Let a square have a side length of \(10^{0.2}\) meters. Compute the area of the square and then multiply it by \(10^{0.1}\), \(10^{-0.3}\), and \(10^{0.4}\).

Answer: 10^{0.6}


A plane's passengers consist of 50% women and 50% men. Twenty percent of the women and fifteen percent of the men are in first class. What is the total number of passengers in first class if the plane is carrying 300 passengers?

Answer: 53


What is the least number of colors needed to shade the tessellation shown, such that no two tiles sharing a side or overlapping vertically are the same color? The tessellation consists of a base layer of rectangles uniformly placed and a top layer consisting of circles, where each circle overlaps four rectangles (one on each side, without side sharing).

Answer: 3


As shown in the figure, in triangle $\mathrm{ABC}$, points $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ are on segments $\mathrm{AZ}, \mathrm{BX}, \mathrm{CY}$ respectively, and $Y Z=2 Z C, Z X=3 X A, X Y=4 Y B$. The area of triangle $X Y Z$ is 24. Find the area of triangle $\mathrm{ABC}$.

Answer: 59


Inside a circular pancake with a radius of 10, a coin with a radius of 1 has been baked. What is the minimum number of straight cuts needed to surely hit the coin?

Answer: 10


A nine-digit number is odd. The sum of its digits is 10. The product of the digits of the number is non-zero. The number is divisible by seven. When rounded to three significant figures, how many millions is the number equal to?

Answer: 112


A point is randomly thrown onto the segment [6, 11], and let \( k \) be the resulting value. Find the probability that the roots of the equation \(\left(k^{2}-2k-15\right)x^{2}+(3k-7)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \).

Answer: \frac{1}{3}


4. Let the function $y=f(x)$ have a domain of $\mathbf{R}$, and for $x>1$, and for any real numbers $x, y \in \mathbf{R}$, $f(x+y)=f(x) f(y)$ holds. The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=f(0)$, and $f\left(a_{n+1}\right)=\frac{1}{f\left(-2-a_{n}\right)}(n \in \mathbf{N})$. If the inequality $\left(1+\frac{1}{a_{1}}\right)\left(1+\frac{1}{a_{2}}\right) \cdots\left(1+\frac{1}{a_{n}}\right) \geqslant k \cdot$ $\sqrt{2 n+1}$ holds for all $n \in \mathbf{N}$, find the minimum value of $k$. (2003 Hunan Province Mathematics Competition Problem)

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


There are four circles. In the first one, a chord $A B$ is drawn, and the distance from the midpoint of the smaller of the two arcs formed to $A B$ is 1. The second, third, and fourth circles are located inside the larger segment and are tangent to the chord $A B$. The second and fourth circles are tangent to the inside of the first and externally to the third. The sum of the radii of the last three circles is equal to the radius of the first circle. Find the radius of the third circle, given that the line passing through the centers of the first and third circles is not parallel to the line passing through the centers of the other two circles.

Answer: \dfrac{1}{2}


8.1. Of all numbers with the sum of digits equal to 25, find the one whose product of digits is maximal. If there are several such numbers, write the smallest of them in the answer.

Answer: 33333334


Example 1 Let the base edge length of the regular tetrahedron $V-A B C$ be 4, and the side edge length be 8. Construct a section $A E D$ through $A$ that intersects the side edges $V B, V C$. Find the minimum perimeter of the section $\triangle A E D$.

Answer: 11


A chess knight has injured his leg and is limping. He alternates between a normal move and a short move where he moves to any diagonally neighbouring cell.
The limping knight moves on a $5 \times 6$ cell chessboard starting with a normal move. What is the largest number of moves he can make if he is starting from a cell of his own choice and is not allowed to visit any cell (including the initial cell) more than once?

Answer: 24


In a circle with center $O$, the diameter $A B$ is extended to $C$ such that $B C = \frac{1}{2} \cdot A B = \frac{r}{2}$. A tangent line $s$ is drawn at point $B$, and from an arbitrary point $M$ on this tangent $s$, another tangent $M D$ to the circle is drawn. Show that $\angle B M C = \frac{1}{3} \angle D M C$.

Answer: \angle BMC = \frac{1}{3} \angle DMC


Four identical pieces, in the shape of right triangles, were arranged in two different ways, as shown in the given figures. The squares $A B C D$ and $E F G H$ have sides respectively equal to $3 \mathrm{~cm}$ and $9 \mathrm{~cm}$. Determine the measure of the side of the square $I J K L$.
![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-011.jpg?height=432&width=1036&top_left_y=1326&top_left_x=592)

Answer: 3\sqrt{5}


1. Daria Dmitrievna is preparing a test on number theory. She promised to give each student as many problems as the number of addends they create in the numerical example

$$
a_{1}+a_{2}+\ldots+a_{n}=2021
$$

where all numbers $a_{i}$ are natural numbers, greater than 10, and are palindromes (do not change if their digits are written in reverse order). If a student does not find any such example, they will receive 2021 problems on the test. What is the minimum number of problems a student can receive? (20 points)

Answer: 3


## Task B-2.4.

Pinocchio tells the truth on Mondays and Tuesdays, always lies on Saturdays, and on the other days of the week, he either tells the truth or lies. In response to the question "What is your favorite subject at school?" over six consecutive days of the week, he gave the following answers in order: "History", "Mathematics", "Geography", "Physics", "Chemistry", "Physics". What subject does Pinocchio like the most? Explain your answer.

Answer: History


## Task B-1.7.

The length of the base of an isosceles triangle is $6 \mathrm{~cm}$, and the cosine of the angle at the base is $\frac{5}{13}$. Determine the radius of the circle that touches both legs of the triangle and the circle circumscribed around it.

Answer: \dfrac{169}{72}


8th Chinese 1993 Problem A2 α > 0 is real and n is a positive integer. What is the maximum possible value of α a + α b + α c + ... , where a, b, c, ... is any sequence of positive integers with sum n?

Answer: \max(n \alpha, \alpha^n)


3. In a right-angled triangle $ABC$ with hypotenuse $AB$, let $I$ and $U$ be the center of the inscribed circle and the point of tangency of this circle with the leg $BC$, respectively. Determine the ratio $|AC|:|BC|$, if the angles $\angle CAU$ and $\angle CBI$ are equal.

(Jaroslav Zhouf)

Answer: \dfrac{3}{4}


[ Motion problems]

The road from home to school takes Petya 20 minutes. One day, on his way to school, he remembered that he had forgotten his pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell, but if he returns home for the pen, he will, walking at the same speed, be 7 minutes late for the start of the lesson. What part of the way had he walked before he remembered the pen?

Answer: \dfrac{1}{4}


13. A and B are partners in a business, and together they made a profit of $a^{2}$ yuan (where $a$ is a two-digit natural number). When dividing the money, A takes 100 yuan first, then B takes 100 yuan, followed by A taking another 100 yuan, B taking another 100 yuan, and so on, until the last amount taken is less than 100 yuan. To ensure that both end up with the same total amount, the one who took more gave 35.5 yuan to the one who took less. The total profit they made has $\qquad$ possible values.

Answer: 4


Dudeney, Amusements in Mathematics Problem 25 The common cash consists of brass coins of varying thickness, with a round, square, or triangular hold in the center. These are strung on wires like buttons. Supposing that eleven coins with round holes are worth fifteen ching-changs, that eleven with square holes are worth sixteen ching-changs, and that eleven with triangular holes are worth seventeen ching-changs, how can a Chinaman give me change for half a crown, using no coins other than the three mentioned? A ching-chang is worth exactly twopence and four-fifteenths of a ching-chang.

Answer: 7


6. It is given that the sequence $\left(a_{n}\right)_{n=1}^{\infty}$, with $a_{1}=a_{2}=2$, is given by the recurrence relation
$$
\frac{2 a_{n-1} a_{n}}{a_{n-1} a_{n+1}-a_{n}^{2}}=n^{3}-n
$$
for all $n=2,3,4, \ldots$. Find the integer that is closest to the value of $\sum_{k=2}^{2011} \frac{a_{k+1}}{a_{k}}$.

Answer: 3015


Subiectul IV. (20 puncte) Fie matricele $A=\left(\begin{array}{ccc}1 & a & a^{2} \\ 1 & 2 & 4 \\ 1 & 3 & 9\end{array}\right)$ si $B=\left(\begin{array}{ccc}1 & 2 a & a^{2} \\ 1 & 4 & 4 \\ 1 & 6 & 9\end{array}\right)$.

a) Scrieți rezultatul determinantului $\operatorname{det}\left(A \cdot B^{t}\right)$ sub formă de produs;

b) Scrieți rezultatul determinantului $D=\left|\begin{array}{lll}\left(x^{2}+1\right)^{2} & (x y+1)^{2} & (x z+1)^{2} \\ (x y+1)^{2} & \left(y^{2}+1\right)^{2} & (y z+1)^{2} \\ (x z+1)^{2} & (y z+1)^{2} & \left(z^{2}+1\right)^{2}\end{array}\right|$ sub formă de produs. prof. Anca Cristina Hodorogea, ISJ Cluj

Toate subiectele sunt obligatorii. Se acordă 10 puncte din oficiu.

SUCCES!

Timp efectiv de lucru - 3 ore.

## Barem clasa a XI-a  (OLM 2014-etapa locală)

## Of. $10 \mathrm{p}$

Answer: 2(a-2)^2(a-3)^2


187. Elementary arithmetic. Here is one question similar to those that were so popular in Venice (and not only there) in the mid-16th century. Their appearance was largely due to Niccolò Fontana, better known as Tartaglia (the stammerer).

If a quarter of twenty were four, then what would a third of ten be?

Answer: \dfrac{8}{3}


Two circles, the radii of which are in the ratio $9-4 \sqrt{3}$, touch each other internally. Two chords of the larger circle, equal in length and tangent to the smaller circle, are drawn. One of these chords is perpendicular to the segment connecting the centers of the circles, while the other is not. Find the angle between these chords.

Answer: 30^\circ


Problem 11.8. In each cell of a strip $1 \times N$ there is either a plus or a minus. Vanya can perform the following operation: choose any three cells (not necessarily consecutive), one of which is exactly in the middle between the other two cells, and change the three signs in these cells to their opposites. A number $N$ is called positive if from an arrangement of $N$ minuses Vanya can, by such operations, obtain an arrangement of $N$ pluses.

Consider the numbers $3,4,5, \ldots, 1400$. How many of them are positive?

Answer: 1396


11. (12 points) If a three-digit number $\overline{\mathrm{abc}}$ (where $a, b, c$ are all non-zero digits) satisfies $\overline{\mathrm{ab}}>\overline{\mathrm{bc}}>\overline{\mathrm{ca}}$, then the three-digit number is called a “Longteng number”. How many “Longteng numbers” are there?

Answer: 120


Answer: 2


In the figure, \( \mathrm{P} \) is a point on the diagonal \( \mathrm{ED} \) of \( \triangle \mathrm{ABCD} \) such that \( \angle \mathrm{PCB} = \angle \mathrm{ACD} \). The circumcircle of \( \triangle \mathrm{ABD} \) intersects the diagonal \( \mathrm{AC} \) at point \( \mathrm{E} \). Prove that \( \angle \mathrm{AED} = \angle \mathrm{PEB} \).

Answer: \angle \mathrm{AED} = \angle \mathrm{PEB}


Given a triangle \( \triangle ABC \) with orthocenter \( H \), points \( D \) and \( E \) lie on sides \( AB \) and \( AC \) respectively such that \( D \), \( H \), and \( E \) are collinear and \( AD = AE \). Extend \( DE \) to intersect the circumcircle \( \odot O \) of \( \triangle ABC \) at points \( F \) and \( G \). Perpendiculars drawn from \( D \) and \( E \) to \( AB \) and \( AC \) respectively intersect at point \( K \). Extend \( HK \) to intersect \( \odot O \) at point \( L \). Lines \( FL \) and \( GL \) intersect \( BC \) at points \( P \) and \( Q \) respectively. Prove that points \( L \), \( P \), \( O \), and \( Q \) are concyclic.

Answer: L, P, O, Q \text{ are concyclic}


Janka drew a rectangle with a perimeter of $22 \mathrm{~cm}$ and side lengths expressed in centimeters as whole numbers. Then she divided the rectangle without any remainder into three rectangles, one of which had dimensions $2 \mathrm{~cm} \times 6 \mathrm{~cm}$. The sum of the perimeters of all three rectangles was $18 \mathrm{~cm}$ greater than the perimeter of the original rectangle. What could be the dimensions of the original rectangle? Find all solutions.

(M. Dillingerová)

Idea. Determine how Janka could have divided the rectangle; for each individual possibility, then express the given difference in perimeters using the lengths of the dividing lines.

Answer: 3 \times 8


In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2+10$, $10+2$, and $3+2+2+5$ are three such ways.

Answer: 70


4.9 On a $10 \times 10$ grid paper, there are 11 horizontal grid lines and 11 vertical grid lines. A line segment connecting two adjacent nodes on the same straight line is called a "link segment". How many link segments need to be erased at least, so that at most 3 link segments remain at each node?

Answer: 41


Given a divisor (non-zero) and a dividend, there is always a quotient and a remainder such that:
\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]
where \( 0 \leq \text{Remainder} < \text{Divisor} \). This is known as division with remainder. When the remainder is 0, the divisor is said to divide the dividend exactly, or the divisor is called a factor of the dividend (the dividend is a multiple of the divisor).

Please list all natural numbers greater than 1 and not exceeding 88 that can be exactly divided by 6.

Answer: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84


5. In an arbitrary triangular pyramid $A B C D$, a section is made by a plane intersecting the edges $A B, D C$, and $D B$ at points $M, N, P$ respectively. Point $M$ divides edge $A B$ in the ratio $A M: M B=1: 2$. Point $N$ divides edge $D C$ in the ratio $D N: N C=3: 2$. Point $P$ divides edge $D B$ in the ratio $D P: P B=2$. Find the ratio $A Q: Q C$.

Answer: \dfrac{3}{8}


Problem 4. Each cell of a $3 \times 3$ table is painted in one of three colors such that cells sharing a side have different colors. Among all possible such colorings, find the proportion of those in which exactly two colors are used.

Answer: \dfrac{1}{41}


Values for $A,B,C,$ and $D$ are to be selected from $\{1, 2, 3, 4, 5, 6\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, the choices $A=3, B=2, C=4, D=1$ is considered the same as the choices $A=4, B=1, C=3, D=2.$) Find the number of ways.

Answer: 90


Let \(ABCD\) be an isosceles trapezoid having parallel bases \(\overline{AB}\) and \(\overline{CD}\) with \(AB>CD.\) Line segments from a point inside \(ABCD\) to the vertices divide the trapezoid into four triangles whose areas are \(2, 3, 4,\) and \(5\) starting with the triangle with base \(\overline{CD}\) and moving clockwise as shown in the diagram below. The original answer for the ratio \(\frac{AB}{CD}\) is in the form \(k+\sqrt{m}\). Please find the value of \(k+m\).

Answer: 4


A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number? Express your answer in the form \(\frac{k}{m}\), where k and m are integers with no common factors other than 1. Find the value of k + m.

Answer: 21


For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. Find the number of values of $n \le 50$ for which $f_{50}(n) = 12$.

Answer: 10


Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$, and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\sqrt{3}$. The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\sqrt{p}$, where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find the value of $m + n + p$.

Answer: 55


Inside a right circular cone with base radius \(5\) and height \(12\) are three congruent spheres with radius \(r\). Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. Find the radius \(r\) of each sphere. The original answer is in the form \(\frac{k-n\sqrt{3}}{m}\). Please give the value of \(k+n+m\).

Answer: 141


Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Please calculate the value of \(m + n\).

Answer: 471


Each of the $5$ sides and the $5$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color? Provide your answer as a fraction in simplest form, \(\frac{k}{m}\), and find the value of \(k + m\).

Answer: 509


Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find the value of $a+b.$

Answer: 85


Problem 11.7. Natural numbers $a$ and $b$ are such that $a^{a}$ is divisible by $b^{b}$, but $a$ is not divisible by $b$. Find the smallest possible value of the number $a+b$, given that the number $b$ is coprime with 210.

Answer: 374


A: There are 7 boxes arranged in a row and numbered 1 through 7 . You have a stack of 2015 cards, which you place one by one in the boxes. The first card is placed in box \#1, the second in box \#2, and so forth up to the seventh card which is placed in box \#7. You then start working back in the other direction, placing the eighth card in box \#6, the ninth in box \#5, up to the thirteenth card being placed in box \#1. The fourteenth card is then placed in box \#2, and this continues until every card is distributed. What box will the last card be placed in?

Answer: 3


3. The fields of a board with 25 rows and 125 columns are alternately colored black and white (like the fields on a chessboard), and knight figures are placed on it, which attack according to special rules. A knight on a white field attacks all white fields in the same row and all black fields in the same column, while a knight on a black field attacks all black fields in the same row and all white fields in the same column (even if there is another figure in between). Determine the maximum number of knights that can be placed on the board so that they do not attack each other.

Answer: 50


The sequence $\left\{a_{n}\right\}$ is defined as follows: $a_{1}=0, a_{2}=1, a_{n}=\frac{1}{2} n a_{n-1} + \frac{1}{2} n(n-1) a_{n-2} + (-1)^{n}\left(1-\frac{n}{2}\right)$ for $n \geqslant 3$. Find the simplest expression for $f_n = a_n + 2 \binom{n}{1} a_{n-1} + 3 \binom{n}{2} a_{n-2} + \cdots + (n-1) \binom{n}{n-2} a_2 + n \binom{n}{n-1} a_1$.

Answer: 2n! - n - 1


The sides of the blue and green equilateral triangles are parallel. The perimeter of the blue triangle is 4, and the perimeter of the green triangle is 5. Find the perimeter of the hexagon formed by the intersection of these triangles.

Answer: 3


4. At the end of a chess tournament, it was determined that each participant had earned exactly half of their points playing against the competitors who finished in the last three places. How many participants were there in the tournament?

(Each participant played one game against each of the other participants. A win earns one point, a draw earns half a point, and a loss earns no points.)

Answer: 9


2. In an acute triangle $ABC$, $AD$, $DE$, $CF$ are altitudes, $H$ is the orthocenter of the triangle. If $EF$ bisects the area of triangle $ABC$, then $DE^2 + EF^2 + FD^2 =$ $\qquad$ (Given the side lengths of the triangle are $3$, $2\sqrt{2}$, $\sqrt{5}$)

Answer: 5


15.8 (League 2001) $D, E$ are points on side $BC$ of $\triangle ABC$, $F$ is a point on the extension of $BA$, $\angle DAE = \angle CAF$.
(1) Determine the positional relationship between the circumcircle of $\triangle ABD$ and the circumcircle of $\triangle AEC$, and prove your conclusion.
(2) If the circumradius of $\triangle ABD$ is twice that of $\triangle AEC$, $BC=6, AB=4$, find the length of $BE$.

---

(1) To determine the positional relationship between the circumcircle of $\triangle ABD$ and the circumcircle of $\triangle AEC$, we need to analyze the given angles and the geometric properties of the circles.

Given:
- $\angle DAE = \angle CAF$
- $D$ and $E$ are on $BC$
- $F$ is on the extension of $BA$

Since $\angle DAE = \angle CAF$, we can infer that the points $A, D, E, F$ are concyclic. This means that the circumcircle of $\triangle ADE$ passes through $F$.

Now, consider the circumcircle of $\triangle ABD$ and the circumcircle of $\triangle AEC$. Since $\angle DAE = \angle CAF$, the points $A, D, E, F$ lie on a common circle. Therefore, the circumcircle of $\triangle ABD$ and the circumcircle of $\triangle AEC$ intersect at points $A$ and $E$.

Thus, the circumcircle of $\triangle ABD$ and the circumcircle of $\triangle AEC$ are **intersecting** at points $A$ and $E$.

(2) Given that the circumradius of $\triangle ABD$ is twice that of $\triangle AEC$, and $BC = 6$, $AB = 4$, we need to find the length of $BE$.

Let the circumradius of $\triangle AEC$ be $R$ and the circumradius of $\triangle ABD$ be $2R$.

Using the circumradius formula for a triangle:
\[ R = \frac{abc}{4K} \]
where $a, b, c$ are the sides of the triangle and $K$ is the area of the triangle.

For $\triangle AEC$:
\[ R = \frac{AE \cdot AC \cdot EC}{4K_{AEC}} \]

For $\triangle ABD$:
\[ 2R = \frac{AB \cdot AD \cdot BD}{4K_{ABD}} \]

Since $AB = 4$ and $BC = 6$, let $BE = x$ and $EC = 6 - x$.

Using the fact that the circumradius of $\triangle ABD$ is twice that of $\triangle AEC$, we can set up the following relationship:
\[ 2 \left( \frac{AE \cdot AC \cdot EC}{4K_{AEC}} \right) = \frac{4 \cdot AD \cdot BD}{4K_{ABD}} \]

Simplifying, we get:
\[ 2 \left( \frac{AE \cdot AC \cdot (6 - x)}{4K_{AEC}} \right) = \frac{4 \cdot AD \cdot (x + 4)}{4K_{ABD}} \]

Since the areas $K_{AEC}$ and $K_{ABD}$ are proportional to the product of the sides, we can simplify further:
\[ 2 \left( \frac{AE \cdot AC \cdot (6 - x)}{4K_{AEC}} \right) = \frac{4 \cdot AD \cdot (x + 4)}{4K_{ABD}} \]

Given that the circumradius relationship holds, we can solve for $x$:
\[ 2 \left( \frac{AE \cdot AC \cdot (6 - x)}{4K_{AEC}} \right) = \frac{4 \cdot AD \cdot (x + 4)}{4K_{ABD}} \]

Since the areas are proportional, we can simplify to:
\[ 2(6 - x) = x + 4 \]

Solving for $x$:
\[ 12 - 2x = x + 4 \]
\[ 12 - 4 = 3x \]
\[ 8 = 3x \]
\[ x = \frac{8}{3} \]

Thus, the length of $BE$ is:
\[ \boxed{\frac{8}{3}} \]

Answer: \dfrac{8}{3}


Ivan the Tsarevich fought with the three-headed and three-tailed Zmey Gorynych. With one blow, he could cut off either one head, one tail, two heads, or two tails. But if he cut off one tail, two would grow back; if he cut off two tails, a head would grow back; if he cut off a head, a new head would grow back, and if he cut off two heads, nothing would grow back. How should Ivan the Tsarevich act to cut off all the heads and all the tails of the Zmey as quickly as possible

Answer: 9


1. How many pairs of numbers $(a, b)$ with $a<b$ can you choose from the set $1,2,3, \ldots, 2005$ such that $a+b$ is a multiple of 5?

Answer: 401802


C1. The positive integer $N$ has six digits in increasing order. For example, 124689 is such a number.
However, unlike 124689 , three of the digits of $N$ are 3,4 and 5 , and $N$ is a multiple of 6 . How many possible six-digit integers $N$ are there?

Answer: 3


3. Find the smallest natural number $n$, such that among any $n$ irrational numbers, there always exist 3 numbers, the sum of any two of which is still an irrational number.

Answer: 5


11. Given positive integers $a, b, c, d$ satisfy $a^{2}+2 a=c d+11 c+d+10, b^{2}-2 b=c d-11 c+d-12$, then the sum of all possible values of $d$ is $\qquad$.

Answer: 72


99***. A rod is broken into two parts at a randomly chosen point; then the larger of the two resulting parts is again broken into two parts at a randomly chosen point. What is the probability that a triangle can be formed from the three resulting pieces?

Answer: 2 \ln 2 - 1


3. On a wooden ruler, three marks are made: 0, 7, and 11 centimeters. How can you measure a segment of (a) 8 cm; (b) 5 cm using it?

Answer: 8


3. In a dark room, a drawer contains socks of two colors and two sizes, 4 pairs in total (one pair of each color and size). How many socks must be taken out to ensure that there are two pairs of different colors and different sizes?

Answer: 7


Example 35 (1999 Shanghai High School Competition Question) Let $a, b, c, d$ be four distinct real numbers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4$, and $a c=b d$. Find the maximum value of $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}$.

Answer: -12


2. Two circles touch each other externally and each touches a larger circle internally. The radius of one is half, and that of the other is one-third of the radius of the largest circle. Find the ratio of the length of the segment of the common internal tangent to the smaller circles, contained within the largest, to its diameter.

#

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


4. Squirrel Pusistik and Lohmatik ate a basket of berries and a bag of seeds, which contained more than 50 but less than 65 seeds, starting and finishing at the same time. At first, Pusistik ate berries, and Lohmatik ate seeds, then (at some point) they switched. Lohmatik ate berries six times faster than Pusistik, and seeds only three times faster. How many seeds did Lohmatik eat if Lohmatik ate twice as many berries as Pusistik?

Answer: 54


25. Let $\dot{P}$ be a 40 -sided convex polygon. Find the number of triangles $S$ formed by the vertices of $P$ such that any two vertices of $S$ are separated by at least two other vertices of $P$.

Answer: 7040


As points $A, B, C, D$ move in the plane, $A D=B C=2$ and $A C=B D=4$ always hold, furthermore, segments $A C$ and $B D$ intersect. How does the distance $C D$ depend on the distance $A B$?

Answer: \dfrac{12}{AB}


1. In parallelograms $A B C D$ and $A E F D$, heights $B P=7$ and $F Q=10$ are drawn to the side $A D$. Find the length of the perpendicular from point $C$ to $E F$. If there are multiple possible values, list them in any order separated by a semicolon.

Answer: 3;17


A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold:  (i) each triangle from $T$ is inscribed in $\omega$;  (ii) no two triangles from $T$ have a common interior point.  Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.  (South Africa)

A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold:  (i) each triangle from $T$ is inscribed in $\omega$;  (ii) no two triangles from $T$ have a common interior point.  Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.  (South Africa)

Answer: 4


## 139. Math Puzzle $12 / 76$

Ralf and Marion had set their watches, real "vintages," at the beginning of a longer hike. At the destination, Marion's watch shows 13:46 and Ralf's 14:13.

What time is it really, if Marion's watch gains 2 minutes daily and Ralf's old alarm clock loses 4 minutes daily?

Answer: 13:55


## Zadatak B-2.5.

Duljina osnovice jednakokračnog trokuta $A B C$ iznosi $|B C|=30 \mathrm{~cm}$, a duljina kraka $|A C|=17 \mathrm{~cm}$. U točki $A$ povučena je okomica na pravac $A C$. Odredite omjer u kojem ova okomica dijeli osnovicu $\overline{B C}$.

Answer: 161:289


C2. We consider a triangle $A B C$ and a point $D$ on the extension of $A B$ towards $B$. The point $E$ lies on the side $A C$ such that angle $\angle D B C$ and angle $\angle D E C$ are equal. The intersection of $D E$ and $B C$ is $F$. Suppose that $|B F|=2,|B D|=3,|A E|=4$ and $|A B|=5$. (Note: the image below is not to scale.)

(a) Show that the triangles $\triangle A B C$ and $\triangle A E D$ are similar.

(b) Determine $|C F|$.

![](https://cdn.mathpix.com/cropped/2024_04_17_88f9acd303931df7c94dg-2.jpg?height=485&width=417&top_left_y=2296&top_left_x=1482)

Answer: 8


4. We will call a number $\mathrm{X}$ "20-supporting" if for any 20 real numbers $a_{1}, \ldots, a_{20}$, the sum of which is an integer, there exists at least one for which $\left|a_{i}-\frac{1}{2}\right| \geq X$.

In your answer, specify the largest 20-supporting $X$, rounded to the thousandths according to standard mathematical rules.

Answer: 0.025


11. Choose three different digits from $0,1, \cdots, 9$ to form a four-digit number (one of the digits can appear twice), such as 5 224. Then the total number of such four-digit numbers is $\qquad$.

Answer: 3888


5. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, the angle between $A B_{1}$ and $A_{1} D$ is $\alpha$, the angle between $A C$ and $B C_{1}$ is $\beta$, and the angle between $A_{1} C_{1}$ and $C D_{1}$ is $\gamma$. Therefore, $\alpha+\beta+\gamma=$ $\qquad$ .

Answer: 180^\circ


6. Given the sets $A=\{1,2,3,4,5,6\}, B=$ $\{6,7,8,9\}$, by selecting 3 elements from $A$ and 2 elements from $B$, we can form $\qquad$ new sets with 5 elements.

Answer: 90


Two flagpoles with heights of 5 meters and 3 meters stand on a horizontal ground. If the coordinates of the bases of the two flagpoles are determined as \( A(-5,0) \) and \( B(5,0) \) respectively, then the locus of points on the ground where the angles of elevation to the tops of the poles are equal is $\quad$ .

Answer: \left(x - \frac{85}{8}\right)^2 + y^2 = \left(\frac{75}{8}\right)^2


Find all quadruples $(a, b, c, d)$ of non-negative integers such that $a b = 2(1 + c d)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.

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

Answer: (1, 2, 0, 1)


I Noticed that 2023 China Western Mathematical Olympiad has not added to collection yet so I made it [url=https://artofproblemsolving.com/community/c3513189]here[/url].
I am not sure if other problems were posted. If it was, please add the link under here, thanks!

Answer: 2023


1.74 Suppose the network lines drawn on the paper consist of $n$ horizontal lines and $n$ vertical lines. How many different closed broken lines of $2 n$ segments can be drawn along these network lines, such that each broken line passes through each horizontal line and each vertical line?

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


\section*{Problem 1}

A book contains 30 stories. Each story has a different number of pages under 31. The first story starts on page 1 and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?

Answer: 23


## 27. Large Families

The Martens have more children than the Duponts. Suppose the difference of the squares of these two numbers is 24 and that both families have more than one child. How many children do the Martens have?

70

Answer: 7


Example 2. Solve the equation $y^{\prime \prime \prime}+y^{\prime}=x^{4}$.

Answer: y = C_1 + C_2 \cos x + C_3 \sin x + \frac{1}{5}x^5 - 4x^3 + 24x


15. (4b, 8-11) In Anchuria, a checkers championship is being held in several rounds. The days and cities for the rounds are determined by a draw. According to the championship rules, no two rounds can take place in the same city, and no two rounds can take place on the same day. Among the fans, a lottery is organized: the main prize goes to the person who correctly guesses before the championship begins in which cities and on which days all the rounds will take place. If no one guesses correctly, the main prize will go to the championship organizing committee. There are eight cities in Anchuria, and the championship is allocated a total of eight days. How many rounds should the organizing committee schedule to have the highest probability of winning the main prize themselves?

Answer: 6


4 $[\underline{\text { Tangent Spheres }}]$

A cone contains five equal spheres. Four of them lie on the base of the cone, each touching two others lying on the base and the lateral surface of the cone. The fifth sphere touches the lateral surface and the other four spheres. Find the volume of the cone if the radius of each sphere is $r$.

Answer: \dfrac{(25 + 22\sqrt{2})}{3} \pi r^3


6. On the table, there are three cones standing on their bases, touching each other. The radii of their bases are 10, 15, and 15. A truncated cone is placed on the table with its smaller base down, and it shares a generatrix with each of the other cones. Find the area of the smaller base of the truncated cone.

Answer: 4\pi


3. Let the line $y=-\frac{1}{2} x+1$ intersect the $x$-axis and $y$-axis at points $B$ and $A$, respectively. Point $C$ is the reflection of point $B$ over the $y$-axis. Construct an isosceles right triangle $\triangle ACD$ with $AC$ as one of the legs in the second quadrant. Draw $DE \perp x$-axis at point $E$. If the line $y=kx-2k$ divides the quadrilateral $OADE$ into two regions of equal area, then $k=$ . $\qquad$

Answer: -\dfrac{3}{14}


8. Given that $A$, $B$, and $C$ are three points on the parabola $y^{2}=x$, and $\left(2, \frac{4}{3}\right)$ is the centroid of $\triangle A B C$. Then the maximum value of the area of $\triangle A B C$ is $\qquad$ .

Answer: \dfrac{\sqrt{3}}{9}


Exercise 2. In the figure below, the triangles ABO, BCO, CDO, DEO, and FEO are right isosceles triangles. We assume that $\mathrm{OA}=8 \mathrm{~cm}$. Determine the area of $A O F$ in $\mathrm{cm}^{2}$.

![](https://cdn.mathpix.com/cropped/2024_06_04_ff4f5d9cabdcb92d54e8g-2.jpg?height=626&width=1095&top_left_y=1178&top_left_x=496)

Answer: 4


7.5. One hundred non-zero integers are written in a circle such that each number is greater than the product of the two numbers following it in a clockwise direction. What is the maximum number of positive numbers that can be among these 100 written numbers?

Answer: 50


\section*{Problem 6B - 131236B}

Let \(M\) be the set of all points \(P(x, y)\) in a plane rectangular Cartesian coordinate system, where \(x, y\) are rational integers, and \(0 \leq x \leq 4\) and \(0 \leq y \leq 4\).

Determine the probability that the distance between two different points chosen arbitrarily from \(M\) is a rational integer (the unit of measurement being the unit of the coordinate system).

Hint: If \(n\) is the number of different ways to choose two points and \(m\) is the number of ways in which the distance is a rational integer, then the quotient \(\frac{m}{n}\) is the probability to be determined. Two selection possibilities are considered different if and only if the sets of points (each consisting of two points) selected in them are different.

Answer: \dfrac{9}{25}


6. On Pandora, the population sizes of each country are different. The 24 countries with the smallest populations account for $45 \%$ of the global population, while the 13 countries with the largest populations account for $26 \%$ of the global population. Therefore, Pandora has $\qquad$ countries.

Answer: 52


In quadrilateral $A B C D$, it is known that $D O=4, B C=5, \angle A B D=45^{\circ}$, where $O$ is the point of intersection of the diagonals. Find $B O$, if the area of quadrilateral $A B C D$ is equal to $\frac{1}{2}(A B \cdot C D+B C \cdot A D)$.

Answer: 3


Example 2 - (83 College Entrance Examination
Mathematics for Science Students, Question 7)
As shown in the figure, it is known that the length of the major axis of the ellipse

is $\left|A_{1} A_{2}\right|=6$, and the focal distance
$\left|\mathbf{F}_{\mathbf{1}} \mathbf{F}_{\mathbf{2}}\right|=4 \sqrt{\mathbf{2}}$. A line is drawn through
the focus $\mathrm{F}_{1}$ of the ellipse,
intersecting the ellipse at points $\mathrm{M}$ and $\mathrm{N}$. Let $\angle \mathrm{F}_{2} \mathrm{~F}_{1} \mathrm{M}=\alpha(0 \leqslant \alpha \leqslant \pi)$,
for what value of $\alpha$ is $|\mathrm{MN}|$ equal to the length of the minor axis of the ellipse?

Answer: \dfrac{\pi}{6}


Task B-2.5. The ages of a father and his two children (they are not twins) are powers of the same prime number. A year ago, the ages of each of them were prime numbers. How old is the father, and how old is each of his two children?

Answer: 32


[ [Ratio of areas of triangles with a common base or common height] Rearrangement of areas Pentagons

Given a convex pentagon $A B C D E$. The area of each of the triangles $A B C, B C D, C D E, D E A, E A B$ is $S$. Find the area of the given pentagon.

#

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


A quadrilateral has two adjacent sides of lengths 2 and 1 units, respectively, and the included angle between them is $60^{\circ}$. The quadrilateral is both a cyclic and a tangential quadrilateral. What are the lengths of the other two sides of the quadrilateral?

Answer: \dfrac{\sqrt{33} - 3}{6}


Problem 9.7. The numbers $1, 2, 3, \ldots, 57$ are written on a board. What is the maximum number of these numbers that can be chosen so that no two chosen numbers differ exactly by a factor of 2.5?

Answer: 48


12th Balkan 1995 Problem 1 Define a n by a 3 = (2 + 3)/(1 + 6), a n = (a n-1 + n)/(1 + n a n-1 ). Find a 1995 . Solution

Answer: \dfrac{1991009}{1991011}


Problem 9.8. A group of kids decided to play a computer game. Any two people either play together or against each other; moreover, if player $A$ plays together with $B$, and $B$ plays against $C$, then $A$ also plays against $C$. How many kids are in the group if each player had exactly 15 opponents? List all possible options.

Answer: 16


Problem 8.8. In how many ways can all natural numbers from 1 to 200 be painted in red and blue so that the sum of any two different numbers of the same color is never equal to a power of two?

Answer: 256


2. Decipher the ciphertext message LFDPH LVDZL FRQTX HUHG that has been enciphered using the Caesar cipher.

Answer: I CAME I SAW I CONQUERED


11.2. On the board, there are 4 numbers. Vasya multiplied the first of these numbers by $\sin \alpha$, the second - by $\cos \alpha$, the third - by $\operatorname{tg} \alpha$, and the fourth - by $\operatorname{ctg} \alpha$ (for some angle $\alpha$) and obtained a set of the same 4 numbers (possibly in a different order). What is the maximum number of different numbers that could have been written on the board?

Answer: 3


8. In a certain World Cup, the Brazilian team has a probability of $\frac{1}{2}$ to win against each opponent, a probability of $\frac{1}{3}$ to draw, and a probability of $\frac{1}{6}$ to lose. Winning a match earns 3 points, drawing earns 1 point, and losing earns 0 points. It is known that each team plays three matches in the group stage, and 4 points or more are required to advance from the group. In the knockout stage, if the teams draw, they go to a penalty shootout. It is known that Brazil's probability of winning a penalty shootout is $\frac{3}{5}$. What is the probability that Brazil will win the championship and have exactly one penalty shootout in the four knockout matches? $\qquad$

Answer: \dfrac{1}{12}


Example 2. In Rt $\triangle A B C$, $\angle C=90^{\circ}, \angle A B C$ $=66^{\circ}, \triangle A B C$ is rotated around $C$ to the position of $\triangle A^{\prime} B^{\prime} C^{\prime}$, with vertex $B$ on the hypotenuse $A^{\prime} B^{\prime}$, and $A^{\prime} C$ intersects $A B$ at $D$. Find $\angle B D C$. (1993, Huanggang Region, Hubei Province Junior High School Mathematics Competition)

Answer: 72^\circ


Find the number of permutations of $1, 2, 3, 4, 5, 6$ such that for each $k$ with $1$ $\leq$ $k$ $\leq$ $5$ , at least one of the first $k$ terms of the permutation is greater than $k$ .
Please give the answer directly without any intermediate steps.

Answer: 461


Let \(\omega_{1}\) and \(\omega_{2}\) be two circles with radii \(r_{1}\) and \(r_{2}\), tangent at a point \(T\), with \(\omega_{1}\) inside \(\omega_{2}\). Let \(K \in \omega_{1}\) be different from \(T\). The tangent to \(\omega_{1}\) at \(K\) intersects \(\omega_{2}\) at points \(A\) and \(B\). Let \(S\) be the midpoint of the arc \(\widehat{AB}\) of \(\omega_{2}\) that does not contain \(T\).

Show that the radius of the circumcircle of \(\triangle AKS\) does not depend on \(K\), and express it in terms of \(r_{1}\) and \(r_{2}\).

Answer: \sqrt{r_2(r_2 - r_1)}


Example 2. The lengths of the three sides of $\triangle ABC$ are $BC=17$, $CA=18$, and $AB=19$. Through a point $P$ inside $\triangle ABC$, perpendiculars $PD$, $PE$, and $PF$ are drawn to the three sides of $\triangle ABC$ ($D$, $E$, and $F$ are the feet of the perpendiculars), and $BD + CE + AF = 27$. Find the length of $BD + BF$. (5th National Junior High School Mathematics Correspondence Competition)

Answer: 18


Three numbers \( x, y, z \) are given. It is known that each of the numbers \( 2x - y \), \( 3y - 2z \), and \( 4z - 3x \) is negative. Prove that each of the numbers \( x, y, z \) is also negative?

Answer: x < 0,\ y < 0,\ z < 0


20. On the table, there are param 1 externally identical coins. It is known that among them, there are exactly param 2 counterfeit ones. You are allowed to point to any two coins and ask whether it is true that both these coins are counterfeit. What is the minimum number of questions needed to guarantee getting at least one "Yes" answer?

| param1 | param2 |  |
| :---: | :---: | :---: |
| 105 | 53 |  |

| 129 | 65 |  |
| :---: | :---: | :---: |
| 167 | 84 |  |
| 125 | 63 |  |
| 207 | 104 |  |

Answer: 54


3.1. (16 points) A square with a side of 36 cm was cut into three rectangles such that the areas of all three rectangles are equal and any two rectangles share a common boundary segment. What is the total length (in cm) of the cuts made?

Answer: 60


1st ASU 1961 problems Problem 6 (a) A and B move clockwise with equal angular speed along circles center P and Q respectively. C moves continuously so that AB=BC=CA. Establish C's locus and speed. *(b) ABC is an equilateral triangle and P satisfies AP=2, BP=3. Establish the maximum possible value of CP. Solution (a) Represent A, B as complex numbers z 1 + w 1 e it , z 2 + w 2 e it . Then C is (z 1 + w 1 e it ) + (z 2 + w 2 e it - z 1 - w 1 e it ) e i π/3 , which is also of the form z + w e it . However, there is one subtlety. There are actually two circles possible for C depending on which side of AB we place it. The continuity requirement means that C is normally confined to one of the circles. However, if A and B ever coincide then C may be able to switch to the other circle. If we regard "moves continuously" as allowing a discontinuous velocity, then a switch is always possible (provided A and B coincide). (b)

Answer: 5


3. Ivan Semenov is taking a Unified State Exam (USE) in mathematics. The exam consists of three types of tasks: A, B, and C. For each task of type A, there are four answer options, only one of which is correct. There are 10 such tasks in total. Tasks of types B and C require detailed answers. Since Ivan constantly skipped classes, his knowledge of mathematics is shallow. He answers tasks of type A by guessing. The first task of type B, Ivan solves with a probability of $\frac{1}{3}$. He cannot solve anything else. For a correct answer to one task of type A, 1 point is awarded, and for a task of type B, 2 points. What is the probability that Ivan will score more than 5 points?

Take the tasks of type A from the 2008 USE practice version. (http://ege.edu.ru/demo/math.zip) and conduct the experiment of randomly selecting answers 10 times. Compare the result with the theoretical one (for 5 correct answers). Ensure that the results do not differ significantly.

#

Answer: 0.088


Consider three diameters of a sphere that are mutually perpendicular. Divide each diameter into 4 equal parts. Through each division point, draw a plane perpendicular to the corresponding diameter. How many regions does the sphere’s surface get divided into by the 9 circles formed by these 9 planes?

- How many regions does the sphere’s surface get divided into if each of the 3 diameters is divided into 6 equal parts, and we proceed similarly?

Answer: 56


In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$

Answer: 135^\circ


11. (2002 National Competition Question) Real numbers $a, b, c$ and a positive number $\lambda$ make $f(x)=x^{3}+a x^{2}+b x+c$ have three real roots $x_{1}, x_{2}, x_{3}$, and satisfy
(1) $x_{2}-x_{1}=\lambda$; (2) $x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)$.
Find the maximum value of $\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}$.

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


Four. (18 points) As shown in Figure 6, given $\triangle A B C$, and $S_{\triangle A B C}=$ 1. $D, E$ are moving points on $A C, A B$ respectively, $B D$ and $C E$ intersect at point $P$, such that $S_{\text {quadrilateral } B C D E}=$ $\frac{16}{9} S_{\triangle B P C}$. Find the maximum value of $S_{\triangle D E P}$.

Answer: \dfrac{1}{18}


4. On a glade, two firs, each 30 meters tall, grow 20 meters apart from each other. The branches of the firs grow very densely, and among them are some that are directed straight towards each other, and the length of each branch is half the distance from it to the top. A spider can crawl up or down the trunk (strictly vertically), along the branches (strictly horizontally), or descend vertically down a web from one branch to another. What is the shortest distance the spider will have to crawl to get from the top of one fir to the top of the other?

Answer: 60


3. Let $E$ be a point on the side $AB$ of square $ABCD$ such that $|AE|=3|EB|$, and let $F$ be a point on the side $DA$ such that $|AF|=5|FD|$. Denote the intersection of segments $DE$ and $FC$ by $K$, the intersection of $DE$ and $BF$ by $L$, and the intersection of $FB$ and $EC$ by $M$. Let $p_{1}$ be the sum of the areas of triangles $EML$ and $DKC$, and $p_{2}$ be the sum of the areas of triangles $FLK$ and $MBC$. Determine the ratio $p_{1}: p_{2}$.

Answer: 1:1


[ $\left[\begin{array}{ll}\text { Cylinder } & \\ \text { Cone } & \end{array}\right]$ [ Tangent spheres $]$
A cylinder with a base radius of 1 and height $\frac{12}{3+2 \sqrt{3}}$ has three identical spheres inscribed in it such that the spheres touch the upper base of the cylinder, its lateral surface, and each other. Find the volume of the cone whose base coincides with the lower base of the cylinder and which touches all three spheres.

Answer: \dfrac{4\pi}{9}


An iguana writes the number 1 on the blackboard. Every minute afterwards, if the number $x$ is written, the iguana erases it and either writes $\frac{1}{x}$ or $x+1$. Can the iguana eventually write the number $\frac{20}{17}$ ?

Answer: \dfrac{20}{17}


15. This year, our country's team will be participating in the 59th International Mathematical Olympiad, to be held in Cluj-Napoca, Romania. The IMO, which was first held in 1959 also in Romania, has been held annually except in 1980 , when it was cancelled due to internal strife in its host country. Which East Asian country was supposed to host the 1980 IMO?

Answer: Mongolia


12.444 The base of the pyramid is an equilateral triangle. One of the lateral faces of the pyramid is perpendicular to the plane of the base. Find the cosine of the angle between the other two lateral faces, if they form an angle \(\alpha\) with the plane of the base.

Answer: \dfrac{1 - 3\cos^2 \alpha}{2}


8,9  [

In triangle $ABC$, angle $B$ is equal to $\arccos \frac{15}{17}$. On side $AC$, a point $K$ is taken such that $AK=12$, $KC=4$. Find the radius of the circle passing through vertex $B$, touching side $AC$ at point $K$, and touching the circumcircle of triangle $ABC$.

#

Answer: 12


3. Determine all prime numbers $p$ for which $\frac{p^{2}-p-2}{2}$ is a cube of a natural number.

Answer: 2


Consider all triangular pyramids whose base edges are $1, 1, c$, and the side edges opposite these, respectively, are $1, c, c$. Determine the range of possible values for $c$.

Answer: \left( \dfrac{\sqrt{5} - 1}{2}, \dfrac{\sqrt{5} + 1}{2} \right)


3. The range of the function $f(x)=[2 \sin x \cdot \cos x]+[\sin x+\cos x]$ is $\qquad$ (where $[x]$ denotes the greatest integer less than or equal to $x$). Answer: $\{-1,0,1,2\}$.

Answer: \{-1, 0, 1, 2\}


Let there be two different media I and II on either side of the $x$-axis (Figure 2), with the speed of light in media I and II being $c_{1}$ and $c_{2}$, respectively. If a light ray starts from point A in medium I and reaches point B in medium II, what path should the light take to minimize the time of travel?

Answer: \frac{\sin \theta_1}{\sin \theta_2} = \frac{c_1}{c_2}


Suppose $ABCD$ is a trapezoid with $AB\parallel CD$ and $AB\perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB=10$ and $CD=15$, compute the maximum possible value of $XY$.

Answer: 6


Two circles meet at A and B and touch a common tangent at C and D. Show that triangles ABC and ABD have the same area.

Answer: \text{Triangles } ABC \text{ and } ABD \text{ have the same area.}


4. As shown in Figure 3, $A B$ and $C D$ are two chords of $\odot O$, $\angle A O B$ and $\angle O C D$ are supplementary, $\angle O A B$ and $\angle C O D$ are equal. Then the value of $\frac{A B}{C D}$ is $\qquad$

Answer: \dfrac{3 + \sqrt{5}}{2}


The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$.  Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$.  Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.

[i]Proposed by Evan Chen[/i]

Answer: 336


N1. Find all pairs $(k, n)$ of positive integers for which $7^{k}-3^{n}$ divides $k^{4}+n^{2}$.

Answer: (2, 4)


69. The area of $\triangle A B C$ is $1, D, E$ are points on sides $A B, A C$ respectively, $B E, C D$ intersect at point $P$, and the area of quadrilateral $B C E D$ is twice the area of $\triangle P B C$. Find the maximum value of the area of $\triangle P D E$.

---

The area of $\triangle A B C$ is $1, D, E$ are points on sides $A B, A C$ respectively, $B E, C D$ intersect at point $P$, and the area of quadrilateral $B C E D$ is twice the area of $\triangle P B C$. Find the maximum value of the area of $\triangle P D E$.

Answer: 5\sqrt{2} -7


Problem 4. Given an isosceles triangle $ABC$, where $\overline{AC}=\overline{BC}$. On the leg $AC$ are chosen points $M$ and $N$ such that $\measuredangle M B A=\measuredangle C B N$ and $\overline{M N}=\overline{B M}$, with point $M$ between points $A$ and $N$. Determine $\measuredangle N B A$.

Answer: 60^\circ


1. In the room, there are knights who always tell the truth, and liars who always lie. 10 of them said: "In this room, there are more knights than liars." 12 said: "In this room, there are more liars than knights." The remaining 22 said: "In this room, there are an equal number of liars and knights." How many liars could there be in the room? If there are multiple possible answers, write them in any order separated by a semicolon.

Answer: 22


On the sides \( AB, AC \), and \( BC \) of the equilateral triangle \( ABC \), points \( C_{1}, B_{1} \), and \( A_{1} \) are located respectively, such that the triangle \( A_{1} B_{1} C_{1} \) is equilateral. The segment \( BB_{1} \) intersects the side \( C_{1} A_{1} \) at point \( O \), with \( \frac{BO}{OB_{1}}=k \). Find the ratio of the area of triangle \( ABC \) to the area of triangle \( A_{1} B_{1} C_{1} \).

Answer: 1 + 3k


4. The numbers $x$ and $y$ are such that the equations $\sin y + \sin x + \cos 3x = 0$ and $\sin 2y - \sin 2x = \cos 4x + \cos 2x$ are satisfied. What is the greatest value that the sum $\cos y + \cos x$ can take?

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


134. Someone said to his friend: "Give me 100 rupees, and I will be twice as rich as you," to which the latter replied: "If you give me just 10 rupees, I will be six times richer than you." The question is: how much did each have?

Answer: 40


Let $P$ be the set of all $2012$ tuples $(x_1, x_2, \dots, x_{2012})$, where $x_i \in \{1,2,\dots 20\}$ for each $1\leq i \leq 2012$. The set $A \subset P$ is said to be decreasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in A$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \leq x_i (1\leq i \leq 2012)$ also belongs to $A$. The set $B \subset P$ is said to be increasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in B$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \geq x_i (1\leq i \leq 2012)$ also belongs to $B$. Find the maximum possible value of $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$, where $A$ and $B$ are nonempty decreasing and increasing sets ($\mid \cdot \mid$ denotes the number of elements of the set).

Answer: (1/20)^{2012}


9. (Adapted from the 1st "Hope Cup" Senior High School Competition) Let the function $f(n)=k$, where $n$ is a natural number, and $k$ is the digit at the $n$-th position after the decimal point of the irrational number $\pi=3.1415926535 \cdots$, with the rule that $f(0)=3$. Let $F_{n}=$ $\underbrace{f\{f\{f\{f\{f}(n)\} \cdots\}\}$, then $F[f(1990)+f(5)+f(13)]=$ $\qquad$.

Answer: 1


3. Each square on an $8 \times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of 3 , the number of checkers in each column is a multiple of 5 .
Assuming the top left corner of the board is shown below, how many checkers are used in total?

Answer: 30


1. A seagull is being fed from a moving boat. A piece of bread is thrown down, the seagull takes 3 seconds to pick up the piece from the sea surface, and then it takes 12 seconds to catch up with the boat. Upon entering the bay, the boat reduces its speed by half. How much time will it now take the seagull to catch up with the boat after picking up the piece of bread?

Om vem: 2 seconds.

Answer: 2


For integers $n$ (where $n \geq 2$), let real numbers $x_1, x_2, \ldots, x_n$ satisfy the conditions:
$$
\begin{array}{l}
x_1 + x_2 + \cdots + x_n = 0, \\
x_1^2 + x_2^2 + \cdots + x_n^2 = 1.
\end{array}
$$

For any set $A \subseteq \{1, 2, \ldots, n\}$, define $S_A = \sum_{i \in A} x_i$ (if $A$ is an empty set, then $S_A = 0$).

Prove: For any positive real number $\lambda$, the number of sets $A$ satisfying $S_A \geq \lambda$ is at most $\frac{2^{n-3}}{\lambda^2}$, and determine all $x_1, x_2, \ldots, x_n$ for which equality holds.

Answer: \dfrac{2^{n-3}}{\lambda^2}


8. Given $a_{k}$ as the number of integer terms in $\log _{2} k, \log _{3} k, \cdots, \log _{2018} k$. Then $\sum_{k=1}^{2018} a_{k}=$ $\qquad$

Answer: 4102


The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D, CA$ at $E$ and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16, r_B = 25$ and $r_C = 36$, determine the radius of $\Gamma$.

Answer: 74


The equation  $\sqrt[3]{\sqrt[3]{x - \frac38}  - \frac38} = x^3+ \frac38$  has exactly two real positive solutions  $r$  and  $s$ . Compute  $r + s$ .

Answer: \frac{1+\sqrt{13}}{4}


Complex numbers  $a$ ,  $b$  and  $c$  are the zeros of a polynomial  $P(z) = z^3+qz+r$ , and  $|a|^2+|b|^2+|c|^2=250$ .  The points corresponding to  $a$ ,  $b$ , and  $c$  in the complex plane are the vertices of a right triangle with hypotenuse  $h$ .  Find  $h^2$ .

Answer: 125


Round 1**1.1.** A classroom has  $29$  students. A teacher needs to split up the students into groups of at most  $4$ . What is the minimum number of groups needed?**1.2.** On his history map quiz, Eric recalls that Sweden, Norway and Finland are adjacent countries, but he has
forgotten which is which, so he labels them in random order. The probability that he labels all three countries
correctly can be written as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .**1.3.** In a class of  $40$  sixth graders, the class average for their final test comes out to be  $90$  (out of a  $100$ ). However, a student brings up an issue with problem  $5$ , and  $10$  students receive credit for this question, bringing the class average to a  $90.75$ . How many points was problem  $5$  worth?

Round 2**2.1.** Compute  $1 - 2 + 3 - 4 + ... - 2022 + 2023$ .**2.2.** In triangle  $ABC$ ,  $\angle ABC = 75^o$ . Point  $D$  lies on side  $AC$  such that  $BD = CD$  and  $\angle BDC$  is a right angle. Compute the measure of  $\angle A$ .**2.3.** Joe is rolling three four-sided dice each labeled with positive integers from  $1$  to  $4$ . The probability the sum of the numbers on the top faces of the dice is  $6$  can be written as  $\frac{p}{q}$  where  $p$  and  $q$  are relatively prime integers. Find  $p + q$ .

Round 3**3.1.** For positive integers  $a, b, c, d$  that satisfy  $a + b + c + d = 23$ , what is the maximum value of  $abcd$ ?**3.2.** A buckball league has twenty teams. Each of the twenty teams plays exactly five games with each of the other teams. If each game takes 1 hour and thirty minutes, then how many total hours are spent playing games?**3.3.** For a triangle  $\vartriangle ABC$ , let  $M, N, O$  be the midpoints of  $AB$ ,  $BC$ ,  $AC$ , respectively. Let  $P, Q, R$  be points on  $AB$ ,  $BC$ ,  $AC$  such that  $AP =\frac13 AB$ ,  $BQ =\frac13 BC$ , and  $CR =\frac13 AC$ . The ratio of the areas of  $\vartriangle MNO$  and  $\vartriangle P QR$  can be expressed as  $\frac{m}{n}$  , where  $ m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .

Round 4**4.1.**  $2023$  has the special property that leaves a remainder of  $1$  when divided by  $2$ ,  $21$  when divided by  $22$ , and  $22$  when divided by  $23$ . Let  $n$  equal the lowest integer greater than  $2023$  with the above properties. What is  $n$ ?**4.2.** Ants  $A, B$  are on points  $(0, 0)$  and  $(3, 3)$  respectively, and ant A is trying to get to  $(3, 3)$  while ant  $B$  is trying to get to  $(0, 0)$ . Every second, ant  $A$  will either move up or right one with equal probability, and ant  $B$  will move down or left one with equal probability. The probability that the ants will meet each other be  $\frac{a}{b}$ , where  $a$  and  $b$  are relatively prime positive integers. Find  $a + b$ .**4.3.** Find the number of trailing zeros of  $100!$  in base  $ 49$ .

PS. You should  use hide for answers. Rounds 5-9 have been posted [here](https://artofproblemsolving.com/community/c3h3129723p28347714). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Answer: 8


In  $\triangle ABC$ ,  $AB= 425$ ,  $BC=450$ , and  $AC=510$ . An interior point  $P$  is then drawn, and segments are drawn through  $P$  parallel to the sides of the triangle. If these three segments are of an equal length  $d$ , find  $d$ .

Answer: 306


Find all positive integers  $ n $  with the following property: It is possible to fill a  $ n \times n $  chessboard with one of arrows  $ \uparrow, \downarrow, \leftarrow, \rightarrow $  such that

1. Start from any grid, if we follows the arrows, then we will eventually go back to the start point.

2. For every row, except the first and the last, the number of  $ \uparrow $  and the number of  $ \downarrow $  are the same.

3. For every column, except the first and the last, the number of  $ \leftarrow $  and the number of  $ \rightarrow $  are the same.

Answer: 2


<u>Set 4</u>**4.1** Triangle  $T$  has side lengths  $1$ ,  $2$ , and  $\sqrt7$ . It turns out that one can arrange three copies of triangle  $T$  to form two equilateral triangles, one inside the other, as shown below. Compute the ratio of the area of the outer equilaterial triangle to the area of the inner equilateral triangle.
![Image](https://cdn.artofproblemsolving.com/attachments/0/a/4a3bcf4762b97501a9575fc6972e234ffa648b.png)**4.2**  Let  $T$  be the answer from the previous problem. The diagram below features two concentric circles of radius  $1$  and  $T$  (not necessarily to scale). Four equally spaced points are chosen on the smaller circle, and rays are drawn from these points to the larger circle such that all of the rays are tangent to the smaller circle and no two rays intersect. If the area of the shaded region can be expressed as  $k\pi$  for some integer  $k$ , find  $k$ .
![Image](https://cdn.artofproblemsolving.com/attachments/a/5/168d1aa812210fd9d60a3bb4a768e8272742d7.png)**4.3** Let  $T$  be the answer from the previous problem.  $T^2$  congruent squares are arranged in the configuration below (shown for  $T = 3$ ), where the squares are tilted in alternating fashion such that they form congruent rhombuses between them. If all of the rhombuses have long diagonal twice the length of their short diagonal, compute the ratio of the total area of all of the rhombuses to the total area of all of the squares. (Hint: Rather than waiting for  $T$ , consider small cases and try to find a general formula in terms of  $T$ , such a formula does exist.)
![Image](https://cdn.artofproblemsolving.com/attachments/1/d/56ef60c47592fa979bfedd782e5385e7d139eb.png)

PS. You should use hide for answers.

Answer: 7


Let  $A$  be a finite set, and  $A_1,A_2,\cdots, A_n$  are subsets of  $A$  with the following conditions:
(1)  $|A_1|=|A_2|=\cdots=|A_n|=k$ , and  $k>\frac{|A|}{2}$ ;
(2) for any  $a,b\in A$ , there exist  $A_r,A_s,A_t\,(1\leq r<s<t\leq n)$  such that $a,b\in A_r\cap A_s\cap A_t$ ;
(3) for any integer  $i,j\, (1\leq i<j\leq n)$ ,  $|A_i\cap A_j|\leq 3$ .
Find all possible value(s) of  $n$  when  $k$  attains maximum among all possible systems  $(A_1,A_2,\cdots, A_n,A)$ .

Answer: 11


3. Given a point on the plane that does not coincide with the origin. How many different points can be obtained from it by sequentially applying symmetries relative to the $O y$ axis and the line $y=-x$ (in any order and any number of times)? If the point itself can also be obtained, it should be counted. If different points can have different answers, list them in any order separated by a semicolon.

Answer: 4;8


1. Each of the three equal containers can hold 600 liters of liquid and each is filled exactly to half. From the first container to the second, we pour $18 \%$ of the liquid. Then from the second container to the third, we pour $2 / 3$ of the liquid. After that, from the third container to the first, we pour $3 / 8$ of the liquid and an additional 5 liters of liquid. How many liters of liquid need to be poured from the container with the most liquid to the container with the least liquid so that the amount of liquid in these containers is equal?

Answer: 167


4、A mover needs to transport 200 buns from the kitchen to the construction site (he is currently in the kitchen), and he can carry 40 buns each time. However, since he is very greedy, he will eat 1 bun whether he is walking from the kitchen to the construction site or from the construction site back to the kitchen. How many buns can this mover transport to the construction site at most?

Answer: 191


The diagram shows a parabola, a line perpendicular to the parabola's axis of symmetry, and three similar isosceles triangles each with a base on the line and vertex on the parabola. The two smaller triangles are congruent and each have one base vertex on the parabola and one base vertex shared with the larger triangle. The ratio of the height of the larger triangle to the height of the smaller triangles is  $\tfrac{a+\sqrt{b}}{c}$  where  $a$ ,  $b$ , and  $c$  are positive integers, and  $a$  and  $c$  are relatively prime. Find  $a + b + c$ .

[asy]
size(200);
real f(real x) {return 1.2*exp(2/3*log(16-x^2));}
path Q=graph(f,-3.99999,3.99999);
path [] P={(-4,0)--(-2,0)--(-3,f(-3))--cycle,(-2,0)--(2,0)--(0,f(0))--cycle,(4,0)--(2,0)--(3,f(3))--cycle};
for(int k=0;k<3;++k)
{
fill(P[k],grey); draw(P[k]);
}
draw((-6,0)--(6,0),linewidth(1)); draw(Q,linewidth(1));[/asy]

Answer: 20


Let  $\triangle ABC$  be a triangle with  $AB = 7$ ,  $AC = 8$ , and  $BC = 3$ . Let  $P_1$  and  $P_2$  be two distinct points on line  $AC$  ( $A, P_1, C, P_2$  appear in that order on the line) and  $Q_1$  and  $Q_2$  be two distinct points on line  $AB$  ( $A, Q_1, B, Q_2$  appear in that order on the line) such that  $BQ_1 = P_1Q_1 = P_1C$  and  $BQ_2 = P_2Q_2 = P_2C$ . Find the distance between the circumcenters of  $BP_1P_2$  and  $CQ_1Q_2$ .

Answer: 3


p1. Suppose the number  $\sqrt[3]{2+\frac{10}{9}\sqrt3} + \sqrt[3]{2-\frac{10}{9}\sqrt3}$  is integer. Calculate it.

p2. A house A is located  $300$  m from the bank of a  $200$  m wide river.  $600$  m above and  $500$  m from the opposite bank is another house  $B$ . A bridge has been built over the river, that allows you to go from one house to the other covering the minimum distance. What distance is this and in what place of the riverbank is at the bridge?

p3. Show that it is not possible to find  $1992$  positive integers  $x_1, x_2, ..., x_{1992}$  that satisfy the equation:   $$ \sum_{j=1}^{1992}2^{j-1}x_j^{1992}=2^{1992}\prod_{j=2}^{1992}x_j $$  p4. The following division of positive integers with remainder has been carried out. Every cross represents one digit, and each question mark indicates that there are digits, but without specifying how many.  Fill in each cross and question mark with the missing digits.
![Image](https://cdn.artofproblemsolving.com/attachments/8/e/5658e77025210a46723b6d85abf5bdc3c7ad51.png)

p5. Three externally tangent circles have, respectively, radii  $1$ ,  $2$ ,  $3$ . Determine the radius of the circumference that passes through the three points of tangency.

p6.  $\bullet$  It is possible to draw a continuous curve that cuts each of the segments of the figure exactly one time? It is understood that cuts should not occur at any of the twelve vertices. $\bullet$   What happens if the figure is on a sphere?
![Image](https://cdn.artofproblemsolving.com/attachments/a/d/b998f38d4bbe4121a41f1223402c394213944f.png)

p7. Given any  $\vartriangle ABC$ , and any point  $P$ , let  $X_1$ ,  $X_2$ ,  $X_3$  be the midpoints of the sides  $AB$ ,  $BC$ ,  $AC$  respectively, and  $Y_1$ ,  $Y_2$ ,  $Y_3$  midpoints of  $PC$ ,  $PA$ ,  $PB$  respectively. Prove that the segments  $X_1Y_1$ ,  $X_2Y_2$ ,  $X_3Y_3$  concur.

Answer: 2


In the space are given  $2006$  distinct points, such that no  $4$  of them are coplanar. One draws a segment between each pair of points.
A natural number  $m$  is called *good* if one can put on each of these segments a positive integer not larger than  $m$ , so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on.
Find the minimum value of a *good* number  $m$ .

Answer: 11


Let  $ABCDE$  be a pentagon inscribed in a circle such that  $AB=CD=3$ ,  $BC=DE=10$ , and  $AE=14$ . The sum of the lengths of all diagonals of  $ABCDE$  is equal to  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. What is  $m+n$ ? $\textbf{(A) }129\qquad
\textbf{(B) }247\qquad
\textbf{(C) }353\qquad
\textbf{(D) }391\qquad
\textbf{(E) }421\qquad$

Answer: 391


The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number.  The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is  $m/n,$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n.$

Answer: 85


A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled  $0, 1,\ldots, 2017$  are marked in a circumference and each couple is placed on a different point. For  $i\geq1$ , let  $s_i=i\ (\textrm{mod}\ 2018)$  and  $r_i=2i\ (\textrm{mod}\ 2018)$ . The dance begins at minute  $0$ . On the  $i$ -th minute, the couple at point  $s_i$  (if there's any) moves to point  $r_i$ , the couple on point  $r_i$  (if there's any) drops out, and the dance continues with the remaining couples. The dance ends after  $2018^2$  minutes. Determine how many couples remain at the end.

Note: If  $r_i=s_i$ , the couple on  $s_i$  stays there and does not drop out.

Answer: 0


In cyclic quadrilateral  $ABCD$ ,  $\angle DBC = 90^\circ$  and  $\angle CAB = 30^\circ$ . The diagonals of  $ABCD$  meet at  $E$ . If  $\frac{BE}{ED} = 2$  and  $CD = 60$ , compute  $AD$ . (Note: a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle.)

Answer: \frac{30}{\sqrt{7}}


Answer: 84


The diagonals of parallelogram  $ABCD$  intersect at  $E$  . The bisectors of  $\angle DAE$  and  $\angle EBC$  intersect at  $F$ . Assume  $ECFD$  is a parellelogram . Determine the ratio  $AB:AD$ .

Answer: \sqrt{3}


p1. Given the set  $H = \{(x, y)|(x -y)^2 + x^2 - 15x + 50 = 0$  where x and y are natural numbers  $\}$ .
Find the number of subsets of  $H$ .

p2. A magician claims to be an expert at guessing minds with following show. One of the viewers was initially asked to hidden write a five-digit number, then subtract it with the sum of the digits that make up the number, then name four of the five digits that make up the resulting number (in order of any). Then the magician can guess the numbers hidden. For example, if the audience mentions four numbers result:  $0, 1, 2, 3$ , then the magician will know that the hidden number is  $3$ .
a. Give an example of your own from the above process.
b. Explain mathematically the general form of the process.

p3. In a fruit basket there are  $20$  apples,  $18$  oranges,  $16$  mangoes,  $10$  pineapples and  $6$  papayas. If someone wants to take  $10$  pieces from the basket. After that, how many possible compositions of fruit are drawn?

p4. Inside the Equator Park, a pyramid-shaped building will be made with base of an equilateral triangle made of translucent material with a side length of the base  $8\sqrt3$  m long and  $8$  m high. A globe will be placed in a pyramid the. Ignoring the thickness of the pyramidal material, determine the greatest possible length of the radius of the globe that can be made.

p5. What is the remainder of  $2012^{2012} + 2014^{2012}$  divided by  $2013^2$ ?

Answer: 64


**p1.**  $17.5\%$  of what number is  $4.5\%$  of  $28000$ ?**p2.** Let  $x$  and  $y$  be two randomly selected real numbers between  $-4$  and  $4$ . The probability that  $(x - 1)(y - 1)$  is positive can be written in the form  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $m + n$ .**p3.** In the  $xy$ -plane, Mallen is at  $(-12, 7)$  and Anthony is at  $(3,-14)$ . Mallen runs in a straight line towards Anthony, and stops when she has traveled  $\frac23$  of the distance to Anthony. What is the sum of the  $x$  and  $y$  coordinates of the point that Mallen stops at?**p4.** What are the last two digits of the sum of the first  $2021$  positive integers?**p5.** A bag has  $19$  blue and  $11$  red balls. Druv draws balls from the bag one at a time, without replacement. The probability that the  $8$ th ball he draws is red can be written in the form  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $m + n$ .**p6.** How many terms are in the arithmetic sequence  $3$ ,  $11$ ,  $...$ ,  $779$ ?**p7.** Ochama has  $21$  socks and  $4$  drawers. She puts all of the socks into drawers randomly, making sure there is at least  $1$  sock in each drawer. If  $x$  is the maximum number of socks in a single drawer, what is the difference between the maximum and minimum possible values of  $x$ ?**p8.** What is the least positive integer  $n$  such that  $\sqrt{n + 1} - \sqrt{n} < \frac{1}{20}$ ?**p9.** Triangle  $\vartriangle ABC$  is an obtuse triangle such that  $\angle ABC > 90^o$ ,  $AB = 10$ ,  $BC = 9$ , and the area of  $\vartriangle ABC$  is  $36$ . Compute the length of  $AC$ .
![Image](https://cdn.artofproblemsolving.com/attachments/a/c/b648d0d60c186d01493fcb4e21b5260c46606e.png)**p10.** If  $x + y - xy = 4$ , and  $x$  and  $y$  are integers, compute the sum of all possible values of $ x + y$ .**p11.** What is the largest number of circles of radius  $1$  that can be drawn inside a circle of radius  $2$  such that no two circles of radius  $1$  overlap?**p12.**  $22.5\%$  of a positive integer  $N$  is a positive integer ending in  $7$ . Compute the smallest possible value of  $N$ .**p13.** Alice and Bob are comparing their ages. Alice recognizes that in five years, Bob's age will be twice her age. She chuckles, recalling that five years ago, Bob's age was four times her age. How old will Alice be in five years?**p14.** Say there is  $1$  rabbit on day  $1$ . After each day, the rabbit population doubles, and then a rabbit dies. How many rabbits are there on day  $5$ ?**15.** Ajit draws a picture of a regular  $63$ -sided polygon, a regular  $91$ -sided polygon, and a regular  $105$ -sided polygon. What is the maximum number of lines of symmetry Ajit's picture can have?**p16.** Grace, a problem-writer, writes  $9$  out of  $15$  questions on a test. A tester randomly selects  $3$  of the  $15$  questions, without replacement, to solve. The probability that all  $3$  of the questions were written by Grace can be written in the form  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $m + n$ .**p17.** Compute the number of anagrams of the letters in  $BMMTBMMT$  with no two  $M$ 's adjacent.**p18.** From a  $15$  inch by  $15$  inch square piece of paper, Ava cuts out a heart such that the heart is a square with two semicircles attached, and the arcs of the semicircles are tangent to the edges of the piece of paper, as shown in the below diagram. The area (in square inches) of the remaining pieces of paper, after the heart is cut out and removed, can be written in the form  $a-b\pi$ , where  $a$  and  $b$  are positive integers. Compute  $a + b$ .**p19.** Bayus has  $2021$  marbles in a bag. He wants to place them one by one into  $9$  different buckets numbered  $1$  through  $9$ . He starts by putting the first marble in bucket  $1$ , the second marble in bucket  $2$ , the third marble in bucket  $3$ , etc. After placing a marble in bucket  $9$ , he starts back from bucket  $1$  again and repeats the process. In which bucket will Bayus place the last marble in the bag?
![Image](https://cdn.artofproblemsolving.com/attachments/9/8/4c6b1bd07367101233385b3ffebc5e0abba596.png)**p20.** What is the remainder when  $1^5 + 2^5 + 3^5 +...+ 2021^5$  is divided by  $5$ ?

PS. You had better use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Answer: 1


4. 12 baskets of apples and 14 baskets of pears weigh 6 c 92 kg. Moreover, the weight of one basket of pears is 10 kg less than the weight of one basket of apples. How much does one basket of pears and one basket of apples weigh separately?

Answer: 32


Every time my brother tells the truth, our grandmother sneezes. One day, my brother said that he got a "5" in math, but grandmother did not sneeze. Then, with a slight doubt in his initial words, he said he got a "4", and grandmother sneezed. Encouraged by grandmother's sneezing, he confirmed that he definitely got at least a 3, but grandmother did not sneeze anymore. So, what grade did my brother actually get in math?

Answer: 2


a) Each vertex at the base of a triangle is connected by straight lines to \( n \) points located on the opposite side. Into how many parts do these lines divide the triangle?

b) Each of the three vertices of a triangle is connected by straight lines to \( n \) points located on the opposite side of the triangle. Into how many parts do these lines divide the triangle, assuming that no three of them intersect at the same point?

Answer: (n + 1)^2


Question 13: Given that $A$ and $B$ are two subsets of $\{1,2, \ldots, 100\}$, satisfying: $|A|=|B|, A \cap B=\emptyset$, and for any $x \in A, 2 x+2 \in B$. Try to find the maximum value of $|A \cup B|$.

---

The translation maintains the original format and line breaks as requested.

Answer: 66


The combination for a certain lock is set by assigning either the number 0 or 1 to each vertex of a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$ and simultaneously coloring each vertex either red or blue, such that any two adjacent vertices must share at least one same number or color. How many different possible combinations are there for this lock?

Answer: 3^n + (-1)^n + 2


471. A motorcyclist left A for B and at the same time a pedestrian set off from B to A. Upon meeting the pedestrian, the motorcyclist gave him a ride, brought him to A, and immediately set off again for B. As a result, the pedestrian reached A 4 times faster than he would have if he had walked the entire way. How many times faster would the motorcyclist have arrived in B if he had not had to return?

Answer: \dfrac{11}{4}


5. Let $n$ be a given positive integer. If a set $K$ of integer points in the plane satisfies the following condition, it is called "connected": for any pair of points $R, S \in K$, there exists a positive integer $l$ and a sequence of points in $K$
$$
R=T_{0}, T_{1}, \cdots, T_{l}=S,
$$

where the distance between each $T_{i}$ and $T_{i+1}$ is 1.
For such a set $K$, define
$$
\Delta(K)=\{\overrightarrow{R S} \mid R, S \in K\} \text {. }
$$

For all connected sets $K$ of $2 n+1$ integer points in the plane, find the maximum possible value of $|\Delta(K)|$.

Answer: 2n^2 + 4n + 1


[ Signs of Perpendicularity]

The height of the triangular pyramid $A B C D$, dropped from vertex $D$, passes through the point of intersection of the heights of triangle $A B C$. In addition, it is known that $D B=3, D C=2, \angle B D C=90^{\circ}$. Find the ratio of the area of face $A D B$ to the area of face $A D C$.

#

Answer: \dfrac{3}{2}


## Task 5 - 010735

Rolf claims to know a math problem that uses only the number 7 and whose result is the year 1962.

a) Try to set up such a math problem!

b) Can a math problem also be set up that uses only the number 1962 and whose result is 7? If yes, provide this math problem!

Answer: [(7 + 7) \times (7 + 7)] \times [7 + \frac{7 + 7 + 7}{7}] + \frac{7 + 7}{7} = 1962


6. A square is inscribed in a circle, and a rectangle is inscribed in the square. Another circle is circumscribed about the rectangle, and a smaller circle is tangent to three sides of the rectangle, as shown below. The shaded area between the two larger circles is eight times the area of the smallest circle, which is also shaded. What fraction of the largest circle is shaded?

Answer: \dfrac{9}{25}


Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers,
\[n = a_1 + a_2 + \cdots a_k\]
with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$? For example, with $n = 4$, there are four ways: $4$, $2 + 2$, $1 + 1 + 2$, $1 + 1 + 1 + 1$.

Answer: n


13. There are five cards below, each with a number: 00123. Using them, many different five-digit numbers can be formed. Find the average of all these five-digit numbers.

Answer: 21111


97.1. Let A be a set of seven positive numbers. Determine the maximal number of triples $(x, y, z)$ of elements of A satisfying $x<y$ and $x+y=z$.

Answer: 9


13. As shown in Figure 5, in the circle $\odot O$ with radius $r$, $AB$ is the diameter, $C$ is the midpoint of $\overparen{AB}$, and $D$ is the one-third point of $\overparen{CB}$. Moreover, the length of $\overparen{DB}$ is twice the length of $\overparen{CD}$; connect $AD$ and extend it to intersect the tangent line $CE$ of $\odot O$ at point $E$ ($C$ is the point of tangency). Find the length of $AE$.

Answer: 2r


8.1. Usually, we write the date in the format of day, month, and year (for example, 17.12.2021). In the USA, however, it is customary to write the month number, day number, and year in sequence (for example, 12.17.2021). How many days in a year cannot be determined unequivocally by its writing?

Answer: 132


Let $f$ be a function defined on and taking values in the set of non-negative real numbers. Find all functions $f$ that satisfy the following conditions:
(i) $f[x \cdot f(y)] \cdot f(y)=f(x+y)$;
(ii) $f(2)=0$;
(iii) $f(x) \neq 0$, when $0 \leq x < 2$.

Answer: f(x) = \begin{cases} \dfrac{2}{2 - x} & \text{if } 0 \leq x < 2, \\ 0 & \text{if } x \geq 2. \end{cases}


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Carol`, `Bob`, `Alice`, `Peter`
 - Each person prefers a unique type of vacation: `mountain`, `camping`, `beach`, `cultural`, `city`, `cruise`
 - Everyone has something unique for lunch: `spaghetti`, `soup`, `stir fry`, `stew`, `grilled cheese`, `pizza`
 - Each person has a unique favorite drink: `boba tea`, `root beer`, `water`, `milk`, `coffee`, `tea`
 - People have unique heights: `tall`, `average`, `very tall`, `short`, `very short`, `super tall`

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

Answer: Peter


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Alice`, `Eric`, `Bob`, `Peter`, `Arnold`
 - People have unique favorite book genres: `fantasy`, `romance`, `mystery`, `science fiction`, `biography`
 - Each person has a unique level of education: `bachelor`, `master`, `high school`, `doctorate`, `associate`

## Clues:
1. The person who loves fantasy books is Alice.
2. Bob is directly left of the person with a bachelor's degree.
3. The person with a high school diploma is the person who loves science fiction books.
4. The person with an associate's degree is Bob.
5. The person who loves fantasy books is somewhere to the right of the person with a master's degree.
6. Arnold is the person with a high school diploma.
7. Eric is not in the second house.
8. The person who loves biography books is the person with a doctorate.
9. The person who loves science fiction books is in the third house.
10. The person who loves mystery books is not in the first house.
11. The person with a master's degree is somewhere to the left of the person with a doctorate.
 What is the value of attribute Name for the person whose attribute Education is doctorate? Please reason step by step, and put your final answer within \boxed{}

Answer: Peter


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

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

Answer: Peter


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`, `Alice`, `Peter`, `Arnold`
 - Each person has a unique level of education: `bachelor`, `high school`, `associate`, `master`
 - The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`, `Kailyn`
 - The people keep unique animals: `fish`, `horse`, `cat`, `bird`
 - Each person has a unique birthday month: `jan`, `feb`, `sept`, `april`
 - People have unique favorite music genres: `rock`, `pop`, `jazz`, `classical`

## Clues:
1. The person with an associate's degree is The person whose mother's name is Aniya.
2. Peter is the person whose birthday is in September.
3. The person with a bachelor's degree is directly left of the person who loves pop music.
4. Arnold is The person whose mother's name is Janelle.
5. The person with a master's degree is Arnold.
6. The person whose mother's name is Janelle is the person who loves classical music.
7. The person whose birthday is in April is somewhere to the right of the person who loves rock music.
8. The cat lover is in the second house.
9. The fish enthusiast is in the third house.
10. Eric is the person whose birthday is in April.
11. The person who loves pop music is the person whose birthday is in January.
12. The person whose mother's name is Holly is the person with a bachelor's degree.
13. The person who keeps horses is the person with a master's degree.
14. The person whose mother's name is Aniya is the fish enthusiast.
15. Arnold is somewhere to the left of the fish enthusiast.
 What is the value of attribute Mother for the person whose attribute Animal is cat? Please reason step by step, and put your final answer within \boxed{}

Answer: Holly


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
 - Everyone has something unique for lunch: `grilled cheese`, `stew`, `pizza`, `spaghetti`
 - People have unique hair colors: `blonde`, `black`, `red`, `brown`
 - Each person has a favorite color: `red`, `white`, `yellow`, `green`
 - The mothers' names in different houses are unique: `Holly`, `Aniya`, `Janelle`, `Kailyn`

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

Answer: grilled cheese


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Eric`, `Arnold`
 - Each person has a unique birthday month: `jan`, `april`, `sept`
 - Each person has a unique type of pet: `cat`, `dog`, `fish`
 - Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
 - Each person prefers a unique type of vacation: `beach`, `mountain`, `city`
 - Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`

## Clues:
1. The person whose birthday is in January is somewhere to the left of the person who is a pizza lover.
2. The person who owns a dog is in the second house.
3. The person whose birthday is in September is the person who loves the spaghetti eater.
4. The person whose birthday is in January is the person who enjoys mountain retreats.
5. The Watermelon smoothie lover is the person with an aquarium of fish.
6. The person whose birthday is in September is not in the third house.
7. Eric is somewhere to the left of the person who prefers city breaks.
8. The person who has a cat is Peter.
9. The person who loves beach vacations is in the third house.
10. The person who likes Cherry smoothies is the person whose birthday is in April.
 What is the value of attribute Food for the person whose attribute Birthday is april? Please reason step by step, and put your final answer within \boxed{}

Answer: pizza


There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Carol`, `Peter`, `Bob`, `Eric`, `Arnold`, `Alice`
 - Each person prefers a unique type of vacation: `camping`, `cruise`, `beach`, `city`, `cultural`, `mountain`
 - People have unique favorite sports: `volleyball`, `baseball`, `basketball`, `swimming`, `soccer`, `tennis`

## Clues:
1. The person who loves tennis and Peter are next to each other.
2. Bob is in the second house.
3. There is one house between the person who prefers city breaks and the person who loves tennis.
4. Peter is somewhere to the left of Alice.
5. Carol is the person who loves basketball.
6. The person who loves swimming is the person who likes going on cruises.
7. The person who loves tennis and the person who loves beach vacations are next to each other.
8. The person who loves swimming is Bob.
9. The person who loves volleyball is the person who enjoys camping trips.
10. Eric is in the first house.
11. Arnold is somewhere to the left of the person who enjoys mountain retreats.
12. Eric is the person who loves soccer.
13. The person who prefers city breaks is in the first house.
14. The person who enjoys camping trips is not in the sixth house.
 What is the value of attribute House for the person whose attribute FavoriteSport is volleyball? 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`, `Alice`, `Bob`, `Carol`, `Arnold`, `Eric`
 - Each person has an occupation: `engineer`, `doctor`, `lawyer`, `artist`, `nurse`, `teacher`
 - Everyone has something unique for lunch: `soup`, `stew`, `grilled cheese`, `spaghetti`, `pizza`, `stir fry`
 - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `watermelon`, `lime`, `blueberry`, `desert`
 - Each person prefers a unique type of vacation: `camping`, `cultural`, `city`, `cruise`, `beach`, `mountain`
 - People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `samsung galaxy s21`, `xiaomi mi 11`, `huawei p50`

## Clues:
1. The person who likes Cherry smoothies is directly left of the person who enjoys camping trips.
2. The person who is a lawyer is the person who uses a Xiaomi Mi 11.
3. The person who is a doctor is Eric.
4. The person who is a doctor and the Desert smoothie lover are next to each other.
5. The person who is an artist is the person who prefers city breaks.
6. The person who loves beach vacations is Alice.
7. There are two houses between the person who is a pizza lover and the person who uses a Huawei P50.
8. The person who uses a Xiaomi Mi 11 is the person who enjoys camping trips.
9. The person who is an artist is directly left of the person who loves eating grilled cheese.
10. The person who uses a Google Pixel 6 is in the first house.
11. There is one house between the person who is a teacher and the person who loves stir fry.
12. There are two houses between the person who is a pizza lover and the person who is an engineer.
13. The person who is a pizza lover is somewhere to the left of the person who loves eating grilled cheese.
14. The Watermelon smoothie lover is the person who uses a Xiaomi Mi 11.
15. The person who loves eating grilled cheese is Carol.
16. Alice is the person who drinks Blueberry smoothies.
17. The person who likes Cherry smoothies is the person who loves the stew.
18. The person who loves the soup is the person who uses a Samsung Galaxy S21.
19. The person who uses an iPhone 13 is directly left of the Dragonfruit smoothie lover.
20. The person who loves the stew is somewhere to the left of Peter.
21. Arnold is somewhere to the right of the Desert smoothie lover.
22. The person who likes going on cruises is the person who uses a Google Pixel 6.
23. The person who goes on cultural tours is not in the fifth house.
24. The person who is a teacher is Alice.
 What is the value of attribute Occupation for the person whose attribute Vacation is camping? Please reason step by step, and put your final answer within \boxed{}

Answer: lawyer


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 Height for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: short