Found problems: 85335
2010 China Team Selection Test, 3
Given integer $n\geq 2$ and real numbers $x_1,x_2,\cdots, x_n$ in the interval $[0,1]$. Prove that there exist real numbers $a_0,a_1,\cdots,a_n$ satisfying the following conditions:
(1) $a_0+a_n=0$;
(2) $|a_i|\leq 1$, for $i=0,1,\cdots,n$;
(3) $|a_i-a_{i-1}|=x_i$, for $i=1,2,\cdots,n$.
2005 Harvard-MIT Mathematics Tournament, 8
Compute \[ \displaystyle\sum_{n=0}^{\infty} \dfrac {n}{n^4 + n^2 + 1}. \]
II Soros Olympiad 1995 - 96 (Russia), 10.2
Without using a calculator, find out what is greater: $\sin 28^o$ or $tg21^o$?
LMT Team Rounds 2010-20, 2020.S11
Let set $\mathcal{S}$ contain all positive integers less than or equal to $2020$ that can be written in the form $n(n+1)$ for some positive integer $n$. Compute the number of ordered pairs $(a,b)$ such that $a, b\in \mathcal{S}$ and $a-b$ is a power of two.
2006 Purple Comet Problems, 23
We have two positive integers both less than $1000$. The arithmetic mean and the geometric mean of these numbers are consecutive odd integers. Find the maximum possible value of the difference of the two integers.
1997 Brazil National Olympiad, 6
$f$ is a plane map onto itself such that points at distance 1 are always taken at point at distance 1.
Show that $f$ preserves distances.
2019 Tournament Of Towns, 5
Basil has an unrestricted supply of straight bricks $1 \times 1 \times 3$ and Γ-shape bricks made of three cubes $1\times 1\times 1$. Basil filled a whole box $m \times n \times k$ with these bricks, where $m, n$ and $k$ are integers greater than $1$. Prove that it was sufficient to use only Γ-shape bricks.
(Mikhail Evdokimov)
2016 SDMO (Middle School), 2
Let $AB$ be a diameter of a circle and let $C$ be a point on $AB$ with $2\cdot AC=BC$. Let $D$ and $E$ be points on the circle such that $DC\perp AB$ and $DE$ is a second diameter. What is the ratio of the area of $\triangle{DCE}$ to the area of $\triangle{ABD}$?
1961 All-Soviet Union Olympiad, 3
Consider $n$ points, some of them connected by segments. These segments do not intersect each other. You can reach every point from any every other one in exactly one way by traveling along the segments. Prove that the total number of segments is $n-1$.
2023 Korea Junior Math Olympiad, 3
Positive integers $a_1, a_2, \dots, a_{2023}$ satisfy the following conditions.
[list]
[*] $a_1 = 5, a_2 = 25$
[*] $a_{n + 2} = 7a_{n+1}-a_n-6$ for each $n = 1, 2, \dots, 2021$
[/list]
Prove that there exist integers $x, y$ such that $a_{2023} = x^2 + y^2.$
1981 USAMO, 3
If $A,B,C$ are the angles of a triangle, prove that
\[-2 \le \sin{3A}+\sin{3B}+\sin{3C} \le \frac{3\sqrt{3}}{2}\]
and determine when equality holds.
2008 Bosnia and Herzegovina Junior BMO TST, 3
Point $ M$ is given in the interior of parallelogram $ ABCD$, and the point $ N$ inside triangle $ AMD$ is chosen so that $ < MNA \plus{} < MCB \equal{} MND \plus{} < MBC \equal{} 180^0$. Prove that $ MN$ is parallel to $ AB$.
2010 Indonesia TST, 4
Let $n$ be a positive integer with $n = p^{2010}q^{2010}$ for two odd primes $p$ and $q$. Show that there exist exactly $\sqrt[2010]{n}$ positive integers $x \le n$ such that $p^{2010}|x^p - 1$ and $q^{2010}|x^q - 1$.
II Soros Olympiad 1995 - 96 (Russia), 11.5
The space is filled in the usual way with unit cubes. (Each cube is adjacent to $6$ others that have a common face with it.) On three edges of one of the cubes emerging from one vertex, points are marked at a distance of $1/19$, $1/9$ and $1/7$ from it, respectively. A plane is drawn through these points. Let's consider the many different polygons formed when this plane intersects with the cubes filling the space. How many different (unequal) polygons are there in this set?
2006 Switzerland Team Selection Test, 3
Let $n$ be natural number. Each of the numbers $\in\{1,2,\ldots ,n\}$ is coloured in black or white. When we choose a number, we flip it's colour and the colour of all the numbers which have at least one common divider with the chosen number. At the beginning all the numbers were coloured white. For which $n$ are all the numbers black after a finite number of changes?
Denmark (Mohr) - geometry, 1997.2
Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area.
[img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]
2023 Junior Balkan Team Selection Tests - Romania, P4
Let $M \geq 1$ be a real number. Determine all natural numbers $n$ for which there exist distinct natural numbers $a$, $b$, $c > M$, such that
$n = (a,b) \cdot (b,c) + (b,c) \cdot (c,a) + (c,a) \cdot (a,b)$
(where $(x,y)$ denotes the greatest common divisor of natural numbers $x$ and $y$).
2000 Kazakhstan National Olympiad, 1
Two guys are playing the game "Sea Battle-2000". On the board $ 1 \times 200 $, they take turns placing the letter "$ S $" or "$ O $" on the empty squares of the board. The winner is the one who gets the word "$ SOS $" first. Prove that the second player wins when played correctly.
2009 Thailand Mathematical Olympiad, 8
Let $a, b, c$ be side lengths of a triangle, and define $s =\frac{a+b+c}{2}$. Prove that
$$\frac{2a(2a-s)}{b + c}+\frac{2b(2b - s)}{c + a}+\frac{2c(2c - s)}{a + b}\ge s.$$
MathLinks Contest 7th, 3.1
Let $ p$ be a prime and let $ d \in \left\{0,\ 1,\ \ldots,\ p\right\}$. Prove that
\[ \sum_{k \equal{} 0}^{p \minus{} 1}{\binom{2k}{k \plus{} d}}\equiv r \pmod{p},
\]where $ r \equiv p\minus{}d \pmod 3$, $ r\in\{\minus{}1,0,1\}$.
2022 USAMO, 5
A function $f: \mathbb{R}\to \mathbb{R}$ is [i]essentially increasing[/i] if $f(s)\leq f(t)$ holds whenever $s\leq t$ are real numbers such that $f(s)\neq 0$ and $f(t)\neq 0$.
Find the smallest integer $k$ such that for any 2022 real numbers $x_1,x_2,\ldots , x_{2022},$ there exist $k$ essentially increasing functions $f_1,\ldots, f_k$ such that \[f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.\]
2011 Vietnam National Olympiad, 3
Let $AB$ be a diameter of a circle $(O)$ and let $P$ be any point on the tangent drawn at $B$ to $(O).$ Define $AP\cap (O)=C\neq A,$ and let $D$ be the point diametrically opposite to $C.$ If $DP$ meets $(O)$ second time in $E,$ then,
[b](i)[/b] Prove that $AE, BC, PO$ concur at $M.$
[b](ii)[/b] If $R$ is the radius of $(O),$ find $P$ such that the area of $\triangle AMB$ is maximum, and calculate the area in terms of $R.$
1991 Poland - Second Round, 5
$ P_1, P_2, \ldots, P_n $ are different two-element subsets of $ \{1,2,\ldots,n\} $. The sets $ P_i $, $ P_j $ for $ i\neq j $ have a common element if and only if the set $ \{i,j\} $ is one of the sets $ P_1, P_2, \ldots, P_n $. Prove that each of the numbers $ 1,2,\ldots,n $ is a common element of exactly two sets from $ P_1, P_2, \ldots, P_n $.
1977 Spain Mathematical Olympiad, 1
Given the determinant of order $n$
$$\begin{vmatrix}
8 & 3 & 3 & \dots & 3 \\
3 & 8 & 3 & \dots & 3 \\
3 & 3 & 8 & \dots & 3 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
3 & 3 & 3 & \dots & 8
\end{vmatrix}$$
Calculate its value and determine for which values of $n$ this value is a multiple of $10$.
2020-21 KVS IOQM India, 23
Find the largest positive integer $N$ such that the number of integers In the set ${1,2,3,...,N}$ which are divisible by $3$ is equal to the number of integers which are divisible by $5$ or $7$ (or both),