Found problems: 85335
2023 LMT Spring, 10
The sequence $a_0,a_1,a_2,...$ is defined such that $a_0 = 2+ \sqrt3$, $a_1 =\sqrt{5-2\sqrt5}$, and
$$a_n a_{n-1}a_{n-2} - a_n + a_{n-1} + a_{n-2} = 0.$$
Find the least positive integer $n$ such that $a_n = 1$.
2008 Baltic Way, 5
Suppose that Romeo and Juliet each have a regular tetrahedron to the vertices of which some positive real numbers are assigned. They associate each edge of their tetrahedra with the product of the two numbers assigned to its end points. Then they write on each face of their tetrahedra the sum of the three numbers associated to its three edges. The four numbers written on the faces of Romeo's tetrahedron turn out to coincide with the four numbers written on Juliet's tetrahedron. Does it follow that the four numbers assigned to the vertices of Romeo's tetrahedron are identical to the four numbers assigned to the vertices of Juliet's tetrahedron?
1997 Brazil National Olympiad, 1
Given $R, r > 0$. Two circles are drawn radius $R$, $r$ which meet in two points. The line joining the two points is a distance $D$ from the center of one circle and a distance $d$ from the center of the other. What is the smallest possible value for $D+d$?
2017 Kosovo Team Selection Test, 2
Prove that there doesn't exist any function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that :
$f(f(n-1)=f(n+1)-f(n)$, for every natural $n\geq2$
2015 Balkan MO Shortlist, G1
In an acute angled triangle $ABC$ , let $BB' $ and $CC'$ be the altitudes. Ray $C'B'$ intersects the circumcircle at $B''$ andl let $\alpha_A$ be the angle $\widehat{ABB''}$. Similarly are defined the angles $\alpha_B$ and $\alpha_C$. Prove that $$\displaystyle\sin \alpha _A \sin \alpha _B \sin \alpha _C\leq \frac{3\sqrt{6}}{32}$$
(Romania)
2017 Novosibirsk Oral Olympiad in Geometry, 7
A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$
2022 Dutch Mathematical Olympiad, 5
Kira has $3$ blocks with the letter $A$, $3$ blocks with the letter $B$, and $3$ blocks with the letter $C$. She puts these $9$ blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence $ABCAABCBC$ the blocks with the letter A have distances $1, 3$, and $4$ between one another, the blocks with the letter $B$ have distances $2, 4$, and $6$ between one another, and the blocks with the letter $C$ have distances $2, 4$, and $6$ between one another. Altogether, we got distances of $1, 2, 3, 4$, and $6$; these are $5$ distinct distances. What is the maximum number of distinct distances that can occur?
2021 China Team Selection Test, 5
Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.
2021 Yasinsky Geometry Olympiad, 3
In the triangle $ABC$, $h_a, h_b, h_c$ are the altitudes and $p$ is its half-perimeter. Compare $p^2$ with $h_ah_b + h_bh_c + h_ch_a$.
(Gregory Filippovsky)
1998 Balkan MO, 4
Prove that the following equation has no solution in integer numbers: \[ x^2 + 4 = y^5. \] [i]Bulgaria[/i]
2022 German National Olympiad, 3
Let $M$ and $N$ be the midpoints of segments $BC$ and $AC$ of a triangle $ABC$, respectively. Let $Q$ be a point on the line through $N$ parallel to $BC$ such that $Q$ and $C$ are on opposite sides of $AB$ and $\vert QN\vert \cdot \vert BC\vert=\vert AB\vert \cdot \vert AC\vert$.
Suppose that the circumcircle of triangle $AQN$ intersects the segment $MN$ a second time in a point $T \ne N$.
Prove that there is a circle through points $T$ and $N$ touching both the side $BC$ and the incircle of triangle $ABC$.
2012 Today's Calculation Of Integral, 831
Let $n$ be a positive integer. Answer the following questions.
(1) Find the maximum value of $f_n(x)=x^{n}e^{-x}$ for $x\geq 0$.
(2) Show that $\lim_{x\to\infty} f_n(x)=0$.
(3) Let $I_n=\int_0^x f_n(t)\ dt$. Find $\lim_{x\to\infty} I_n(x)$.
2021 Malaysia IMONST 1, 16
Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the area of this octagon?
2022 JHMT HS, 4
For a positive integer $n$, let $p(n)$ denote the product of the digits of $n$, and let $s(n)$ denote the sum of the digits of $n$. Find the sum of all positive integers $n$ satisfying $p(n)s(n)=8$.
2018 PUMaC Algebra B, 1
Find the sum of the solutions to $\dfrac{1}{1+\dfrac{1}{|x-25|}}=\frac{49}{50}$.
2010 Indonesia TST, 1
Given $ a,b, c $ positive real numbers satisfying $ a+b+c=1 $.
Prove that \[ \dfrac{1}{\sqrt{ab+bc+ca}}\ge \sqrt{\dfrac{2a}{3(b+c)}} +\sqrt{\dfrac{2b}{3(c+a)}} + \sqrt{\dfrac{2c}{3(a+b)}} \ge \sqrt{a} +\sqrt{b}+\sqrt{c} \]
1969 IMO Shortlist, 65
$(USS 2)$ Prove that for $a > b^2,$ the identity ${\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+\cdots}}}}=\sqrt{a-\frac{3}{4}b^2}-\frac{1}{2}b}$
2006 Hanoi Open Mathematics Competitions, 2
Find the last three digits of the sum
$2005^{11}$ + $2005^{12}$ + ... + $2005^{2006}$
1972 Miklós Schweitzer, 6
Let $ P(z)$ be a polynomial of degree $ n$ with complex coefficients, \[ P(0)\equal{}1, \;\textrm{and}\ \;|P(z)|\leq M\ \;\textrm{for}\ \;|z| \leq 1\ .\] Prove that every root of $ P(z)$ in the closed unit disc has multiplicity at most $ c\sqrt{n}$, where $ c\equal{}c(M) >0$ is a constant depending only on $ M$.
[i]G. Halasz[/i]
1981 Swedish Mathematical Competition, 3
Find all polynomials $p(x)$ of degree $5$ such that $p(x) + 1$ is divisible by $(x-1)^3$ and $p(x) - 1$ is divisible by $(x+1)^3$.
2022 China Team Selection Test, 6
Let $m$ be a positive integer, and $A_1, A_2, \ldots, A_m$ (not necessarily different) be $m$ subsets of a finite set $A$. It is known that for any nonempty subset $I$ of $\{1, 2 \ldots, m \}$,
\[ \Big| \bigcup_{i \in I} A_i \Big| \ge |I|+1. \]
Show that the elements of $A$ can be colored black and white, so that each of $A_1,A_2,\ldots,A_m$ contains both black and white elements.
2008 AMC 10, 23
Two subsets of the set $ S\equal{}\{a,b,c,d,e\}$ are to be chosen so that their union is $ S$ and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?
$ \textbf{(A)}\ 20 \qquad
\textbf{(B)}\ 40 \qquad
\textbf{(C)}\ 60 \qquad
\textbf{(D)}\ 160 \qquad
\textbf{(E)}\ 320$
2010 Contests, 4
Prove that
\[ a^2b^2(a^2+b^2-2) \geq (a+b)(ab-1) \]
for all positive real numbers $a$ and $b.$
2019 India PRMO, 25
A village has a circular wall around it, and the wall has four gates pointing north, south, east and west. A tree stands outside the village, $16 \, \mathrm{m}$ north of the north gate, and it can be [i]just[/i] seen appearing on the horizon from a point $48 \, \mathrm{m}$ east of the south gate. What is the diamter in meters, of the wall that surrounds the village?
V Soros Olympiad 1998 - 99 (Russia), 10.9
Six cities are located at the vertices of a convex hexagon, all angles of which are equal. Three sides of this hexagon have length $a$, and the remaining three have length $b$ ($a \le b$). It is necessary to connect these cities with a network of roads so that from each city you can drive to any other (possibly through other cities). Find the shortest length of such a road network.