Found problems: 85335
2014 Hanoi Open Mathematics Competitions, 8
Let $ABC$ be a triangle. Let $D,E$ be the points outside of the triangle so that $AD=AB,AC=AE$ and $\angle DAB =\angle EAC =90^o$. Let $F$ be at the same side of the line $BC$ as $A$ such that $FB = FC$ and $\angle BFC=90^o$. Prove that the triangle $DEF$ is a right- isosceles triangle.
2022 MIG, 22
How many ways are there to color each of the $8$ cells below red or blue such that no two blue cells are adjacent?
[asy]
size(3cm);
draw((0,0)--(4,0)--(4,1)--(0,1)--(0,0));
draw((1,-1)--(1,2)--(3,2)--(3,-1)--(1,-1));
draw((2,-1)--(2,2));
[/asy]
$\textbf{(A) }48\qquad\textbf{(B) }50\qquad\textbf{(C) }52\qquad\textbf{(D) }54\qquad\textbf{(E) }56$
1993 Putnam, B1
What is the smallest integer $n > 0$ such that for any integer m in the range $1, 2, 3, ... , 1992$ we can always find an integral multiple of $\frac{1}{n}$ in the open interval $(\frac{m}{1993}, \frac{m+1}{1994})$?
2012 Pan African, 2
Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(x^2 - y^2) = (x+y)(f(x) - f(y))$ for all real numbers $x$ and $y$.
2020 Princeton University Math Competition, A7
Suppose that $p$ is the unique monic polynomial of minimal degree such that its coefficients are rational numbers and one of its roots is $\sin \frac{2\pi}{7} + \cos \frac{4\pi}{7}$. If $p(1) = \frac{a}{b}$, where $a, b$ are relatively prime integers, find $|a + b|$.
1994 ITAMO, 1
Show that there exists an integer $N$ such that for all $n \ge N$ a square can be partitioned into $n$ smaller squares.
2022 China Second Round A2, 3
$S=\{1,2,...,N\}$. $A_1,A_2,A_3,A_4\subseteq S$, each having cardinality $500$. $\forall x,y\in S$, $\exists i\in\{1,2,3,4\}$, $x,y\in A_i$. Determine the maximal value of $N$.
2018 Online Math Open Problems, 7
A quadrilateral and a pentagon (both not self-intersecting) intersect each other at $N$ distinct points, where $N$ is a positive integer. What is the maximal possible value of $N$?
[i]Proposed by James Lin
1998 National Olympiad First Round, 17
In triangle $ ABC$, internal bisector of angle $ A$ intersects with $ BC$ at $ D$. Let $ E$ be a point on $ \left[CB\right.$ such that $ \left|DE\right|\equal{}\left|DB\right|\plus{}\left|BE\right|$. The circle through $ A$, $ D$, $ E$ intersects $ AB$ at $ F$, again. If $ \left|BE\right|\equal{}\left|AC\right|\equal{}7$, $ \left|AD\right|\equal{}2\sqrt{7}$ and $ \left|AB\right|\equal{}5$, then $ \left|BF\right|$ is
$\textbf{(A)}\ \frac {7\sqrt {5} }{5} \qquad\textbf{(B)}\ \sqrt {7} \qquad\textbf{(C)}\ 2\sqrt {2} \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \sqrt {10}$
2000 Irish Math Olympiad, 4
The sequence $ a_1<a_2<...<a_M$ of real numbers is called a weak arithmetic progression of length $ M$ if there exists an arithmetic progression $ x_0,x_1,...,x_M$ such that:
$ x_0 \le a_1<x_1 \le a_2<x_2 \le ... \le a_M<x_M.$
$ (a)$ Prove that if $ a_1<a_2<a_3$ then $ (a_1,a_2,a_3)$ is a weak arithmetic progression.
$ (b)$ Prove that any subset of $ \{ 0,1,2,...,999 \}$ with at least $ 730$ elements contains a weak arithmetic progression of length $ 10$.
1978 Germany Team Selection Test, 5
Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds:
(i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$
(ii) some plane contains exactly three points from $E.$
2007 Stars of Mathematics, 2
Let be a structure formed by $ n\ge 4 $ points in space, four by four noncoplanar, and two by two connected by a wire. If we cut the $ n-1 $ wires that connect a point to the others, the remaining point is said to be [i]isolated.[/i] The structure is said to be [i]disconnected[/i] if there are at least two points for which there isn´t a chain of wires connecting them. So, initially, it´s not disconnected.
$ \text{(1)} $ Prove that, by cutting a number smaller or equal with $ n-2, $ the structure won´t become disconnected.
$ \text{(2)} $ Determine the minimum number of wires that needs to be cut so that the remaining structure is disconnected, yet every point, not isloated.
2018 AMC 10, 22
Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$?
$\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}$
1976 Czech and Slovak Olympiad III A, 4
Determine all solutions of the linear system of equations
\begin{align*}
&x_1& &-x_2& &-x_3& &-\cdots& &-x_n& &= 2a, \\
-&x_1& &+3x_2& &-x_3& &-\cdots& &-x_n& &= 4a, \\
-&x_1& &-x_2& &+7x_3& &-\cdots& &-x_n& &= 8a, \\
&&&&&&&&&&&\vdots \\
-&x_1& &-x_2& &-x_3& &-\cdots& &+\left(2^n-1\right)x_n& &= 2^na,
\end{align*}
with unknowns $x_1,\ldots,x_n$ and a real parameter $a.$
2013 IFYM, Sozopol, 7
Let $T$ be a set of natural numbers, each of which is greater than 1. A subset $S$ of $T$ is called “good”, if for each $t\in T$ there exists $s\in S$, for which $gcd(t,s)>1$. Prove that the number of "good" subsets of $T$ is odd.
2011 International Zhautykov Olympiad, 3
Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $K.$ The midpoints of diagonals $AC$ and $BD$ are $M$ and $N,$ respectively. The circumscribed circles $ADM$ and $BCM$ intersect at points $M$ and $L.$ Prove that the points $K ,L ,M,$ and $ N$ lie on a circle. (all points are supposed to be different.)
2022 Iranian Geometry Olympiad, 1
Four points $A$, $B$, $C$ and $D$ lie on a circle $\omega$ such that $AB=BC=CD$. The tangent line to $\omega$ at point $C$ intersects the tangent line to $\omega$ at $A$ and the line $AD$ at $K$ and $L$. The circle $\omega$ and the circumcircle of triangle $KLA$ intersect again at $M$. Prove that $MA=ML$.
[i]Proposed by Mahdi Etesamifard[/i]
2015 JBMO Shortlist, NT4
Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\]
Proposed by Moldova
CNCM Online Round 2, 2
There is a rectangle $ABCD$ such that $AB=12$ and $BC=7$. $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$. Call $X$ the intersection of $AF$ and $DE$. What is the area of pentagon $BCFXE$?
Proposed by Minseok Eli Park (wolfpack)
2004 Korea Junior Math Olympiad, 2
For $n\geq3$ define $S_n=\{1, 2, ..., n\}$. $A_1, A_{2}, ..., A_{n}$ are given subsets of $S_n$, each having an even number of elements. Prove that there exists a set $\{i_1, i_2, ..., i_t\}$, a nonempty subset of $S_n$ such that
$$A_{i_1} \Delta A_{i_2} \Delta \ldots \Delta A_{i_t}=\emptyset$$
(For two sets $A, B$, we define $\Delta$ as $A \Delta B=(A\cup B)-(A\cap B)$)
1969 IMO Shortlist, 13
$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?
1966 Miklós Schweitzer, 6
A sentence of the following type if often heard in Hungarian weather reports: "Last night's minimum temperatures took all values between $ \minus{}3$ degrees and $ \plus{}5$ degrees." Show that it would suffice to say, "Both $ \minus{}3$ degrees and $ \plus{}5$ degrees occurred among last night's minimum temperatures." (Assume that temperature as a two-variable function of place and time is continuous.)
[i]A.Csaszar[/i]
Today's calculation of integrals, 767
For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$
Evaluate $\int_0^1 f(t)dt.$
1990 IMO Longlists, 53
Find the real solution(s) for the system of equations
\[\begin{cases}x^3+y^3 &=1\\x^5+y^5 &=1\end{cases}\]
2012 Kazakhstan National Olympiad, 2
Function $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(xf(y))=yf(x)$ for any $x,y$ are real numbers. Prove that $f(-x) = -f(x)$ for all real numbers $x$.