Found problems: 85335
2019 Oral Moscow Geometry Olympiad, 2
On the side $AC$ of the triangle $ABC$ in the external side is constructed the parallelogram $ACDE$ . Let $O$ be the intersection point of its diagonals, $N$ and $K$ be midpoints of BC and BA respectively. Prove that lines $DK, EN$ and $BO$ intersect at one point.
2014 Contests, Problem 3
Given $n\geq2$, let $\mathcal{A}$ be a family of subsets of the set $\{1,2,\dots,n\}$ such that, for any $A_1,A_2,A_3,A_4 \in \mathcal{A}$, it holds that $|A_1 \cup A_2 \cup A_3 \cup A_4| \leq n -2$.
Prove that $|\mathcal{A}| \leq 2^{n-2}.$
2000 May Olympiad, 4
There are pieces in the shape of an equilateral triangle with sides $1, 2, 3, 4, 5$ and $6$ ($50$ pieces of each size). You want to build an equilateral triangle of side $7$ using some of these pieces, without gaps or overlaps. What is the least number of pieces needed?
1973 AMC 12/AHSME, 16
If the sum of all the angles except one of a convex polygon is $ 2190^{\circ}$, then the number of sides of the polygon must be
$ \textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 15 \qquad
\textbf{(C)}\ 17 \qquad
\textbf{(D)}\ 19 \qquad
\textbf{(E)}\ 21$
2019 Danube Mathematical Competition, 2
Find all nondecreasing functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation
$$ f\left( f\left( x^2 \right) +y+f(y) \right) =x^2+2f(y) , $$
for any real numbers $ x,y. $
2020 Ukrainian Geometry Olympiad - April, 2
Inside the triangle $ABC$ is point $P$, such that $BP > AP$ and $BP > CP$. Prove that $\angle ABC$ is acute.
Ukrainian From Tasks to Tasks - geometry, 2014.9
On a circle with diameter $AB$ we marked an arbitrary point $C$, which does not coincide with $A$ and $B$. The tangent to the circle at point $A$ intersects the line $BC$ at point $D$. Prove that the tangent to the circle at point $C$ bisects the segment $AD$.
2006 Germany Team Selection Test, 2
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]
2012 IMO Shortlist, N8
Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.
2014 All-Russian Olympiad, 4
In a country of $n$ cities, an express train runs both ways between any two cities. For any train, ticket prices either direction are equal, but for any different routes these prices are different. Prove that the traveler can select the starting city, leave it and go on, successively, $n-1$ trains, such that each fare is smaller than that of the previous fare. (A traveler can enter the same city several times.)
1991 Czech And Slovak Olympiad IIIA, 1
Prove that for any real numbers $p,q,r,\phi$,:
$$\cos^2\phi+q \sin \phi \cos \phi +r\sin^2 \phi \ge \frac12 (p+r-\sqrt{(p-r)^2+q^2})$$
2015 Iran Team Selection Test, 3
$a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ are $2n$ positive real numbers such that $a_1,a_2,\cdots ,a_n$ aren't all equal. And assume that we can divide $a_1,a_2,\cdots ,a_n$ into two subsets with equal sums.similarly $b_1,b_2,\cdots ,b_n$ have these two conditions. Prove that there exist a simple $2n$-gon with sides $a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ and parallel to coordinate axises Such that the lengths of horizontal sides are among $a_1,a_2,\cdots ,a_n$ and the lengths of vertical sides are among $b_1,b_2,\cdots ,b_n$.(simple polygon is a polygon such that it doesn't intersect itself)
2004 Putnam, B1
Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers
$c_nr, c_nr^2+c_{n-1}r, c_nr^3+c_{n-1}r^2+c_{n-1}r, \dots, c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$
are all integers.
2010 China Western Mathematical Olympiad, 4
Let $a_1,a_2,..,a_n,b_1,b_2,...,b_n$ be non-negative numbers satisfying the following conditions simultaneously:
(1) $\displaystyle\sum_{i=1}^{n} (a_i + b_i) = 1$;
(2) $\displaystyle\sum_{i=1}^{n} i(a_i - b_i) = 0$;
(3) $\displaystyle\sum_{i=1}^{n} i^2(a_i + b_i) = 10$.
Prove that $\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}$ for all $1 \le k \le n$.
2022 China Team Selection Test, 6
Let $m$ be a positive integer, and $A_1, A_2, \ldots, A_m$ (not necessarily different) be $m$ subsets of a finite set $A$. It is known that for any nonempty subset $I$ of $\{1, 2 \ldots, m \}$,
\[ \Big| \bigcup_{i \in I} A_i \Big| \ge |I|+1. \]
Show that the elements of $A$ can be colored black and white, so that each of $A_1,A_2,\ldots,A_m$ contains both black and white elements.
2002 AMC 10, 1
The ratio $ \frac{2^{2001}\cdot3^{2003}}{6^{2002}}$ is
$ \textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{2}{3} \qquad
\textbf{(E)}\ \frac{3}{2}$
2024 Putnam, B4
Let $n$ be a positive integer. Set $a_{n,0}=1$. For $k\geq 0$, choose an integer $m_{n,k}$ uniformly at random from the set $\{1,\,\ldots,\,n\}$, and let
\[
a_{n,k+1}=
\begin{cases}
a_{n,k}+1, & \text{if $m_{n,k}>a_{n,k}$;}\\
a_{n,k}, & \text{if $m_{n,k}=a_{n,k}$;}\\
a_{n,k}-1, & \text{if $m_{n,k}<a_{n,k}$.}
\end{cases}
\]
Let $E(n)$ be the expected value of $a_{n,n}$. Determine $\lim_{n\to\infty}E(n)/n$.
2005 Junior Balkan MO, 2
Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$.
Prove that the lines $AP$ and $CS$ are parallel.
2025 Poland - Second Round, 4
Let $n\ge 2$ be an integer. Consider a $2n+1\times 2n+1$ board. All cells lying both in an even row and an even column have been removed. The remaining cells form a [i]labyrinth[/i]. An ant takes a walk in the labyrinth. A single step of the ant consists of moving to a neighbouring cell. Determine, in terms of $n$, the smallest possible number of steps so that every cell of the labirynth is visited by the ant. The ant chooses the start cell. The start cell and the end cell are considered visited. Each cell could be visited several times.
The picture depicts the labyrinth for $n=3$ and possible steps of the ant in its four locations.
1950 AMC 12/AHSME, 12
As the number of sides of a polygon increases from $3$ to $ n$, the sum of the exterior formed by extending each side in succession:
$\textbf{(A)}\ \text{Increases} \qquad
\textbf{(B)}\ \text{Decreases} \qquad
\textbf{(C)}\ \text{Remains constant} \qquad
\textbf{(D)}\ \text{Cannot be predicted} \qquad\\
\textbf{(E)}\ \text{Becomes }(n-3)\text{ straight angles}$
2018 Bosnia And Herzegovina - Regional Olympiad, 5
It is given $2018$ points in plane. Prove that it is possible to cover them with circles such that:
$i)$ sum of lengths of all diameters of all circles is not greater than $2018$
$ii)$ distance between any two circles is greater than $1$
2000 Poland - Second Round, 5
Decide whether exists function $f: \mathbb{N} \rightarrow \mathbb{N}$, such that for each $n \in \mathbb{N}$ is $f(f(n) )= 2n$.
2005 Postal Coaching, 23
Let $\Gamma$ be the incircle of an equilateral triangle $ABC$ of side length $2$ units.
(a) Show that for all points $P$ on $\Gamma$, $PA^2 +PB^2 +PC^2 = 5$.
(b) Show that for all points $P$ on $\Gamma$, it is possible to construct a triangle of sides equal to $PA,PB,PC$ and whose area is equal to $\frac{\sqrt{3}}{4}$ units.
2012 Online Math Open Problems, 12
Let $a_1,a_2,\ldots$ be a sequence defined by $a_1 = 1$ and for $n\ge1$, $a_{n+1} = \sqrt{a_n^2 -2a_n + 3} + 1$. Find $a_{513}$.
[i]Ray Li.[/i]
2007 Purple Comet Problems, 3
Square $ABCD$ has side length $36$. Point $E$ is on side $AB$ a distance $12$ from $B$, point $F$ is the midpoint of side $BC$, and point $G$ is on side $CD$ a distance $12$ from $C$. Find the area of the region that lies inside triangle $EFG$ and outside triangle $AFD$.