Found problems: 85335
2012 Irish Math Olympiad, 4
There exists an infinite set of triangles with the following properties:
(a) the lengths of the sides are integers with no common factors, and
(b) one and only one angle is $60^\circ$.
One such triangle has side lengths $5,7,8$. Find two more.
1969 IMO Shortlist, 40
$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
2016 USAMTS Problems, 3:
Find all positive integers $n$ for which $(x^n+y^n+z^n)/2$ is a perfect square whenever $x$, $y$, and $z$ are integers such that $x+y+z=0$.
2008 Singapore Senior Math Olympiad, 4
There are $11$ committees in a club. Each committee has $5$ members and every two committees have a member in common. Show that there is a member who belongs to $4$ committees.
1982 IMO Shortlist, 9
Let $ABC$ be a triangle, and let $P$ be a point inside it such that $\angle PAC = \angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$, respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM.$
2009 Korea - Final Round, 3
2008 white stones and 1 black stone are in a row. An 'action' means the following: select one black stone and change the color of neighboring stone(s).
Find all possible initial position of the black stone, to make all stones black by finite actions.
Kvant 2023, M2733
A convex 51-gon is given. For each of its vertices and each diagonal that does not contain this vertex, we mark in red a point symmetrical to the vertex relative to the middle of the diagonal. Prove that strictly inside the polygon there are no more than 20400 red dots.
[i]Proposed by P. Kozhevnikov[/i]
2010 Postal Coaching, 6
Let $n > 1$ be an integer.
A set $S \subseteq \{ 0, 1, 2, \cdots , 4n - 1 \}$ is called ’sparse’ if for any $k \in \{ 0, 1, 2, \cdots , n - 1 \}$ the following two conditions are satisfied:
$(a)$ The set $S \cap \{4k - 2, 4k - 1, 4k, 4k + 1, 4k + 2 \}$ has at most two elements;
$(b)$ The set $S \cap \{ 4k +1, 4k +2, 4k +3 \}$ has at most one element.
Prove that there are exactly $8 \cdot 7^{n-1}$ sparse subsets.
MOAA Individual Speed General Rounds, 2023.10
If $x,y,z$ satisfy the system of equations
\[xy+yz+zx=23\]
\[\frac{y}{x+y}+\frac{z}{y+z}+\frac{x}{z+x}=-1\]
\[\frac{z^2x}{x+y}+\frac{x^2y}{y+z}+\frac{y^2z}{z+x}=202\]
Find the value of $x^2+y^2+z^2$.
[i]Proposed by Harry Kim[/i]
1991 All Soviet Union Mathematical Olympiad, 557
The real numbers $x_1, x_2, ... , x_{1991}$ satisfy $$|x_1 - x_2| + |x_2 - x_3| + ... + |x_{1990} - x_{1991}| = 1991$$ What is the maximum possible value of $$|s_1 - s_2| + |s_2 - s_3| + ... + |s_{1990} - s_{1991}|$$ where $$s_n = \frac{x_1 + x_2 + ... + x_n}{n}?$$
1982 Tournament Of Towns, (016) 2
The lengths of all sides and both diagonals of a quadrilateral are less than $1$ metre.
Prove that it may be placed in a circle of radius $0.9$ metres.
1994 All-Russian Olympiad Regional Round, 9.6
Point $ P$ is taken inside a right angle $ KLM$. A circle $ S_1$ with center $ O_1$ is tangent to the rays $ LK,LP$ of angle $ KLP$ at $ A,D$ respectively. A circle $ S_2$ with center $ O_2$ is tangent to the rays of angle $ MLP$, touching $ LP$ at $ B$. Suppose $ A,B,O_1$ are collinear. Let $ O_2D,KL$ meet at $ C$. Prove that $ BC$ bisects angle $ ABD$.
2018 Stanford Mathematics Tournament, 6
In $\vartriangle AB$C, $AB = 3$, $AC = 6,$ and $D$ is drawn on $BC$ such that $AD$ is the angle bisector of $\angle BAC$. $D$ is reflected across $AB$ to a point $E$, and suppose that $AC$ and $BE$ are parallel. Compute $CE$.
2024 Korea Summer Program Practice Test, 6
Find all possible values of $C\in \mathbb R$ such that there exists a real sequence $\{a_n\}_{n=1}^\infty$ such that
$$a_na_{n+1}^2\ge a_{n+2}^4 +C$$
for all $n\ge 1$.
2014 Harvard-MIT Mathematics Tournament, 2
[4] Let $x_1,x_2,\ldots,x_{100}$ be defined so that for each $i$, $x_i$ is a (uniformly) random integer between $1$ and $6$ inclusive. Find the expected number of integers in the set $\{x_1,x_1+x_2,\ldots,x_1+x_2+\cdots+x_{100}\}$ that are multiples of $6$.
1983 IMO Longlists, 71
Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$
2009 IMO Shortlist, 2
Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that:
\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]
[i]Proposed by Juhan Aru, Estonia[/i]
2021 AMC 10 Fall, 1
What is the value of $1234+2341+3412+4123$?
$\textbf{(A) } 10,000 \qquad \textbf{(B) }10,010 \qquad \textbf{(C) }10,110 \qquad \textbf{(D) }11,000 \qquad \textbf{(E) }11,110$
2013 CHMMC (Fall), 6
Let $a_1 < a_2 < a_3 < ... < a_n < ...$ be positive integers such that, for $n = 1, 2, 3, ...,$ $$a_{2n} = a_n + n.$$
Given that if $a_n$ is prime, then $n$ is also, find $a_{2014}$.
2004 National Olympiad First Round, 9
What is the area of the region determined by the points outside a triangle with perimeter length $\pi$ where none of these points has a distance greater than $1$ to any corner of the triangle?
$
\textbf{(A)}\ 4\pi
\qquad\textbf{(B)}\ 3\pi
\qquad\textbf{(C)}\ \dfrac{5\pi}2
\qquad\textbf{(D)}\ 2\pi
\qquad\textbf{(E)}\ \dfrac{3\pi}2
$
1985 IMO Longlists, 42
Prove that the product of two sides of a triangle is always greater than the product of the diameters of the inscribed circle and the circumscribed circle.
KoMaL A Problems 2018/2019, A. 749
Given are two polyominos, the first one is an L-shape consisting of three squares, the other one contains at least two squares. Prove that if $n$ and $m$ are coprime then at most one of the $n\times n$ and $m\times m$ boards can be tiled by translated copies of the two polyominos.
[i]Proposed by: András Imolay, Dávid Matolcsi, Ádám Schweitzer and Kristóf Szabó, Budapest[/i]
2014 Contests, 1
Find the triplets of primes $(a,\ b,\ c)$ such that $a-b-8$ and $b-c-8$ are primes.
2013 Tournament of Towns, 1
There are $100$ red, $100$ yellow and $100$ green sticks. One can construct a triangle using any three sticks all of different colours (one red, one yellow and one green). Prove that there is a colour such that one can construct a triangle using any three sticks of this colour.
Kyiv City MO 1984-93 - geometry, 1993.8.3
In the triangle $ABC$, $\angle .ACB = 60^o$, and the bisectors $AA_1$ and $BB_1$ intersect at the point $M$. Prove that $MB_1 = MA_1$.