Found problems: 85335
2021 India National Olympiad, 3
Betal marks $2021$ points on the plane such that no three are collinear, and draws all possible segments joining these. He then chooses any $1011$ of these segments, and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using [b]only[/b] a straightedge. Can Vikram always complete this challenge?
[i]Note.[/i] A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points.
[i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]
2010 Contests, 1
Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$, there is a circle passing through all four or a line passing through some three.
[i]Carl Lian.[/i]
2018 Israel National Olympiad, 4
The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?
Russian TST 2016, P4
A regular $n{}$-gon and a regular $m$-gon with distinct vertices are inscribed in the same circle. The vertices of these polygons divide the circle into $n+m$ arcs. Is it always possible to inscribe a regular $(n+m)$-gon in the same circle so that exactly one of its vertices is on each of these arcs?
2011 N.N. Mihăileanu Individual, 1
[b]a)[/b] Prove that $ 4040100 $ divides $ 2009\cdot 2011^{2011}+1. $
[i]Gabriel Iorgulescu[/i]
[b]b)[/b] Let be three natural numbers $ x,y,z $ with the property that $ (1+\sqrt 2)^x=y^2+2z^2+2yz\sqrt 2. $ Show that $ x $ is even.
[i]Marius Cavachi[/i]
2011 District Olympiad, 3
A positive integer $N$ has the digits $1, 2, 3, 4, 5, 6$ and $7$, so that each digit $i$, $i \in \{1, 2, 3, 4, 5, 6, 7\}$ occurs $4i$ times in the decimal representation of $N$. Prove that $N$ is not a perfect square.
Denmark (Mohr) - geometry, 2000.2
Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom?
[img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]
2024 Al-Khwarizmi IJMO, 2
For how many $x \in \{1,2,3,\dots, 2024\}$ is it possible that [i]Bekhzod[/i] summed $2024$ non-negative consecutive integers, [i]Ozod[/i] summed $2024+x$ non-negative consecutive integers and they got the same result?
[i]Proposed by Marek Maruin, Slovakia[/i]
2021 Baltic Way, 12
Let $I$ be the incentre of a triangle $ABC$. Let $F$ and $G$ be the projections of $A$ onto the lines $BI$ and $CI$, respectively. Rays $AF$ and $AG$ intersect the circumcircles of the triangles $CFI$ and $BGI$ for the second time at points $K$ and $L$, respectively. Prove that the line $AI$ bisects the segment $KL$.
2022 AMC 10, 17
One of the following numbers is not divisible by any prime number less than 10. Which is it?
(A) $2^{606} - 1 \ \ $ (B) $2^{606} + 1 \ \ $ (C) $2^{607} - 1 \ \ $ (D) $2^{607} + 1 \ \ $ (E) $2^{607} + 3^{607} \ \ $
2006 AIME Problems, 5
The number \[ \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006} \] can be written as $a\sqrt{2}+b\sqrt{3}+c\sqrt{5},$ where $a, b,$ and $c$ are positive integers. Find $a\cdot b\cdot c$.
2016 Purple Comet Problems, 26
Find the sum of all values of $a$ such that there are positive integers $a$ and $b$ satisfying $(a - b)\sqrt{ab} = 2016$.
2001 Czech-Polish-Slovak Match, 1
Let $n\ge2$ be a natural number, and $a_i$ be positive numbers, where $i=1,2,\cdots,n.$ Show that
\[\left(a_1^3+1\right)\left(a_2^3+1\right)\cdots\left(a_n^3+1\right) \geq \left(a_1^2a_2+1\right)\left(a_2^2a_3+1\right)\cdots\left(a_n^2a_1+1\right)\]
2000 Brazil Team Selection Test, Problem 1
Consider a triangle $ABC$ and $I$ its incenter. The line $(AI)$ meets the circumcircle of $ABC$ in $D$. Let $E$ and $F$ be the orthogonal projections of $I$ on $(BD)$ and $(CD)$ respectively. Assume that $IE+IF=\frac{1}{2}AD$. Calculate $\angle{BAC}$.
[color=red][Moderator edited: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=5088 .][/color]
1966 AMC 12/AHSME, 37
Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. Then $h$ equals:
$\text{(A)}\ \dfrac{5}{2}\qquad
\text{(B)}\ \frac{3}{2}\qquad
\text{(C)}\ \dfrac{4}{3}\qquad
\text{(D)}\ \dfrac{5}{4}\qquad
\text{(E)}\ \dfrac{3}{4}$
2021 China National Olympiad, 3
Let $n$ be positive integer such that there are exactly 36 different prime numbers that divides $n.$ For $k=1,2,3,4,5,$ $c_n$ be the number of integers that are mutually prime numbers to $n$ in the interval $[\frac{(k-1)n}{5},\frac{kn}{5}] .$ $c_1,c_2,c_3,c_4,c_5$ is not exactly the same.Prove that$$\sum_{1\le i<j\le 5}(c_i-c_j)^2\geq 2^{36}.$$
1999 Spain Mathematical Olympiad, 5
The distances from the centroid $G$ of a triangle $ABC$ to its sides $a,b,c$ are denoted $g_a,g_b,g_c$ respectively. Let $r$ be the inradius of the triangle. Prove that:
a) $g_a,g_b,g_c \ge \frac{2}{3}r$
b) $g_a+g_b+g_c \ge 3r$
2023 Lusophon Mathematical Olympiad, 3
An integer $n$ is called $k$-special, with $k$ a positive integer, if it's the sum of the squares of $k$ consecutive integers. For example, $13$ is $2$-special, since $13=2^2+3^2$, and $2$ is $3$-special, since $2=(-1)^2+0^2+1^2$.
a) Prove that there's no perfect square that is $4$-special.
b) Find a perfect square that is $I^2$-special, for some odd positive integer $I$ with $I\ge 3$.
2019 Chile National Olympiad, 1
A square of $3 \times 3$ is subdivided into 9 small squares of $1 \times 1$. It is desired to distribute the nine digits $1, 2, . . . , 9$ in each small square of $1 \times 1$, a number in each small square. Find the number of different distributions that can be formed in such a way that the difference of the digits in cells that share a side in common is less than or equal to three. Two distributions are distinct even if they differ by rotation and/or reflection.
2018 Junior Balkan Team Selection Tests - Romania, 1
Prove that the equation $x^2+y^2+z^2 = x+y+z+1$ has no rational solutions.
2017-2018 SDML (Middle School), 6
In the figure, a circle is located inside a trapezoid with two right angles so that a point of tangency of the circle is the midpoint of the side perpendicular to the two bases. The circle also has points of tangency on each base of the trapezoid. The diameter of the circle is $\frac{2}{3}$ the length of $EF$. If the area of the circle is $9\pi$ square units, what is the area of the trapezoid?
[asy]
draw((0,0) -- (11, 0) -- (7,6) -- (0,6) -- cycle);
draw((0,3) -- (9,3));
draw(circle((3,3), 3));
draw(rightanglemark((1,0),(0,0),(0,1),12));
draw(rightanglemark((0,0),(0,6),(6,6), 12));
label("E", (0,3), W);
label("F", (9,3), E);
[/asy]
2020 Macedonian Nationаl Olympiad, 2
Let $x_1, ..., x_n$ ($n \ge 2$) be real numbers from the interval $[1, 2]$. Prove that
$|x_1 - x_2| + ... + |x_n - x_1| \le \frac{2}{3}(x_1 + ... + x_n)$,
with equality holding if and only if $n$ is even and the $n$-tuple $(x_1, x_2, ..., x_{n - 1}, x_n)$ is equal to $(1, 2, ..., 1, 2)$ or $(2, 1, ..., 2, 1)$.
Gheorghe Țițeica 2024, P1
Find all continuous functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any sequences $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ such that the sequence $(a_n+b_n)_{n\geq 1}$ is convergent, the sequence $(f(a_n)+g(b_n))_{n\geq 1}$ is also convergent.
1994 AMC 8, 21
A gumball machine contains $9$ red, $7$ white, and $8$ blue gumballs. The least number of gumballs a person must buy to be sure of getting four gumballs of the same color is
${\text{(A)}\ 8 \qquad \text{(B)}\ 9 \qquad \text(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 18$
2009 China Northern MO, 5
Assume : $x,y,z>0$ , $ x^2+y^2+z^2 = 3 $ . Prove the following inequality :
$${\frac{x^{2009}-2008(x-1)}{y+z}+\frac{y^{2009}-2008(y-1)}{x+z}+\frac{z^{2009}-2008(z-1)}{x+y}\ge\frac{1}{2}(x+y+z)}$$