Found problems: 85335
Estonia Open Junior - geometry, 2005.2.3
The vertices of the square $ABCD$ are the centers of four circles, all of which pass through the center of the square. Prove that the intersections of the circles on the square $ABCD$ sides are vertices of a regular octagon.
1976 AMC 12/AHSME, 13
If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk?
$\textbf{(A) }\frac{x(x+2)(x+5)}{(x+1)(x+3)}\qquad\textbf{(B) }\frac{x(x+1)(x+5)}{(x+2)(x+3)}\qquad$
$\textbf{(C) }\frac{(x+1)(x+3)(x+5)}{x(x+2)}\qquad\textbf{(D) }\frac{(x+1)(x+3)}{x(x+2)(x+5)}\qquad \textbf{(E) }\text{none of these}$
2020 Caucasus Mathematical Olympiad, 1
By one magic nut, Wicked Witch can either turn a flea into a beetle or a spider into a bug; while by one magic acorn, she can either turn a flea into a spider or a beetle into a bug. In the evening Wicked Witch had spent 20 magic nuts and 23 magic acorns. By these actions, the number of beetles increased by 5. Determine what was the change in the number of spiders. (Find all possible answers and prove that the other answers are impossible.)
2019-2020 Winter SDPC, 6
Fix a positive integer $n$. Let $a_1, a_2, \ldots$ be a sequence of positive integers such that for all $1 \leq j \leq n$, $a_j=j$, and for all $j>n$, $a_j$ is the largest value of $\min(a_i,a_{j-i})$ among $i=1,2, \ldots j-1$. For example, if $n=3$, we have $a_1=1$, $a_2=2$, $a_3=3$, and $a_4=2$ since $\min(a_1,a_3)=1$, $\min(a_2,a_2)=2$, and $\min(a_3,a_1)=1$. We will determine the values of $a_k$ for sufficiently large $k$.
(a) Show that $a_i \in \{1,2,3, \ldots n\}$ for all $i$.
(b) Show that if $a_x \geq n-1$ and $a_y \geq n-1$, $a_{x+y} \geq n-1$.
(c) Show that for some positive integer $N$, $a_k \in \{n-1,n\}$ for all $k \geq N$.
(d) Show that $a_k = n$ if and only if $n \mid k$.
1918 Eotvos Mathematical Competition, 2
Find three distinct natural numbers such that the sum of their reciprocals is an integer.
2004 All-Russian Olympiad, 2
A country has 1001 cities, and each two cities are connected by a one-way street. From each city exactly 500 roads begin, and in each city 500 roads end. Now an independent republic splits itself off the country, which contains 668 of the 1001 cities. Prove that one can reach every other city of the republic from each city of this republic without being forced to leave the republic.
2019 LIMIT Category A, Problem 4
From a point $P$ outside of a circle with centre $O$, tangent segments $\overline{PA}$ and $\overline{PB}$ are drawn. If $\frac1{\left|\overline{OA}\right|^2}+\frac1{\left|\overline{PA}\right|^2}=\frac1{16}$, then $\left|\overline{AB}\right|=$?
$\textbf{(A)}~4$
$\textbf{(B)}~6$
$\textbf{(C)}~8$
$\textbf{(D)}~10$
MIPT student olimpiad spring 2022, 3
Prove that for any two linear subspaces $V, W \subset R^n$ the same
dimension there is an orthogonal transformation $A:R^n\to R^n$, such that $A(V )=W$ and $A(W) = V$
2015 Sharygin Geometry Olympiad, 8
A perpendicular bisector of side $BC$ of triangle $ABC$ meets lines $AB$ and $AC$ at points $A_B$ and $A_C$ respectively. Let $O_a$ be the circumcenter of triangle $AA_BA_C$. Points $O_b$ and $O_c$ are defined similarly. Prove that the circumcircle of triangle $O_aO_bO_c$ touches the circumcircle of the original triangle.
2016 NIMO Summer Contest, 13
The area of the region in the $xy$-plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is $k\pi$, for some integer $k$. Find $k$.
[i]Proposed by Michael Tang[/i]
2014 239 Open Mathematical Olympiad, 3
A natural number is called [i]good[/i] if it can be represented as sum of two coprime natural numbers, the first of which decomposes into odd number of primes (not necceserily distinct) and the second to even. Prove that there exist infinity many $n$ with $n^4$ being good.
1984 AIME Problems, 15
Determine $w^2+x^2+y^2+z^2$ if
\[ \begin{array}{l} \displaystyle \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 \\ \displaystyle \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 \\ \displaystyle \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 \\ \displaystyle \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 \\ \end{array} \]
2016 Middle European Mathematical Olympiad, 3
Let $ABC$ be an acute triangle such that $\angle BAC > 45^{\circ}$ with circumcenter $O$. A point $P$ is chosen inside triangle $ABC$ such that $A, P, O, B$ are concyclic and the line $BP$ is perpendicular to the line $CP$. A point $Q$ lies on the segment $BP$ such that the line $AQ$ is parallel to the line $PO$.
Prove that $\angle QCB = \angle PCO$.
2005 India Regional Mathematical Olympiad, 4
Find the number of 5-digit numbers that each contains the block '15' and is divisible by 15.
1998 IMC, 5
$S$ is a family of balls in $\mathbb{R}^{n}$ ($n > 1$) such that the intersection of any two contains at most one point. Show that the set of points belonging to at least two members of $S$ is countable.
2016 Junior Balkan Team Selection Test, 4
Let $a,b,c\in \mathbb{R}^+$, prove that: $$\frac{2a}{\sqrt{3a+b}}+\frac{2b}{\sqrt{3b+c}}+\frac{2c}{\sqrt{3c+a}}\leq \sqrt{3(a+b+c)}$$
2018 Junior Regional Olympiad - FBH, 1
Price of some item has decreased by $5\%$. Then price increased by $40\%$ and now it is $1352.06\$$ cheaper than doubled original price. How much did the item originally cost?
2006 Alexandru Myller, 1
For an odd prime $ p, $ show that $ \sum_{k=1}^{p-1} \frac{k^p-k}{p}\equiv \frac{1+p}{2}\pmod p . $
2024 Israel TST, P3
For a set $S$ of at least $3$ points in the plane, let $d_{\text{min}}$ denote the minimal distance between two different points in $S$ and $d_{\text{max}}$ the maximal distance between two different points in $S$.
For a real $c>0$, a set $S$ will be called $c$-[i]balanced[/i] if
\[\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S|\]
Prove that there exists a real $c>0$ so that for every $c$-balanced set of points $S$, there exists a triangle with vertices in $S$ that contains at least $\sqrt{|S|}$ elements of $S$ in its interior or on its boundary.
2015 Stars Of Mathematics, 2
Prove that there exist an infinite number of odd natural numbers $m_1<m_2<...$ and an infinity of natural numbers $n_1<n_2<...$ ,such that $(m_k,n_k)=1$ and $m_k^4-2n_k^4$ is a perfect square,for all $k\in\mathbb{N}$.
2012 Belarus Team Selection Test, 2
Determine the greatest possible value of n that satisfies the following condition:
for any choice of $n$ subsets $M_1, ...,M_n$ of the set $M = \{1,2,...,n\}$ satisfying the conditions
i) $i \in M_i$ and
ii) $i \in M_j \Leftrightarrow j \notin M_i$ for all $i \ne j$,
there exist $M_k$ and $M_l$ such that $M_k \cup M_l = M$.
(Moscow regional olympiad,adopted)
1949 Moscow Mathematical Olympiad, 162
Given a set of $4n$ positive numbers such that any distinct choice of ordered foursomes of these numbers constitutes a geometric progression. Prove that at least $4$ numbers of the set are identical.
2013 Irish Math Olympiad, 10
Let $a,b,c $ be real numbers and let $x=a+b+c,y=a^2+b^2+c^2,z=a^3+b^3+c^3$
and $S=2x^3-9xy+9z .$
$(a)$ Prove that $S$ is unchanged when $a,b,c$ are replaced by $a+t,b+t,c+t $ , respectively , for any real number $t$.
$(b)$ Prove that $ (3y-x^2)^3\ge S^2 .$
1911 Eotvos Mathematical Competition, 2
Let $Q$ be any point on a circle and let $P_1P_2P_3...P_8$ be a regular inscribed octagon. Prove that the sum of the fourth powers of the distances from $Q$ to the diameters $P_1P_5$, $P_2P_6$, $P_3P_7$, $P_4P_8$ is independent of the position of $Q$.
May Olympiad L2 - geometry, 2003.4
Bob plotted $2003$ green points on the plane, so all triangles with three green vertices have area less than $1$.
Prove that the $2003$ green points are contained in a triangle $T$ of area less than $4$.