Found problems: 85335
$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$
Let $n$ be a positive integer. The numbers $1,2,3,\dots, 4n$ are written in a board. Olive wants to make some "couples" of numbers, such that the product of the numbers in each couple is a perfect square. Each number is in, at most, one couple and the two numbers in each couple are distincts.
Determine, for each positive integer $n$, the maximum number of couples that Olive can write.
The Tokyo Metro system is one of the most efficient in the world. There is some odd positive integer $k$ such that each metro line passes through exactly $k$ stations, and each station is serviced by exactly $k$ metro lines. One can get from any station to any otherstation using only one metro line - but this connection is unique. Furthermore, any two metro lines must share exactly one station. David is planning an excursion for the IMO team, and wants to visit a set $S$ of $k$ stations. He remarks that no three of the stationsin $S$ are on a common metro line. Show that there is some station not in $S$, which is connected to every station in $S$ by a different metro line.
Find the smallest prime $p$ for which there exist positive integers $a,b$ such that
\[
a^{2} + p^{3} = b^{4}.
\]
$\triangle{ABC}$ can cover a convex polygon $M$.Prove that there exsit a triangle which is congruent to $\triangle{ABC}$ such that it can also cover $M$ and has one side line paralel to or superpose one side line of $M$.
Let $ S$ be a finite family of squares on a plane such that every point on that plane is contained in at most $ k$ squares in $ S$. Prove that $ P$ can be divided into $ 4(k\minus{}1)\plus{}1$ sub-family such that in each sub-family, each pair of squares are disjoint.
Determine all polynomials $P$ with integer coefficients satisfying
\[P(P(P(P(P(x)))))=x^{28}\cdot P(P(x))\qquad \forall x\in\mathbb{R}\]
Prove that, for every integer $n \ge 2$, it is possible to divide a regular hexagon into $n$ quadrilaterals such that any two of them are similar.
Clarification: Two quadrilaterals are similar if they have their corresponding sides proportional and their corresponding angles are equal, that is, the quadrilaterals $ABCD$ and $EFGH$ are similar if $\frac{AB}{EF}= \frac{BC}{FG}= \frac{CD}{GH} = \frac{DA}{HE}$, $\angle ABC = \angle EFG$, $\angle BCD = \angle FGH$, $\angle CDA = \angle GHE$ and $\angle DAB = \angle HEF$.
In the country of Óxisz the minister of finance observed at the end of the tax census that the sum of properties of any two neighboring city counted in dinar is divisible by $1000$, and she also observed that the sum of properties of all cities is also divisible by $1000$. What is the least sum of properties of all cities if the map of the cities looks as follows?
[img]https://cdn.artofproblemsolving.com/attachments/0/5/274730ebfdd52d0c3642dfbd0596fe587eb211.png[/img]
[i]Remark: The cities may have non-integer properties, but it is also positive. On the map the points are the cities, and two cities are neighboring if there is a direct connection between them.[/i]
The roots of the equation
\[ x^3-3ax^2+bx+18c=0 \]
form a non-constant arithmetic progression and the roots of the equation
\[ x^3+bx^2+x-c^3=0 \]
form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.
There are $5$ people left in a game of Among Us, $4$ of whom are crewmates and the last is the impostor. None of the crewmates know who the impostor is. The person with the most votes is ejected, unless there is a tie in which case no one is ejected. Each of the $5$ remaining players randomly votes for someone other than themselves. The probability the impostor is ejected can be expressed as $\frac{m}{n}$. Find $m+n$.
[i]Proposed by Sammy Charney[/i]
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$\displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,}$$ for all $x \in \mathbb{R}.$ Find all possible values of $f(2).$
Find the number of partitions of the set $\{1, 2, \cdots, n\}$ into three subsets $A_1,A_2,A_3$, some of which may be empty, such that the following conditions are satisfied:
$(i)$ After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity.
$(ii)$ If $A_1,A_2,A_3$ are all nonempty, then in exactly one of them the minimal number is even .
[i]Proposed by Poland.[/i]
Garfield and Odie are situated at $(0,0)$ and $(25,0)$, respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as $\frac{m}{\sqrt{n}}$ for relatively prime positive integers $m$ and $n$. Find $m+n$.
[i]Proposed by [b] Th3Numb3rThr33 [/b][/i]
We say that a rook is "attacking" another rook on a chessboard if the two rooks are in the same row or column of the chessboard and there is no piece directly between them. Let $n$ be the maximum number of rooks that can be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other. How many ways can $n$ rooks be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other?
Let $ ABCD$ be a convex quadrilateral. Points $ P,Q$ and $ R$ are the feets of the perpendiculars from point $ D$ to lines $ BC, CA$ and $ AB$, respectively. Prove that $ PQ\equal{}QR$ if and only if the bisectors of the angles $ ABC$ and $ ADC$ meet on segment $ AC$.
If $a, b, c$, and $d$ are pairwise distinct positive integers that satisfy $lcm (a, b, c, d) < 1000$ and $a+b = c+d$, compute the largest possible value of $a + b$.
Let $f:\mathbb{Q}\rightarrow \mathbb{Q}$ a monotonic bijective function.
a)Prove that there exist a unique continuous function $F:\mathbb{R}\rightarrow \mathbb{R}$ such that $F(x)=f(x),\ (\forall)x\in \mathbb{Q}$.
b)Give an example of a non-injective polynomial function $G:\mathbb{R}\rightarrow \mathbb{R}$ such that $G(\mathbb{Q})\subset \mathbb{Q}$ and it's restriction defined on $\mathbb{Q}$ is injective.
Let $n$ be the answer to this problem. The polynomial $x^n+ax^2+bx+c$ has real coefficients and exactly $k$ real roots. Find the sum of the possible values of $k$.
Congruent circles $\Gamma_1$ and $\Gamma_2$ have radius $2012,$ and the center of $\Gamma_1$ lies on $\Gamma_2.$ Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. The line through $A$ perpendicular to $AB$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$, respectively. Find the length of $CD$.
[i]Author: Ray Li[/i]
The functions $f_0, f_1, f_2, ...$ are defined on the reals by $f_0(x) = 8$ for all $x$, $f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)}$. For all $n$ solve the equation $f_n(x) = 2x$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
$$f(xf(y)-y^2)=(y+1)f(x-y)$$
holds for all $x,y\in\mathbb{R}$.
Convex quadrilaterals \(ABCD\), \(A_1B_1C_1D_1\), and \(A_2B_2C_2D_2\) are similar with vertices in order. Points \(A\), \(A_1\), \(B_2\), \(B\) are collinear in order, points \(B\), \(B_1\), \(C_2\), \(C\) are collinear in order, points \(C\), \(C_1\), \(D_2\), \(D\) are collinear in order, and points \(D\), \(D_1\), \(A_2\), \(A\) are collinear in order. Diagonals \(AC\) and \(BD\) intersect at \(P\), diagonals \(A_1C_1\) and \(B_1D_1\) intersect at \(P_1\), and diagonals \(A_2C_2\) and \(B_2D_2\) intersect at \(P_2\). Prove that points \(P\), \(P_1\), and \(P_2\) are collinear.
[i]Proposed by Holden Mui[/i]
A modern artist paints all of his paintings by dividing his $3$ ft by $5$ ft canvas into $21$ random regions. He then colours some of the regions, and leaves some of them white. If the smallest region has area $a = 10$ square inches, and the probability that any given region with area $a_i$ is left white is $\frac{a}{a_i}$, then what is the probability that any given point on the canvas is left white? ($1$ ft $= 12$ in)
Two circles $C_1$ and $C_2$ intersect at $S$.
The tangent in $S$ to $C_1$ intersects $C_2$ in $A$ different from $S$.
The tangent in $S$ to $C_2$ intersects $C_1$ in $B$ different from $S$.
Another circle $C_3$ goes through $A, B, S$.
The tangent in $S$ to $C_3$ intersects $C_1$ in $P$ different from $S$ and $C_2$ in $Q$ different from $S$.
Prove that the distance $PS$ is equal to the distance $QS$.