Found problems: 85335
Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$
(The [i]excircle[/i] of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)
[i]Proposed by Evangelos Psychas, Greece[/i]
Is it possible to color points in the Cartesian Plane $(x,y)$ with integer coordinates with three colors, such that each color appears infinitely many times on infinitely many lines parallel to the $x$-axis and that any three points, each of a different color, are not in a line? Justify your answer.
On the circle $O,$ six points $A,B,C,D,E,F$ are on the circle counterclockwise. $BD$ is the diameter of the circle and it is perpendicular to $CF.$ Also, lines $CF, BE, AD$ is concurrent. Let $M$ be the foot of altitude from $B$ to $AC$ and let $N$ be the foot of altitude from $D$ to $CE.$ Prove that the area of $\triangle MNC$ is less than half the area of $\square ACEF.$
Find the biggest real number $ k$ such that for each right-angled triangle with sides $ a$, $ b$, $ c$, we have
\[ a^{3}\plus{}b^{3}\plus{}c^{3}\geq k\left(a\plus{}b\plus{}c\right)^{3}.\]
Given $I=[0,1]$ and $G=\{(x,y)|x,y \in I\}$, find all functions $f:G\rightarrow I$, such that $\forall x,y,z \in I$ we have:
i. $f(f(x,y),z)=f(x,f(y,z))$;
ii. $f(x,1)=x, f(1,y)=y$;
iii. $f(zx,zy)=z^kf(x,y)$.
($k$ is a positive real number irrelevant to $x,y,z$.)
Let $ A$ be a set of positive integers such that no positive integer greater than 1 divides all the elements of $ A.$ Prove that any sufficiently large positive integer can be written as a sum of elements of $ A.$ (Elements may occur several times in the sum.)
Let a ,b and c be positive real numbers such that $a^2+b^2+c^2=3$.
Prove that for whatever positive real numbers x y and z, the inequality below holds.
$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge \sqrt{xy}+\sqrt{yz}+\sqrt{zx}$
At first I noticed $\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\le \sqrt{x+y+z}\sqrt{x+y+z}=x+y+z$, so perhaps the next move is to prove
$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge x+y+z$, but I don't see how to do that, the best thing that I can do with the LHS of this inequality is to prove it by AM-GM in the way that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge 3\left(\frac{xyz}{abc}\right)^{\frac{1}{3}}\ge 3(xyz)^{\frac{1}{3}}$, but this isn't going to be helpful...
An aquarium has a rectangular base that measures $ 100$ cm by $ 40$ cm and has a height of $ 50$ cm. It is filled with water to a height of $ 40$ cm. A brick with a rectangular base that measures $ 40$ cm by $ 20$ cm and a height of $ 10$ cm is placed in the aquarium. By how many centimeters does the water rise?
$ \textbf{(A)}\ 0.5 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 1.5 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 2.5$
A circle $\omega$ passes through the two vertices $B$ and $C$ of a triangle $ABC.$ Furthermore, $\omega$ intersects segment $AC$ in $D\ne C$ and segment $AB$ in $E\ne B.$ On the ray from $B$ through $D$ lies a point $K$ such that $|BK| = |AC|,$ and on the ray from $C$ through $E$ lies a point $L$ such that $|CL| = |AB|.$ Show that the circumcentre $O$ of triangle $AKL$ lies on $\omega$.
Each vertex of a convex polyhedron lies on exactly three edges, at least two of which have the same length. Prove that the polyhedron has three edges of the same length.
Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=10$. What is the maximum value of $A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A$?
$\text{(A)}\ 49 \qquad\text{(B)}\ 59 \qquad\text{(C)}\ 69 \qquad\text{(D)}\ 79\qquad\text{(E)}\ 89$
Points $B$,$D$ , and $J$ are midpoints of the sides of right triangle $ACG$ . Points $K$, $E$, $I$ are midpoints of the sides of triangle , etc. If the dividing and shading process is done 100 times (the first three are shown) and $ AC=CG=6 $, then the total area of the shaded triangles is nearest
[asy]
draw((0,0)--(6,0)--(6,6)--cycle);
draw((3,0)--(3,3)--(6,3));
draw((4.5,3)--(4.5,4.5)--(6,4.5));
draw((5.25,4.5)--(5.25,5.25)--(6,5.25));
fill((3,0)--(6,0)--(6,3)--cycle,black);
fill((4.5,3)--(6,3)--(6,4.5)--cycle,black);
fill((5.25,4.5)--(6,4.5)--(6,5.25)--cycle,black);
label("$A$",(0,0),SW);
label("$B$",(3,0),S);
label("$C$",(6,0),SE);
label("$D$",(6,3),E);
label("$E$",(6,4.5),E);
label("$F$",(6,5.25),E);
label("$G$",(6,6),NE);
label("$H$",(5.25,5.25),NW);
label("$I$",(4.5,4.5),NW);
label("$J$",(3,3),NW);
label("$K$",(4.5,3),S);
label("$L$",(5.25,4.5),S);[/asy]
$ \text{(A)}\ 6\qquad\text{(B)}\ 7\qquad\text{(C)}\ 8\qquad\text{(D)}\ 9\qquad\text{(E)}\ 10 $
Determine all totally multiplicative and non-negative functions $f\colon\mathbb{Z}\rightarrow \mathbb{Z}$ with the property that if $a, b\in \mathbb{Z}$ and $b\neq 0$, then there exist integers $q$ and $r$ such that $a-qb+r$ and $f(r)<f(b)$.
Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.
Determine all polynomials $f (x)$ with real coeffcients that satisfy
\[f (x^{2}-2x) = f^{2}(x-2)\]
for all $x.$
Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$
[b]1.[/b] Find the groups every generating system of which contains a basis. (A basis is a set of elements of the group such that the direct product of the cyclic groups generated by them is the group itself.) [b](A. 14)[/b]
Let $ABC$ be a triangle with $AB=5$, $AC=8$, and $\angle BAC=60^\circ$. Let $UVWXYZ$ be a regular hexagon that is inscribed inside $ABC$ such that $U$ and $V$ lie on side $BA$, $W$ and $X$ lie on side $AC$, and $Z$ lies on side $CB$. What is the side length of hexagon $UVWXYZ$?
[i]Proposed by Ryan Kim.[/i]
Find all ordered pairs $(a,b)$ of complex numbers with $a^2+b^2\neq 0$, $a+\tfrac{10b}{a^2+b^2}=5$, and $b+\tfrac{10a}{a^2+b^2}=4$.
Find all positive integers $n$ satisfying the following: there exists a way to fill in $1, \cdots, n^2$ into a $n \times n$ grid so that each block has exactly one number, each number appears exactly once, and:
1. For all positive integers $1 \leq i < n^2$, $i$ and $i + 1$ are neighbors (two numbers neighbor each other if and only if their blocks share a common edge.)
2. Any two numbers among $1^2, \cdots, n^2$ are not in the same row or the same column.
Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $p$ is an odd prime such that $p \equiv \pm 1(\bmod 5)$. Prove that:
a) ${F_{p + 1}} \equiv 0(\bmod p).$
b) $k(p)|2p+2.$
c) $k(p)$ is divisible by $4.$
Problem 2. Prove that for every n≥3, there exists a convex polygon with n sides, such that one can divide it into n-2 triangles that are all similar, but pairwise non-congruent.
[color=#00f]Moved to HSO. ~ oVlad[/color]
Given $ 0< c< 1$, we define $f(x) =
\begin{cases}
\frac{x}{c} & x \in [0,c] \\
\frac{1-x}{1-c} & x \in [c, 1]
\end{cases}
$
Let $f^{n}(x)=f(f(...f(x)))$ . Show that for each positive integer $n$, $f^{n}$ has a non-zero finite nunber of fixed points which aren't fixed points of $f^k$ for $k< n$.
Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$.
If $n \geq 5$ and $e(G_{n}) \geq \frac{n^{2}}{4}+2$, prove that $G_{n}$ contains two triangles that share exactly one vertex.
Two circles intersect at $A$ and $B$. A line through $A$ meets the first circle again at $C$ and the second circle again at $D$. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ not containing $A$, and let $K$ be the midpoint of the segment $CD$. Show that $\angle MKN =\pi/2$.
(You may assume that $C$ and $D$ lie on opposite sides of $A$.)
[i]D. Tereshin[/i]