Found problems: 85335
Let $n$ be a positive integer. Let $S$ be a subset of points on the plane with these conditions:
$i)$ There does not exist $n$ lines in the plane such that every element of $S$ be on at least one of them.
$ii)$ for all $X \in S$ there exists $n$ lines in the plane such that every element of $S - {X} $ be on at least one of them.
Find maximum of $\mid S\mid$.
[i]Proposed by Erfan Salavati[/i]
Let $I$ be the incenter of an acute triangle $ABC$. Assume that $K_1$ is the point such that $AK_1 \perp BC$ and the circle with center $K_1$ of radius $K_1A$ is internally tangent to the incircle of triangle $ABC$ at $A_1$. The points $B_1, C_1$ are defined similarly.
a) Prove that $AA_1, BB_1, CC_1$ are concurrent at a point $P$.
b) Let $\omega_1,\omega_2,\omega_3$ be the excircles of triangle $ABC$ with respect to $A, B, C$, respectively. The circles $\gamma_1,\gamma_2\gamma_3$ are the reflections of $\omega_1,\omega_2,\omega_3$ with respect to the midpoints of $BC, CA, AB$, respectively. Prove that P is the radical center of $\gamma_1,\gamma_2,\gamma_3$.
There are $2016$ costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.)
Find the maximal $k$ such that the following holds:
There are $k$ customers such that either all of them were in the shop at a speci c time instance or no two of them were both in the shop at any time instance.
Let $BB_1$ and $CC_1$ be the altitudes of the acute-angled triangle $ABC$. From the point $B_1$ the perpendiculars $B_1E$ and $B_1F$ are drawn on the sides $AB$ and $BC$ of the triangle, respectively, and from the point $C_1$ the perpendiculars $C_1 K$ and $C_1L$ on the sides $AC$ and $BC$, respectively. It turned out that the lines $EF$ and $KL$ are perpendicular. Find the measure of the angle $A$ of the triangle $ABC$.
(Alexander Dunyak)
Let $ a_1$, $ \ldots$, $ a_k$, $ a_{k\plus{}1}$, $ \ldots$, $ a_n$ be $ n$ positive numbers ($ k<n$). Suppose that the values of $ a_{k\plus{}1}$, $ a_{k\plus{}2}$, $ \ldots$, $ a_n$ are fixed. Choose the values of $ a_1$, $ a_2$, $ \ldots$, $ a_k$ that minimize the sum $ \sum_{i, j, i\neq j}\frac{a_i}{a_j}$
Find the locus of points $M$ inside the rhombus $ABCD$ such that the sum of angles $AMB$ and $CMD$ equals $180^o$ .
Let $ABC$ be a triangle and let the points $D, E$ be on the rays $AB$, $AC$ such that $BCED$ is cyclic. Prove that the following two statements are equivalent:
$\bullet$ There is a point $X$ on the circumcircle of $ABC$ such that $BDX$, $CEX$ are tangent to each other.
$\bullet$ $AB \cdot AD \le 4R^2$, where $R$ is the radius of the circumcircle of $ABC$.
How many paths are there from$ A$ to the line $BC$ if the path does not go through any vertex twice and always moves to the left?
[img]https://cdn.artofproblemsolving.com/attachments/e/6/a4bc3a9decc06eaeed6f7e99cb58f7b2524471.jpg[/img]
In a $44\times 44$ board, some of the cells are blue, and the rest are red. No blue cells borders another blue cell on the side. The red cells, on the other hand, form a connected component (one may get from any red cell to any other red cell only by traversing edge-adjacent red cells). Prove that less than one third of the cells on the board are blue.
[i]Proposed by B. Frenkin[/i]
Several zeros and ones are written down in a row. Consider all pairs of digits (not necessarily adjacent) such that the left digit is $1$ while the right digit is $0$. Let $M$ be the number of the pairs in which $1$ and $0$ are separated by an even number of digits (possibly zero), and let $N$ be the number of the pairs in which $1$ and $0$ are separated by an odd number of digits. Prove that $M \ge N$.
Let $ \{m_1,m_2,\dots\}$ be a (finite or infinite) set of positive integers. Consider the system of congruences
(1) $ x\equiv 2m_i^2 \pmod{2m_i\minus{}1}$ ($ i\equal{}1,2,...$ ).
Give a necessary and sufficient condition for the system (1) to be solvable.
Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products
\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]
form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
Determine the quadratic functions $f(x) = ax^2 + bx + c$ for which there exists an interval $(h, k)$ such that for all $x \in (h, k)$ it holds that $f(x)f(x + 1) < 0$ and $f(x)f(x -1) < 0$.
$a,b$ are real numbers, satisfying that $\frac{1}{a}+\frac{1}{b}=1$. Prove that for any $n\in\mathbb{Z}_+$, $(a+b)^{2n}-a^n-b^n\geq2^{2n}-2^{n+1}$.
The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $\gcd (F_{m}, F_{n})=F_{\gcd (m, n)}$ for all $m, n \in \mathbb{N}$.
Baron Münchausen boasts that he knows a remarkable quadratic triniomial with positive coefficients. The trinomial has an integral root; if all of its coefficients are increased by $1$, the resulting trinomial also has an integral root; and if all of its coefficients are also increased by $1$, the new trinomial, too, has an integral root. Can this be true?
$\text{ABCD}$ is a rectangle, $\text{D}$ is the center of the circle, and $\text{B}$ is on the circle. If $\text{AD}=4$ and $\text{CD}=3$, then the area of the shaded region is between
[asy]
pair A,B,C,D;
A=(0,4); B=(3,4); C=(3,0); D=origin;
draw(circle(D,5));
fill((0,5)..(1.5,4.7697)..B--A--cycle,black);
fill(B..(4,3)..(5,0)--C--cycle,black);
draw((0,5)--D--(5,0));
label("A",A,NW);
label("B",B,NE);
label("C",C,S);
label("D",D,SW);
[/asy]
$\text{(A)}\ 4\text{ and }5 \qquad \text{(B)}\ 5\text{ and }6 \qquad \text{(C)}\ 6\text{ and }7 \qquad \text{(D)}\ 7\text{ and }8 \qquad \text{(E)}\ 8\text{ and }9$
From a point $P$, two tangent lines are drawn to a circle $\Gamma$, which touch it at points $A$ and $B$. A circle $\Phi$ is drawn with center at $P$ and passes through $A$ and $B$ and is taken a point $R$ that is on the circumference $\Phi$ and in the interior of $\Gamma$. The straight line $PR$ intersects $\Gamma$ at the points $S$ and $Q$. The straight lines $AR$ and $BR$ meet $\Gamma$ again at points $C$ and $D$, respectively. Prove that $CD$ passes through the midpoint of $SQ$.
In a country, there are ${}N{}$ cities and $N(N-1)$ one-way roads: one road from $X{}$ to $Y{}$ for each ordered pair of cities $X \neq Y$. Every road has a maintenance cost. For each $k = 1,\ldots, N$ let's consider all the ways to select $k{}$ cities and $N - k{}$ roads so that from each city it is possible to get to some selected city, using only selected roads.
We call such a system of cities and roads with the lowest total maintenance cost $k{}$[i]-optimal[/i]. Prove that cities can be numbered from $1{}$ to $N{}$ so that for each $k = 1,\ldots, N$ there is a $k{}$-optimal system of roads with the selected cities numbered $1,\ldots, k$.
[i]Proposed by V. Buslov[/i]
A closed polygonal line is drawn on a unit squared paper so that its vertices lie at lattice points and its sides have odd lengths. Prove that its number of sides is divisible by $4$.
A sequence whose first term is positive has the property that any given term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term.
Let $X,Y,Z$ be the points outside the $\triangle ABC$ such that $\angle BAZ=\angle CAY,\angle CBX=\angle ABZ,\angle ACY=\angle BCX.$ Prove that the lines $AX, BY, CZ$ are concurrent.
Solve the equation
$$2\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)(x+5)}}}}=x$$
Find the largest positive integer $N$ such that the equation $99x + 100y + 101z = N$ has an unique solution in the positive integers $x, y, z$.
[i](7 points)[/i]
Let $\mathcal{V}_c$ denote the infinite parallel ruler with the parallel edges being at distance $c$ from each other. The following construction steps are allowed using ruler $\mathcal V_c$:
[list]
[*] the line through two given points;
[*] line $\ell'$ parallel to a given line $\ell $at distance $c$ (there are two such lines, both of which can be constructed using this step);
[*] for given points $A$ and $B$ with $|AB|\ge c$ two parallel lines at distance $c$ such that one of them passes through $A$, and the other one passes through $B$ (if $|AB|>c$, there exists two such pairs of parallel lines, and both can be constructed using this step).
[/list]
On the perimeter of a circular piece of paper three points are given that form a scalene triangle. Let $n$ be a given positive integer. Prove that based on the three points and $n$ there exists $C>0$ such that for any $0<c\le C$ it is possible to construct $n$ points using only $\mathcal V_c$ on one of the excircles of the triangle.
[i]We are not allowed to draw anything outside our circular paper. We can construct on the boundary of the paper; it is allowed to take the intersection point of a line with the boundary of the paper.[/i]
[i]Proposed by Áron Bán-Szabó[/i]