Olympiad problems

2008 F = Ma, 25

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Two satellites are launched at a distance $R$ from a planet of negligible radius. Both satellites are launched in the tangential direction. The first satellite launches correctly at a speed $v_\text{0}$ and enters a circular orbit. The second satellite, however, is launched at a speed $\frac{1}{2}v_\text{0}$. What is the minimum distance between the second satellite and the planet over the course of its orbit? (a) $\frac{1}{\sqrt{2}}R$ (b) $\frac{1}{2}R$ (c) $\frac{1}{3}R$ (d) $\frac{1}{4}R$ (e) $\frac{1}{7}R$

1990 Baltic Way, 3

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Given $a_0 > 0$ and $c > 0$, the sequence $(a_n)$ is defined by \[a_{n+1}=\frac{a_n+c}{1-ca_n}\quad\text{for }n=1,2,\dots\] Is it possible that $a_0, a_1, \dots , a_{1989}$ are all positive but $a_{1990}$ is negative?

1986 IMO Shortlist, 21

Tags: geometry , 3d geometry , tetrahedron , inradius , geometric inequality

Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that \[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\] where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.

2022 CCA Math Bonanza, L3.3

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Determine the sum of all positive integers $n<100$ satisfying the following expression. \[\sum_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod \;{10^{k+1})}-n \;(\bmod \;{10^k)}\right)=\prod_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod\; 10^{k+1})-n \;(\bmod\; 10^k)\right).\] Here, $\textstyle\sum$ and $\textstyle\prod$ represent sum and product, respectively. [i]2022 CCA Math Bonanza Lightning Round 3.3[/i]

2021 IMC, 3

Tags: number theory

We say that a positive real number $d$ is $good$ if there exists an infinite squence $a_1,a_2,a_3,...\in (0,d)$ such that for each $n$, the points $a_1,a_2,...,a_n$ partition the interval $[0,d]$ into segments of length at most $\frac{1}{n}$ each . Find $\text{sup}\{d| d \text{is good}\}$.

1998 Iran MO (3rd Round), 1

Tags: combinatorics

A one-player game is played on a $m \times n$ table with $m \times n$ nuts. One of the nuts' sides is black, and the other side of them is white. In the beginning of the game, there is one nut in each cell of the table and all nuts have their white side upwards except one cell in one corner of the table which has the black side upwards. In each move, we should remove a nut which has its black side upwards from the table and reverse all nuts in adjacent cells (i.e. the cells which share a common side with the removed nut's cell). Find all pairs $(m,n)$ for which we can remove all nuts from the table.

2018 Regional Olympiad of Mexico Southeast, 1

Tags: winning strategy , polygon , combinatorics

Lalo and Sergio play in a regular polygon of $n\geq 4$ sides. In his turn, Lalo paints a diagonal or side of pink, and in his turn Sergio paint a diagonal or side of orange. Wins the game who achieve paint the three sides of a triangle with his color, if none of the players can win, they game tie. Lalo starts playing. Determines all natural numbers $n$ such that one of the players have winning strategy.

2001 Junior Balkan Team Selection Tests - Romania, 2

Tags: trapezoid , circumcircle , geometry

Let $ABCDEF$ be a hexagon with $AB||DE,\ BC||EF,\ CD||FA$ and in which the diagonals $AD,BE$ and $CF$ are congruent. Prove that the hexagon can be inscribed in a circle.

1982 IMO Shortlist, 2

Tags: geometry , analytic geometry , convex polygon , area , geometric inequality

Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that \[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\] where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$

2021 China Second Round A1, 2

Tags: number theory , function

Find a necessary and sufficient condition of $a,b,n\in\mathbb{N^*}$ such that for $S=\{a+bt\mid t=0,1,2,\cdots,n-1\}$, there exists a one-to-one mapping $f: S\to S$ such that for all $x\in S$, $\gcd(x,f(x))=1$.

1968 AMC 12/AHSME, 25

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Ace runs with constant speed and Flash runs $x$ times as fast, $x>1$. Flash gives Ace a head start of $y$ yards, and, at a given signal, they start off in the same direction. Then the number of yards Flash must run to catch Ace is: $\textbf{(A)}\ xy \qquad\textbf{(B)}\ \frac{y}{x+y} \qquad\textbf{(C)}\ \frac{xy}{x-1} \qquad\textbf{(D)}\ \frac{x+y}{x+1} \qquad\textbf{(E)}\ \frac{x+y}{x-1}$

2015 Federal Competition For Advanced Students, 4

Tags: number theory

A [i]police emergency number[/i] is a positive integer that ends with the digits $133$ in decimal representation. Prove that every police emergency number has a prime factor larger than $7$. (In Austria, $133$ is the emergency number of the police.) (Robert Geretschläger)

2022 JHMT HS, 8

Tags: geometry , circumcircle

In equilateral $\triangle ABC$, point $D$ lies on $\overline{BC}$ such that the radius of the circumcircle $\Gamma_1$ of $\triangle ACD$ is $7$ and the radius of the incircle $\Gamma_2$ of $\triangle{ABD}$ is $2$. Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at points $X$ and $Y$. Find $XY$.

2018 District Olympiad, 1

Tags: number theory , fractional part

Prove that $\left\{ \frac{m}{n}\right\}+\left\{ \frac{n}{m}\right\} \ne 1$ , for any positive integers $m, n$.

2014 Baltic Way, 12

Tags: geometry , circumcircle , trapezoid , symmetry , geometric transformation , reflection , parallelogram

Triangle $ABC$ is given. Let $M$ be the midpoint of the segment $AB$ and $T$ be the midpoint of the arc $BC$ not containing $A$ of the circumcircle of $ABC.$ The point $K$ inside the triangle $ABC$ is such that $MATK$ is an isosceles trapezoid with $AT\parallel MK.$ Show that $AK = KC.$

2007 AMC 10, 23

Tags: algebra , difference of squares , special factorizations

How many ordered pairs $ (m,n)$ of positive integers, with $ m > n$, have the property that their squares differ by $ 96$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 12$

2013 Romania National Olympiad, 4

Tags: function , inequalities , real analysis

a) Consider\[f\text{:}\left[ \text{0,}\infty \right)\to \left[ \text{0,}\infty \right)\] a differentiable and convex function .Show that $f\left( x \right)\le x$, for every $x\ge 0$, than ${f}'\left( x \right)\le 1$ ,for every $x\ge 0$ b) Determine \[f\text{:}\left[ \text{0,}\infty \right)\to \left[ \text{0,}\infty \right)\] differentiable and convex functions which have the property that $f\left( 0 \right)=0\,$, and ${f}'\left( x \right)f\left( f\left( x \right) \right)=x$, for every $x\ge 0$

2020 OMpD, 1

Tags: algebra , factoring polynomials

Let $a, b, c$ be real numbers such that $a + b + c = 0$. Given that $a^3 + b^3 + c^3 \neq 0$, $a^2 + b^2 + c^2 \neq 0$, determine all possible values for: $$\frac{a^5 + b^5 + c^5}{(a^3 + b^3 + c^3)(a^2 + b^2 + c^2)}$$

2013 Greece Junior Math Olympiad, 3

Tags: combinatorics

Let $A=\overline{abcd}$ be a four-digit positive integer with digits $a, b, c, d$, such that $a\ge7$ and $a>b>c>d>0$. Consider the positive integer $B=\overline{dcba}$ , that comes from number $A$ by reverting the order of it's digits. Given that the number $A+B$ has all it's digits odd, find all possible values of number $A$.

2017 SDMO (High School), 4

Tags: number theory , relatively prime

For each positive integer $n$, let $\tau\left(n\right)$ be the number of positive divisors of $n$. It is well-known that if $a$ and $b$ are relatively prime positive integers then $\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right)$. Does the converse hold? That is, if $a$ and $b$ are positive integers such that $\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right)$, then is it necessarily true that $a$ and $b$ are relatively prime? Either give a proof, or find a counter-example.

Ukraine Correspondence MO - geometry, 2013.9

Tags: geometry , cyclic quadrilateral , equal circles

Let $E$ be the point of intersection of the diagonals of the cyclic quadrilateral $ABCD$, and let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$, respectively. Prove that the radii of the circles circumscribed around the triangles $KLE$ and $MNE$ are equal.

2001 Moldova National Olympiad, Problem 7

Tags: set , number theory

Let $n$ be a positive integer. We denote by $S$ the sum of elements of the set $M=\{x\in\mathbb N|(n-1)^2\le x<(n+1)^2\}$. (a) Show that $S$ is divisible by $6$. (b) Find all $n\in\mathbb N$ for which $S+(1-n)(1+n)=2001$.

2004 Mexico National Olympiad, 4

Tags: combinatorics , tournament

At the end of a soccer tournament in which any pair of teams played between them exactly once, and in which there were not draws, it was observed that for any three teams $A, B$ and C, if $A$ defeated $B$ and $B$ defeated $C$, then $A$ defeated $C$. Any team calculated the difference (positive) between the number of games that it won and the number of games it lost. The sum of all these differences was $5000$. How many teams played in the tournament? Find all possible answers.

2009 Iran MO (3rd Round), 7

Tags: 3d geometry , sphere , geometry

A sphere is inscribed in polyhedral $P$. The faces of $P$ are coloured with black and white in a way that no two black faces share an edge. Prove that the sum of surface of black faces is less than or equal to the sum of the surface of the white faces. Time allowed for this problem was 1 hour.

2008 ISI B.Stat Entrance Exam, 4

Tags: geometry

Suppose $P$ and $Q$ are the centres of two disjoint circles $C_1$ and $C_2$ respectively, such that $P$ lies outside $C_2$ and $Q$ lies outside $C_1$. Two tangents are drawn from the point $P$ to the circle $C_2$, which intersect the circle $C_1$ at point $A$ and $B$. Similarly, two tangents are drawn from the point $Q$ to the circle $C_1$, which intersect the circle $C_2$ at points $M$ and $N$. Show that $AB=MN$