Found problems: 18
2022 Bolivia Cono Sur TST, P2
On $\triangle ABC$ if there existed a point $D$ in $AC$ such that $\angle CBD=\angle ABD+60$ and $\angle BDC=30$ and $AB \cdot BC=BD^2$, then find the angles inside the triangle $\triangle ABC$
2021 Bolivian Cono Sur TST, 1
Find the sum of all positive integers $n$ such that
$$\frac{n+11}{\sqrt{n-1}}$$
is an integer.
2021 Bolivian Cono Sur TST, 3
Let $ABCD$ be a rectangle with sides $AB,BC,CD$ and $DA$. Let $K,L$ be the midpoints of the sides $BC,DA$ respectivily. The perpendicular from $B$ to $AK$ hits $CL$ at $M$. Find
$$\frac{[ABKM]}{[ABCL]}$$
2022 Bolivia Cono Sur TST, P3
Is it possible to complete the following square knowning that each row and column make an aritmetic progression?
2021 Bolivian Cono Sur TST, 1
[b]a)[/b] Among $9$ apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making $2$ weighing in a two-course balance?
[b]b)[/b] Find the least necessary number of weighing that must be done to cover a false currency between $27$ coins if all the others are true.
2021 Bolivian Cono Sur TST, 2
Let $n$ be a posititve integer and let $M$ the set of all all integer cordinates $(a,b,c)$ such that $0 \le a,b,c \le n$. A frog needs to go from the point $(0,0,0)$ to the point $(n,n,n)$ with the following rules:
$\cdot$ The frog can jump only in points of $M$
$\cdot$ The frog can't jump more than $1$ time over the same point.
$\cdot$ In each jump the frog can go from $(x,y,z)$ to $(x+1,y,z)$, $(x,y+1,z)$, $(x,y,z+1)$ or $(x,y,z-1)$
In how many ways the Frog can make his target?
2021 Bolivia Ibero TST, 1
Let $n$ be a posititve integer. On a $n \times n$ grid there are $n^2$ unit squares and on these we color the sides with blue such that every unit square has exactly one side with blue.
[b]a)[/b] Find the maximun number of blue unit sides we can have on the $n \times n$ grid.
[b]b)[/b] Find the minimun number of blue unit sides we can have on the $n \times n$ grid.
2021 Bolivian Cono Sur TST, 1
Inside a rhombus $ABCD$ with $\angle BAD=60$, points $F,H,G$ are choosen on lines $AD,DC,AC$ respectivily such that $DFGH$ is a paralelogram. Show that $BFH$ is a equilateral triangle.
2021 Bolivian Cono Sur TST, 2
The numbers $1,2,...,100$ are written in a board. We are allowed to choose any two numbers from the board $a,b$ to delete them and replace on the board the number $a+b-1$.
What are the possible numbers u can get after $99$ consecutive operations of these?
2021 Bolivia Ibero TST, 2
Let $f: \mathbb Z^+ \to \mathbb Z$ be a function such that
[b]a)[/b] $f(p)=1$ for every prime $p$.
[b]b)[/b] $f(xy)=xf(y)+yf(x)$ for every pair of positive integers $x,y$
Find the least number $n \ge 2021$ such that $f(n)=n$
2022 Bolivia Cono Sur TST, P6
On $\triangle ABC$ let points $D,E$ on sides $AB,BC$ respectivily such that $AD=DE=EC$ and $AE \ne DC$. Let $P$ the intersection of lines $AE, DC$, show that $\angle ABC=60$ if $AP=CP$.
2021 Bolivia Ibero TST, 4
On a isosceles triangle $\triangle ABC$ with $AB=BC$ let $K,M$ be the midpoints of $AB,AC$ respectivily. Let $(CKB)$ intersect $BM$ at $N \ne M$, the line through $N$ parallel to $AC$ intersects $(ABC)$ at $A_1,C_1$. Show that $\triangle A_1BC_1$ is equilateral.
2022 Bolivia Cono Sur TST, P4
Find all right triangles with integer sides and inradius 6.
2022 Bolivia Cono Sur TST, P5
Find the sum of all even numbers greater than 100000, that u can make only with the digits 0,2,4,6,8,9 without any digit repeating in any number.
2022 Bolivia IMO TST, P3
On $\triangle ABC$, let $M$ the midpoint of $AB$ and $N$ the midpoint of $CM$. Let $X$ a point such that $\angle XMC=\angle MBC$ and $\angle XCM=\angle MCB$ with $X,B$ in opposite sides of line $CM$. Let $\Omega$ the circumcircle of triangle $\triangle AMX$
[b]a)[/b] Show that $CM$ is tangent to $\Omega$
[b]b)[/b] Show that the lines $NX$ and $AC$ meet at $\Omega$
2021 Bolivia Ibero TST, 3
Let $p=ab+bc+ac$ be a prime number where $a,b,c$ are different two by two, show that $a^3,b^3,c^3$ gives different residues modulo $p$
2022 Bolivia IMO TST, P1
Find all possible values of $\frac{1}{x}+\frac{1}{y}$, if $x,y$ are real numbers not equal to $0$ that satisfy
$$x^3+y^3+3x^2y^2=x^3y^3$$
2021 Bolivian Cono Sur TST, 2
Find all posible pairs of positive integers $x,y$ such that $$\text{lcm}(x,y+3001)=\text{lcm}(y,x+3001)$$