Found problems: 85335
1998 Italy TST, 2
In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.
2022 Serbia National Math Olympiad, P1
Let $k$ be incircle of acute triangle $ABC$, $AC\neq BC$, and $l$ be excircle that touches $AB$. Line $p$ through the $C$ is orthogonal to $AB$, $p\cap k = \{X, Y\}$ , $p\cap l = \{Z, T\}$ and the point arrangement is $X-Y-Z-T$. Circle $m$ through $X$ and $Z$ intersects $AB$ at $D$ and $E$. Prove that points $D,Y,E,T$ are concyclic.
2008 Grigore Moisil Intercounty, 2
Given a convex quadrilateral $ ABCD, $ find the locus of points $ X $ that verify the qualities:
$$ XA^2+XB^2+CD^2=XB^2+XC^2+DA^2=XC^2+XD^2+AB^2=XD^2+XA^2+BC^2 $$
[i]Maria Pop[/i]
2019 Regional Olympiad of Mexico West, 4
Let $ABC$ be a triangle. $M$ the midpoint of $AB$ and $L$ the midpoint of $BC$. We denote by $G$ the intersection of $AL$ with $CM$ and we take $E$ a point such that $G$ is the midpoint of the segment $AE$. Prove that the quadrilateral $MCEB$ is cyclic if and only if $MB = BG$.
2002 Tuymaada Olympiad, 4
A real number $a$ is given. The sequence $n_{1}< n_{2}< n_{3}< ...$ consists of all the positive integral $n$ such that $\{na\}< \frac{1}{10}$. Prove that there are at most three different numbers among the numbers $n_{2}-n_{1}$, $n_{3}-n_{2}$, $n_{4}-n_{3}$, $\ldots$.
[i]A corollary of a theorem from ergodic theory[/i]
2006 Germany Team Selection Test, 3
Does there exist a set $ M$ of points in space such that every plane intersects $ M$ at a finite but nonzero number of points?
2016 AIME Problems, 4
A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60^\circ$. Find $h^2$.
1992 India National Olympiad, 1
In a triangle $ABC$, $\angle A = 2 \cdot \angle B$. Prove that $a^2 = b (b+c)$.
2013 Korea National Olympiad, 7
For positive integer $k$, define integer sequence $\{ b_n \}, \{ c_n \} $ as follows:
\[ b_1 = c_1 = 1 \]
\[ b_{2n} = kb_{2n-1} + (k-1)c_{2n-1}, c_{2n} = b_{2n-1} + c_{2n-1} \]
\[ b_{2n+1} = b_{2n} + (k-1)c_{2n}, c_{2n+1} = b_{2n} + kc_{2n} \]
Let $a_k = b_{2014} $. Find the value of
\[ \sum_{k=1}^{100} { (a_k - \sqrt{{a_k}^2-1} )^{ \frac{1}{2014}} }\]
May Olympiad L1 - geometry, 2019.4
You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.
2019 Romania Team Selection Test, 3
Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.
2019 Philippine TST, 3
Determine all ordered triples $(a, b, c)$ of real numbers such that whenever a function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c$$ for all real numbers $x$ and $y$, then $f$ must be a constant function.
2008 Purple Comet Problems, 19
One side of a triangle has length $75$. Of the other two sides, the length of one is double the length of the other. What is the maximum possible area for this triangle
2017 CMIMC Algebra, 4
It is well known that the mathematical constant $e$ can be written in the form $e = \tfrac{1}{0!}+\tfrac{1}{1!}+\tfrac{1}{2!}+\cdots$. With this in mind, determine the value of
\[\sum_{j=3}^\infty\dfrac{j}{\lfloor\frac j2\rfloor!}.\]
Express your answer in terms of $e$.
2019 IFYM, Sozopol, 8
Find all polynomials $f\in Z[X],$ such that for each odd prime $p$ $$f(p)|(p-3)!+\frac{p+1}{2}.$$
2020 Sharygin Geometry Olympiad, 24
Let $I$ be the incenter of a tetrahedron $ABCD$, and $J$ be the center of the exsphere touching the face $BCD$ containing three remaining faces (outside these faces). The segment $IJ$ meets the circumsphere of the tetrahedron at point $K$. Which of two segments $IJ$ and $JK$ is longer?
1970 IMO Longlists, 2
Prove that the two last digits of $9^{9^{9}}$ and $9^{9^{9^{9}}}$ are the same in decimal representation.
2008 Sharygin Geometry Olympiad, 1
(B.Frenkin) An inscribed and circumscribed $ n$-gon is divided by some line into two inscribed and circumscribed polygons with different numbers of sides. Find $ n$.
2009 AMC 10, 2
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could [i]not[/i] be the total value of the four coins, in cents?
$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 25 \qquad
\textbf{(C)}\ 35 \qquad
\textbf{(D)}\ 45 \qquad
\textbf{(E)}\ 55$
2010 239 Open Mathematical Olympiad, 5
Given three natural numbers greater than $100$, that are pairwise coprime and such that the square of the difference of any two of them is divisible by the third and any of them is less than the product of the other two. Prove that these numbers are squares of some natural numbers.
2020 Brazil Cono Sur TST, 1
Maria have $14$ days to train for an olympiad. The only conditions are that she cannot train by $3$ consecutive days and she cannot rest by $3$ consecutive days. Determine how many configurations of days(in training) she can reach her goal.
2017 Auckland Mathematical Olympiad, 4
The positive integers from $ 1$ to $n$ inclusive are written on a whiteboard. After one number is erased, the average (arithmetic mean) of the remaining $n - 1$ numbers is $22$. Knowing that $n$ is odd, determine $n$ and the number that was erased. Explain your reasoning.
2011 Thailand Mathematical Olympiad, 10
Does there exists a function $f : \mathbb{N} \longrightarrow \mathbb{N}$
\begin{align*} f \left( m+ f(n) \right) = f(m) +f(n) + f(n+1) \end{align*}
for all $m,n \in \mathbb{N}$ ?
2010 ISI B.Math Entrance Exam, 4
If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that
$\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$
2020 Thailand Mathematical Olympiad, 2
There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf $E_n$ takes $n$ minutes to travel $1 \text{ km}$ for each $n = 1,2,...,63$ , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?