Found problems: 85335
The inscribed sphere of a tetrahedron $ABCD$ touches $ABC,ABD,ACD$ and $BCD$ at $D_1,C_1,B_1$ and $A_1$ respectively. Consider the plane equidistant from $A$ and plane $B_1C_1D_1$ (parallel to $B_1C_1D_1$) and the three planes defined analogously for the vertices $B,C,D$. Prove that the circumcenter of the tetrahedron formed by these four planes coincides with the circumcenter of tetrahedron of $ABCD$.
let $ABCD$ be a isosceles trapezium having an incircle with $AB$ parallel to $CD$.
let $CE$ be the perpendicular from $C$ on $AB$
prove that
$ CE^2 = AB. CD $
Suppose we have a square $ABCD$ and a point $S$ in the interior of this square.
Under homothety with centre $S$ and ratio of magnification $k > 1$, this square becomes another square $A'B'C'D'$.
Prove that the sum of the areas of the two quadrilaterals $A'ABB'$ and $C'CDD'$ are equal to the sum of the areas of the two quadrilaterals $B'BCC'$ and $D'DAA'$.
[asy]
unitsize(3 cm);
pair[] A, B, C, D;
pair S;
A[1] = (0,1);
B[1] = (0,0);
C[1] = (1,0);
D[1] = (1,1);
S = (0.3,0.6);
A[0] = interp(S,A[1],2/3);
B[0] = interp(S,B[1],2/3);
C[0] = interp(S,C[1],2/3);
D[0] = interp(S,D[1],2/3);
draw(A[0]--B[0]--C[0]--D[0]--cycle);
draw(A[1]--B[1]--C[1]--D[1]--cycle);
draw(A[1]--S, dashed);
draw(B[1]--S, dashed);
draw(C[1]--S, dashed);
draw(D[1]--S, dashed);
dot("$A$", A[0], N);
dot("$B$", B[0], SE);
dot("$C$", C[0], SW);
dot("$D$", D[0], SE);
dot("$A'$", A[1], NW);
dot("$B'$", B[1], SW);
dot("$C'$", C[1], SE);
dot("$D'$", D[1], NE);
dot("$S$", S, dir(270));
[/asy]
Given are real numbers $a_1, a_2,...,a_{101}$ from the interval $[-2,10]$ such that their sum is $0$. Prove that the sum of their squares is smaller than $2020$.
An isosceles trapezoid $ABCD$ $(AB \parallel CD)$ is given. Points $E$ and $F$ lie on the
sides $BC$ and $AD$, and the points $M$ and $N$ lie on the segment $EF$ such that $DF = BE$ and
$FM = NE$. Let $K$ and $L$ be the foot of perpendicular lines from $M$ and $N$ to $AB$ and $CD$, respectively. Prove that $EKFL$ is a parallelogram.
[i]Proposed by Mahdi Etesamifard[/i]
A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?
Determine all the possible positive integer $n,$ such that $3^n+n^2+2019$ is a perfect square.
In Sixcountry there are $ 12$ months, but each month consists of $6$ weeks. The month are named the same way we do, from January to December, but in each month the weeks have different lengths. In the $k$-th month the weeks consist of $6^{k-1}$ days. What is the number of days of the spring (March, April, May together)?
Compute the sum of the positive integers $n \le 100$ for which the polynomial $x^n + x + 1$ can be written as the product of at least $2$ polynomials of positive degree with integer coefficients.
Compute
$$\int_0^4\frac{x^4-4x+4}{1+2017^{x-2}}dx.$$
Let $ n$ be a positive integer, and let $ M\equal{}\left\{ 1,2,...,n\right\}$. Two players take turns at the following game: Each player, at his turn, has to select an element of $ M$ and remove all divisors of this element (including this element itself) from the set $ M$.
[b]a)[/b] Assume that the player who cannot move anymore (because the set $ M$ is empty when it's his move) wins. For which values of $ n$ does the first player have a winning strategy?
[b]b)[/b] Assume that the player who cannot move anymore (because the set $ M$ is empty when it's his move) loses. For which values of $ n$ does the first player have a winning strategy?
Prove that
$$S_n=\frac1{1^2}+\frac1{2^2}+\ldots+\frac1{n^2}<2$$for every $n\in\mathbb N$.
Andy has 2010 square tiles, each of which has a side length of one unit. He plans to arrange
the tiles in an m x n rectangle, where mn = 2010. Compute the sum of the perimeters of all
of the different possible rectangles he can make. Two rectangles are considered to be the same
if one can be rotated to become the other, so, for instance, a 1 x 2010 rectangle is considered
to be the same as a 2010 x 1 rectangle.
Given a segment $AB$ in the plane. Let $C$ be another point in the same plane,$H,I,G$ denote the orthocenter,incenter and centroid of triangle $ABC$. Find the locus of $M$ for which $A,B,H,I$ are concyclic.
Let $ABC$ be a non-obtuse triangle satisfying $AB>AC$ and $\angle B=45^{\circ}$. The circumcentre $O$ and incentre $I$ of triangle $ABC$ are such that $\sqrt{2}\ OI=AB-AC$. Find the value of $\sin A$.
A [i]quadratic[/i] number is a real root of the equations $ax^2 + bx + c = 0$ where $|a|,|b|,|c|\in\{1,2,\ldots,10\}$. Find the smallest positive integer $n$ for which at least one of the intervals$$\left(n-\dfrac{1}{3}, n\right)\quad \text{and}\quad\left(n, n+\dfrac{1}{3}\right)$$does not contain any quadratic number.
Given that $a,b,c$ are distinct real numbers, show that the equation
\[\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}=0\]
has a real root.
Source: 2018 Canadian Open Math Challenge Part B Problem 1
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Let $(1+\sqrt2)^5 = a+b\sqrt2$, where $a$ and $b$ are positive integers. Determine the value of $a+b.$
Let $p\ge 3$ be a prime number. Each side of a triangle is divided into $p$ equal parts, and we draw a line from each division point to the opposite vertex. Find the maximum number of regions, every two of them disjoint, that are formed inside the triangle.
For a positive integer $n \ge 2$, define a sequence $a_1, a_2, \cdots ,a_n$ as the following.
$$ a_1 = \frac{n(2n-1)(2n+1)}{3}$$ $$a_k = \frac{(n+k-1)(n-k+1)}{2(k-1)(2k+1)}a_{k-1}, \text{ } (k=2,3, \cdots n)$$
(a) Show that $a_1, a_2, \cdots a_n$ are all integers.
(b) Prove that there are exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n-1$ and exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n+1$ if and only if $2n-1$ and $2n+1$ are all primes.
Celeste has an unlimited amount of each type of $n$ types of candy, numerated type 1, type 2, ... type n. Initially she takes $m>0$ candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:
$1.$ She eats a candy of type $k$, and in its position in the row she places one candy type $k-1$ followed by one candy type $k+1$ (we consider type $n+1$ to be type 1, and type 0 to be type $n$).
$2.$ She chooses two consecutive candies which are the same type, and eats them.
Find all positive integers $n$ for which Celeste can leave the table empty for any value of $m$ and any configuration of candies on the table.
$\textit{Proposed by Federico Bach and Santiago Rodriguez, Colombia}$
Let $O$ be the circumcentre of triangle $ABC$. Lines $AO$ and $BC$ intersect at point $D$. Let $S$ be a point on line $BO$ such that $DS \parallel AB$ and lines $AS$ and $BC$ intersect at point $T$. Prove that if $O, D, S$ and $T$ lie on the same circle, then $ABC$ is an isosceles triangle.
Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$
Prove that all real numbers $x \ne -1$, $y \ne -1$ with $xy = 1$ satisfy the following inequality:
$$\left(\frac{2+x}{1+x}\right)^2 + \left(\frac{2+y}{1+y}\right)^2 \ge \frac92$$
(Karl Czakler)
Prove or disprove the existence of a function $f : S \to R$ such that for all $x \ne y \in S$ we have $|f(x) - f(y)| \ge \frac{1}{x^2 + y^2}$, in each of the cases:
a) $S = R$
b) $S = Q$.