Found problems: 85335
2022 Belarusian National Olympiad, 11.1
A sequence of positive integer numbers $a_1,a_2,\ldots$ for $i \geq 3$ satisfies $$a_{i+1}=a_i+gcd(a_{i-1},a_{i-2})$$
Prove that there exist two positive integer numbers $N, M$, such that $a_{n+1}-a_n=M$ for all $n \geq N$
2017 CCA Math Bonanza, L2.4
Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$. Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$.
[i]2017 CCA Math Bonanza Lightning Round #2.4[/i]
2005 Sharygin Geometry Olympiad, 14
Let $P$ be an arbitrary point inside the triangle $ABC$. Let $A_1, B_1$ and $C_1$ denote the intersection points of the straight lines $AP, BP$ and $CP$, respectively, with the sides $BC, CA$ and $AB$. We order the areas of the triangles $AB_1C_1,A_1BC_1,A_1B_1C$. Denote the smaller by $S_1$, the middle by $S_2$, and the larger by $S_3$. Prove that $\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3}$ ,where $S$ is the area of the triangle $A_1B_1S_1$.
2024 ELMO Shortlist, N6
Given a positive integer whose base-$10$ representation is $\overline{d_k\ldots d_0}$ for some integer $k \geq 0$, where $d_k \neq 0$, a move consists of selecting some integers $0 \leq i \leq j \leq k$, such that the digits $d_j,\ldots,d_i$ are not all $0$, erasing them from $n$, and replacing them with a divisor of $\overline{d_j\ldots d_i}$ (this divisor need not have the same number of digits as $\overline{d_j\ldots d_i}$).
Prove that for all sufficiently large even integers $n$, we may apply some sequence of moves to $n$ to transform it into $2024$.
[i]Allen Wang[/i]
2011 Korea - Final Round, 3
There is a chessboard with $m$ columns and $n$ rows. In each blanks, an integer is given. If a rectangle $R$ (in this chessboard) has an integer $h$ satisfying the following two conditions, we call $R$ as a 'shelf'.
(i) All integers contained in $R$ are bigger than $h$.
(ii) All integers in blanks, which are not contained in $R$ but meet with $R$ at a vertex or a side, are not bigger than $h$.
Assume that all integers are given to make shelves as much as possible. Find the number of shelves.
2013 Estonia Team Selection Test, 4
Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?
1998 Greece National Olympiad, 3
Prove that for any non-zero real numbers $a, b, c,$
\[\frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac 35.\]
1977 All Soviet Union Mathematical Olympiad, 237
a) Given a circle with two inscribed triangles $T_1$ and $T_2$. The vertices of $T_1$ are the midpoints of the arcs with the ends in the vertices of $T_2$. Consider a hexagon -- the intersection of $T_1$ and $T_2$. Prove that its main diagonals are parallel to $T_1$ sides and are intersecting in one point.
b) The segment, that connects the midpoints of the arcs $AB$ and $AC$ of the circle circumscribed around the $ABC$ triangle, intersects $[AB]$ and $[AC]$ sides in $D$ and $K$ points. Prove that the points $A,D,K$ and $O$ -- the centre of the circle -- are the vertices of a diamond.
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.
2012 IFYM, Sozopol, 2
In $\Delta ABC$ with $AC=10$ and $BC=15$ the points $G$ and $I$ are its centroid and the center of its inscribed circle respectively. Find $AB$, if $\angle GIC=90^\circ$.
2006 China Team Selection Test, 3
Given $n$ real numbers $a_1$, $a_2$ $\ldots$ $a_n$. ($n\geq 1$). Prove that there exists real numbers $b_1$, $b_2$ $\ldots$ $b_n$ satisfying:
(a) For any $1 \leq i \leq n$, $a_i - b_i$ is a positive integer.
(b)$\sum_{1 \leq i < j \leq n} (b_i - b_j)^2 \leq \frac{n^2-1}{12}$
2010 HMNT, 2
$16$ progamers are playing in another single elimination tournament. Each round, each of the remaining progamers plays against another and the loser is eliminated. Additionally, each time a progamer wins, he will have a ceremony to celebrate. A player's rst ceremony is ten seconds long, and afterward each ceremony is ten seconds longer than the last. What is the total length in seconds of all the ceremonies over the entire tournament?
2014 Bosnia And Herzegovina - Regional Olympiad, 4
How namy subsets with $3$ elements of set $S=\{1,2,3,...,19,20\}$ exist, such that their product is divisible by $4$.
2022 Harvard-MIT Mathematics Tournament, 3
Let $ABCD$ and $AEF G$ be unit squares such that the area of their intersection is $\frac{20}{21}$ . Given that $\angle BAE < 45^o$, $\tan \angle BAE$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.
2011 Bosnia Herzegovina Team Selection Test, 3
Numbers $1,2, ..., 2n$ are partitioned into two sequences $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$. Prove that number
\[W= |a_1-b_1|+|a_2-b_2|+...+|a_n-b_n|\]
is a perfect square.
2008 JBMO Shortlist, 11
Determine the greatest number with $n$ digits in the decimal representation which is divisible by $429$ and has the sum of all digits less than or equal to $11$.
2015 HMNT, 6
Consider a $6 \times 6$ grid of squares. Edmond chooses four of these squares uniformly at random. What is the probability that the centers of these four squares form a square?
2021 Durer Math Competition (First Round), 3
The floor plan of a contemporary art museum is a (not necessarily convex) polygon and its walls are solid. The security guard guarding the museum has two favourite spots (points $A$ and $B$) because one can see the whole area of the museum standing at either point. Is it true that from any point of the $AB$ section one can see the whole museum?
2025 Ukraine National Mathematical Olympiad, 8.7
Find the smallest real number \(a\) such that for any positive integer number \(n > 2\) and any arrangement of the numbers from 1 to \(n\) on a circle, there exists a pair of adjacent numbers whose ratio (when dividing the larger number by the smaller one) is less than \(a\).
[i]Proposed by Mykhailo Shtandenko[/i]
2024 Indonesia TST, A
Given real numbers $x,y,z$ which satisfies
$$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$
Show that $max\{ |x|,|y|,|z|\} \le 1$.
2006 AIME Problems, 13
How many integers $ N$ less than 1000 can be written as the sum of $ j$ consecutive positive odd integers from exactly 5 values of $ j\ge 1$?
2006 Stanford Mathematics Tournament, 1
A college student is about to break up with her boyfriend, a mathematics major who is apparently more interested in math than her. Frustrated, she cries, ”You mathematicians have no soul! It’s all numbers and equations! What is the root of your incompetence?!” Her boyfriend assumes she means the square root of himself, or the square root of i. What two answers should he give?
2003 Pan African, 3
Find all functions $f: R\to R$ such that:
\[ f(x^2)-f(y^2)=(x+y)(f(x)-f(y)), x,y \in R \]
2013 National Olympiad First Round, 8
How many kites are there such that all of its four vertices are vertices of a given regular icosagon ($20$-gon)?
$
\textbf{(A)}\ 105
\qquad\textbf{(B)}\ 100
\qquad\textbf{(C)}\ 95
\qquad\textbf{(D)}\ 90
\qquad\textbf{(E)}\ 85
$
2015 Saudi Arabia GMO TST, 1
Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Prove that $$2 \left( \frac{ab}{a + b} +\frac{bc}{b + c} +\frac{ca}{c+ a}\right)+ 1 \ge 6(ab + bc + ca)$$
Trần Nam Dũng