very short`, `short`
 - The mothers' names in different houses are unique: `Janelle`, `Kailyn`, `Holly`, `Aniya`

## Clues:
1. The person who is very short is the person who loves the stew.
2. The person who is a pizza lover is Alice.
3. The person who loves the spaghetti eater is the person who owns a Honda Civic.
4. The person who owns a Toyota Camry is somewhere to the left of the person who is a pizza lover.
5. The person whose birthday is in February is not in the fourth house.
6. Eric is the person who owns a Tesla Model 3.
7. The person who owns a Toyota Camry is somewhere to the right of The person whose mother's name is Janelle.
8. The person who is tall is in the fourth house.
9. The person whose birthday is in September is Arnold.
10. The person who has an average height is somewhere to the right of Arnold.
11. The person whose mother's name is Aniya and Alice are next to each other.
12. The person whose birthday is in April and The person whose mother's name is Holly are next to each other.
13. The person who is short is in the second house.
14. The person who loves the stew and The person whose mother's name is Janelle are next to each other.
 What is the value of attribute House for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


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`, `Eric`, `Carol`, `Alice`
 - People own unique car models: `bmw 3 series`, `ford f150`, `tesla model 3`, `toyota camry`, `honda civic`, `chevrolet silverado`
 - Each person has a unique hobby: `woodworking`, `painting`, `knitting`, `photography`, `gardening`, `cooking`

## Clues:
1. The person who loves cooking is not in the first house.
2. The person who owns a Tesla Model 3 is in the fifth house.
3. The person who owns a Honda Civic is in the third house.
4. Alice and the person who owns a Honda Civic are next to each other.
5. There is one house between the person who owns a Chevrolet Silverado and Carol.
6. The photography enthusiast is directly left of the person who paints as a hobby.
7. Peter is somewhere to the right of the person who owns a Tesla Model 3.
8. The person who owns a Honda Civic is directly left of Bob.
9. The person who enjoys knitting is somewhere to the right of the person who owns a Toyota Camry.
10. The person who enjoys knitting is somewhere to the left of Eric.
11. The woodworking hobbyist is directly left of the person who owns a BMW 3 Series.
12. The person who paints as a hobby is in the third house.
 What is the value of attribute CarModel for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: honda civic


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`
 - The people keep unique animals: `horse`, `bird`, `fish`, `cat`
 - Each person has a unique level of education: `high school`, `bachelor`, `associate`, `master`
 - Each person has a unique favorite drink: `tea`, `water`, `coffee`, `milk`
 - People have unique favorite book genres: `fantasy`, `science fiction`, `romance`, `mystery`
 - Each person has a unique birthday month: `april`, `feb`, `sept`, `jan`

## Clues:
1. Peter is the person whose birthday is in February.
2. The person with a master's degree is the bird keeper.
3. The person who keeps horses is not in the second house.
4. The person whose birthday is in January is the person who loves science fiction books.
5. The person who loves mystery books is Alice.
6. There is one house between the person who likes milk and Alice.
7. The person whose birthday is in April is the person who loves fantasy books.
8. The person with a high school diploma is not in the second house.
9. Eric is directly left of the person who likes milk.
10. The fish enthusiast is the person whose birthday is in February.
11. The tea drinker is somewhere to the left of Peter.
12. The one who only drinks water is the person with a master's degree.
13. The person with a bachelor's degree is somewhere to the right of the person whose birthday is in January.
14. Peter is in the third house.
15. The person who loves science fiction books is Arnold.
 What is the value of attribute Drink for the person whose attribute Animal is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: milk


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Eric`, `Arnold`, `Peter`, `Alice`
 - People have unique favorite sports: `tennis`, `baseball`, `basketball`, `soccer`, `swimming`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Meredith`, `Timothy`, `Samantha`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Penny`, `Janelle`, `Aniya`
 - The people keep unique animals: `horse`, `cat`, `dog`, `bird`, `fish`

## Clues:
1. The person whose mother's name is Penny is in the fourth house.
2. Eric is The person whose mother's name is Penny.
3. The person who keeps horses is the person who loves swimming.
4. The person who loves baseball is the person's child is named Meredith.
5. The dog owner is directly left of the person who loves basketball.
6. The person who loves soccer is the bird keeper.
7. The person who loves swimming is directly left of the person's child is named Fred.
8. The person who loves tennis is The person whose mother's name is Kailyn.
9. The person who is the mother of Timothy is The person whose mother's name is Aniya.
10. Eric is somewhere to the right of the person's child is named Samantha.
11. The person who loves basketball is Bob.
12. The person who is the mother of Timothy is the fish enthusiast.
13. The person's child is named Meredith is somewhere to the right of Peter.
14. The person who loves basketball is somewhere to the left of the person who keeps horses.
15. The bird keeper is Alice.
16. The person whose mother's name is Holly is the bird keeper.
17. Bob is not in the third house.
 What is the value of attribute Mother for the person whose attribute Children is Samantha? Please reason step by step, and put your final answer within \boxed{}

Answer: Kailyn


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Arnold`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`, `Samantha`
 - People have unique hair colors: `red`, `brown`, `black`, `blonde`
 - They all have a unique favorite flower: `carnations`, `roses`, `daffodils`, `lilies`
 - The people keep unique animals: `cat`, `bird`, `fish`, `horse`

## Clues:
1. Alice is somewhere to the right of the person who has red hair.
2. The person who loves a bouquet of daffodils is directly left of Arnold.
3. The fish enthusiast is the person's child is named Samantha.
4. Alice is directly left of the bird keeper.
5. The cat lover is the person who loves the rose bouquet.
6. The person who loves a carnations arrangement is somewhere to the right of the person who has brown hair.
7. The person who loves a bouquet of daffodils is directly left of the bird keeper.
8. Eric is the person's child is named Meredith.
9. There are two houses between the person who has red hair and the person who has black hair.
10. Eric is in the fourth house.
11. The person's child is named Bella is the person who loves a bouquet of daffodils.
 What is the value of attribute Children for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: Samantha


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `Eric`, `Alice`, `Carol`
 - People have unique hair colors: `red`, `gray`, `auburn`, `blonde`, `brown`, `black`
 - People have unique favorite book genres: `historical fiction`, `biography`, `romance`, `fantasy`, `mystery`, `science fiction`
 - People have unique heights: `very short`, `very tall`, `tall`, `short`, `super tall`, `average`
 - Each person has a unique level of education: `associate`, `doctorate`, `high school`, `trade school`, `bachelor`, `master`
 - Each person has a unique favorite drink: `boba tea`, `milk`, `water`, `root beer`, `coffee`, `tea`

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

Answer: trade school


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Bob`, `Peter`, `Arnold`
 - People have unique favorite book genres: `fantasy`, `mystery`, `science fiction`, `biography`, `romance`
 - The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Kailyn`, `Janelle`
 - People use unique phone models: `oneplus 9`, `iphone 13`, `samsung galaxy s21`, `google pixel 6`, `huawei p50`
 - The people are of nationalities: `dane`, `brit`, `german`, `swede`, `norwegian`
 - Each person lives in a unique style of house: `ranch`, `modern`, `craftsman`, `victorian`, `colonial`

## Clues:
1. The person in a ranch-style home is somewhere to the right of the person residing in a Victorian house.
2. The Dane and the person who uses a Samsung Galaxy S21 are next to each other.
3. The person who uses a Google Pixel 6 is not in the second house.
4. There are two houses between the person who loves fantasy books and the person in a modern-style house.
5. The person whose mother's name is Kailyn is the Swedish person.
6. The Swedish person is Arnold.
7. The British person is somewhere to the left of the person who uses an iPhone 13.
8. The Dane is The person whose mother's name is Penny.
9. The person who uses a Samsung Galaxy S21 is the Norwegian.
10. The Norwegian is The person whose mother's name is Holly.
11. The person who loves fantasy books is Bob.
12. Alice is directly left of the Swedish person.
13. The person who loves romance books is Arnold.
14. The person who loves science fiction books is Peter.
15. The person whose mother's name is Aniya is directly left of the person who loves science fiction books.
16. The person who loves science fiction books is the person who uses a OnePlus 9.
17. The person residing in a Victorian house is the person who loves biography books.
18. The person who loves fantasy books is somewhere to the left of the German.
19. The person who uses an iPhone 13 is the person in a Craftsman-style house.
 What is the value of attribute HouseStyle for the person whose attribute BookGenre is biography? Please reason step by step, and put your final answer within \boxed{}

Answer: victorian


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - People have unique favorite music genres: `jazz`, `rock`, `classical`, `pop`

## Clues:
1. Alice is directly left of Peter.
2. The person who loves classical music is directly left of Peter.
3. Arnold is the person who loves jazz music.
4. The person who loves rock music is not in the second house.
5. The person who loves rock music is directly left of the person who loves pop music.
 What is the value of attribute House for the person whose attribute MusicGenre is classical? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `Eric`, `Alice`
 - Everyone has something unique for lunch: `stew`, `stir fry`, `spaghetti`, `grilled cheese`, `pizza`
 - Everyone has a favorite smoothie: `watermelon`, `lime`, `desert`, `dragonfruit`, `cherry`
 - People have unique favorite book genres: `fantasy`, `biography`, `mystery`, `science fiction`, `romance`
 - Each person has a unique hobby: `photography`, `knitting`, `cooking`, `gardening`, `painting`

## Clues:
1. The person who loves stir fry is the Desert smoothie lover.
2. The person who loves the spaghetti eater is the person who enjoys gardening.
3. Bob is somewhere to the left of the person who loves mystery books.
4. The person who paints as a hobby is directly left of Alice.
5. The Desert smoothie lover is the person who loves romance books.
6. The photography enthusiast and the person who loves eating grilled cheese are next to each other.
7. The person who loves cooking is somewhere to the left of the Desert smoothie lover.
8. The person who likes Cherry smoothies is Alice.
9. There is one house between the person who drinks Lime smoothies and the person who enjoys gardening.
10. The Watermelon smoothie lover is Eric.
11. The person who loves eating grilled cheese is directly left of the Dragonfruit smoothie lover.
12. The person who is a pizza lover is Alice.
13. The person who loves romance books is Peter.
14. The person who loves fantasy books is the Watermelon smoothie lover.
15. Alice is the person who loves biography books.
16. Eric is not in the third house.
 What is the value of attribute House for the person whose attribute Smoothie is desert? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Bob`, `Carol`, `Arnold`
 - Everyone has a unique favorite cigar: `dunhill`, `blends`, `blue master`, `yellow monster`, `pall mall`, `prince`
 - People have unique favorite book genres: `fantasy`, `romance`, `science fiction`, `mystery`, `historical fiction`, `biography`

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

Answer: yellow monster


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Arnold`, `Peter`, `Eric`
 - Each person has a unique level of education: `doctorate`, `bachelor`, `associate`, `master`, `high school`
 - People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `google pixel 6`, `iphone 13`, `huawei p50`
 - People have unique favorite music genres: `rock`, `classical`, `jazz`, `pop`, `hip hop`
 - The people keep unique animals: `bird`, `dog`, `horse`, `cat`, `fish`
 - Each person has a unique hobby: `painting`, `cooking`, `knitting`, `gardening`, `photography`

## Clues:
1. The person who loves hip-hop music is in the second house.
2. The person who loves classical music is not in the fifth house.
3. The person who loves classical music is the fish enthusiast.
4. The person who uses an iPhone 13 is the person with a doctorate.
5. The person with a master's degree is not in the second house.
6. The person who enjoys knitting is not in the first house.
7. The person with a high school diploma is somewhere to the left of the photography enthusiast.
8. The person who paints as a hobby is somewhere to the right of the person who loves rock music.
9. Arnold is the bird keeper.
10. The person with a bachelor's degree is the person who uses a Google Pixel 6.
11. The person with a bachelor's degree is somewhere to the right of the person who paints as a hobby.
12. Bob is not in the fourth house.
13. The person who loves cooking is the person who loves classical music.
14. The person who loves cooking is the person who uses a Huawei P50.
15. The person who keeps horses is somewhere to the right of Arnold.
16. The person who paints as a hobby is the person who uses a OnePlus 9.
17. The person who uses a OnePlus 9 is somewhere to the left of the person with an associate's degree.
18. The person who uses an iPhone 13 is the dog owner.
19. The person with a high school diploma is Alice.
20. Peter is the person who enjoys gardening.
21. The person who paints as a hobby is the person who loves jazz music.
 What is the value of attribute Hobby for the person whose attribute PhoneModel is samsung galaxy s21? Please reason step by step, and put your final answer within \boxed{}

Answer: knitting


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Alice`, `Eric`, `Arnold`, `Peter`
 - People have unique hair colors: `red`, `brown`, `black`, `blonde`, `gray`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `tulips`, `roses`, `lilies`

## Clues:
1. The person who loves the vase of tulips is somewhere to the left of Eric.
2. Bob and Eric are next to each other.
3. The person who has black hair and the person who has blonde hair are next to each other.
4. Arnold is the person who loves the rose bouquet.
5. Arnold is the person who has blonde hair.
6. The person who has brown hair is not in the first house.
7. Arnold is somewhere to the right of the person who loves the boquet of lilies.
8. The person who has red hair is Eric.
9. There is one house between the person who loves the boquet of lilies and the person who has brown hair.
10. Bob is in the third house.
11. The person who has black hair is not in the third house.
12. Alice is the person who loves a carnations arrangement.
 What is the value of attribute Flower for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: roses


Kalea, a native Hawaiian, lives next door and loves sharing her knowledge of the island and its traditions. She invites you to help her prepare leis for the annual Aloha Festival. Kalea plans to make leis using two different types of flowers: 5 hibiscus flowers and 3 plumeria flowers for each lei. She wants to make 12 leis in total. How many hibiscus and plumeria flowers does Kalea need altogether to make all the leis for the festival?

Answer: 96


A publishing agent is thrilled to release the memoir of an archaeologist, detailing their incredible discoveries. The agent plans to print 2,500 copies of the book initially. The archaeologist's memoir is divided into 10 chapters, with each printed book containing 200 pages. Each chapter describes a different archaeological discovery, and the agent decides to feature 5 illustrations per chapter to showcase these discoveries. If each illustration takes up one full page, how many pages in total will the text of the book occupy across all 2,500 copies?

Answer: 375000


Jamie, a knowledgeable local and vegan enthusiast, loves taking visitors on a tour of the best vegan-friendly spots in their city. On a sunny Saturday, Jamie decides to take a group of 5 friends on a vegan food tour. They start the day at a vegan bakery, where each person buys 2 vegan pastries costing $3 each. Next, they visit a vegan cafe where everyone orders a smoothie for $5. Finally, they end the tour at a vegan restaurant, where the group collectively orders 3 large vegan pizzas costing $12 each to share. How much money did the group spend in total on the vegan food tour?

Answer: 102


Alex, a sociology graduate and community service enthusiast, is investigating the impact of telecommunications on community connectivity. Alex learned that a local telecommunications company offers a special plan for families that costs $15 per month per person. If a family of 4 subscribes to this plan, how much would they pay in total for 6 months of service? Additionally, if the company offers a 10% discount on the total cost if the family pays for all 6 months upfront, how much would the family save with the discount?

Answer: 36


As a life coach, you have a group of 12 people who you are guiding through a series of personal development workshops. You plan to hold a total of 4 workshops, and each workshop requires 2 hours of preparation time and 3 hours of actual workshop time. After each workshop, you spend 1 hour reflecting and making notes on each participant's progress. How many total hours will you spend on preparation, conducting the workshops, and reflecting on your group's progress by the time all workshops are completed?

Answer: 68


Sarah, a conscientious resident of Milton, loves to maintain her fitness by jogging around the local park. She jogs 3 times a week and each session she runs 2.5 miles. After jogging, she likes to do strength training at the gym for 45 minutes. If Sarah burns approximately 100 calories per mile while jogging and 8 calories per minute during strength training, how many total calories does she burn in one week from both jogging and strength training?

Answer: 1830


Alex is an editor for an online encyclopedia and loves collecting trivia and obscure facts. One day, Alex finds out that there are exactly 1,200 trivia questions in the trivia section of the encyclopedia. Each day, they add 8 new trivia questions and edit 3 existing ones to ensure accuracy. After 15 days, how many trivia questions will there be in total in the trivia section?

Answer: 1320