Found problems: 85335
Let the $f:\mathbb{N}\rightarrow\mathbb{N}$ be the function such that
(i) For all positive integers $n,$ $f(n+f(n))=f(n)$
(ii) $f(n_o)=1$ for some $n_0$
Prove that $f(n)\equiv 1.$
Three students $A, B$ and $C$ are traveling from a location on the National Highway No.$5$ on direction to Hanoi for participating the HOMC $2018$. At beginning, $A$ takes $B$ on the motocycle, and at the same time $C$ rides the bicycle. After one hour and a half, $B$ switches to a bicycle and immediately continues the trip to Hanoi, while $A$ returns to pick up $C$. Upon meeting, $C$ continues the travel on the motocycle to Hanoi with $A$. Finally, all three students arrive in Hanoi at the same time. Suppose that the average speed of the motocycle is $50$ km per hour and of the both bicycles are $10$ km per hour. Find the distance from the starting point to Hanoi.
Let $f(x)$ be a real-valued function satisfying $af(x)+bf(-x)=px^2+qx+r$. $a$ and $b$ are distinct real numbers and $p,q,r$ are non-zero real numbers. Then $f(x)=0$ will have real solutions when
(A)$\left(\frac{a+b}{a-b}\right)\leq\frac{q^2}{4pr}$
(B)$\left(\frac{a+b}{a-b}\right)\leq\frac{4pr}{q^2}$
(C)$\left(\frac{a+b}{a-b}\right)\geq\frac{q^2}{4pr}$
(D)$\left(\frac{a+b}{a-b}\right)\geq\frac{4pr}{q^2}$
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. At the beginning of a turn there are n ≥ 1 marbles on the table, then the player whose turn is removes k marbles, where k ≥ 1 either is an even number with $k \le \frac{n}{2}$ or an odd number with $ \frac{n}{2}\le k \le n$. A player wins the game if she removes the last marble from the table.
Determine the smallest number $N\ge100000$ which Berta has wining strategy.
[i]proposed by Gerhard Woeginger[/i]
Compute $\int^9_{-9}17x^3 \cos (x^2) dx.$
Let $ACD$ and $BCD$ be acute-angled triangles located in different planes. Let $G$ and $H$ be the centroid and the orthocenter respectively of the $BCD$ triangle; Similarly let $G'$ and $H'$ be the centroid and the orthocenter of the $ACD$ triangle. Knowing that $HH'$ is perpendicular to the plane $(ACD)$, show that $GG' $ is perpendicular to the plane $(BCD)$.
Let $n\geq 2$. Find all matrices $A\in\mathcal{M}_n(\mathbb{C})$ such that $$\text{rank}(A^2)+\text{rank}(B^2)\geq 2\text{rank}(AB),$$ for all $B\in\mathcal{M}_n(\mathbb{C})$.
[i]Cristi Săvescu[/i]
Two circles with radii $71$ and $100$ are externally tangent. Compute the largest possible area of a right triangle whose vertices are each on at least one of the circles.
Find all real functions $f$ definited on positive integers and satisying:
(a) $f(x+22)=f(x)$,
(b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$
for all positive integers $x$ and $y$.
Let $a, b, c$ be the lengths of sides of a triangle. Prove the inequality
$$(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$$
Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards.
Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells.
[*]Prove that every minimal blocking set containing at most $3m^2$ cells.
A circle contains the points $(0, 11)$ and $(0, -11)$ on its circumference and contains all points $(x, y)$ with $x^2+y^2<1$ in its interior. Compute the largest possible radius of the circle.
Compute the number of permutations $(a,b,c,x,y,z)$ of $(1,2,3,4,5,6)$ which satisfy the five inequalities
\[ a < b < c, \quad x < y < z, \quad a < x, \quad b < y, \quad\text{and}\quad c < z. \]
[i]Proposed by Evan Chen[/i]
How many
a) bishops
b) horses
can be positioned on a chessboard at most, so that no one threatens another?
In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$
Let us consider $a,b$ two integers. Prove that there exists and it is unique a pair of integers $(x,y)$ such that: \[(x+2y-a)^{2}+(2x-y-b)^{2}\leq 1.\]
Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains $ 8$ unit squares. The second ring contains $ 16$ unit squares. If we continue this process, the number of unit squares in the $ 100^\text{th}$ ring is
$ \textbf{(A)}\ 396 \qquad
\textbf{(B)}\ 404 \qquad
\textbf{(C)}\ 800 \qquad
\textbf{(D)}\ 10,\!000 \qquad
\textbf{(E)}\ 10,\!404$
[asy]unitsize(3mm);
defaultpen(linewidth(1pt));
fill((2,2)--(2,7)--(7,7)--(7,2)--cycle, mediumgray);
fill((3,3)--(6,3)--(6,6)--(3,6)--cycle, gray);
fill((4,4)--(5,4)--(5,5)--(4,5)--cycle, black);
for(real i=0; i<=9; ++i)
{
draw((i,0)--(i,9));
draw((0,i)--(9,i));
}[/asy]
A line through the point $(-a,0)$ cuts from the second quadrant a triangular region with area $T$. The equation of the line is:
$ \textbf{(A)}\ 2Tx+a^2y+2aT=0 \qquad\textbf{(B)}\ 2Tx-a^2y+2aT=0 \qquad$
${{\textbf{(C)}\ 2Tx+a^2y-2aT=0 \qquad\textbf{(D)}\ 2Tx-a^2y-2aT=0 }\qquad\textbf{(E)}\ \text{none of these} } $
In the center of each cell of a checkered rectangle $M{}$ there is a point-like light bulb. All the light bulbs are initially switched off. In one turn it is allowed to choose a straight line not intersecting any light bulbs such that on one side of it all the bulbs are switched off, and to switch all of them on. In each turn at least one bulb should be switched on. The task is to switch on all the light bulbs using the largest possible number of turns. What is the maximum number of turns if:
[list=a]
[*]$M$ is a square of size $21 \times 21$;
[*]$M$ is a rectangle of size $20 \times 21$?
[/list]
[i]Alexandr Shapovalov[/i]
Evaluate $2\times 0+2\times 1+ 2+0\times 2 +1$.
[i]Proposed by Nathan Xiong[/i]
Let $ ABCDE$ be convex pentagon such that $ S(ABC) \equal{} S(BCD) \equal{} S(CDE) \equal{} S(DEA) \equal{} S(EAB)$. Prove that there is a point $ M$ inside pentagon such that $ S(MAB) \equal{} S(MBC) \equal{} S(MCD) \equal{} S(MDE) \equal{} S(MEA)$.
Let $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}$, where $a_{0},\ldots,a_{n}$ are integers, $a_{n}>0$, $n\geq 2$. Prove that there exists a positive integer $m$ such that $P(m!)$ is a composite number.
Let $M$ be the set of integer coordinate points situated on the line $d$ of real numbers. We color the elements of M in black or white. Show that at least one of the following statements is true:
(a) there exists a finite subset $F \subset M$ and a point $M \in d$ so that the elements of the set $M - F$ that are lying on one of the rays determined by $M$ on $d$ are all white, and the elements of $M - F$ that are situated on the opposite ray are all black,
(b) there exists an infinite subset $S \subset M$ and a point $T \in d$ so that for each $A \in S$ the reflection of A about $T$ belongs to $S$ and has the same color as $A$
Consider the function
\[f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).\]
Find the period of $f$ and sketch diagram of $f$ in one period. Also prove that $\lim_{x \to 1} f(x)$ does not exist.
Find all real numbers $ x$ which satisfy: $ \frac{x^2}{(x\plus{}1\minus{}\sqrt{x\plus{}1})^2}<\frac{x^2\plus{}3x\plus{}18}{(x\plus{}1)^2}.$