Olympiad problems

2023 MIG, 5

Tags:

In the regular hexagon shown below, how many diagonals are longer than the red diagonal? [asy] size(2cm); draw((0,0)--(2,0)--(3,1.732)--(2,3.464)--(0,3.464)--(-1,1.732)--cycle); draw((-1,1.732)--(2,0),red); [/asy] $\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$

1990 Federal Competition For Advanced Students, P2, 6

Tags: trigonometry , geometry

A convex pentagon $ ABCDE$ is inscribed in a circle. The distances of $ A$ from the lines $ BC,CD,DE$ are $ a,b,c,$ respectively. Compute the distance of $ A$ from the line $ BE$.

1977 IMO Longlists, 6

Tags: induction , function , algebra , inequalities , trigonometry

Let $x_1, x_2, \ldots , x_n \ (n \geq 1)$ be real numbers such that $0 \leq x_j \leq \pi, \ j = 1, 2,\ldots, n.$ Prove that if $\sum_{j=1}^n (\cos x_j +1) $ is an odd integer, then $\sum_{j=1}^n \sin x_j \geq 1.$

2024 Canadian Open Math Challenge, B1

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For any positive integer number $k$, the factorial $k!$ is defined as a product of all integers between $1$ and $k$ inclusive: $k!=k\times{(k-1)}\times\dots\times{1}$. Let $s(n)$ denote the sum of the first $n$ factorials, i.e. $$s(n)=\underbrace{n\times{(n-1)}\times\dots\times{1}}_{n!}+\underbrace{(n-1)\times{(n-2)}\times\dots\times{1}}_{(n-1)!}+\cdots +\underbrace{2\times{1}}_{2!}+\underbrace{1}_{1!}$$ Find the remainder when $s(2024)$ is divided by $8$

2008 Tournament Of Towns, 3

Tags: perpendicular bisector , geometry , perimeter

A $30$-gon $A_1A_2\cdots A_{30}$ is inscribed in a circle of radius $2$. Prove that one can choose a point $B_k$ on the arc $A_kA_{k+1}$ for $1 \leq k \leq 29$ and a point $B_{30}$ on the arc $A_{30}A_1$, such that the numerical value of the area of the $60$-gon $A_1B_1A_2B_2 \dots A_{30}B_{30}$ is equal to the numerical value of the perimeter of the original $30$-gon.

1999 Irish Math Olympiad, 1

Tags: system of equations , algebra

Solve the system of equations: $ y^2\equal{}(x\plus{}8)(x^2\plus{}2),$ $ y^2\minus{}(8\plus{}4x)y\plus{}(16\plus{}16x\minus{}5x^2)\equal{}0.$

2018 Iran Team Selection Test, 4

Tags:

Let $ABC$ be a triangle ($\angle A\neq 90^\circ$). $BE,CF$ are the altitudes of the triangle. The bisector of $\angle A$ intersects $EF,BC$ at $M,N$. Let $P$ be a point such that $MP\perp EF$ and $NP\perp BC$. Prove that $AP$ passes through the midpoint of $BC$. [i]Proposed by Iman Maghsoudi, Hooman Fattahi[/i]

2022 Iran MO (3rd Round), 3

Tags: spiral similarity , khi point , geometry

The point $M$ is the middle of the side $BC$ of the acute-angled triangle $ABC$ and the points $E$ and $F$ are respectively perpendicular foot of $M$ to the sides $AC$ and $AB$. The points $X$ and $Y$ lie on the plane such that $\triangle XEC\sim\triangle CEY$ and $\triangle BYF\sim\triangle XBF$(The vertices of triangles with this order are corresponded in the similarities) and the points $E$ and $F$ [u]don't[/u][neither] lie on the line $XY$. Prove that $XY\perp AM$.

1979 IMO Longlists, 28

Tags: recurrence relation , counting , linear recurrence , combinatorics

Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]

2013 India IMO Training Camp, 1

Tags: rearrangement inequality , induction , inequalities

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. If $n$ is a positive integer then prove that \[ \frac{(3a)^n}{(b + 1)(c + 1)} + \frac{(3b)^n}{(c + 1)(a + 1)} + \frac{(3c)^n}{(a + 1)(b + 1)} \ge \frac{27}{16} \,. \]

1996 Greece National Olympiad, 1

Tags: recurrence relation , algebra , geometric progression , sequence , geometric sequence

Let $a_n$ be a sequence of positive numbers such that: i) $\dfrac{a_{n+2}}{a_n}=\dfrac{1}{4}$, for every $n\in\mathbb{N}^{\star}$ ii) $\dfrac{a_{k+1}}{a_k}+\dfrac{a_{n+1}}{a_n}=1$, for every $ k,n\in\mathbb{N}^{\star}$ with $|k-n|\neq 1$. (a) Prove that $(a_n)$ is a geometric progression. (n) Prove that exists $t>0$, such that $\sqrt{a_{n+1}}\leq \dfrac{1}{2}a_n+t$

1994 Poland - First Round, 7

Tags: function

(a) Find out, whether there exists a differentiable function $f: R \longrightarrow R$, not equaling $0$ for all $x \in R$, satisfying the conditions $2f(f(x)) = f(x) \geq 0$ for all $x \in R$. (b) Find out, whether there exists a differentiable function $f: R \longrightarrow R$, not equaling $0$ for all $x \in R$, satisfying the conditions $-1 \leq 2f(f(x)) = f(x) \leq 1$ for all $x \in R$.

2023 Purple Comet Problems, 16

Tags: combinatorics

A sequence of $28$ letters consists of $14$ of each of the letters $A$ and $B$ arranged in random order. The expected number of times that $ABBA$ appears as four consecutive letters in that sequence is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2017 Czech-Polish-Slovak Junior Match, 5

Tags: numbers in a table , table , combinatorics

In each square of the $100\times 100$ square table, type $1, 2$, or $3$. Consider all subtables $m \times n$, where $m = 2$ and $n = 2$. A subtable will be called [i]balanced [/i] if it has in its corner boxes of four identical numbers boxes . For as large a number $k$ prove, that we can always find $k$ balanced subtables, of which no two overlap, i.e. do not have a common box.

1994 IMC, 1

Tags: real analysis

Let $f\in C^1[a,b]$, $f(a)=0$ and suppose that $\lambda\in\mathbb R$, $\lambda >0$, is such that $$|f'(x)|\leq \lambda |f(x)|$$ for all $x\in [a,b]$. Is it true that $f(x)=0$ for all $x\in [a,b]$?

2006 China Team Selection Test, 2

Tags: inequalities , function

Given three positive real numbers $ x$, $ y$, $ z$ such that $ x \plus{} y \plus{} z \equal{} 1$, prove that $ \frac {xy}{\sqrt {xy \plus{} yz}} \plus{} \frac {yz}{\sqrt {yz \plus{} zx}} \plus{} \frac {zx}{\sqrt {zx \plus{} xy}} \le \frac {\sqrt {2}}{2}$.

2013 International Zhautykov Olympiad, 3

Tags: inequalities

Let $a, b, c$, and $d$ be positive real numbers such that $abcd = 1$. Prove that \[\frac{(a-1)(c+1)}{1+bc+c} + \frac{(b-1)(d+1)}{1+cd+d} + \frac{(c-1)(a+1)}{1+da+a} + \frac{(d-1)(b+1)}{1+ab+b} \geq 0.\] [i]Proposed by Orif Ibrogimov, Uzbekistan.[/i]

2019 USMCA, 12

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Determine the number of 10-letter strings consisting of $A$s, $B$s, and $C$s such that there is no $B$ between any two $A$s.

2004 Iran MO (3rd Round), 13

Tags: algebra , integer polynomial , integer sequence

Suppose $f$ is a polynomial in $\mathbb{Z}[X]$ and m is integer .Consider the sequence $a_i$ like this $a_1=m$ and $a_{i+1}=f(a_i)$ find all polynomials $f$ and alll integers $m$ that for each $i$: \[ a_i | a_{i+1}\]

Kvant 2022, M2715

Tags: board , combinatorics

A lame rook lies on a $9\times 9$ chessboard. It can move one cell horizontally or vertically. The rook made $n{}$ moves, visited each cell at most once, and did not make two moves consecutively in the same direction. What is the largest possible value of $n{}$? [i]From the folklore[/i]

2013 National Chemistry Olympiad, 52

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If $\text{A}$ represents the central atom, in which molecule is the $\text{F-A-F}$ angle the smallest? $ \textbf{(A) } \ce{BF3} \qquad\textbf{(B) }\ce{CF4} \qquad\textbf{(C) }\ce{NF3} \qquad\textbf{(D) }\ce{OF2} \qquad $

1989 Tournament Of Towns, (233) 1

Tags: combinatorics

Ten friends send greeting cards to each other, each sending $5$ cards. Prove that at least two of them sent cards to each other. (Folklore)

2014 Math Prize For Girls Problems, 13

Tags: probability

Deepali has a bag containing 10 red marbles and 10 blue marbles (and nothing else). She removes a random marble from the bag. She keeps doing so until all of the marbles remaining in the bag have the same color. Compute the probability that Deepali ends with exactly 3 marbles remaining in the bag.

2023 AMC 12/AHSME, 12

Tags: complex number

For complex numbers $u=a+bi$ and $v=c+di$, define the binary operation $\otimes$ by \[u\otimes v=ac+bdi.\] Suppose $z$ is a complex number such that $z\otimes z=z^{2}+40$. What is $|z|$? $\textbf{(A)}~\sqrt{10}\qquad\textbf{(B)}~3\sqrt{2}\qquad\textbf{(C)}~2\sqrt{6}\qquad\textbf{(D)}~6\qquad\textbf{(E)}~5\sqrt{2}$

2009 Indonesia TST, 2

Tags: algebra , logarithm , parameterization , inequalities

Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]