Found problems: 9
2022 Azerbaijan Junior National Olympiad, C4
There is a $8*8$ board and the numbers $1,2,3,4,...,63,64$. In all the unit squares of the board, these numbers are places such that only $1$ numbers goes to only one unit square. Prove that there is atleast $4$ $2*2$ squares such that the sum of the numbers in $2*2$ is greater than $100$.
2022 Azerbaijan Junior National Olympiad, N2
If $x,y,z \in\mathbb{N}$ and $2x^2+3y^3=4z^4$, Prove that $6|x,y,z$
2015 Azerbaijan JBMO TST, 1
With the conditions $a,b,c\in\mathbb{R^+}$ and $a+b+c=1$, prove that \[\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\geq\frac{69}{4}\]
2022 Azerbaijan Junior National Olympiad, G5
Let $ABC$ be an acute triangle and $G$ be the intersection of the meadians of triangle $ABC$. Let $D $be the foot of the altitude drawn from $A$ to $BC$. Draw a parallel line such that it is parallel to $BC$ and one of the points of it is $A$.Donate the point $S$ as the intersection of the parallel line and circumcircle $ABC$. Prove that $S,G,D$ are co-linear
[asy]
size(6cm);
defaultpen(fontsize(10pt));
pair A = dir(50), S = dir(130), B = dir(200), C = dir(-20), G = (A+B+C)/3, D = foot(A, B, C);
draw(A--B--C--cycle, black+linewidth(1));
draw(A--S^^A--D, magenta);
draw(S--D, red+dashed);
draw(circumcircle(A, B, C), heavymagenta);
string[] names = {"$A$", "$B$", "$C$","$D$", "$G$","$S$"};
pair[] points = {A, B, C,D,G,S};
pair[] ll = {A, B, C,D, G,S};
int pt = names.length;
for (int i=0; i<pt; ++i)
dot(names[i], points[i], dir(ll[i]));
[/asy]
2022 Azerbaijan JBMO TST, C4
$n$ is a natural number. Given $3n \cdot 3n$ table, the unit cells are colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen $2 \cdot 2$ square's white cells are colored orange, orange are colored black and black are colored white. Find all $n$ such that with finite steps, all the white cells in the table turns to black, and all black cells in the table turns to white. ( From starting point)
2016 Azerbaijan Junior Mathematical Olympiad, 6
For all reals $x,y,z$ prove that $$\sqrt {x^2+\frac {1}{y^2}}+ \sqrt {y^2+\frac {1}{z^2}}+ \sqrt {z^2+\frac {1}{x^2}}\geq 3\sqrt {2}. $$
2014 JBMO Shortlist, 4
With the conditions $a,b,c\in\mathbb{R^+}$ and $a+b+c=1$, prove that \[\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\geq\frac{69}{4}\]
2022 Azerbaijan National Mathematical Olympiad, 1
Find the minimum positive value of $ 1*2*3*4*...*2020*2021*2022$ where you can replace $*$ as $+$ or $-$
2022 Azerbaijan Junior National Olympiad, A1
Find the minimum positive value of $ 1*2*3*4*...*2020*2021*2022$ where you can replace $*$ as $+$ or $-$