$7 \cdot 71$ A certain truck can only carry $L$ liters of gasoline, with which it can travel $a$ kilometers. Now it needs to travel $d=\frac{4}{3} a$ kilometers to a certain place, with no refueling stations along the way, but it can transport gasoline to any location along the road and store it for later use. Assuming there is only this one truck, how should it travel to reach the destination and use the least amount of gasoline? If the distance to the destination is $d=\frac{23}{15} a$ kilometers, how should it proceed?

Answer: 2L


8.4. Can the numbers from 1 to 9 be placed in the cells of a $3 \times 3$ square so that the sum of any two numbers in adjacent cells (cells sharing a side) is a prime number?

Answer: \text{No}


B2. Given is a quadrilateral $A B C D$

with sides $A B=16, B C=21, C D=2$ and $D A=28$.

Furthermore, $A B$ is parallel to $C D$. Two lines that are parallel to $A B$ and $C D$ divide quadrilateral $A B C D$ into three similar quadrilaterals. Calculate the perimeter of the smallest of these three quadrilaterals.

Answer: 13


12. As shown in the figure, rectangle $ABCD$ is a cargo yard with a length of 1000 meters and a width of 600 meters, with entrances at $A$ and $D$. A toll booth $P$ is to be built inside the cargo yard, and a loading platform $H$ is to be built on the railway segment $BC$. The minimum sum of the lengths of the three roads $AP$, $DP$, and $PH$ is $\qquad$ meters.

Answer: 600 + 500\sqrt{3}


8. In $\triangle A B C$, $a, b, c$ are the sides opposite to angles $A, B, C$ respectively. Given $a+c=2 b, A-C=\frac{\pi}{3}, \sin B=$ $\qquad$ .

Answer: \dfrac{\sqrt{39}}{8}


Example 9 The ellipse $x^{2}+2 y^{2}=2$ intersects the line $x+2 y-1=0$ at points $B$ and $C$. Point $A$ has coordinates $(2,2)$. Find the equation of the circle passing through the three points $A$, $B$, and $C$.

Answer: 6x^2 + 6y^2 - 9x - 14y - 2 = 0


13. (25 points) The equation of line $m$ is $y=k x+1, A$ and $B$ are two points on line $m$, whose x-coordinates are exactly the two different negative real roots of the quadratic equation in $x$
$$
\left(1-k^{2}\right) x^{2}-2 k x-2=0
$$

The line $l$ passes through point $P(-2,0)$ and the midpoint of segment $A B$, $C D$ is a moving line segment on the $y$-axis. Considering all possible lines $l$, when $l$ and segment $C D$ have no common points, does the maximum length of $C D$ exist? If it exists, find the maximum value; if not, explain the reason.

Answer: 4 + \sqrt{2}


107. Misunderstanding. An American reader asked me to find a number composed of any number of digits, for which division by 2 can be performed by moving the last digit to the front. Apparently, this problem arose for him after he became acquainted with a previously misstated problem. If it were required to move the first digit to the end, the answer would be the number 315789473684210526, and from this, it would be easy to find a solution starting with any digit. But if it is required to move the digit from the end to the front, there is no solution for the divisor 2. However, there is a solution for the divisor 3. Could you find it?

Answer: 428571


A regular tetrahedron $SABC$ of volume $V$ is given. The midpoints $D$ and $E$ are taken on $SA$ and $SB$ respectively and the point $F$ is taken on the edge $SC$ such that $SF: FC = 1: 3$. Find the volume of the pentahedron $FDEABC$.

Answer: \dfrac{15}{16}V


The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is $\tfrac{a-\sqrt{2}}{b}$ , where $a$ and $b$ are positive integers. What is $a+b$
[asy] real x=.369; draw((0,0)--(0,1)--(1,1)--(1,0)--cycle); filldraw((0,0)--(0,x)--(x,x)--(x,0)--cycle, gray); filldraw((0,1)--(0,1-x)--(x,1-x)--(x,1)--cycle, gray); filldraw((1,1)--(1,1-x)--(1-x,1-x)--(1-x,1)--cycle, gray); filldraw((1,0)--(1,x)--(1-x,x)--(1-x,0)--cycle, gray); filldraw((.5,.5-x*sqrt(2)/2)--(.5+x*sqrt(2)/2,.5)--(.5,.5+x*sqrt(2)/2)--(.5-x*sqrt(2)/2,.5)--cycle, gray); [/asy]

Answer: 11


C5 (8-1, USSR) In a middle school mathematics competition, three problems $A, B, C$ were given, with 25 students participating. Each student solved at least one of the problems. Among the students who did not solve problem $A$, the number of students who solved problem $B$ is twice the number of students who solved problem $C$. Among the students who solved problem $A$, the number of students who only solved $A$ is one more than the number of students who solved $A$ and at least one other problem. Additionally, among the students who only solved one problem, half did not solve problem $A$. How many students only solved problem $B$?

Answer: 6


We form a decimal code of $21$ digits. the code may start with $0$. Determine the probability that the fragment $0123456789$ appears in the code.

Answer: \dfrac{12}{10^{10}} - \dfrac{3}{10^{20}}


67. $x$ is a positive rational number, $(x)$ represents the number of prime numbers not exceeding $x$, for example, $(5)=3$, meaning there are 3 prime numbers not exceeding 5, which are 2, 3, 5. Therefore, $(x)$ defines an operation on $x$. Then $((20) \times(1)+(7))$ is $\qquad$

Answer: 2


4. The board for playing "Battleship" is $10 \times 10$. What is the maximum number of ships of size $1 \times 4$ that can be placed on it?

Answer: 24


Problem 8.1. Let $s(n)$ denote the sum of all odd digits of the number $n$. For example, $s(4)=0$, $s(173)=11, s(1623)=4$.

Calculate the value of the sum $s(1)+s(2)+s(3)+\ldots+s(321)$.

Answer: 1727


# Task No. 5.1

## Condition:

A Dog, a Cat, and a Mouse are running around a circular lake. They all started simultaneously in the same direction from the same point and finished at the same time, each running at a constant speed. The Dog ran 12 laps, the Cat ran 6 laps, and the Mouse ran 4 laps. How many total overtakes were made from the start to the finish?

If two or more overtakes occur simultaneously, each overtake is counted separately. The start and finish moments are not counted as overtakes.

Answer: 13


Eva inscribed a circle in a given triangle. Then she drew three segments that touched the inscribed circle and formed three smaller triangles within the original triangle, see the figure. The perimeters of these three smaller triangles were $12 \mathrm{~cm}, 14 \mathrm{~cm}$, and $16 \mathrm{~cm}$.

Determine the perimeter of the original triangle.

(E. Novotná)

![](https://cdn.mathpix.com/cropped/2024_04_17_25f31dcc0efac6090fc3g-3.jpg?height=440&width=440&top_left_y=1208&top_left_x=814)

Answer: 42


All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of $k$ for which the same $k$-digit number can be found in at least two different places in this string?

Answer: 6


Let $f(n)$ be a function that fulfills the following properties:

- For each natural $n$, $f(n)$ is an integer greater than or equal to $0$.
- $f(n) = 2010$, if $n$ ends in $7$. For example, $f(137) = 2010$.
- If $a$ is a divisor of $b$, then: $f\left(\frac{b}{a}\right) = |f(b) - f(a)|$.

Find $\displaystyle f(2009^{2009^{2009}})$ and justify your answer.

Answer: 0


Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
- In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.
- In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line.
- In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.

Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

Answer: 0


A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row infect the people at the ends of the rows in front of and behind them that are at the same side of the row as they are. Given that two people are chosen at random to be infected with COVID at the beginning of day 1, what is the earliest possible day that the last uninfected person will be infected with COVID?

Answer: 2019


In rectangle $ABCD$, diagonal $AC$ is intersected by the angle bisector from $B$ at point $B'$ and by the angle bisector from $D$ at point $D'$. Similarly, diagonal $BD$ is intersected by the angle bisector from $A$ at point $A'$ and by the angle bisector from $C$ at point $C'$. The area of quadrilateral $A'B'C'D'$ is $\frac{9}{16}$ of the area of rectangle $ABCD$. Determine the ratio of the longer side to the shorter side of rectangle $ABCD$.

Answer: 7


A $8 \times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.

Answer: 8


Circle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. Circle $C_2$ has radius larger than 1 and is tangent to line $A$ and both circles $B$ and $C_1$. For $n > 1$, circle $C_n$ is tangent to line $A$ and both circles $B$ and $C_{n-1}$. Find the largest value of $n$ such that this sequence of circles can be constructed through circle $C_n$ where the $n$ circles are all tangent to line $A$ at points to the right of $P$.

Answer: 45


Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each step, if you are currently at meter $n$, you have a $\frac{n}{100}$ probability of $1$ meter forward, to meter $n+1$, and a $\frac{100-n}{100}$ of going $1$ meter backward, to meter $n-1$. What is the expected value of the number of steps it takes for you to step off the bridge (i.e., to get to meter $0$ or $100$)?

Answer: 2500


In an exam, $6$ problems were proposed. Every problem was solved by exactly $1000$ students, but in no case has it happened that two students together have solved the $6$ problems. Determine the smallest number of participants that could have been in said exam.

Answer: 2000


It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest.

Answer: 5


There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number $k$ such that no matter how I select and color $k$ points, you can always color the remaining $100-k$ points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines intersect.

Answer: 50


In a school, there are $1200$ students. Each student is part of exactly $k$ clubs. For any group of $23$ students, they are part of a common club. Furthermore, there is no club to which all students belong. Determine the smallest possible value of $k$.

Answer: 23


A tram ticket costs $1$ Tug, which is equivalent to $100$ tugriks. There are $20$ passengers, each having only coins in denominations of $2$ and $5$ tugriks. The conductor starts with no money at all. Despite this, all passengers manage to pay the fare and receive the necessary change. What is the smallest total number of passengers that the tram could have?

Answer: 20


Kobar and Borah are playing a game on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board as follows:

1. From $P$ and $Q$ to $2P-Q$ and $2Q-P$, or
2. From $P$ and $Q$ to $5P-4Q$ and $5Q-4P$.

The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner.

At the beginning of the game, the two numbers on the board are $2024$ and $A$. It is known that Kobar does not lose on his first move. Determine the largest possible value of $A$ so that Borah can win this game.

Answer: 2179


Draw a regular hexagon. Then, construct a square on each edge of the hexagon. Form equilateral triangles by connecting the vertices of neighboring squares. If this pattern is extended symmetrically to infinity, what is the ratio of the number of triangles to the number of squares?

Answer: 1


Let $n$ be a positive integer. If $S$ is a nonempty set of positive integers, then we say $S$ is $n$-complete if all elements of $S$ are divisors of $n$, and if $d_1$ and $d_2$ are any elements of $S$, then $n$ divides $d_1$ and gcd$(d_1, d_2)$ are in $S$. How many $2310$-complete sets are there?

Answer: 32


An edge of an inclined parallelepiped is equal to $l$. It is bounded by two adjacent faces, whose areas are $m^{2}$ and $n^{2}$, and the planes of these faces form an angle of $30^{\circ}$. Calculate the volume of the parallelepiped.

Answer: \frac{m^2 n^2}{2 l}


A smooth ball with a radius of 1 cm was dipped in red paint and then placed between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm respectively (the ball ended up outside the smaller sphere but inside the larger one). When the ball comes into contact with both spheres, it leaves a red mark. During its motion, the ball traced a closed path, resulting in a region outlined by a red contour on the smaller sphere, with an area of 47 square cm. Find the area of the region outlined by the red contour on the larger sphere. Give the answer in square centimeters, rounded to hundredths if necessary.

Answer: 105.75 \text{ cm}^2


We move on a grid by taking only one-unit steps to the right or upward. What is the number of ways to go from point \((0,0)\) to point \((m, n)\)?

Answer: \binom{m+n}{m}


Three hedgehogs are dividing three pieces of cheese with masses of 5 g, 8 g, and 11 g. A fox comes to help them. She can simultaneously cut 1 g of cheese from any two pieces and eat it. Can the fox leave the hedgehogs with equal pieces of cheese?

Answer: Yes


Vasya throws three dice (each die has numbers from 1 to 6 on its faces) and sums up the rolled numbers. Furthermore, if all three rolled numbers are different, he can roll all three dice again and add the rolled numbers to the already accumulated sum. This process continues until at least two of the three rolled numbers are the same. What is the expected value of Vasya's total result?

Answer: 23.625


Find all polynomials \( P \in \mathbb{R}[X] \) such that \( P(X^2 + 1) = P(X)^2 + 1 \) with \( P(0) = 0 \).

Bonus (difficult): Remove the condition \( P(0) = 0 \).

Answer: P(X) = X


We colored every side of unit squares using one of the colors red, yellow, blue, and green in such a way that each square has all four sides in different colors. The squares colored in this way can be assembled along their sides with the same color.

Can a $99 \times 99$ square and a $100 \times 100$ square be assembled from such colored squares such that the outer sides of the assembled square are single-colored and each side has a different color?

Answer: \text{Yes}


Suppose I have a closed cardboard box in the shape of a cube. By cutting 7 out of its 12 edges with a razor blade (it must be exactly 7 edges), I can unfold the box onto a plane, and the unfolding can take various forms. For instance, if I cut along the edges shown by the bold lines in the diagram and along the invisible edge indicated by a dashed line, I will get unfolding $A$. By cutting the box differently, one can obtain unfolding $B$ or $C$. It is easy to see that unfolding $D$ is simply a flipped version of unfolding $C$, so we consider these two unfoldings to be identical.

How many distinct unfoldings can be obtained in this way?

Answer: 11


Divide each three-digit (natural) number by the sum of its digits in your mind. What is the smallest resulting quotient?

Answer: \frac{199}{19}


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


With a number, it is permissible to perform one of two actions: square it or add one to it. Given the numbers 19 and 98, is it possible to obtain equal numbers from them using the same number of actions?

Answer: \text{No}


Each interior angle of an \( n \)-sided regular polygon is \( 160^{\circ} \). Find \( n \).

The \( n \)-th day of May in a year is Friday. The \( k \)-th day of May in the same year is Tuesday, where \( 20 < k < 26 \). Find \( k \).

Answer: 22


Let \(\left(u_{n}\right)\) be the sequence defined by \( u_{0}=5 \) and \( u_{n+1}=u_{n}+\frac{1}{u_{n}} \) for \( n \in \mathbb{N} \). Show that \( u_{1000}>45 \).

Answer: u_{1000} > 45


The most common dice is a six-sided die, which is a cube. The 6 faces have 1 to 6 points on them, with the sum of points on opposite faces equal to 7. From a single point in space, you can see multiple faces of the die. The minimum visible sum of points is 1, and the maximum is 15 (15 = 4 + 5 + 6). Determine which sums in the range from 1 to 15 are not possible to see.

Answer: 13


A carpenter wants to cut a wooden cube with a side length of 3 inches into 27 smaller cubes with a side length of 1 inch each. The question is: "What is the minimum number of cuts required if the pieces can be rearranged arbitrarily during the cutting process, given that the carpenter can make 6 cuts to split the cube into smaller cubes while keeping the pieces together so they don't fall apart?"

Given that without rearranging the pieces, the minimum number of cuts would be 6 because each face needs to be cut to separate the inner cube, what is the minimum number of cuts required if the inner cube is missing and pieces can be rearranged arbitrarily during the process?

Answer: 6


A car starts from point \( O \) and moves along a straight road with a constant speed \( v \). At the same time, a person on a bicycle, located at a distance \( a \) from point \( O \) and \( b \) from the road, begins to ride with the intention of delivering a letter to the car driver. What is the minimum speed the cyclist must maintain to achieve this goal?

Answer: \frac{v b}{a}


We roll a die $n$ times. What is the probability that there are two equal numbers among the rolled numbers?

Answer: 1 - \dfrac{6!}{(6 - n)! \cdot 6^n}


1. Solve the inequality $\frac{\sqrt{\frac{x}{\gamma}+(\alpha+2)}-\frac{x}{\gamma}-\alpha}{x^{2}+a x+b} \geqslant 0$.

In your answer, specify the number equal to the number of integer roots of this inequality. If there are no integer roots, or if there are infinitely many roots, enter the digit 0 in the answer sheet.

$$
\alpha=3, \gamma=1, a=-15, b=54 .
$$

Answer: 7


10. Fill the nine blanks in Figure 2 with the numbers $1,2, \cdots, 9$, requiring that each row from left to right and each column from top to bottom increase sequentially. When 3 and 4 are fixed in the positions shown in the figure, the number of ways to fill the blanks is . $\qquad$

Answer: 6


Equilateral triangle $ABC$ has side length $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the “lune” that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p}-\frac{q\pi}{r}$, where $p, q$, and $ r$ are positive integers such that $q$ and $r$ are relatively prime. Compute $p + q + r$.
[img]https://cdn.artofproblemsolving.com/attachments/7/7/f349a807583a83f93ba413bebf07e013265551.png[/img]

Answer: 10


1. All gnomes are divided into liars and knights. Liars always lie, and knights always tell the truth. On each cell of a $4 \times 4$ board, there is a gnome. It is known that among them, there are both liars and knights. Each gnome stated: “Among my neighbors (by side) there are an equal number of liars and knights.” How many liars are there in total?

Answer: 12


Bogdanov I.I.

In a certain country, there are 100 cities (consider the cities as points on a plane). In a directory, for each pair of cities, there is a record of the distance between them (a total of 4950 records).

a) One record has been erased. Can it always be uniquely restored from the remaining ones?

b) Suppose $k$ records have been erased, and it is known that no three cities in this country lie on the same straight line. For what largest $k$ can the erased records always be uniquely restored?

#

Answer: 96


$\mathbb{\text { Frankin B.P. }}$

In a row, 39 numbers, not equal to zero, are written. The sum of any two adjacent numbers is positive, while the sum of all numbers is negative.

What is the sign of the product of all numbers

#

Answer: positive


I4.4 Let $x$ and $y$ be real numbers satisfying the equation $(x-\sqrt{p})^{2}+(y-\sqrt{p})^{2}=1$.
If $k=\frac{y}{x-3}$ and $q$ is the least possible values of $k^{2}$, find the value of $q$.

Answer: 8


4. Determine all natural numbers $n$ that have the following properties: $n$ is divisible by 2 but not by 4, the sum of the digits of $n$ is equal to 6, the number of divisors of $n$ is equal to 8, the sum of the divisors of $n$ is divisible by 10, and $n$ does not give a remainder of 12 or 14 when divided by 16.

Answer: 114


The sequence \(\{a_n\}\) is defined as follows: \(a_n = \lfloor \sqrt{2} n \rfloor\), for \(n = 0, 1, 2, \cdots\). Prove that this sequence contains infinitely many perfect squares.

Answer: \text{The sequence } \{a_n\} \text{ contains infinitely many perfect squares.}


The chords  $AB$  and  $CD$  of a circle intersect at  $M$ , which is the midpoint of the chord  $PQ$ . The points  $X$  and  $Y$  are the intersections of the segments  $AD$  and  $PQ$ , respectively, and  $BC$  and  $PQ$ , respectively. Show that  $M$  is the midpoint of  $XY$ .

Answer: M \text{ is the midpoint of } XY


4. When a die is rolled three times, the probability that the three numbers facing up can form the side lengths of a triangle whose perimeter is divisible by 3 is $\qquad$ .

Answer: \dfrac{11}{72}


## Task B-3.6.

A rectangle is given with side lengths $a$ and $b$, where $a = \frac{4}{7} b$. Determine the probability that a randomly selected point within the given rectangle is closer to the shorter side and is less than $\frac{a}{8}$ away from it.

Answer: \dfrac{1}{8}


Let the functions \( f(x) \) and \( g(x) \) be defined on the set of real numbers \(\mathbf{R}\) such that:
(1) \( f(0)=0 \);
(2) For any real numbers \( x \) and \( y \), \( g(x-y) \geqslant f(x)f(y) + g(x)g(y) \).

Prove that \( f^{2008}(x) + g^{2008}(x) \leqslant 1 \).

Answer: f^{2008}(x) + g^{2008}(x) \leqslant 1


30. Arrange the natural numbers 1 to 7 in a row, such that each number is either greater than all the numbers before it or less than all the numbers before it (for example: 4356271 satisfies the condition, starting from the second position, 3 is less than the first 4; 5 is greater than the first two 4, 3; 6 is greater than the first three 4, 3, 5; 2 is less than the first four 4, 3, 5, 6; 7 is greater than the first five 4, 3, 5, 6, 2; 1 is less than the first six 4, 3, 5, 6, 2, 7). Then, among the arrangements that satisfy the condition, the number of arrangements where 7 is not in the fourth position is  kinds.

Answer: 60


250. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $a$. On the line $A A_{1}$, a point $M$ is taken, and on the line $B C$ - a point $N$ such that the line $M N$ intersects the edge $C_{1} D_{1}$. Find the minimum value of the magnitude | $M N \mid$.

Answer: 3a


## Task 6B - 221246B

In the investigation of frequency distributions in mathematical statistics, functions appear that are defined for finitely many natural numbers and are required to satisfy so-called functional equations (equations between different function values).

An example of this is the following:

Given a natural number $n \geq 2$ and a real number $p$ with $0<p<1$.

Determine (in dependence on $n$ and $p$) those functions $f$ with the set $\{1,2, \ldots, n\}$ as the domain, which for $k=1,2, \ldots, n$ satisfy the following equation (1):

$$
\sum_{x=0}^{n} x \cdot(x-1) \cdot(x-2) \cdot \ldots \cdot(x-k+1) \cdot f(x)=n \cdot(n-1) \cdot(n-2) \cdot \ldots \cdot(n-k+1) \cdot p^{k}
$$

Hint: For $k=1$, equation (1) is to be interpreted as

$$
\sum_{x=0}^{n} x \cdot f(x)=n \cdot p
$$

Answer: f(x) = \binom{n}{x} p^x (1 - p)^{n - x}


Let \( g(n) \) denote the sum of all the positive divisors of the integer \( n \) including twice the greatest prime divisor of \( n \). How many integers \( j \) exist such that \( 1 \leq j \leq 5000 \) and \( g(j) = j + 2\sqrt{j} + 1 \)?

Answer: 19


A square region has quarter-circular arcs constructed at each of its corners. If the side of the square measures $4/\pi$, what is the perimeter of this region?

Answer: 4


At a different university, the department of natural sciences consists of the physics, chemistry, and biology departments. Each department has three male and three female professors. A committee is to be formed with four men and four women. Additionally, the committee must include:
- At least one professor from each gender in each department,
- Exactly three professors from any one department.
Find the number of possible committees that can be formed following these conditions.

Answer: 4374


Seventy percent of a cruise ship's passengers are women, and fifteen percent of those women are in luxury suites. What is the number of women in luxury suites if the cruise ship is carrying 250 passengers?

Answer: 26


A cylinder with a height of `15` and a radius of `5` is topped with a cone that has the same radius and a height of `8`. Calculate the total surface area of this combined shape.

Answer: 175\pi + 5\pi \sqrt{89}


Let's say that James and Sara each think of a polynomial. Each of their polynomials is monic, has degree 4, and has the same positive coefficient for the term $z^3$ and the same positive constant term. The product of their polynomials results in \[z^8 + 4z^7 + 5z^6 + 7z^5 + 9z^4 + 8z^3 + 6z^2 + 8z + 9.\] Determine the constant term of Sara's polynomial.

Answer: 3


Let $G'$ be a point inside triangle $ABC$ such that $G' = \frac{1}{4}A + \frac{1}{4}B + \frac{1}{2}C$. Let $Q$ be another arbitrary point. Determine the constant $k'$ such that
\[ QA^2 + QB^2 + QC^2 = k' \cdot QG'^2 + GA^2 + GB^2 + GC^2. \]

Answer: 4


Peter has two numbers, $a$ and $b$. When he divides $a$ by 80, he gets a remainder of 75. When he divides $b$ by 90, he gets a remainder of 85. What remainder does he get when he divides $a+b$ by 40?

Answer: 0


There exists a constant $c$, so that among all chords $\overline{AB}$ of the circle $(x-1)^2 + (y-1)^2 = 4$ passing through $C = (0,c),$
\[ t = \sqrt{d_A^2 + d_B^2} \]
where $d_A$ and $d_B$ are the distances from $C$ to $A$ and $B$, respectively, is a fixed constant. Find the constant $t$.

Answer: 2


Answer: \frac{1}{2}


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
 - People have unique favorite sports: `basketball`, `swimming`, `soccer`, `tennis`
 - People have unique hair colors: `black`, `brown`, `blonde`, `red`
 - Each person has a favorite color: `yellow`, `red`, `white`, `green`
 - The people are of nationalities: `dane`, `norwegian`, `brit`, `swede`
 - Each person has a unique birthday month: `april`, `feb`, `jan`, `sept`

## Clues:
1. The person who has brown hair is in the first house.
2. The person who loves soccer is directly left of the person who loves tennis.
3. The person who has brown hair is directly left of Eric.
4. The person who has blonde hair is somewhere to the left of the British person.
5. The person who loves white is directly left of the Norwegian.
6. The person who has blonde hair and Arnold are next to each other.
7. The person whose birthday is in January is the Swedish person.
8. The person whose birthday is in April is the British person.
9. The person who loves basketball is somewhere to the right of the person who has black hair.
10. The person who has brown hair and the person who loves yellow are next to each other.
11. The person who loves swimming is Peter.
12. The person whose favorite color is red is Peter.
13. The person who has blonde hair is the Swedish person.
14. Eric is the person whose birthday is in February.
 What is the value of attribute Color for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: green


There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`
 - People have unique favorite music genres: `pop`, `rock`
 - Each person has a unique type of pet: `dog`, `cat`
 - Everyone has something unique for lunch: `pizza`, `grilled cheese`
 - The people are of nationalities: `brit`, `dane`
 - Each person has a unique birthday month: `april`, `sept`

## Clues:
1. Arnold is the person who loves eating grilled cheese.
2. The person whose birthday is in April is in the first house.
3. Eric is the person who owns a dog.
4. The person who loves rock music is the person who owns a dog.
5. The British person is in the second house.
6. The person who is a pizza lover is directly left of the person who loves eating grilled cheese.
 What is the value of attribute Birthday for the person whose attribute Nationality is brit? Please reason step by step, and put your final answer within \boxed{}

Answer: sept


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - Each person has a unique level of education: `associate`, `bachelor`, `high school`
 - Each person has a unique favorite drink: `water`, `tea`, `milk`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`
 - Each person has a unique hobby: `photography`, `cooking`, `gardening`
 - People own unique car models: `ford f150`, `tesla model 3`, `toyota camry`

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

Answer: mystery


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`, `Bob`, `Alice`, `Peter`, `Arnold`
 - Each person has a favorite color: `yellow`, `green`, `red`, `white`, `blue`
 - Each mother is accompanied by their child: `Samantha`, `Fred`, `Meredith`, `Bella`, `Timothy`
 - People have unique favorite book genres: `biography`, `mystery`, `romance`, `science fiction`, `fantasy`
 - Each person has a unique birthday month: `april`, `feb`, `sept`, `jan`, `mar`
 - People have unique favorite sports: `soccer`, `basketball`, `swimming`, `baseball`, `tennis`

## Clues:
1. The person who loves swimming is the person who loves mystery books.
2. Eric is somewhere to the left of the person who is the mother of Timothy.
3. The person who loves basketball is somewhere to the left of the person who is the mother of Timothy.
4. The person who loves soccer is the person whose birthday is in September.
5. Eric is the person who loves biography books.
6. The person who loves baseball is the person who loves white.
7. The person who loves science fiction books is the person whose birthday is in March.
8. The person who loves swimming and the person whose favorite color is red are next to each other.
9. The person who loves biography books is not in the first house.
10. The person who loves baseball is somewhere to the left of the person whose favorite color is red.
11. The person who loves fantasy books is Bob.
12. The person's child is named Bella is the person whose birthday is in February.
13. The person who loves tennis and the person's child is named Samantha are next to each other.
14. The person's child is named Bella is not in the first house.
15. The person's child is named Meredith is the person whose birthday is in March.
16. The person who loves blue is directly left of the person who loves yellow.
17. There is one house between Alice and the person whose birthday is in March.
18. The person whose favorite color is red is directly left of the person whose birthday is in January.
19. The person's child is named Meredith is not in the first house.
20. Arnold is somewhere to the right of the person who loves swimming.
21. The person who loves fantasy books is the person who is the mother of Timothy.
 What is the value of attribute FavoriteSport for the person whose attribute Birthday is feb? Please reason step by step, and put your final answer within \boxed{}

Answer: swimming


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 CarModel for the person whose attribute Food is grilled cheese? Please reason step by step, and put your final answer within \boxed{}

Answer: toyota camry


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
 - People have unique favorite music genres: `rock`, `jazz`, `pop`, `classical`
 - Everyone has a favorite smoothie: `dragonfruit`, `watermelon`, `desert`, `cherry`
 - People have unique hair colors: `red`, `black`, `brown`, `blonde`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`

## Clues:
1. The person who has red hair is not in the third house.
2. There is one house between Eric and the person who likes Cherry smoothies.
3. The person who loves science fiction books is Peter.
4. The person who loves fantasy books is Alice.
5. The Dragonfruit smoothie lover is the person who loves pop music.
6. The Dragonfruit smoothie lover is the person who has red hair.
7. The Watermelon smoothie lover is somewhere to the right of the person who loves classical music.
8. The person who has black hair is the person who likes Cherry smoothies.
9. There is one house between the person who loves pop music and the person who loves fantasy books.
10. The person who loves romance books is somewhere to the right of the person who loves jazz music.
11. The person who has red hair is directly left of the person who has black hair.
12. The person who has brown hair and the Dragonfruit smoothie lover are next to each other.
13. The person who loves science fiction books is the Dragonfruit smoothie lover.
 What is the value of attribute House for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`, `knitting`
 - People have unique favorite sports: `swimming`, `tennis`, `soccer`, `baseball`, `basketball`
 - Each person lives in a unique style of house: `ranch`, `craftsman`, `victorian`, `modern`, `colonial`
 - Each mother is accompanied by their child: `Timothy`, `Samantha`, `Bella`, `Meredith`, `Fred`
 - People have unique heights: `average`, `very tall`, `very short`, `short`, `tall`

## Clues:
1. The person who has an average height is the person's child is named Meredith.
2. The person who is tall is in the second house.
3. Peter is directly left of the person residing in a Victorian house.
4. Alice is the person who is tall.
5. The person who loves baseball is the person who is very tall.
6. The person's child is named Meredith and the person who is the mother of Timothy are next to each other.
7. Bob is the person who paints as a hobby.
8. The person who enjoys gardening is in the second house.
9. The person who is very short is somewhere to the right of Eric.
10. The person who loves tennis is the person's child is named Samantha.
11. The person who loves soccer is not in the first house.
12. The person's child is named Samantha is the person in a modern-style house.
13. The person in a Craftsman-style house is the person who has an average height.
14. The person's child is named Fred is the person residing in a Victorian house.
15. The person who is short is the person who loves basketball.
16. Peter is the person who is very tall.
17. The person in a ranch-style home is somewhere to the left of the person who loves cooking.
18. The person who enjoys knitting and the person who enjoys gardening are next to each other.
19. The person in a modern-style house is the person who loves cooking.
20. The person residing in a Victorian house is in the fifth house.
 What is the value of attribute Children for the person whose attribute Hobby is cooking? 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: `Arnold`, `Eric`, `Bob`, `Alice`, `Carol`, `Peter`
 - The people are of nationalities: `german`, `brit`, `dane`, `chinese`, `swede`, `norwegian`
 - People have unique favorite music genres: `pop`, `jazz`, `hip hop`, `classical`, `country`, `rock`
 - Each person has a unique level of education: `doctorate`, `associate`, `master`, `bachelor`, `high school`, `trade school`

## Clues:
1. The Chinese is the person who loves pop music.
2. The person with a bachelor's degree is directly left of Arnold.
3. The person with a master's degree is in the first house.
4. Carol is somewhere to the left of Peter.
5. The Dane is the person with a doctorate.
6. The person who loves classical music is the person with a master's degree.
7. The person with a bachelor's degree is in the fifth house.
8. Eric is somewhere to the right of Bob.
9. Carol is the British person.
10. The person with a doctorate is somewhere to the right of the Norwegian.
11. The person with a doctorate is the person who loves rock music.
12. The Norwegian is somewhere to the right of the German.
13. The person who attended trade school is somewhere to the left of the person with a high school diploma.
14. The person who attended trade school and the Chinese are next to each other.
15. The person who loves jazz music is Arnold.
16. The person who loves hip-hop music is directly left of Alice.
 What is the value of attribute Nationality for the person whose attribute MusicGenre is jazz? Please reason step by step, and put your final answer within \boxed{}

Answer: swede


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`, `Bob`, `Alice`, `Arnold`, `Peter`
 - Each person has an occupation: `engineer`, `doctor`, `artist`, `lawyer`, `teacher`
 - Everyone has a favorite smoothie: `desert`, `dragonfruit`, `lime`, `cherry`, `watermelon`
 - Each mother is accompanied by their child: `Fred`, `Bella`, `Meredith`, `Samantha`, `Timothy`

## Clues:
1. Arnold is somewhere to the right of the person who is a teacher.
2. The person's child is named Bella is in the third house.
3. The person who is the mother of Timothy is the person who is an artist.
4. Alice is in the third house.
5. The person who is a doctor is directly left of the person who is an engineer.
6. The Desert smoothie lover is the person who is an engineer.
7. The person's child is named Fred is the person who is a teacher.
8. The person who is an engineer is not in the third house.
9. Bob is the Dragonfruit smoothie lover.
10. Peter is somewhere to the right of the person who is an engineer.
11. The person's child is named Meredith is the person who drinks Lime smoothies.
12. The person's child is named Meredith is in the first house.
13. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
 What is the value of attribute House for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}

Answer: 2


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Alice`, `Bob`, `Arnold`, `Carol`
 - People have unique favorite book genres: `mystery`, `fantasy`, `romance`, `historical fiction`, `science fiction`, `biography`
 - Everyone has a favorite smoothie: `cherry`, `desert`, `lime`, `watermelon`, `blueberry`, `dragonfruit`
 - The people keep unique animals: `fish`, `rabbit`, `bird`, `cat`, `horse`, `dog`
 - People have unique favorite music genres: `classical`, `hip hop`, `country`, `jazz`, `rock`, `pop`
 - Everyone has a unique favorite cigar: `prince`, `dunhill`, `blends`, `pall mall`, `blue master`, `yellow monster`

## Clues:
1. The person who smokes many unique blends is the person who loves fantasy books.
2. The person who loves biography books is the Desert smoothie lover.
3. There are two houses between the person who loves pop music and the person who loves country music.
4. The rabbit owner is in the third house.
5. Arnold is the bird keeper.
6. The person who loves country music is in the fifth house.
7. The person who loves mystery books is not in the first house.
8. The person who drinks Lime smoothies is not in the third house.
9. The fish enthusiast is somewhere to the right of the person who loves rock music.
10. The Prince smoker is somewhere to the left of the person who likes Cherry smoothies.
11. Bob is the cat lover.
12. Eric is the person who likes Cherry smoothies.
13. The person who smokes Yellow Monster is in the sixth house.
14. The Prince smoker is the Dragonfruit smoothie lover.
15. Carol is directly left of Arnold.
16. The person who smokes Blue Master is directly left of the person who loves classical music.
17. The person who loves classical music is the person who loves historical fiction books.
18. The Dunhill smoker and the person who loves science fiction books are next to each other.
19. The person who keeps horses is directly left of Peter.
20. The Watermelon smoothie lover is the cat lover.
21. The person who smokes many unique blends is directly left of the person who loves jazz music.
22. The cat lover is in the first house.
23. The Desert smoothie lover is the person who loves jazz music.
24. The person who drinks Lime smoothies is directly left of the fish enthusiast.
25. The person who smokes Blue Master is not in the fifth house.
 What is the value of attribute MusicGenre for the person whose attribute Animal is rabbit? Please reason step by step, and put your final answer within \boxed{}

Answer: jazz


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Eric`, `Peter`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `lilies`, `roses`
 - Everyone has a unique favorite cigar: `dunhill`, `blue master`, `pall mall`, `prince`
 - Each mother is accompanied by their child: `Meredith`, `Samantha`, `Bella`, `Fred`
 - The mothers' names in different houses are unique: `Janelle`, `Aniya`, `Holly`, `Kailyn`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `samsung galaxy s21`

## Clues:
1. The person whose mother's name is Kailyn is the Prince smoker.
2. The person whose mother's name is Holly is the person who loves a bouquet of daffodils.
3. Arnold is in the fourth house.
4. The person's child is named Bella is somewhere to the left of the person who uses a OnePlus 9.
5. The person's child is named Fred is not in the second house.
6. The person's child is named Meredith is directly left of The person whose mother's name is Janelle.
7. The person who loves a bouquet of daffodils is Eric.
8. The person who loves the boquet of lilies is the person partial to Pall Mall.
9. The person whose mother's name is Janelle is the person who loves a carnations arrangement.
10. Alice is somewhere to the right of the person's child is named Fred.
11. The person who uses an iPhone 13 is directly left of the Dunhill smoker.
12. The person who loves the rose bouquet is somewhere to the right of the Dunhill smoker.
13. The person who loves the rose bouquet is somewhere to the left of the person who uses a Google Pixel 6.
 What is the value of attribute Mother for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`, `Bob`
 - Each person has a unique level of education: `associate`, `bachelor`, `high school`, `master`, `doctorate`
 - People have unique favorite sports: `swimming`, `basketball`, `tennis`, `baseball`, `soccer`
 - Each person prefers a unique type of vacation: `camping`, `cruise`, `city`, `mountain`, `beach`
 - People use unique phone models: `google pixel 6`, `huawei p50`, `oneplus 9`, `samsung galaxy s21`, `iphone 13`

## Clues:
1. The person who uses a Huawei P50 is not in the second house.
2. The person with a master's degree is the person who uses a Google Pixel 6.
3. The person who loves swimming is the person with a high school diploma.
4. The person with a bachelor's degree is directly left of the person who loves tennis.
5. The person who enjoys camping trips is the person who loves tennis.
6. Bob is the person who uses a Samsung Galaxy S21.
7. The person with a high school diploma is not in the third house.
8. The person who loves swimming is directly left of the person who loves beach vacations.
9. The person who likes going on cruises is Arnold.
10. Bob is the person with an associate's degree.
11. The person who uses a OnePlus 9 is directly left of the person who loves basketball.
12. The person who enjoys camping trips and the person who prefers city breaks are next to each other.
13. Eric is the person who loves swimming.
14. The person who loves basketball is the person who enjoys mountain retreats.
15. The person who loves tennis is somewhere to the left of Alice.
16. Arnold is the person who loves soccer.
 What is the value of attribute Education for the person whose attribute PhoneModel is huawei p50? Please reason step by step, and put your final answer within \boxed{}

Answer: high school


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Arnold`, `Eric`, `Alice`, `Bob`, `Peter`
 - Each person has a unique level of education: `bachelor`, `high school`, `master`, `associate`, `trade school`, `doctorate`
 - Each person has a favorite color: `purple`, `yellow`, `white`, `red`, `blue`, `green`
 - The people are of nationalities: `swede`, `dane`, `norwegian`, `chinese`, `brit`, `german`
 - Each person has a unique birthday month: `feb`, `may`, `sept`, `april`, `jan`, `mar`

## Clues:
1. The person who loves yellow is somewhere to the right of the person whose birthday is in January.
2. Bob is the person who loves white.
3. The person who loves blue is the person whose birthday is in April.
4. The person with an associate's degree is Eric.
5. The person whose favorite color is green is the person whose birthday is in May.
6. The person who loves yellow is not in the fifth house.
7. The person with an associate's degree is directly left of the Swedish person.
8. The person with a doctorate is the Dane.
9. The person whose birthday is in January is somewhere to the right of the person whose favorite color is green.
10. There are two houses between the British person and the person whose birthday is in February.
11. The person whose birthday is in February is in the fourth house.
12. Alice is the person with a high school diploma.
13. Carol is in the first house.
14. Bob is the German.
15. The person whose birthday is in May is directly left of the Norwegian.
16. The person whose favorite color is red is not in the fifth house.
17. The person whose birthday is in September is in the second house.
18. There are two houses between the person whose birthday is in May and the person with a master's degree.
19. The Norwegian is Peter.
20. The person who attended trade school is the person who loves purple.
 What is the value of attribute Color for the person whose attribute Nationality is brit? Please reason step by step, and put your final answer within \boxed{}

Answer: green


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Bob`, `Peter`, `Arnold`, `Eric`
 - People have unique hair colors: `black`, `red`, `brown`, `gray`, `blonde`
 - Each person has a unique level of education: `bachelor`, `associate`, `high school`, `master`, `doctorate`
 - Each mother is accompanied by their child: `Bella`, `Timothy`, `Samantha`, `Meredith`, `Fred`
 - The people keep unique animals: `horse`, `cat`, `fish`, `bird`, `dog`

## Clues:
1. The person who has red hair is directly left of the bird keeper.
2. Peter is the person with a doctorate.
3. The fish enthusiast is somewhere to the right of the person who has brown hair.
4. The cat lover is the person with a bachelor's degree.
5. The person's child is named Samantha is somewhere to the right of the person who has black hair.
6. The fish enthusiast is the person's child is named Bella.
7. Arnold and the person with an associate's degree are next to each other.
8. The person who keeps horses is the person who is the mother of Timothy.
9. The person who has brown hair is somewhere to the right of Bob.
10. The person who has blonde hair is the bird keeper.
11. The person with a master's degree is the fish enthusiast.
12. The person with a high school diploma is the person's child is named Fred.
13. The person with a high school diploma is Eric.
14. The person with a bachelor's degree is the person who has red hair.
15. The person who has gray hair is the person with an associate's degree.
 What is the value of attribute House for the person whose attribute Children is Samantha? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Bob`, `Alice`, `Carol`
 - The people keep unique animals: `rabbit`, `dog`, `fish`, `bird`, `cat`, `horse`
 - Each person has a unique hobby: `cooking`, `knitting`, `painting`, `woodworking`, `photography`, `gardening`
 - People have unique favorite sports: `basketball`, `soccer`, `swimming`, `tennis`, `baseball`, `volleyball`
 - Each person prefers a unique type of vacation: `cruise`, `cultural`, `city`, `camping`, `mountain`, `beach`
 - Each person has a unique favorite drink: `water`, `root beer`, `boba tea`, `tea`, `milk`, `coffee`

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

Answer: gardening


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`, `Alice`
 - Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`, `stew`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`
 - Each person has a unique hobby: `painting`, `cooking`, `photography`, `gardening`

## Clues:
1. Peter is the person who is a pizza lover.
2. The person who is a pizza lover and the person who loves the spaghetti eater are next to each other.
3. The person who loves eating grilled cheese is directly left of Eric.
4. Eric is the person who loves cooking.
5. The person who uses a Google Pixel 6 is in the fourth house.
6. The photography enthusiast is in the second house.
7. The person who is a pizza lover is the person who uses a OnePlus 9.
8. The person who uses a Samsung Galaxy S21 is in the third house.
9. Arnold is directly left of the person who enjoys gardening.
 What is the value of attribute PhoneModel for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}

Answer: iphone 13


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
 - They all have a unique favorite flower: `carnations`, `tulips`, `lilies`, `roses`, `daffodils`, `iris`
 - People have unique favorite book genres: `science fiction`, `fantasy`, `romance`, `mystery`, `biography`, `historical fiction`
 - People use unique phone models: `huawei p50`, `oneplus 9`, `iphone 13`, `xiaomi mi 11`, `samsung galaxy s21`, `google pixel 6`
 - People have unique hair colors: `black`, `gray`, `red`, `auburn`, `blonde`, `brown`
 - Each person has a favorite color: `yellow`, `green`, `red`, `white`, `blue`, `purple`

## Clues:
1. There is one house between the person who has gray hair and the person who uses an iPhone 13.
2. The person who loves the rose bouquet is the person who has blonde hair.
3. There is one house between the person who has gray hair and the person who loves yellow.
4. Carol is somewhere to the right of the person who loves a bouquet of daffodils.
5. The person who loves white is somewhere to the left of Peter.
6. The person who loves white is somewhere to the left of the person who loves science fiction books.
7. Arnold is directly left of the person who loves historical fiction books.
8. The person who uses a Huawei P50 is the person who loves a bouquet of daffodils.
9. The person who loves purple is somewhere to the left of the person who loves fantasy books.
10. Eric is the person who loves science fiction books.
11. The person who loves a bouquet of daffodils is the person who has auburn hair.
12. Alice is the person whose favorite color is green.
13. The person who loves fantasy books is in the second house.
14. The person who uses a Samsung Galaxy S21 is Carol.
15. The person who uses a Xiaomi Mi 11 is not in the fourth house.
16. Eric is the person who has auburn hair.
17. The person who loves the rose bouquet is somewhere to the left of the person who has red hair.
18. The person who loves historical fiction books is the person who loves the rose bouquet.
19. The person who uses a Xiaomi Mi 11 is somewhere to the right of the person who loves white.
20. The person who has black hair is in the fourth house.
21. The person whose favorite color is red is the person who loves mystery books.
22. The person who loves historical fiction books and the person who loves the vase of tulips are next to each other.
23. The person who has brown hair is the person who uses a OnePlus 9.
24. The person who loves romance books is somewhere to the left of the person who loves fantasy books.
25. The person who loves the boquet of lilies is in the second house.
26. The person who loves the boquet of iris is somewhere to the right of the person whose favorite color is green.
 What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Peter`, `Alice`, `Arnold`, `Bob`, `Carol`
 - People have unique hair colors: `brown`, `red`, `gray`, `black`, `blonde`, `auburn`

## Clues:
1. Arnold is not in the sixth house.
2. The person who has blonde hair and the person who has brown hair are next to each other.
3. Bob is the person who has blonde hair.
4. The person who has black hair is in the fourth house.
5. Eric is the person who has black hair.
6. Carol is somewhere to the left of Eric.
7. Peter is in the first house.
8. The person who has auburn hair is not in the third house.
9. Bob is not in the sixth house.
10. Carol is somewhere to the right of the person who has red hair.
11. The person who has blonde hair is not in the third house.
12. There are two houses between the person who has brown hair and the person who has auburn hair.
 What is the value of attribute House for the person whose attribute Name is Peter? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Alice`, `Peter`
 - People have unique heights: `tall`, `short`, `average`, `very short`

## Clues:
1. Alice is somewhere to the left of the person who is very short.
2. The person who is tall is somewhere to the right of the person who has an average height.
3. The person who has an average height is Eric.
4. Peter is not in the fourth house.
5. There is one house between the person who is short and Peter.
6. The person who is very short is not in the third house.
 What is the value of attribute Height for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}

Answer: short


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Bob`, `Peter`, `Eric`, `Alice`, `Carol`
 - Each person has a unique birthday month: `april`, `feb`, `mar`, `jan`, `may`, `sept`
 - Each person prefers a unique type of vacation: `city`, `beach`, `mountain`, `cruise`, `cultural`, `camping`
 - Each person has a favorite color: `blue`, `white`, `green`, `purple`, `red`, `yellow`

## Clues:
1. The person whose birthday is in January is the person who loves blue.
2. The person whose birthday is in January is directly left of the person who loves white.
3. The person who enjoys camping trips is in the fourth house.
4. Eric is in the sixth house.
5. There are two houses between the person who likes going on cruises and the person whose birthday is in February.
6. The person whose birthday is in May is not in the sixth house.
7. Peter is directly left of the person who loves yellow.
8. The person who goes on cultural tours is Eric.
9. The person whose birthday is in September is directly left of the person whose favorite color is green.
10. Arnold and the person who loves purple are next to each other.
11. The person who loves beach vacations is the person whose birthday is in March.
12. The person whose favorite color is green is somewhere to the left of Carol.
13. The person whose birthday is in May is somewhere to the right of Peter.
14. The person who loves blue is Bob.
15. The person who enjoys mountain retreats is in the fifth house.
16. The person who loves purple is in the third house.
17. Carol is in the fourth 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: 5


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

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

Answer: very short


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Bob`, `Alice`, `Eric`, `Arnold`, `Peter`
 - Each person has a unique type of pet: `hamster`, `cat`, `fish`, `rabbit`, `dog`, `bird`
 - Each person has a favorite color: `blue`, `purple`, `yellow`, `white`, `red`, `green`

## Clues:
1. The person who has a cat is not in the third house.
2. Arnold is not in the first house.
3. The person who owns a dog is the person whose favorite color is green.
4. The person who loves yellow is somewhere to the left of Bob.
5. The person who loves blue is the person who has a cat.
6. There are two houses between the person with an aquarium of fish and Eric.
7. The person who owns a dog is Carol.
8. The person who loves yellow is the person with a pet hamster.
9. The person who loves white and the person who keeps a pet bird are next to each other.
10. The person who owns a rabbit is somewhere to the left of the person who loves white.
11. The person who owns a dog is directly left of the person who loves blue.
12. The person who loves purple is directly left of Carol.
13. Bob is not in the third house.
14. Alice is the person who loves purple.
 What is the value of attribute House for the person whose attribute Pet is dog? 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: `Arnold`, `Eric`, `Alice`, `Peter`
 - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Bella`, `Fred`
 - Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `watermelon`, `desert`
 - Everyone has something unique for lunch: `stew`, `grilled cheese`, `spaghetti`, `pizza`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Janelle`, `Aniya`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `lilies`

## Clues:
1. The Watermelon smoothie lover is the person's child is named Bella.
2. The person whose mother's name is Aniya and the person's child is named Fred are next to each other.
3. Alice is the person's child is named Meredith.
4. Eric is the person who likes Cherry smoothies.
5. The person who loves a bouquet of daffodils is directly left of The person whose mother's name is Janelle.
6. The person who loves the spaghetti eater is in the first house.
7. The person who likes Cherry smoothies is the person who loves the boquet of lilies.
8. The person's child is named Meredith is The person whose mother's name is Kailyn.
9. There is one house between The person whose mother's name is Kailyn and The person whose mother's name is Janelle.
10. Arnold is in the second house.
11. The person who loves eating grilled cheese is somewhere to the right of The person whose mother's name is Aniya.
12. The person whose mother's name is Janelle is somewhere to the right of The person whose mother's name is Kailyn.
13. The person who loves the stew is the person who loves a carnations arrangement.
14. Eric is somewhere to the left of the Dragonfruit smoothie lover.
 What is the value of attribute Mother for the person whose attribute Smoothie is cherry? Please reason step by step, and put your final answer within \boxed{}

Answer: Janelle


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Bob`, `Alice`, `Carol`, `Peter`
 - The people are of nationalities: `german`, `brit`, `dane`, `chinese`, `swede`, `norwegian`
 - People have unique favorite music genres: `pop`, `jazz`, `hip hop`, `classical`, `country`, `rock`
 - Each person has a unique level of education: `doctorate`, `associate`, `master`, `bachelor`, `high school`, `trade school`

## Clues:
1. The Chinese is the person who loves pop music.
2. The person with a bachelor's degree is directly left of Arnold.
3. The person with a master's degree is in the first house.
4. Carol is somewhere to the left of Peter.
5. The Dane is the person with a doctorate.
6. The person who loves classical music is the person with a master's degree.
7. The person with a bachelor's degree is in the fifth house.
8. Eric is somewhere to the right of Bob.
9. Carol is the British person.
10. The person with a doctorate is somewhere to the right of the Norwegian.
11. The person with a doctorate is the person who loves rock music.
12. The Norwegian is somewhere to the right of the German.
13. The person who attended trade school is somewhere to the left of the person with a high school diploma.
14. The person who attended trade school and the Chinese are next to each other.
15. The person who loves jazz music is Arnold.
16. The person who loves hip-hop music is directly left of Alice.
 What is the value of attribute House for the person whose attribute Nationality is norwegian? 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: `Arnold`, `Alice`, `Eric`, `Bob`, `Peter`
 - Each person has a unique birthday month: `mar`, `feb`, `jan`, `sept`, `april`
 - They all have a unique favorite flower: `tulips`, `lilies`, `carnations`, `daffodils`, `roses`
 - Each person has a unique level of education: `high school`, `bachelor`, `doctorate`, `associate`, `master`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`, `dragonfruit`, `lime`

## Clues:
1. There is one house between the Desert smoothie lover and the person whose birthday is in April.
2. The person with a bachelor's degree is Peter.
3. The person who loves a bouquet of daffodils is the person with a high school diploma.
4. Eric is not in the second house.
5. The person who drinks Lime smoothies and the person who loves the rose bouquet are next to each other.
6. Alice is in the third house.
7. The person whose birthday is in February is Peter.
8. The person with a master's degree is the person whose birthday is in March.
9. The person whose birthday is in February is in the fourth house.
10. The person whose birthday is in March is the person who drinks Lime smoothies.
11. Peter is the Dragonfruit smoothie lover.
12. The person with an associate's degree is not in the second house.
13. The person who loves a carnations arrangement is not in the second house.
14. The Watermelon smoothie lover is Arnold.
15. Alice is the person whose birthday is in September.
16. The person who loves the boquet of lilies is not in the second house.
17. The person with an associate's degree is Arnold.
18. The person who loves the vase of tulips is Peter.
19. The person who loves the boquet of lilies is not in the first house.
 What is the value of attribute Flower for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: roses


In the figure on the right, $O$ is the center of the circle. The lines $OK$ and $OA$ are perpendicular to each other. $M$ is the midpoint of $OK$, and $BN$ is parallel to $OK$. Given that $\angle AMN = \angle NMO$, determine the measure of $\angle ABN$ in degrees.

```asy
defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, A=dir(90), K=dir(180), M=0.5*dir(180), N=2/5*dir(90), B=dir(degrees((2/5, sqrt(21/25)))+90);
draw(K--O--A--M--N--B--A^^Circle(origin,1));
label("$A$", A, dir(O--A));
label("$K$", K, dir(O--K));
label("$B$", B, dir(O--B));
label("$N$", N, E);
label("$M$", M, S);
label("$O$", O, SE);
```

Note: The problem contains geometric constructs and angle relationships that need to be solved to find the measure of the specified angle. Ensure the Asymptote (asy) code is correctly rendered to visualize the problem's diagram.

Answer: 36


In a certain foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies within the government. Friendship and enmity are symmetric relations. A triplet of ministers is called **uniform** if all three are either mutual friends or mutual enemies. How many uniform triplets are there?

Answer: 40


The diagrams below show a $2 \times 2$ grid made up of four $1 \times 1$ squares. Two paths are illustrated on the grid, starting from the lower left corner to the upper right corner, with lengths $4$ and $6$ respectively. A path may not intersect itself by moving to a point where it has already been. Find the sum of the lengths of all possible paths from the lower left corner to the upper right corner of the grid.

Answer: 64


Let $A$ and $B$ be unit hexagons that share a center. Let $\mathcal{P}$ be the set of points contained in at least one of the hexagons. If the maximum possible area of $\mathcal{P}$ is $X$ and the minimum possible area of $\mathcal{P}$ is $Y$, then the value of $Y-X$ can be expressed as $\frac{a\sqrt{b}-c}{d}$, where $a, b, c, d$ are positive integers such that $b$ is square-free and $\gcd(a, c, d) = 1$. Find $a + b + c + d$.

Answer: 62


Answer: 540


In quadrilateral $ABCD$, where $AC = BD$ and $\angle B = 60^\circ$, denote by $M$ and $N$ the midpoints of $\overline{AB}$ and $\overline{CD}$, respectively. Given that $MN = 12$ and the area of quadrilateral $ABCD$ is 420, compute $AC$.

Answer: 37


Let $\triangle ABC$ have $AB = 15$, $AC = 20$, and $BC = 21$. Suppose $\omega$ is a circle passing through $A$ that is tangent to segment $BC$. Let point $D \neq A$ be the second intersection of $AB$ with $\omega$, and let point $E \neq A$ be the second intersection of $AC$ with $\omega$. Suppose $DE$ is parallel to $BC$. If $DE = \frac{a}{b}$, where $a$, $b$ are relatively prime positive integers, find $a + b$.

Answer: 361


Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Company $XYZ$ wants to locate their base at the point $P$ in the plane minimizing the total distance to their workers, who are located at vertices $A$, $B$, and $C$. There are $1$, $5$, and $4$ workers at $A$, $B$, and $C$, respectively. Find the minimum possible total distance Company $XYZ$'s workers have to travel to get to $P$.

Answer: 69


Find the number of ordered pairs of integers $(a,b) \in \{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a quadratic residue modulo $x^2+1$ and $35$. This means there exists a polynomial $f(x)$ with integer coefficients such that either of the following equivalent conditions holds:

- There exist polynomials $P(x)$, $Q(x)$ with integer coefficients such that $f(x)^2 - (ax+b) = (x^2+1)P(x) + 35Q(x)$.
- The remainder when the polynomial $f(x)^2 - (ax+b)$ is divided by the polynomial $x^2+1$ is a polynomial with integer coefficients, all divisible by $35$.

Answer: 225


A positive integer has exactly $81$ divisors, which are arranged in a $9 \times 9$ table. In this table, for any two numbers in the same row or column, one of them divides the other. Find the maximum possible number of distinct prime divisors of $n$.

Answer: 2


Triangle $ABC$ satisfies $\angle ABC = \angle ACB = 78^\circ$. Points $D$ and $E$ lie on $AB$ and $AC$ respectively, and satisfy $\angle BCD = 24^\circ$ and $\angle CBE = 51^\circ$. If $\angle BED = x^\circ$, find $x$.

Answer: 12


Tim is participating in the following three math contests. On each contest, his score is the number of correct answers.
- The Local Area Inspirational Math Exam consists of 15 problems.
- The Further Away Regional Math League has 10 problems.
- The Distance-Optimized Math Open has 50 problems.

For every positive integer $n$, Tim knows the answer to the $n$th problems on each contest (which are pairwise distinct), if they exist; however, these answers have been randomly permuted so that he does not know which answer corresponds to which contest. Unaware of the shuffling, he competes with his modified answers. Compute the expected value of the sum of his scores on all three contests.

Answer: 50


Let $c$ be a fixed real number. Show that a root of the equation
\[x(x+1)(x+2)\cdots(x+2009)=c\]
can have multiplicity at most $2$. Determine the number of values of $c$ for which the equation has a root of multiplicity $2$.

Answer: 1005


6. For a natural number $n$, $G(n)$ denotes the number of natural numbers $m$ for which $m+n$ divides $m n$. Find $G\left(10^{\mathrm{k}}\right)$.

Answer: 2k(k + 1)


7. In $\triangle A B C$, $A C=B C, \angle A C B=90^{\circ}$, $D 、 E$ are points on side $A B$, $A D=3, B E=4$, $\angle D C E=45^{\circ}$. Then the area of $\triangle A B C$ is $\qquad$

Answer: 36


5. Try to simplify $\sum_{k=0}^{n} \frac{(-1)^{k} C_{n}^{k}}{k^{3}+9 k^{2}+26 k+24}$ into the form $\frac{p(n)}{q(n)}$, where $p(n)$ and $q(n)$ are two polynomials with integer coefficients.

Answer: \dfrac{1}{2n^2 + 14n + 24}


7. For all $x \in\left[-2, \frac{1}{2}\right]$, the inequality
$$
a x^{3}-x^{2}+x+1 \geqslant 0
$$

always holds. Then the range of the real number $a$ is $\qquad$ $\because$

Answer: [-10, -1]


8. For $\triangle A B C$ with side lengths $a, b, c(a \leqslant b \leqslant c)$, the following conditions are satisfied: (1) $a, b, c$ are all integers; (2) $a, b, c$ form a geometric sequence; (3) at least one of $a$ and $c$ equals 100. The number of all possible triples $(a, b, c)$ is $\qquad$

Answer: 10


4.6 Four table tennis players are dressed in 4 different colors for a performance match, with 2 in red, 2 in yellow, 1 in blue, and 1 in black. Each performance selects 3 players to appear, and matches are only played between players with different colored outfits, with the specific rules being:
(1) If the "3-person group" on stage has all different colored outfits, then each pair of players will play 1 match, and players who have already played against each other in different "3-person groups" will still play again if they meet;
(2) If the "3-person group" on stage has 2 players with the same colored outfit, then these 2 players will not play against each other, and only 1 of them will play 1 match against the other player.

According to these rules, when all different "3-person groups" have appeared, a total of $\qquad$ matches have been played.

Answer: 44


66. Given the password: $3 \times \overline{A B C P Q R}=4 \times \overline{P Q R A B C}$, where each letter represents a decimal digit, translate this password into an equation is $\qquad$ .

Answer: 3 \times 571428 = 4 \times 428571


10. (20 points) In the Cartesian coordinate system $x O y$, the vertices of $\triangle A B C$ are $B(-1,0)$, $C(1,0)$, and $A$ is a moving point. $H$ and $G$ are the orthocenter and centroid of $\triangle A B C$, respectively. If the midpoint $K$ of segment $H G$ lies on the line $B C$:
(1) Find the equation of the trajectory $E$ of point $A$;
(2) A line passing through point $C$ with a slope of 1 intersects the curve $E$ at point $D$. If there exist two points $P$ and $Q$ on the curve $E$ such that points $P$, $C$, $Q$, and $D$ are concyclic, find the minimum value of the area of the circumcircle of quadrilateral $P C Q D$.

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


Ezekiel has a rectangular piece of paper with an area of 40 . The width of the paper is more than twice the height. He folds the bottom left and top right corners at $45^{\circ}$ and creates a parallelogram with an area of 24 . What is the perimeter of the original rectangle?

![](https://cdn.mathpix.com/cropped/2024_04_17_45c89f07dc9351e0476cg-2.jpg?height=198&width=415&top_left_y=324&top_left_x=909)

Answer: 28


3. On the sides $AB$ and $CD$ of the inscribed quadrilateral $ABCD$, which is not a trapezoid, points $P$ and $R$ are marked, respectively, such that $AP: PB = CR: RD$. On the segment $PR$, a point $Q$ is chosen such that $PQ: QR = AB: CD$. Find the ratio of the areas of triangles $AQD$ and $BQC$, given that $AD = x$ and $BC = y$.

Answer: \dfrac{x}{y}


1. Two workers with a constant work rate can complete a job in 30 days. After 6 days of joint work, the first worker quits, and the second worker completes the job in 40 days from the day the first worker quits. How many days would it take for the first worker to complete the same job if he continued alone after 6 days of joint work?

Answer: 60


1. On a grid paper, a rectangle 200 cells wide and 100 cells tall is drawn. It is filled cell by cell, starting from the top-left corner and following a spiral path (turning right when reaching an edge or a filled part, see Fig. 8). Which cell will be filled last? (Specify the number of its row and column. For example, the bottom-right cell is in the 100th row and 200th column.)

![](https://cdn.mathpix.com/cropped/2024_05_21_6455f48598646f41a890g-048.jpg?height=164&width=294&top_left_y=1307&top_left_x=951)

Fig. 8

Answer: (51, 50)


69*. Determine the five last digits of the number

$$
\left.N=9^{\left(9^{(} \cdot 9^{\left(9^{9}\right)}\right)} \ldots\right),
$$

written using 1001 nines in a similar manner to the number in problem 68).

Answer: 45289


1.1. From the village to the station along one road, a dachnik A set off on foot and a motorcyclist with a passenger - dachnik B - at the same time. Not reaching the station, the motorcyclist dropped off the passenger and immediately headed back to the village, while dachnik B continued on foot to the station. Meeting dachnik A, the motorcyclist picked him up and brought him to the station. As a result, both dachniks arrived at the station simultaneously. What part of the journey from the village to the station did dachnik A travel by motorcycle, if the dachniks walked at the same speed, 9 times slower than the speed of the motorcycle?

Answer: \dfrac{5}{6}


11. As shown in the figure, positive integers starting from 1 are arranged in the following form, and a "L" shaped pattern composed of 3 squares (which can be rotated) is used to frame three of these numbers (in the figure, the sum of the three numbers framed is $10+11+18=39$). If the sum of the three numbers framed by such a "L" shape is 2015, then the largest number among them is $\qquad$.

Answer: 676


B3. Twelve balls are numbered with the numbers $1,2,3, \ldots, 12$. Each ball is painted either red or green such that the following conditions are met:

(a) if the balls labeled with different numbers $a$ and $b$ are painted red and $a+b<13$, then the ball labeled with the number $a+b$ is also painted red;

(b) if the balls labeled with different numbers $a$ and $b$ are painted green and $a+b<13$, then the ball labeled with the number $a+b$ is also painted green.

In how many ways can the balls be painted?

## Problems for 3rd Grade

Time for solving: 120 minutes. In section A, a correct answer will be worth two points, while a wrong answer will result in a deduction of half a point. Write your answers for section A in the left table.

![](https://cdn.mathpix.com/cropped/2024_06_07_4050f14f84cf4f327f67g-06.jpg?height=160&width=317&top_left_y=548&top_left_x=641)

| B1 | B2 | B3 |
| :--- | :--- | :--- |
|  |  |  |

Answer: 6


9.9. In an alphabet of $n>1$ letters, a word is any finite sequence of letters in which any two adjacent letters are different. A word is called good if it is impossible to erase all letters from it, except for four, so that the remaining sequence is of the form $a a b b$, where $a$ and $b$ are different letters. Find the maximum possible number of letters in a good word.

(D. Khramtsov)

Answer: 2n +1


Let $ABC$ be a triangle such that angle $C$ is obtuse. Let $D \in [AB]$ and $[DC] \perp [BC]$. If $m(\widehat{ABC}) = \alpha$, $m(\widehat{BCA}) = 3\alpha$, and $|AC| - |AD| = 10$, find the length of $|BD|$.

Answer: 20


Let $T$ be the triangle in the coordinate plane with vertices $(0,0)$, $(4,0)$, and $(0,3)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. Find how many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position.

Answer: 12


Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{\left(\frac{n}{2}\right)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, find the sum of the digits of $n$.

Answer: 15


There is an unlimited number of chips of six colors. What is the minimum number of chips that need to be arranged in a row so that for any two different colors in the row, there are two adjacent chips of these colors?

Answer: 18


Consider parallelogram \(ABCD\) with \(AB > BC\). Point \(E\) on \(\overline{AB}\) and point \(F\) on \(\overline{CD}\) are marked such that there exists a circle \(\omega_1\) passing through \(A, D, E, F\) and a circle \(\omega_2\) passing through \(B, C, E, F\). If \(\omega_1\) and \(\omega_2\) partition \(\overline{BD}\) into segments \(\overline{BX}, \overline{XY}, \overline{YD}\) in that order, with lengths 200, 9, and 80, respectively, compute \(BC\).

Answer: 51


In the figure, the number 8 was obtained by summing the two numbers directly below its position. By doing the same to fill in the blank positions, the number 42 is obtained in the indicated position. What is the value of \( x \)?

(a) 7
(b) 3
(c) 5
(d) 4
(e) 6

![Figure](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-008.jpg?height=331&width=311&top_left_y=366&top_left_x=1518)

Answer: 6


Suppose point \(P\) is inside triangle \(ABC\). Let \(AP\), \(BP\), and \(CP\) intersect sides \(BC\), \(CA\), and \(AB\) at points \(D\), \(E\), and \(F\), respectively. Suppose \(\angle APB = \angle BPC = \angle CPA\), \(PD = \frac{1}{4}\), \(PE = \frac{1}{5}\), and \(PF = \frac{1}{7}\). Compute \(AP + BP + CP\).

Answer: \frac{19}{12}


A book is composed of 12 identical notebooks. Each notebook consists of several double sheets nested within one another. The notebooks are stitched together sequentially. All pages in the book are numbered starting from 1. The sum of the numbers of four pages of one of the double sheets of the fourth notebook is 338. How many pages are in this book?

Answer: 288


In triangle \(ABC\) (angle B is obtuse), altitude \(BH\) and angle bisector \(AK\) are drawn. Find the angle \(\angle AKB\) if \(\angle KHC\) is \(45^\circ\).

Answer: 45


A three-digit number is called a "cool number" if there is a digit in one of its places that is half the product of the other two digits. A three-digit number is called a "super cool number" if such digits appear in two or three of its places. How many different "super cool numbers" exist? (Zero is not allowed in the digits of "cool" and "super cool" numbers).

Answer: 25


Let \( P \) be a 30-sided polygon inscribed in a circle. Find the number of triangles whose vertices are the vertices of \( P \) such that any two vertices of each triangle are separated by at least three other vertices of \( P \).

Answer: 1900


In a regular pentagon \(ABCDE\), there is an equilateral triangle \(ABM\). Determine the measure of the angle \(BCM\).

Answer: 66


Let \( S(x) \) denote the sum of the digits of a positive integer \( x \). Find the maximum possible value of \( S(x+2019) - S(x) \).

Answer: 12


A cup is filled with a salt solution at a concentration of 15%. There are three iron balls in large, medium, and small sizes, with their volume ratio being 10:5:3. First, the small ball is placed into the cup of salt solution, causing 10% of the salt solution to overflow. After removing the small ball, the medium ball is placed into the cup and then removed. Next, the large ball is placed into the cup and then removed. Finally, pure water is added to the cup to fill it to the top. What is the final concentration of the salt solution in the cup?

Answer: 10


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Bob`, `Alice`, `Peter`, `Arnold`, `Carol`
 - They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `tulips`, `iris`, `lilies`
 - People have unique heights: `very tall`, `super tall`, `average`, `short`, `very short`, `tall`
 - Each person has a favorite color: `blue`, `yellow`, `purple`, `red`, `green`, `white`
 - Everyone has a unique favorite cigar: `yellow monster`, `pall mall`, `prince`, `blue master`, `blends`, `dunhill`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `xiaomi mi 11`, `samsung galaxy s21`, `huawei p50`, `oneplus 9`

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

Answer: 5


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
 - Each person has a unique hobby: `cooking`, `gardening`, `painting`, `photography`
 - People own unique car models: `honda civic`, `toyota camry`, `tesla model 3`, `ford f150`
 - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `spaghetti`, `stew`
 - Each person has an occupation: `doctor`, `artist`, `engineer`, `teacher`
 - They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `carnations`

## Clues:
1. The person who loves the stew is Eric.
2. The person who is a teacher is directly left of Peter.
3. The person who loves a carnations arrangement is the person who is an artist.
4. The person who paints as a hobby is the person who loves the spaghetti eater.
5. There is one house between the person who is an artist and the photography enthusiast.
6. The person who enjoys gardening is in the first house.
7. The person who loves the spaghetti eater is somewhere to the right of the person who owns a Honda Civic.
8. The person who is a pizza lover is directly left of the photography enthusiast.
9. The person who loves a carnations arrangement is Alice.
10. The person who owns a Tesla Model 3 is directly left of Eric.
11. The person who is an engineer is the person who loves a bouquet of daffodils.
12. The person who loves the boquet of lilies is somewhere to the left of the person who loves the stew.
13. There is one house between the person who enjoys gardening and the person who owns a Toyota Camry.
14. The person who is a teacher is in the third house.
 What is the value of attribute House for the person whose attribute CarModel is toyota camry? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Bob`, `Alice`, `Arnold`, `Eric`, `Peter`
 - People use unique phone models: `samsung galaxy s21`, `google pixel 6`, `iphone 13`, `huawei p50`, `oneplus 9`, `xiaomi mi 11`
 - The people are of nationalities: `swede`, `chinese`, `norwegian`, `dane`, `german`, `brit`
 - Each person has a favorite color: `blue`, `red`, `yellow`, `green`, `white`, `purple`

## Clues:
1. Carol is not in the third house.
2. There is one house between the Dane and the British person.
3. Carol is the person whose favorite color is green.
4. Arnold is directly left of Alice.
5. Alice is the German.
6. The person who uses a OnePlus 9 is the person who loves purple.
7. The person who uses a Huawei P50 is not in the third house.
8. The person who uses a Samsung Galaxy S21 is in the fifth house.
9. The person who loves white is somewhere to the right of the person whose favorite color is red.
10. The person who uses a Samsung Galaxy S21 is Bob.
11. The Dane is the person who loves yellow.
12. The person who uses a Samsung Galaxy S21 is somewhere to the left of Peter.
13. The person who loves blue is Peter.
14. Peter is the British person.
15. The person who uses a Samsung Galaxy S21 is directly left of the person who uses an iPhone 13.
16. The Norwegian is the person who loves purple.
17. The person who uses a Xiaomi Mi 11 is the Chinese.
 What is the value of attribute House for the person whose attribute Nationality is dane? Please reason step by step, and put your final answer within \boxed{}

Answer: 4


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`
 - Each person has a unique favorite drink: `water`, `tea`, `coffee`, `milk`
 - Each person has a unique level of education: `master`, `associate`, `high school`, `bachelor`
 - The mothers' names in different houses are unique: `Kailyn`, `Holly`, `Aniya`, `Janelle`
 - Each person has a favorite color: `yellow`, `white`, `green`, `red`

## Clues:
1. Eric is the person with an associate's degree.
2. The person whose mother's name is Kailyn is the person who loves yellow.
3. The person who likes milk is in the third house.
4. The person who likes milk is somewhere to the right of the person who loves white.
5. The person with a bachelor's degree is in the first house.
6. The person whose mother's name is Aniya is not in the second house.
7. The one who only drinks water is Arnold.
8. The person whose favorite color is green is Peter.
9. The person whose favorite color is red is somewhere to the left of the person who loves white.
10. The person with a high school diploma is directly left of The person whose mother's name is Janelle.
11. The person with a high school diploma is the tea drinker.
 What is the value of attribute Color for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}

Answer: white