Olympiad problems

1963 IMO, 3

In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation \[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \] Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

2016 Czech-Polish-Slovak Match, 2

Tags: combinatorics

Let $m,n > 2$ be even integers. Consider a board of size $m \times n$ whose every cell is colored either black or white. The Guesser does not see the coloring of the board but may ask the Oracle some questions about it. In particular, she may inquire about two adjacent cells (sharing an edge) and the Oracle discloses whether the two adjacent cells have the same color or not. The Guesser eventually wants to find the parity of the number of adjacent cell-pairs whose colors are different. What is the minimum number of inquiries the Guesser needs to make so that she is guaranteed to find her answer?(Czech Republic)

2010 Iran MO (3rd Round), 5

Tags: number theory

prove that if $p$ is a prime number such that $p=12k+\{2,3,5,7,8,11\}$($k \in \mathbb N \cup \{0\}$), there exist a field with $p^2$ elements.($\frac{100}{6}$ points)

2024 AMC 8 -, 2

Tags:

What is the value of this expression in decimal form? \[\dfrac{44}{11}+\dfrac{110}{44}+\dfrac{44}{1100}\] $\textbf{(A) }6.4\qquad\textbf{(B) }6.504\qquad\textbf{(C) }6.54\qquad\textbf{(D) }6.9\qquad\textbf{(E) }6.94$

1992 AMC 12/AHSME, 18

Tags: inequalities

The increasing sequence of positive integers $a_{1},a_{2},a_{3},\ldots$ has the property that $a_{n+2} = a_{n} + a_{n+1}$ for all $n \ge 1$. If $a_{7} = 120$, then $a_{8}$ is $ \textbf{(A)}\ 128\qquad\textbf{(B)}\ 168\qquad\textbf{(C)}\ 193\qquad\textbf{(D)}\ 194\qquad\textbf{(E)}\ 210 $

2017 Azerbaijan BMO TST, 2

Tags: divisibility , rmn

Determine all positive integers $n$ such that all positive integers less than or equal to $n$ and relatively prime to $n$ are pairwise coprime.

2000 Romania Team Selection Test, 2

Tags: circumcircle , geometry

Let $ABC$ be an acute-angled triangle and $M$ be the midpoint of the side $BC$. Let $N$ be a point in the interior of the triangle $ABC$ such that $\angle NBA=\angle BAM$ and $\angle NCA=\angle CAM$. Prove that $\angle NAB=\angle MAC$. [i]Gabriel Nagy[/i]

2003 Alexandru Myller, 3

Tags: algebra , function , analytic geometry , invariant

Let $ S $ be the first quadrant and $ T:S\longrightarrow S $ be a transformation that takes the reciprocal of the coordinates of the points that belong to its domain. Define an [i]S-line[/i] to be the intersection of a line with $ S. $ [b]a)[/b] Show that the fixed points of $ T $ lie on any fixed S-line of $ T. $ [b]b)[/b] Find all fixed S-lines of $ T. $ [i]Gabriel Popa[/i]

1994 Tournament Of Towns, (420) 1

Tags: combinatorics

Several boys and girls are dancing a waltz at a ball. Is it possible that each girl can always get to dance the next dance with a boy who is either more handsome or more clever than for the previous dance, and that each time one of the girls gets to dance the next dance with a boy who is more handsome and more clever? (The numbers of boys and girls are equal and all are dancing.) (AY Belov)

MOAA Individual Speed General Rounds, 2021.8

Tags: speed

Andrew chooses three (not necessarily distinct) integers $a$, $b$, and $c$ independently and uniformly at random from $\{1,2,3,4,5,6,7\}$. Let $p$ be the probability that $abc(a+b+c)$ is divisible by $4$. If $p$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$. [i]Proposed by Andrew Wen[/i]

2020 Simon Marais Mathematics Competition, A3

Tags: analysis , real analysis

Determine the set of real numbers $\alpha$ that can be expressed in the form \[\alpha=\sum_{n=0}^{\infty}\frac{x_{n+1}}{x_n^3}\] where $x_0,x_1,x_2,\dots$ is an increasing sequence of real numbers with $x_0=1$.

PEN S Problems, 31

Tags:

Is there a $3 \times 3$ magic square consisting of distinct Fibonacci numbers (both $f_{1}$ and $f_{2}$ may be used; thus two $1$s are allowed)?

2014 IFYM, Sozopol, 2

Tags: number theory , divisor

Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.

1996 China National Olympiad, 3

Tags: geometry

In the triangle $ABC$, $\angle{C}=90^{\circ},\angle {A}=30^{\circ}$ and $BC=1$. Find the minimum value of the longest side of all inscribed triangles (i.e. triangles with vertices on each of three sides) of the triangle $ABC$.

2015 Online Math Open Problems, 22

Tags:

For a positive integer $n$ let $n\#$ denote the product of all primes less than or equal to $n$ (or $1$ if there are no such primes), and let $F(n)$ denote the largest integer $k$ for which $k\#$ divides $n$. Find the remainder when $F(1) + F(2) +F(3) + \dots + F(2015\#-1) + F(2015\#)$ is divided by $3999991$. [i]Proposed by Evan Chen[/i]

2006 Junior Balkan MO, 1

Tags: floor function , number theory

If $n>4$ is a composite number, then $2n$ divides $(n-1)!$.

2019 Turkey EGMO TST, 5

Tags: geometry

Let $D$ be the midpoint of $\overline{BC}$ in $\Delta ABC$. Let $P$ be any point on $\overline{AD}$. If the internal angle bisector of $\angle ABP$ and $\angle ACP$ intersect at $Q$. Prove that, if $BQ \perp QC$, then $Q$ lies on $AD$

2007 Hong Kong TST, 6

Tags: number theory

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 6 Determine all pairs $(x,y)$ of positive integers such that $\frac{x^{2}y+x+y}{xy^{2}+y+11}$ is an integer.

2015 239 Open Mathematical Olympiad, 2

Tags: binomial coefficient , number theory

Prove that $\binom{n+k}{n}$ can be written as product of $n$ pairwise coprime numbers $a_1,a_2,\dots,a_n$ such that $k+i$ is divisible by $a_i$ for all indices $i$.

1970 AMC 12/AHSME, 15

Tags: analytic geometry , arithmetic sequence

Lines in the xy-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1).$ One of these lines has the equation $\textbf{(A) }3x-2y-1=0\qquad\textbf{(B) }4x-5y+8=0\qquad\textbf{(C) }5x+2y-23=0\qquad$ $\textbf{(D) }x+7y-31=0\qquad \textbf{(E) }x-4y+13=0$

2022 Romania National Olympiad, P4

Tags: ring theory , group theory

Let $(R,+,\cdot)$ be a ring with center $Z=\{a\in\mathbb{R}:ar=ra,\forall r\in\mathbb{R}\}$ with the property that the group $U=U(R)$ of its invertible elements is finite. Given that $G$ is the group of automorphisms of the additive group $(R,+),$ prove that \[|G|\geq\frac{|U|^2}{|Z\cap U|}.\][i]Dragoș Crișan[/i]

1997 IMO Shortlist, 15

Tags: number theory , modular arithmetic , perfect square , perfect cube , arithmetic sequence

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

1980 Putnam, A3

Tags: trigonometry , integration

Evaluate $$\int_{0}^{ \pi \slash 2} \frac{ dx}{1+( \tan x)^{\sqrt{2}} }\;.$$

2011 India National Olympiad, 2

Tags: number theory

Call a natural number $n$ faithful if there exist natural numbers $a<b<c$ such that $a|b,$ and $b|c$ and $n=a+b+c.$ $(i)$ Show that all but a finite number of natural numbers are faithful. $(ii)$ Find the sum of all natural numbers which are not faithful.

2014 Israel National Olympiad, 4

Tags: combinatorics , game , game strategy , strategy

We are given a row of $n\geq7$ tiles. In the leftmost 3 tiles, there is a white piece each, and in the rightmost 3 tiles, there is a black piece each. The white and black players play in turns (the white starts). In each move, a player may take a piece of their color, and move it to an adjacent tile, so long as it's not occupied by a piece of the [u]same color[/u]. If the new tile is empty, nothing happens. If the tile is occupied by a piece of the [u]opposite color[/u], both pieces are destroyed (both white and black). The player who destroys the last two pieces wins the game. Which player has a winning strategy, and what is it? (The answer may depend on $n$)