Found problems: 85335
An $m\times n(m,n\in \mathbb{Z}_+)$ rectangle is divided into some smaller squares. All sides of each square are parallel to the sides of the rectangle, and the length of each side is an integer. Determine the minimum value of the sum of the lengths of sides of these squares.
Let $ f(x)$ be a polynomial with rational coefficients and $ \alpha$ be a real number such that \[ \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}.\] Prove that for each $ n \geq 1,$ \[ \left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992},\] where $ f^{n}(x) \equal{} f(f(\cdots f(x))),$ and $ n$ is a positive integer.
An acute triangle $ABC$ has circumradius $R$, inradius $r$. $A$ is the biggest angle among $A,B,C$. Let $M$ be the midpoint of $BC$, and $X$ be the intersection of two lines that touches circumcircle of $ABC$ and goes through $B,C$ respectively. Prove the following inequality : $ \frac{r}{R} \geq \frac{AM}{AX}$.
Let $n\geq 3$ and $A=\{1,2,\dots ,n\}$. For any function $f:A\rightarrow A$ we define $$A_f=\{|f(1)-f(2)|,|f(2)-f(3)|,\dots ,|f(n-1)-f(n)|,|f(n)-f(1)|\}.$$ Determine the smallest and greatest value of the cardinal of $A_f$ as $f$ goes through all bijective functions from $A$ to $A$.
[i]Silviu Cristea[/i]
Triangle $ABC$ has circumcenter $O$. $D$ is an arbitrary point on $BC$, and $AD$ intersects $\odot(ABC)$ at $E$. $S$ is a point on $\odot(ABC)$ such that $D, O, E, S$ are colinear. $AS$ intersects $BC$ at $P$. $Q$ is a point on $BC$ such that $D, O, A, Q$ are concylic. Prove that $\odot(ABC)$ is tangent to $\odot (APQ)$.
[i]Proposed by chengbilly[/i]
Let $\alpha\in(0,1]$ be a given real number and let a real sequence $\{a_n\}^\infty_{n=1}$ satisfy the inequality
$$a_{n+1}\le\alpha a_n+(1-\alpha)a_{n-1}\qquad\text{for }n=2,3,\ldots$$Prove that if $\{a_n\}$ is bounded, then it must be convergent.
We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of $100$ cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions:
1. The winner only depends on the relative order of the $200$ cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner.
2. If we write the elements of both sets in increasing order as $A =\{ a_1 , a_2 , \ldots, a_{100} \}$ and $B= \{ b_1 , b_2 , \ldots , b_{100} \}$, and $a_i > b_i$ for all $i$, then $A$ beats $B$.
3. If three players draw three disjoint sets $A, B, C$ from the deck, $A$ beats $B$ and $B$ beats $C$ then $A$ also beats $C$.
How many ways are there to define such a rule? Here, we consider two rules as different if there exist two sets $A$ and $B$ such that $A$ beats $B$ according to one rule, but $B$ beats $A$ according to the other.
[i]Proposed by Ilya Bogdanov, Russia[/i]
Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.
Let $ABC$ be a triangle with $\angle ABC = 90^{\circ}$, and let $H$ be the foot of the altitude from $B$. The points $M$ and $N$ are the midpoints of the segments $AH$ and $CH$, respectively. Let $P$ and $Q$ be the second points of intersection of the circumcircle of the triangle $ABC$ with the lines $BM$ and $BN$, respectively. The segments $AQ$ and $CP$ intersect at the point $R$. Prove that the line $BR$ passes through the midpoint of the segment $MN$.
Twenty-five people of different heights stand in a $5\times 5$ grid of squares, with one person in each square. We know that each row has a shortest person, suppose Ana is the tallest of these five people. Similarly, we know that each column has a tallest person, suppose Bev is the shortest of these five people.
Assuming Ana and Bev are not the same person, who is taller: Ana or Bev?
Prove that your answer is always correct.
Let $(a_1,b_1),(a_2,b_2),\ldots ,(a_n,b_n)$ be the vertices of a convex polygon containing the origin in its interior. Prove that there are positive real numbers $x,y$ such that :
\[ (a_1,b_1)x^{a_1}y^{b_1}+(a_2,b_2)x^{a_2}y^{b_2}+\ldots +(a_n,b_n)x^{a_n}y^{b_n}=(0,0) \]
If $a$ and $b$ are positive integers such that $ab \mid a^2+b^2$ prove that $a=b$
For $a,b,c$ be real numbers greater than $1$, prove that \[\frac{a+b+c}{4} \geq \frac{\sqrt{ab-1}}{b+c}+\frac{\sqrt{bc-1}}{c+a}+\frac{\sqrt{ca-1}}{a+b}.\]
There are five positive integers written in a row. Each one except for the first one is the minimal positive integer that is not a divisor of the previous one. Can all these five numbers be distinct?
Boris Frenkin
We wish to place natural numbers around a circle such that the following property is satisfied: the absolute values of the differences of each pair of neighboring numbers are all different.
a) Is it possible to place the numbers from 1 to 2009 satisfying this property
b) Is it possible to suppress one of the numbers from 1 to 2009 in such a way that the remaining 2008 numbers can be placed satisfying the property
In Greifswald there are three schools called $A,B$ and $C$, each of which is attended by at least one student. Among any three students, one from $A$, one from $B$ and one from $C$, there are two knowing each other and two not knowing each other. Prove that at least one of the following holds:
[list]
[*]Some student from $A$ knows all students from $B$.
[*]Some student from $B$ knows all students from $C$.
[*] Some student from $C$ knows all students from $A$.[/list]
Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively.If $D$ is the intersection point of the lines $EF$ and $MC$, prove that \[\angle ADB = \angle EMF.\]
For a positive integer $n$, we will say that a sequence $a_1, a_2, \dots a_n$ where $a_i \in \{1, 2, \dots , n\}$ for all $i$ is $n$[i]-highly divisible[/i] if, for every positive integer $d$ that divides $n$ and every nonnegative integer $k$ less than $\frac{n}{d}$ we have that \[ d\;\Bigg\vert \sum_{i=kd+1}^{(k+1)d} a_i. \] Let $\chi(n)$ be the probability that a sequence $a_1, a_2, \dots, a_n$ where $a_i$ is chosen randomly from $\{1, 2, \dots n\}$ independently for all $i$ is $n$-highly divisible. Suppose that $n$ is a positive integer such that there exists a positive integer $m$ not divisible by 3 such that $3^{40}\chi(n)=\frac{1}{m}$. Compute the sum of all possible values of $n$.
[i]Proposed by Jaedon Whyte[/i]
Nir finds integers $a_0, a_1, ... , a_{208}$ such that $$(x + 2)^{208} = a_0x^0 + a_1x^1 + a_2x^2 +... + a_{208}x^{208}.$$ Let $S$ be the sum of all an such that $n -3$ is divisible by $5$. Compute the remainder when $S$ is divided by $103$.
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hours during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it take William to get to school and back.
[i]Proposed by William Yue[/i]
Smallville is populated by unmarried men and women, some of them are acquainted. Two city’s matchmakers are aware of all acquaintances. Once, one of matchmakers claimed: “I could arrange that every brunette man would marry a woman he was acquainted with”. The other matchmaker claimed “I could arrange that every blonde woman would marry a man she was acquainted with”. An amateur mathematician overheard their conversation and said “Then both arrangements could be done at the same time! ” Is he right?
An $n \times n$ board is given. In the top left corner cell there is a fox, whereas in the bottom left corner cell there is a rabbit. Every minute, the fox and the rabbit jump to a neighbouring cell at the same time. The fox can jump only to neighbouring cells that are below it or on its right, whereas the rabbit can only jump to the cells above it or in its right. They continue like this until they have no possible moves. The fox catches the rabbit if at a certain moment they are in the same cell, otherwise the rabbit gets away. Find all natural numbers $n$ for which the fox has a winning strategy to catch the rabbit.
[i](Note: Two squares are considered neighbours if they have a common side.)[/i]
Let $n$ be a positive integer. Find the number of sequences $x_{1},x_{2},\ldots x_{2n-1},x_{2n}$, where $x_{i}\in\{-1,1\}$ for each $i$, satisfying the following condition: for any integer $k$ and $m$ such that $1\le k\le m\le n$ then the following inequality holds \[\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2\]
Evaluate
\[\left \lfloor \ \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \ \right \rfloor\]
Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Points $D$ and $E$ are chosen on $AC$ and $AB$, respectively, such that quadrilateral $BCDE$ is cyclic and when the triangle is folded along segment $DE$, point $A$ lies on side $BC$. If the length of $DE$ can be expressed as $\tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $100m+n$.
[i]Proposd by Joseph Heerens[/i]