Found problems: 85335
STEMS 2021 CS Cat B, Q1
We are given $k$ colors and we have to assign a single color to every vertex. An edge is [u][b]satisfied[/b][/u] if the vertices on that edge, are of different colors.
[list]
[*]Prove that you can always find an algorithm which assigns colors to vertices so that at least $\frac{k - 1}{k}|E|$ edges are satisfied where \(|E|\) is the cardinality of the edges in the graph.[/*]
[*]Prove that there is a poly time deterministic algorithm for this [/*]
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1972 Yugoslav Team Selection Test, Problem 1
Given non-zero real numbers $u,v,w,x,y,z$, how many different possibilities are there for the signs of these numbers if
$$(u+ix)(v+iy)(w+iz)=i?$$
2019 Federal Competition For Advanced Students, P1, 4
Find all pairs $(a, b)$ of real numbers such that $a \cdot \lfloor b \cdot n\rfloor = b \cdot \lfloor a \cdot n \rfloor$ applies to all positive integers$ n$.
(For a real number $x, \lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.)
1995 Poland - Second Round, 2
Let $ABCDEF$ be a convex hexagon with $AB = BC, CD = DE$ and $EF = FA$.
Prove that the lines through $C,E,A$ perpendicular to $BD,DF,FB$ are concurrent.
2015 BMT Spring, 4
Determine the greatest integer $N$ such that $N$ is a divisor of $n^{13}-n$ for all integers $n$.
2017 AMC 12/AHSME, 18
The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and the line $ED$ is perpendicular to the line $AD$. Segment $\overline{AE}$ intersects the circle at point $C$ between $A$ and $E$. What is the area of $\triangle ABC$?
$\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$
2007 Sharygin Geometry Olympiad, 12
A rectangle $ABCD$ and a point $P$ are given. Lines passing through $A$ and $B$ and perpendicular to $PC$ and $PD$ respectively, meet at a point $Q$. Prove that $PQ \perp AB$.
1992 Miklós Schweitzer, 5
Prove that if the $a_i$'s are different natural numbers, then $\sum_ {j = 1}^n a_j ^ 2 \prod_{k \neq j} \frac{a_j + a_k}{a_j-a_k}$ is a square number.
2012 Indonesia TST, 4
The Fibonacci sequence $\{F_n\}$ is defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$. Determine all triplets of positive integers $(k,m,n)$ such that $F_n = F_m^k$.
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 $.