Found problems: 85335
II Soros Olympiad 1995 - 96 (Russia), 9.6
Without using a calculator (especially a computer), find out what is more:
$$\sqrt[3]{5\sqrt{13}+18}- \sqrt[3]{2\sqrt{13}+5} \,\,\, or \,\,\, 1 $$
MathLinks Contest 7th, 5.2
Let $ A^{\prime}$ be an arbitrary point on the side $ BC$ of a triangle $ ABC$. Denote by $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ the circles simultanously tangent to $ AA^{\prime}$, $ A^{\prime}B$, $ \Gamma$ and $ AA^{\prime}$, $ A^{\prime}C$, $ \Gamma$, respectively, where $ \Gamma$ is the circumcircle of $ ABC$. Prove that $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ are congruent if and only if $ AA^{\prime}$ passes through the Nagel point of triangle $ ABC$.
([i]If $ M,N,P$ are the points of tangency of the excircles of the triangle $ ABC$ with the sides of the triangle $ BC$, $ CA$ and $ AB$ respectively, then the Nagel point of the triangle is the intersection point of the lines $ AM$, $ BN$ and $ CP$[/i].)
2020 Iranian Combinatorics Olympiad, 6
Consider a triangular grid of equilateral triangles with unit sides. Assume that $\mathcal{P}$ is a non-self-intersecting polygon with perimeter 1399 and sides from the grid. Prove that $\mathcal{P}$ has either an internal or an external 120-degree angle.
[i]Proposed by Seyed Hessam Firouzi[/i]
2011 Dutch IMO TST, 5
Find all triples $(a, b, c)$ of positive integers with $a+b+c = 10$ such that there are $a$ red, $b$ blue and $c$ green points (all different) in the plane satisfying the following properties:
$\bullet$ for each red point and each blue point we consider the distance between these two points, the sum of these distances is $37$,
$\bullet$ for each green point and each red point we consider the distance between these two points, the sum of these distances is $30$,
$\bullet$ for each blue point and each green point we consider the distance between these two points, the sum of these distances is $1$.
2020 Centroamerican and Caribbean Math Olympiad, 2
Suppose you have identical coins distributed in several piles with one or more coins in each pile. An action consists of taking two piles, which have an even total of coins among them, and redistribute their coins in two piles so that they end up with the same number of coins.
A distribution is [i]levelable[/i] if it is possible, by means of 0 or more operations, to end up with all the piles having the same number of coins.
Determine all positive integers $n$ such that, for all positive integers $k$, any distribution of $nk$ coins in $n$ piles is levelable.
2014 Online Math Open Problems, 13
Suppose that $g$ and $h$ are polynomials of degree $10$ with integer coefficients such that $g(2) < h(2)$ and
\[ g(x) h(x)
= \sum_{k=0}^{10} \left( \binom{k+11}{k} x^{20-k} - \binom{21-k}{11} x^{k-1} + \binom{21}{11}x^{k-1} \right) \]
holds for all nonzero real numbers $x$. Find $g(2)$.
[i]Proposed by Yang Liu[/i]
2014 Contests, 1
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that
\[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]
2019 Serbia Team Selection Test, P6
A [i]figuric [/i] is a convex polyhedron with $26^{5^{2019}}$ faces. On every face of a figuric we write down a number. When we throw two figurics (who don't necessarily have the same set of numbers on their sides) into the air, the figuric which falls on a side with the greater number wins; if this number is equal for both figurics, we repeat this process until we obtain a winner. Assume that a figuric has an equal probability of falling on any face. We say that one figuric rules over another if when throwing these figurics into the air, it has a strictly greater probability to win than the other figuric (it can be possible that given two figurics, no figuric rules over the other).
Milisav and Milojka both have a blank figuric. Milisav writes some (not necessarily distinct) positive integers on the faces of his figuric so that they sum up to $27^{5^{2019}}$. After this, Milojka also writes positive integers on the faces of her figuric so that they sum up to $27^{5^{2019}}$. Is it always possible for Milojka to create a figuric that rules over Milisav's?
[i]Proposed by Bojan Basic[/i]
2004 IberoAmerican, 2
Given a scalene triangle $ ABC$. Let $ A'$, $ B'$, $ C'$ be the points where the internal bisectors of the angles $ CAB$, $ ABC$, $ BCA$ meet the sides $ BC$, $ CA$, $ AB$, respectively. Let the line $ BC$ meet the perpendicular bisector of $ AA'$ at $ A''$. Let the line $ CA$ meet the perpendicular bisector of $ BB'$ at $ B'$. Let the line $ AB$ meet the perpendicular bisector of $ CC'$ at $ C''$. Prove that $ A''$, $ B''$ and $ C''$ are collinear.
2006 Iran Team Selection Test, 6
Let $G$ be a tournoment such that it's edges are colored either red or blue.
Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.
2018 Math Prize for Girls Olympiad, 4
For all integers $x$ and $y$, let $a_{x, y}$ be a real number. Suppose that $a_{0, 0} = 0$. Suppose that only a finite number of the $a_{x, y}$ are nonzero. Prove that
\[
\sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x,y} ( a_{x, 2x + y} + a_{x + 2y, y} )
\le \sqrt{3} \sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x, y}^2 \, .
\]
2001 Hong kong National Olympiad, 4
There are $212$ points inside or on a given unit circle. Prove that there are at least $2001$ pairs of points having distances at most $1$.
2021 Dutch Mathematical Olympiad, 2
We consider sports tournaments with $n \ge 4$ participating teams and where every pair of teams plays against one another at most one time. We call such a tournament [i]balanced [/i] if any four participating teams play exactly three matches between themselves. So, not all teams play against one another.
Determine the largest value of $n$ for which a balanced tournament with $n$ teams exists.
2015 China Team Selection Test, 6
There are some players in a Ping Pong tournament, where every $2$ players play with each other at most once. Given:
\\(1) Each player wins at least $a$ players, and loses to at least $b$ players. ($a,b\geq 1$)
\\(2) For any two players $A,B$, there exist some players $P_1,...,P_k$ ($k\geq 2$) (where $P_1=A$,$P_k=B$), such that $P_i$ wins $P_{i+1}$ ($i=1,2...,k-1$).
\\Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$, such that $Q_i$ wins $Q_{i+1}$ ($i=1,...,a+b$)
2006 Poland - Second Round, 1
Let $c$ be fixed natural number. Sequence $(a_n)$ is defined by:
$a_1=1$, $a_{n+1}=d(a_n)+c$ for $n=1,2,...$.
where $d(m)$ is number of divisors of $m$. Prove that there exist $k$ natural such that sequence $a_k,a_{k+1},...$ is periodic.
2020 HMNT (HMMO), 1
For how many positive integers $n \le 1000$ does the equation in real numbers $x^{\lfloor x \rfloor } = n$ have a positive solution for $x$?
2014 Chile National Olympiad, 6
Prove that for every set of $2n$ lines in the plane, such that there are no two parallel lines, there are two lines that divide the plane into four quadrants such that in each quadrant the number of unbounded regions is equal to $n$.
[asy]
unitsize(1cm);
pair[] A, B;
pair P, Q, R, S;
A[1] = (0,5.2);
B[1] = (6.1,0);
A[2] = (1.5,5.5);
B[2] = (3.5,0);
A[3] = (6.8,5.5);
B[3] = (1,0);
A[4] = (7,4.5);
B[4] = (0,4);
P = extension(A[2],B[2],A[4],B[4]);
Q = extension(A[3],B[3],A[4],B[4]);
R = extension(A[1],B[1],A[2],B[2]);
S = extension(A[1],B[1],A[3],B[3]);
fill(P--Q--S--R--cycle, palered);
fill(A[4]--(7,0)--B[1]--S--Q--cycle, paleblue);
draw(A[1]--B[1]);
draw(A[2]--B[2]);
draw(A[3]--B[3]);
draw(A[4]--B[4]);
label("Bounded region", (3.5,3.7), fontsize(8));
label("Unbounded region", (5.4,2.5), fontsize(8));
[/asy]
2020 CMIMC Combinatorics & Computer Science, 6
The nation of CMIMCland consists of 8 islands, none of which are connected. Each citizen wants to visit the other islands, so the government will build bridges between the islands. However, each island has a volcano that could erupt at any time, destroying that island and any bridges connected to it. The government wants to guarantee that after any eruption, a citizen from any of the remaining $7$ islands can go on a tour, visiting each of the remaining islands exactly once and returning to their home island (only at the end of the tour). What is the minimum number of bridges needed?
2004 National High School Mathematics League, 4
$O$ is a point inside $\triangle ABC$, and $\overrightarrow{OA}+2\overrightarrow{OB}+3\overrightarrow{OC}=\overrightarrow{0}$, then the ratio of the area of $\triangle ABC$ to $\triangle AOC$ is
$\text{(A)}2\qquad\text{(B)}\frac{3}{2}\qquad\text{(C)}3\qquad\text{(D)}\frac{5}{3}$
2017 Taiwan TST Round 2, 1
There is a $2n\times 2n$ rectangular grid and a chair in each cell of the grid. Now, there are $2n^2$ pairs of couple are going to take seats. Define the distance of a pair of couple to be the sum of column difference and row difference between them. For example, if a pair of couple seating at $(3,3)$ and $(2,5)$ respectively, then the distance between them is $|3-2|+|3-5|=3$. Moreover, define the total distance to be the sum of the distance in each pair. Find the maximal total distance among all possibilities.
2010 BAMO, 1
We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.)
What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.
1989 Putnam, B4
Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?
2017 MMATHS, 1
For any integer $n > 4$, prove that $2^n > n^2$.
1983 IMO, 3
Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that
\[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0.
\]
Determine when equality occurs.
1999 Miklós Schweitzer, 5
Let $\alpha>-2$ , $n\in \mathbb{N}$ and $y_1,\cdots,y_n$ be the solutions to the system of equations:
$\sum_{j=1}^n \frac{y_j}{j+k+\alpha}= \frac{1}{n+1+k+\alpha}$ , $k=1,\cdots,n$
Prove that $y_{j-1}y_{j+1}\leq y_j^2 \,\forall 1<j<n$