Found problems: 85335
1996 North Macedonia National Olympiad, 3
Prove that if $\alpha, \beta, \gamma$ are angles of a triangle, then $\frac{1}{\sin \alpha}+ \frac{1}{\sin \beta} \ge \frac{8}{ 3+2 \ cos\gamma}$
.
1998 Switzerland Team Selection Test, 9
If $x$ and $y$ are positive numbers, prove the inequality $\frac{x}{x^4 +y^2 }+\frac{y}{x^2 +y^4} \le \frac{1}{xy}$
.
2022 Germany Team Selection Test, 1
Given a triangle $ABC$ and three circles $x$, $y$ and $z$ such that $A \in y \cap z$, $B \in z \cap x$ and $C \in x \cap y$.
The circle $x$ intersects the line $AC$ at the points $X_b$ and $C$, and intersects the line $AB$ at the points $X_c$ and $B$.
The circle $y$ intersects the line $BA$ at the points $Y_c$ and $A$, and intersects the line $BC$ at the points $Y_a$ and $C$.
The circle $z$ intersects the line $CB$ at the points $Z_a$ and $B$, and intersects the line $CA$ at the points $Z_b$ and $A$.
(Yes, these definitions have the symmetries you would expect.)
Prove that the perpendicular bisectors of the segments $Y_a Z_a$, $Z_b X_b$ and $X_c Y_c$ concur.
2009 Federal Competition For Advanced Students, P2, 1
If $x,y,K,m \in N$, let us define:
$a_m= \underset{k \, twos}{2^{2^{,,,{^{2}}}}}$, $A_{km} (x)= \underset{k \, twos}{ 2^{2^{,,,^{x^{a_m}}}}}$, $B_k(y)= \underset{m \, fours}{4^{4^{4^{,,,^{4^y}}}}}$,
Determine all pairs $(x,y)$ of non-negative integers, dependent on $k>0$, such that $A_{km} (x)=B_k(y)$
2021 Argentina National Olympiad, 6
Milly chooses a positive integer $n$ and then Uriel colors each integer between $1$ and $n$ inclusive red or blue. Then Milly chooses four numbers $a, b, c, d$ of the same color (there may be repeated numbers). If $a+b+c= d$ then Milly wins. Determine the smallest $n$ Milly can choose to ensure victory, no matter how Uriel colors.
2022 Durer Math Competition (First Round), 4
We want to partition the integers $1, 2, 3, . . . , 100$ into several groups such that within each group either any two numbers are coprime or any two are not coprime. At least how many groups are needed for such a partition?
[i]We call two integers coprime if they have no common divisor greater than $1$.[/i]
2024 Mexican Girls' Contest, 5
Consider the acute-angled triangle \(ABC\). The segment \(BC\) measures 40 units. Let \(H\) be the orthocenter of triangle \(ABC\) and \(O\) its circumcenter. Let \(D\) be the foot of the altitude from \(A\) and \(E\) the foot of the altitude from \(B\). Additionally, point \(D\) divides the segment \(BC\) such that \(\frac{BD}{DC} = \frac{3}{5}\). If the perpendicular bisector of segment \(AC\) passes through point \(D\), calculate the area of quadrilateral \(DHEO\).
2019 Taiwan APMO Preliminary Test, P6
Let $\mathbb{N}$ denote the set of all positive integers.Function $f:\mathbb{N}\cup{0}\rightarrow\mathbb{N}\cup{0}$ satisfies :for any two distinct positive integer $a,b$, we have $$f(a)+f(b)-f(a+b)=2019$$
(1)Find $f(0)$
(2)Let $a_1,a_2,...,a_{100}$ be 100 positive integers (they are pairwise distinct), find $f(a_1)+f(a_2)+...+f(a_{100})-f(a_1+a_2+...+a_{100})$
1965 AMC 12/AHSME, 10
The statement $ x^2 \minus{} x \minus{} 6 < 0$ is equivalent to the statement:
$ \textbf{(A)}\ \minus{} 2 < x < 3 \qquad \textbf{(B)}\ x > \minus{} 2 \qquad \textbf{(C)}\ x < 3$
$ \textbf{(D)}\ x > 3 \text{ and }x < \minus{} 2 \qquad \textbf{(E)}\ x > 3 \text{ and }x < \minus{} 2$
2002 Portugal MO, 5
Consider the three squares indicated in the figure. Show that if the lengths of the sides of the smaller square and the square greater are integers, then adding to the area of the smallest square the area of the inclined square, a perfect square is obtained.
[img]https://1.bp.blogspot.com/-B0QdvZIjOLw/X4URvs3C0ZI/AAAAAAAAMmw/S5zMpPBXBn8Jj39d-OZVtMRUDn3tXbyWgCLcBGAsYHQ/s0/2002%2Bportugal%2Bp5.png[/img]
2019 Greece Team Selection Test, 1
Given an equilateral triangle with sidelength $k$ cm. With lines parallel to it's sides, we split it into $k^2$ small equilateral triangles with sidelength $1$ cm. This way, a triangular grid is created. In every small triangle of sidelength $1$ cm, we place exactly one integer from $1$ to $k^2$ (included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths $1$ cm. We shall name as [i]value [/i] of the hexagon, the sum of the numbers that lie on the $6$ small equilateral triangles that the hexagon consists of . Find (in terms of the integer $k>4$) the maximum and the minimum value of the sum of the values of all hexagons .
2006 Victor Vâlcovici, 1
Prove that for any real numbers $ a,b,c, $ the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as
$$ f(x)=\sqrt{(x-c)^2+b^2} +\sqrt{(x+c)^2+b^2} $$ is decreasing on $ (-\infty ,0] $ and increasing on $ [0,\infty ) . $
2021 Dutch IMO TST, 3
Let $ABC$ be an acute-angled and non-isosceles triangle with orthocenter $H$. Let $O$ be the center of the circumscribed circle of triangle $ABC$ and let $K$ be center of the circumscribed circle of triangle $AHO$. Prove that the reflection of $K$ wrt $OH$ lies on $BC$.
2005 Iran Team Selection Test, 2
Suppose there are $n$ distinct points on plane. There is circle with radius $r$ and center $O$ on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move $O$ to the baricenter of the point in the circle. Prove that location of $O$ is constant after some steps.
2024 AMC 8 -, 18
Three concentric circles centered at $O$ have radii of $1$, $2$, and $3$. Points $B$ and $C$ lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle $BOC$, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of $\angle{BOC}$ in degrees?
[asy]
size(100);
import graph;
draw(circle((0,0),3));
real radius = 3;
real angleStart = -54; // starting angle of the sector
real angleEnd = 54; // ending angle of the sector
label("$O$",(0,0),W);
pair O = (0, 0);
filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, lightgray);
filldraw(circle((0,0),2),lightgray);
filldraw(circle((0,0),1),white);
draw((1.763,2.427)--(0,0)--(1.763,-2.427));
label("$B$",(1.763,2.427),NE);
label("$C$",(1.763,-2.427),SE);
[/asy]
$\textbf{(A)}\ 108 \qquad \textbf{(B)}\ 120 \qquad \textbf{(C)}\ 135 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 150$
Cono Sur Shortlist - geometry, 2021.G4
Let $ABC$ be a triangle and $\Gamma$ the $A$- exscribed circle whose center is $J$ . Let $D$ and $E$ be the touchpoints of $\Gamma$ with the lines $AB$ and $AC$, respectively. Let $S$ be the area of the quadrilateral $ADJE$, Find the maximum value that $\frac{S}{AJ^2}$ has and when equality holds.
2008 Bosnia And Herzegovina - Regional Olympiad, 4
Determine is there a function $a: \mathbb{N} \rightarrow \mathbb{N}$ such that:
$i)$ $a(0)=0$
$ii)$ $a(n)=n-a(a(n))$, $\forall n \in$ $ \mathbb{N}$.
If exists prove:
$a)$ $a(k)\geq a(k-1)$
$b)$ Does not exist positive integer $k$ such that $a(k-1)=a(k)=a(k+1)$.
2001 China Team Selection Test, 1
$E$ and $F$ are interior points of convex quadrilateral $ABCD$ such that $AE = BE$, $CE = DE$, $\angle AEB = \angle CED$, $AF = DF$, $BF = CF$, $\angle AFD = \angle BFC$. Prove that $\angle AFD + \angle AEB = \pi$.
1980 IMO, 1
Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides
\[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\]
are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.
2024 Dutch BxMO/EGMO TST, IMO TSTST, 5
In a triangle $ABC ~(\overline{AB} < \overline{AC})$, points $D (\neq A, B)$ and $E (\neq A, C)$ lies on side $AB$ and $AC$ respectively. Point $P$ satisfies $\overline{PB}=\overline{PD}, \overline{PC}=\overline{PE}$. $X (\neq A, C)$ is on the arc $AC$ of the circumcircle of triangle $ABC$ not including $B$. Let $Y (\neq A)$ be the intersection of circumcircle of triangle $ADE$ and line $XA$. Prove that $\overline{PX} = \overline{PY}$.
2018 Portugal MO, 3
How many ways are there to paint an $m \times n$ board, so that each square is painted blue, white, brown or gold, and in each $2 \times 2$ square there is one square of each color?
2002 Polish MO Finals, 1
Find all the natural numbers $a,b,c$ such that:
1) $a^2+1$ and $b^2+1$ are primes
2) $(a^2+1)(b^2+1)=(c^2+1)$
2022 Rioplatense Mathematical Olympiad, 3
Let $n$ be a positive integer. Given a sequence of nonnegative real numbers $x_1,\ldots ,x_n$ we define the [i]transformed sequence[/i] $y_1,\ldots ,y_n$ as follows: the number $y_i$ is the greatest possible value of the average of consecutive terms of the sequence that contain $x_i$. For example, the transformed sequence of $2,4,1,4,1$ is $3,4,3,4,5/2$.
Prove that
a) For every positive real number $t$, the number of $y_i$ such that $y_i>t$ is less than or equal to $\frac{2}{t}(x_1+\cdots +x_n)$.
b) The inequality $\frac{y_1+\cdots +y_n}{32n}\leq \sqrt{\frac{x_1^2+\cdots +x_n^2}{32n}}$ holds.
1967 IMO Longlists, 9
Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$
2020 Iranian Our MO, 1
Find the maximum number of cells that can be coloured from a $4\times 3000$ board such that no tetromino is formed.
[i]Proposed by Arian Zamani, Matin Yousefi[/i] [b]Rated 5[/b]