Found problems: 85335
2022 Thailand Mathematical Olympiad, 2
Define a function $f:\mathbb{N}\times \mathbb{N}\to\{-1,1\}$ such that
$$f(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} \\ -1 &\text{if }m,n\text{ have different parity}\end{cases}$$
for every positive integers $m,n$. Determine the minimum possible value of
$$\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j)$$
across all permutations $x_1,x_2,x_3,\dots,x_{2565}$ of $1,2,\dots,2565$.
1985 Traian Lălescu, 1.3
Let $ H $ be the orthocenter of $ ABC $ and $ A',B',C', $ the symmetric points of $ A,B,C $ with respect to $ H. $ The intersection of the segments $ BC,CA, AB $ with the circles of diameter $ A'H,B'H, $ respectively, $ C'H, $ consists of $ 6 $ points. Prove that these are concyclic.
1995 Poland - First Round, 11
In a skiing jump competition $65$ contestants take part. They jump with the previously established order. Each of them jumps once. We assume that the obtained results are different and the orders of the contestants after the competition are equally likely. In each moment of the competition by a leader we call a person who is scored the best at this moment. Denote by $p$ the probability that during the whole competition there was exactly one change of the leader. Prove that $p > 1/16$.
1995 AMC 8, 20
Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6$. What is the probability that Diana's number is larger than Apollo's number?
$\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{5}{12} \qquad \text{(C)}\ \dfrac{4}{9} \qquad \text{(D)}\ \dfrac{17}{36} \qquad \text{(E)}\ \dfrac{1}{2}$
2017 Vietnamese Southern Summer School contest, Problem 4
In a summer school, there are $n>4$ students. It is known that, among these students,
i. If two ones are friends, then they don't have any common friends.
ii If two ones are not friends, then they have exactly two common friends.
1. Prove that $8n-7$ must be a perfect square.
2. Determine the smallest possible value of $n$.
2004 Singapore Team Selection Test, 1
Let $D$ be a point in the interior of $\bigtriangleup ABC$ such that $AB = ab$, $AC = ac$, $AD = ad$, $BC = bc$, $BD = bd$ and $CD = cd$. Prove that $\angle ABD + \angle ACD = \frac{\pi}{3}$.
2017 ASDAN Math Tournament, 8
Let $\triangle ABC$ be a right triangle with right angle $\angle ACB$. Square $DEFG$ is contained inside triangle $ABC$ such that $D$ lies on $AB$, $E$ lies on $BC$, $F$ lies on $AC$, $AD=AF$, and $GA=GD=GF$. Suppose that $CE=2$. If $M$ is the area of triangle $ABC$ and $N$ is the area of square $DEFG$, compute $M-N$.
2003 Purple Comet Problems, 21
Let $a_n = \sqrt{1 + (1 - \tfrac{1}{n})^2} + \sqrt{1 + (1 + \tfrac{1}{n})^2}, n \ge 1$. Evaluate $\tfrac{1}{a_1} + \tfrac{1}{a_2} + \ldots + \tfrac{1}{a_{20}}$.
2006 Kyiv Mathematical Festival, 5
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
All the positive integers from 1 till 1000 are written on the blackboard in some order and there is a collection of cards each containing 10 numbers. If there is a card with numbers $1\le a_1<a_2<\ldots<a_{10}\le1000$ in collection then it is allowed to arrange in increasing order the numbers at places $a_1, a_2, \ldots, a_{10},$ counting from left to right. What is the smallest amount of cards in the collection which enables us to arrange in
increasing order all the numbers for any initial arrangement of them?
2013 CentroAmerican, 2
Around a round table the people $P_1, P_2,..., P_{2013}$ are seated in a clockwise order. Each person starts with a certain amount of coins (possibly none); there are a total of $10000$ coins. Starting with $P_1$ and proceeding in clockwise order, each person does the following on their turn:
[list][*]If they have an even number of coins, they give all of their coins to their neighbor to the left.
[*]If they have an odd number of coins, they give their neighbor to the left an odd number of coins (at least $1$ and at most all of their coins) and keep the rest.[/list]
Prove that, repeating this procedure, there will necessarily be a point where one person has all of the coins.
2007 Hanoi Open Mathematics Competitions, 15
Let $p = \overline{abc}$ be the 3-digit prime number. Prove that the equation $ax^2 + bx + c = 0$ has no rational roots.
2007 Today's Calculation Of Integral, 215
For $ a\in\mathbb{R}$, let $ M(a)$ be the maximum value of the function $ f(x)\equal{}\int_{0}^{\pi}\sin (x\minus{}t)\sin (2t\minus{}a)\ dt$.
Evaluate $ \int_{0}^{\frac{\pi}{2}}M(a)\sin (2a)\ da$.
2024 Sharygin Geometry Olympiad, 17
Let $ABC$ be a non-isosceles triangle, $\omega$ be its incircle. Let $D, E, $ and $F$ be the points at which the incircle of $ABC$ touches the sides $BC, CA, $ and $AB$ respectively. Let $M$ be the point on ray $EF$ such that $EM = AB$. Let $N$ be the point on ray $FE$ such that $FN = AC$. Let the circumcircles of $\triangle BFM$ and $\triangle CEN$ intersect $\omega$ again at $S$ and $T$ respectively. Prove that $BS, CT, $ and $AD$ concur.
2006 CentroAmerican, 6
Let $ABCD$ be a convex quadrilateral. $I=AC\cap BD$, and $E$, $H$, $F$ and $G$ are points on $AB$, $BC$, $CD$ and $DA$ respectively, such that $EF \cap GH= I$. If $M=EG \cap AC$, $N=HF \cap AC$, show that \[\frac{AM}{IM}\cdot \frac{IN}{CN}=\frac{IA}{IC}.\]
1963 Bulgaria National Olympiad, Problem 1
Find all three-digit numbers whose remainders after division by $11$ give quotient, equal to the sum of the squares of its digits.
2010 Indonesia TST, 4
For each positive integer $ n$, define $ f(n)$ as the number of digits $ 0$ in its decimal representation. For example, $ f(2)\equal{}0$, $ f(2009)\equal{}2$, etc. Please, calculate \[ S\equal{}\sum_{k\equal{}1}^{n}2^{f(k)},\] for $ n\equal{}9,999,999,999$.
[i]Yudi Satria, Jakarta[/i]
2016 Junior Balkan Team Selection Tests - Romania, 2
Given three colors and a rectangle m × n dice unit, we want to color each segment constituting one side of a square drive with one of three colors so that each square unit have two sides of one color and two sides another color. How many colorings we have?
2002 China Girls Math Olympiad, 5
There are $ n \geq 2$ permutations $ P_1, P_2, \ldots, P_n$ each being an arbitrary permutation of $ \{1,\ldots,n\}.$ Prove that
\[ \sum^{n\minus{}1}_{i\equal{}1} \frac{1}{P_i \plus{} P_{i\plus{}1}} > \frac{n\minus{}1}{n\plus{}2}.\]
Estonia Open Senior - geometry, 2004.1.5
Find the smallest real number $x$ for which there exist two non-congruent triangles with integral side lengths having area $x$.
2004 Baltic Way, 4
Let $x_1$, $x_2$, ..., $x_n$ be real numbers with arithmetic mean $X$. Prove that there is a positive integer $K$ such that for any integer $i$ satisfying $0\leq i<K$, we have $\frac{1}{K-i}\sum_{j=i+1}^{K} x_j \leq X$. (In other words, prove that there is a positive integer $K$ such that the arithmetic mean of each of the lists $\left\{x_1,x_2,...,x_K\right\}$, $\left\{x_2,x_3,...,x_K\right\}$, $\left\{x_3,...,x_K\right\}$, ..., $\left\{x_{K-1},x_K\right\}$, $\left\{x_K\right\}$ is not greater than $X$.)
2015 South Africa National Olympiad, 2
Determine all pairs of real numbers $a$ and $x$ that satisfy the simultaneous equations $$5x^3 + ax^2 + 8 = 0$$ and $$5x^3 + 8x^2 + a = 0.$$
2005 Abels Math Contest (Norwegian MO), 4a
Show that for all positive real numbers $a, b$ and $c$, the inequality $(a+b)(a+c)\ge 2\sqrt{abc(a+b+c)}$ is true.
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 9
We draw a circle with radius 5 on a gridded paper where the grid consists of squares with sides of length 1. The center of the circle is placed in the middle of one of the squares. All the squares through which the circle passes are colored. How many squares are colored? (The figure illustrates this for a smaller circle.)
[img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1995Number9.jpg[/img]
A. 24
B. 32
C. 40
D. 64
E. None of these
2019 China Team Selection Test, 5
Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds.
Swiss NMO - geometry, 2009.7
Points $A, M_1, M_2$ and $C$ are on a line in this order. Let $k_1$ the circle with center $M_1$ passing through $A$ and $k_2$ the circle with center $M_2$ passing through $C$. The two circles intersect at points $E$ and $F$. A common tangent of $k_1$ and $k_2$, touches $k_1$ at $B$ and $k_2$ at $D$. Show that the lines $AB, CD$ and $EF$ intersect at one point.