Found problems: 85335
2020 MMATHS, I7
Suppose that $ABC$ is a triangle with $AB = 6, BC = 12$, and $\angle B = 90^{\circ}$. Point $D$ lies on side $BC$, and point $E$ is constructed on $AC$ such that $\angle ADE = 90^{\circ}$. Given that $DE = EC = \frac{a\sqrt{b}}{c}$ for positive integers $a, b,$ and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, find $a+ b+c$.
[i]Proposed by Andrew Wu[/i]
2017 Bulgaria JBMO TST, 1
Given is a triangle $ABC$ and let $AA_1$, $BB_1$ be angle bisectors. It turned out that $\angle AA_1B=24^{\circ}$ and $\angle BB_1A=18^{\circ}$. Find the ratio $\angle BAC:\angle ACB:\angle ABC$.
1966 IMO Shortlist, 7
For which arrangements of two infinite circular cylinders does their intersection lie in a plane?
May Olympiad L1 - geometry, 2016.4
In a triangle $ABC$, let $D$ and $E$ point in the sides $BC$ and $AC$ respectively. The segments $AD$ and $BE$ intersects in $O$, let $r$ be line (parallel to $AB$) such that $r$ intersects $DE$ in your midpoint, show that the triangle $ABO$ and the quadrilateral $ODCE$ have the same area.
1951 AMC 12/AHSME, 10
Of the following statements, the one that is incorrect is:
$ \textbf{(A)}\ \text{Doubling the base of a given rectangle doubles the area.}$
$ \textbf{(B)}\ \text{Doubling the altitude of a triangle doubles the area.}$
$ \textbf{(C)}\ \text{Doubling the radius of a given circle doubles the area.}$
$ \textbf{(D)}\ \text{Doubling the divisor of a fraction and dividing its numerator by 2 changes the quotient.}$
$ \textbf{(E)}\ \text{Doubling a given quantity may make it less than it originally was.}$
2020 Memorial "Aleksandar Blazhevski-Cane", 2
One positive integer is written in each $1 \times 1$ square of the $m \times n$ board. The following operations are allowed :
(1) In an arbitrarily selected row of the board, all numbers should be reduced by $1$.
(2) In an arbitrarily selected column of the board, double all the numbers.
Is it always possible, after a final number of steps, for all the numbers written on the board to be equal to $-1$?
(Explain the answer.)
2019 Yasinsky Geometry Olympiad, p1
A circle with center at the origin and radius $5$ intersects the abscissa in points $A$ and $B$. Let $P$ a point lying on the line $x = 11$, and the point $Q$ is the intersection point of $AP$ with this circle. We know what is the $Q$ point is the midpoint of the $AP$. Find the coordinates of the point $P$.
2020 LMT Fall, B29
Alicia bought some number of disposable masks, of which she uses one per day. After she uses each of her masks, she throws out half of them (rounding up if necessary) and reuses each of the remaining masks, repeating this process until she runs out of masks. If her masks lasted her $222$ days, how many masks did she start out with?
2014 Contests, 1
Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$
2012 China Team Selection Test, 1
Given two circles ${\omega _1},{\omega _2}$, $S$ denotes all $\Delta ABC$ satisfies that ${\omega _1}$ is the circumcircle of $\Delta ABC$, ${\omega _2}$ is the $A$- excircle of $\Delta ABC$ , ${\omega _2}$ touches $BC,CA,AB$ at $D,E,F$.
$S$ is not empty, prove that the centroid of $\Delta DEF$ is a fixed point.
2022 CMIMC Integration Bee, 2
\[\int_{-2}^2 |1-x^2|\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
1969 Putnam, B2
Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if "two' is replaced by "three'?
1966 Swedish Mathematical Competition, 1
Let $\{x\}$ denote the fractional part of $x$, $x - [x]$. The sequences $x_1, x_2, x_3, ...$ and $y_1, y_2, y_3, ...$ are such that $\lim \{x_n\} = \lim \{y_n\} = 0$. Is it true that $\lim \{x_n + y_n\} = 0$? $\lim \{x_n - y_n\} = 0$?
Ukrainian From Tasks to Tasks - geometry, 2011.14
The lengths of the four sides of an cyclic octagon are $4$ cm, the lengths of the other four sides are $6$ cm. Find the area of the octagon.
2023 Polish MO Finals, 1
Given a sequence of positive integers $a_1, a_2, a_3, \ldots$ such that for any positive integers $k$, $l$ we have $k+l ~ | ~ a_k + a_l$. Prove that for all positive integers $k > l$, $a_k - a_l$ is divisible by $k-l$.
1989 Czech And Slovak Olympiad IIIA, 5
Consider a rectangular table $2 \times n.$ Let every cell be dyed either by black or white color in a way that no $2\times 2$ square is completely black. Denote $P_n$ the number of such colorings. Prove that the number $P_{1989}$ is divisible by three and find the greatest power of three that divides them.
2024 Czech-Polish-Slovak Junior Match, 6
We are given a rectangular table with a positive integer written in each of its cells. For each cell of the table, the number in it is equal to the total number of different values in the cells that are in the same row or column (including itself). Find all tables with this property.
2023-24 IOQM India, 30
Let $d(m)$ denote the number of positive integer divisors of a positive integer $m$. If $r$ is the number of integers $n \leqslant 2023$ for which $\sum_{i=1}^{n} d(i)$ is odd. , find the sum of digits of $r.$
2002 Croatia National Olympiad, Problem 2
Prove that a natural number can be written as a sum of two or more consecutive positive integers if and only if that number is not a power of two.
1997 Junior Balkan MO, 2
Let $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$. Compute the following expression in terms of $k$:
\[ E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}. \]
[i]Ciprus[/i]
2001 Switzerland Team Selection Test, 8
Find two smallest natural numbers $n$ for which each of the fractions
$\frac{68}{n+70},\frac{69}{n+71},\frac{70}{n+72},...,\frac{133}{n+135}$ is irreducible.
2006 Kurschak Competition, 3
We deal $n-1$ cards in some way to $n$ people sitting around a table. From then on, in one move a person with at least $2$ cards gives one card to each of his/her neighbors. Prove that eventually a state will be reached where everyone has at most one card.
2018 AIME Problems, 9
Find the number of four-element subsets of $\{1,2,3,4,\dots, 20\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\{3,5,13,19\}$ and $\{6,10,20,18\}$ are two such subsets.
Oliforum Contest I 2008, 3
Let $ 0 < a_1 < a_2 < a_3 < ... < a_{10000} < 20000$ be integers such that $ gcd(a_i,a_j) < a_i, \forall i < j$ ; is $ 500 < a_1$ [i](always)[/i] true ?
[i](own)[/i] :lol:
2016 Iranian Geometry Olympiad, 3
Suppose that $ABCD$ is a convex quadrilateral with no parallel sides. Make a parallelogram on each two consecutive sides. Show that among these $4$ new points, there is only one point inside the quadrilateral $ABCD$.
by Morteza Saghafian