Olympiad problems

2021 MIG, 18

Tags:

The buttons $\{\times, +, \div\}$ on a calculator have their functions swapped. A button instead performs one of the other two functions; no two buttons have the same function. The calculator claims that $2 + 3 \div 4 = 10$ and $4 \times 2 \div 3 = 5$. What does $4 + 3 \times 2 \div 1$ equal on this calculator? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }10$

1984 National High School Mathematics League, 6

Tags: function

If $F(\frac{1-x}{1+x})=x$, then $\text{(A)}F(-2-x)=-2-F(x)\qquad\text{(B)}F(-x)=F(\frac{1+x}{1-x})$ $\text{(C)}F(\frac{1}{x})=F(x)\qquad\text{(D)}F(F(-x))=-x$

2010 Belarus Team Selection Test, 5.2

Tags: median , inequalities , geometric inequality , geometry

Numbers $a, b, c$ are the length of the medians of some triangle. If $ab + bc + ac = 1$ prove that a) $a^2b + b^2c + c^2a > \frac13$ b) $a^2b + b^2c + c^2a > \frac12$ (I. Bliznets)

2018 Macedonia JBMO TST, 1

Tags: number theory

Determine all positive integers $n>2$, such that $n = a^3 + b^3$, where $a$ is the smallest positive divisor of $n$ greater than $1$ and $b$ is an arbitrary positive divisor of $n$.

2022 Belarusian National Olympiad, 11.1

Tags: sequence , number theory

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

Tags: number theory , least common multiple

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

Tags: geometry , area of a triangle , geometric inequality , triangle inequality , inequalities , area

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

Tags: number theory

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

Tags: geometry , rectangle , combinatorics

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

Tags: right triangle , concyclic , cyclic quadrilateral , square , geometry

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?