Found problems: 85335
2025 Nordic, 4
Denote by $S_{n}$ the set of all permutations of the set $\{1,2,\dots, n\}$. Let $\sigma \in S_{n}$ be a permutation. We define the $\textit{displacement}$ of $\sigma$ to be the number $d(\sigma)=\sum_{i=1}^{n} \vert \sigma(i)-i \vert$. We saw that $\sigma$ is $\textit{maximally}$ $\textit{displacing}$ if $d(\sigma)$ is the largest possible, i.e. if $d(\sigma) \geq d({\pi})$, for all $\pi \in S_{n}$.
$\text{a)}$ Suppose $\sigma$ is a maximally displacing permutation of $\{1,2, \dots, 2024\}$. Prove that $\sigma(i)\neq i$, for all $i \in \{1,2, \dots, 2024.\}$
$\text{b)}$ Does the statement of part a) hold for permutations of $\{1,2, \dots, 2025\}$?
2023 Ukraine National Mathematical Olympiad, 11.7
For a positive integer $n$ consider all its divisors $1 = d_1 < d_2 < \ldots < d_k = n$. For $2 \le i \le k-1$, let's call divisor $d_i$ good, if $d_{i-1}d_{i+1}$ isn't divisible by $d_i$. Find all $n$, such that the number of their good divisors is smaller than the number of their prime distinct divisors.
[i]Proposed by Mykhailo Shtandenko[/i]
1987 Traian Lălescu, 1.1
Consider the parabola $ P:x-y^2-(p+3)y-p=0,p\in\mathbb{R}^*. $ Show that $ P $ intersects the coordonate axis at three points, and that the circle formed by these three points passes through a fixed point.
Kyiv City MO Juniors Round2 2010+ geometry, 2017.9.1
Find the angles of the triangle $ABC$, if we know that its center $O$ of the circumscribed circle and the center $I_A$ of the exscribed circle (tangent to $BC$) are symmetric wrt $BC$.
(Bogdan Rublev)
KoMaL A Problems 2018/2019, A. 738
Consider the following sequence: $a_1 = 1$, $a_2 = 2$, $a_3 = 3$, and
\[a_{n+3} = \frac{a_{n+1}^2 + a_{n+2}^2 - 2}{a_n}\]
for all integers $n \ge 1$. Prove that every term of the sequence is a positive integer.
1996 Romania National Olympiad, 1
Prove that a group $G$ in which exactly two elements other than the identity commute with each other is isomorphic to $\mathbb{Z}/3 \mathbb{Z}$ or $S_3.$
1999 Czech And Slovak Olympiad IIIA, 6
Find all pairs of real numbers $a,b$ for which the system of equations $$ \begin{cases} \dfrac{x+y}{x^2 +y^2} = a \\ \\ \dfrac{x^3 +y^3}{x^2 +y^2} = b \end{cases}$$ has a real solution.
2005 National Olympiad First Round, 21
What is the radius of the circle passing through the center of the square $ABCD$ with side length $1$, its corner $A$, and midpoint of its side $[BC]$?
$
\textbf{(A)}\ \dfrac {\sqrt 3}4
\qquad\textbf{(B)}\ \dfrac {\sqrt 5}4
\qquad\textbf{(C)}\ \sqrt 2
\qquad\textbf{(D)}\ \sqrt 3
\qquad\textbf{(E)}\ \dfrac {\sqrt {10}}4
$
2016 PUMaC Number Theory A, 5
Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$.
Let $S$ be the sum of $\frac{gcd(m,n)}{lcm(m,n)}$ over all ordered pairs of positive integers $(m, n)$ where $mn = k$.
If $S$ can be written in simplest form as $\frac{r}{s}$, compute $r + s$.
2003 JHMMC 8, 7
Yao Ming is $7\text{ ft and }5\text{ in}$ tall. His basketball hoop is $10$ feet from the ground. Given that there are
$12$ inches in a foot, how many inches must Yao jump to touch the hoop with his head?
1988 IMO Longlists, 70
$ABC$ is a triangle, with inradius $r$ and circumradius $R.$ Show that: \[ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{B}{2} \right) + \sin \left( \frac{B}{2} \right) \cdot \sin \left( \frac{C}{2} \right) + \sin \left( \frac{C}{2} \right) \cdot \sin \left( \frac{A}{2} \right) \leq \frac{5}{8} + \frac{r}{4 \cdot R}. \]
1986 IMO Longlists, 45
Given $n$ real numbers $a_1 \leq a_2 \leq \cdots \leq a_n$, define
\[M_1=\frac 1n \sum_{i=1}^{n} a_i , \quad M_2=\frac{2}{n(n-1)} \sum_{1 \leq i<j \leq n} a_ia_j, \quad Q=\sqrt{M_1^2-M_2}\]
Prove that
\[a_1 \leq M_1 - Q \leq M_1 + Q \leq a_n\]
and that equality holds if and only if $a_1 = a_2 = \cdots = a_n.$
2023 Sharygin Geometry Olympiad, 9.1
The ratio of the median $AM$ of a triangle $ABC$ to the side $BC$ equals $\sqrt{3}:2$. The points on the sides of $ABC$ dividing these side into $3$ equal parts are marked. Prove that some $4$ of these $6$ points are concyclic.
2012 Kosovo National Mathematical Olympiad, 1
If
$(x^2-x-1)^n=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0$,
where $a_i,i\in\{0,1,2,..,2n\}$, find $a_1+a_3+...+a_{2n-1}$ and $a_0+a_2+a_4+...+a_{2n}$.
2017 JBMO Shortlist, C1
Consider a regular $2n + 1$-gon $P$ in the plane, where n is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$, at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$, we consider them colorless). Find the largest positive integer for which such a coloring is possible.
2018 Moldova EGMO TST, 6
Let $ x,y\in\mathbb{R}$ , and $ x,y \in $ $ \left(0,\frac{\pi}{2}\right) $, and $ m \in \left(2,+\infty\right) $ such that $ \tan x * \tan y = m $ . Find the minimum value of the expression $ E(x,y) = \cos x + \cos y $.
2015 ISI Entrance Examination, 4
Let $p(x) = x^7 +x^6 + b_5 x^5 + \cdots +b_0 $ and $q(x) = x^5 + c_4 x^4 + \cdots +c_0$ . If $p(i)=q(i)$ for $i=1,2,3,\cdots,6$ . Show that there exists a negative integer r such that $p(r)=q(r)$ .
TNO 2023 Junior, 6
Show that for every integer $n \geq 1$, it is possible to express $5^n$ as the sum of two nonzero squares.
2014 Math Prize For Girls Problems, 14
A triangle has area 114 and sides of integer length. What is the perimeter of the triangle?
2006 Dutch Mathematical Olympiad, 4
Given is triangle $ABC$ with an inscribed circle with center $M$ and radius $r$.
The tangent to this circle parallel to $BC$ intersects $AC$ in $D$ and $AB$ in $E$.
The tangent to this circle parallel to $AC$ intersects $AB$ in $F$ and $BC$ in $G$.
The tangent to this circle parallel to $AB$ intersects $BC$ in $H$ and $AC$ in $K$.
Name the centers of the inscribed circles of triangle $AED$, triangle $FBG$ and triangle $KHC$ successively $M_A, M_B, M_C$ and the rays successively $r_A, r_B$ and $r_C$.
Prove that $r_A + r_B + r_C = r$.
2010 Indonesia MO, 2
Given an acute triangle $ABC$ with $AC>BC$ and the circumcenter of triangle $ABC$ is $O$. The altitude of triangle $ABC$ from $C$ intersects $AB$ and the circumcircle at $D$ and $E$, respectively. A line which passed through $O$ which is parallel to $AB$ intersects $AC$ at $F$. Show that the line $CO$, the line which passed through $F$ and perpendicular to $AC$, and the line which passed through $E$ and parallel with $DO$ are concurrent.
[i]Fajar Yuliawan, Bandung[/i]
2012 Romanian Masters In Mathematics, 5
Given a positive integer $n\ge 3$, colour each cell of an $n\times n$ square array with one of $\lfloor (n+2)^2/3\rfloor$ colours, each colour being used at least once. Prove that there is some $1\times 3$ or $3\times 1$ rectangular subarray whose three cells are coloured with three different colours.
[i](Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov[/i]
1959 IMO, 4
Construct a right triangle with given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.
2016 Baltic Way, 12
Does there exist a hexagon (not necessarily convex) with side lengths $1, 2, 3, 4, 5, 6$ (not necessarily in this order) that can be tiled with a) $31$ b) $32$ equilateral triangles with side length $1?$
2016 Junior Balkan Team Selection Tests - Romania, 5
Let n$\ge$2.Each 1x1 square of a nxn board is colored in black or white such that every black square has at least 3 white neighbors(Two squares are neighbors if they have a common side).What is the maximum number of black squares?