Olympiad problems

2019 HMNT, 10

Tags: combinatorics

An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of $1$ unit either up or to the right. A lattice point $(x, y)$ with $0 \le x, y \le 5$ is chosen uniformly at random. Compute the expected number of up-right paths from $(0, 0)$ to$ (5,5)$ not passing through $(x, y)$.

2015 Saudi Arabia JBMO TST, 3

Tags: number theory

A natural number is called $nice$ if it doesn't contain 0 and if we add the product of its digit to the number, we obtain number with the same product of its digits. Prove that there is a nice 2015-digit number.

MBMT Team Rounds, 2020.17

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$\triangle KWU$ is an equilateral triangle with side length $12$. Point $P$ lies on minor arc $\overarc{WU}$ of the circumcircle of $\triangle KWU$. If $\overline{KP} = 13$, find the length of the altitude from $P$ onto $\overline{WU}$. [i]Proposed by Bradley Guo[/i]

2013 National Chemistry Olympiad, 7

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A solid can be separated from a liquid by all the following means EXCEPT $ \textbf{(A) }\text{decantation} \qquad\textbf{(B) }\text{distillation}\qquad$ $\textbf{(C) }\text{filtration}\qquad\textbf{(D) }\text{hydration}\qquad$

2012 Romanian Master of Mathematics, 3

Tags: function , algebra , additive combinatorics , combinatorics

Each positive integer is coloured red or blue. A function $f$ from the set of positive integers to itself has the following two properties: (a) if $x\le y$, then $f(x)\le f(y)$; and (b) if $x,y$ and $z$ are (not necessarily distinct) positive integers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$. Prove that there exists a positive number $a$ such that $f(x)\le ax$ for all positive integers $x$. [i](United Kingdom) Ben Elliott[/i]

2012 District Olympiad, 3

Tags: superior algebra , function , greatest common divisor

Let $G$ a $n$ elements group. Find all the functions $f:G\rightarrow \mathbb{N}^*$ such that: (a) $f(x)=1$ if and only if $x$ is $G$'s identity; (b) $f(x^k)=\frac{f(x)}{(f(x),k)}$ for any divisor $k$ of $n$, where $(r,s)$ stands for the greatest common divisor of the positive integers $r$ and $s$.

2010 F = Ma, 6

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A projectile is launched across flat ground at an angle $\theta$ to the horizontal and travels in the absence of air resistance. It rises to a maximum height $H$ and lands a horizontal distance $R$ away. What is the ratio $H/R$? (A) $\tan \theta$ (B) $2 \tan \theta$ (C) $\frac{2}{\tan \theta}$ (D) $\frac{1}{2}\tan \theta$ (E) $\frac{1}{4}\tan \theta$

2013 Brazil National Olympiad, 3

Tags: function , induction , limit , polynomial , functional equation , algebra

Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.

2007 Tournament Of Towns, 1

Tags: circumcircle , perpendicular bisector , geometry

Let $ABCD$ be a rhombus. Let $K$ be a point on the line $CD$, other than $C$ or $D$, such that $AD = BK$. Let $P$ be the point of intersection of $BD$ with the perpendicular bisector of $BC$. Prove that $A, K$ and $P$ are collinear.

2007 Mongolian Mathematical Olympiad, Problem 5

Tags: geometry , circumcircle

Given a point $P$ in the circumcircle $\omega$ of an equilateral triangle $ABC$, prove that the segments $PA$, $PB$, and $PC$ form a triangle $T$. Let $R$ be the radius of the circumcircle $\omega$ and let $d$ be the distance between $P$ and the circumcenter. Find the area of $T$.

2021 AMC 10 Spring, 2

Tags: sqrtblocked

What is the value of $\sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^2}$? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 4\sqrt{3}-6 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 4\sqrt{3} \qquad \textbf{(E)}\ 4\sqrt{3}+6$

1962 All-Soviet Union Olympiad, 5

Tags: combinatorics

An $n \times n$ array of numbers is given. $n$ is odd and each number in the array is $1$ or $-1$. Prove that the number of rows and columns containing an odd number of $-1$s cannot total $n$.

1984 IMO Longlists, 19

Tags: function , inequalities

Let $ABC$ be an isosceles triangle with right angle at point $A$. Find the minimum of the function $F$ given by \[F(M) = BM +CM-\sqrt{3}AM\]

2017 Dutch BxMO TST, 3

Tags: geometry

Let $ABC$ be a triangle with $\angle A = 90$ and let $D$ be the orthogonal projection of $A$ onto $BC$. The midpoints of $AD$ and $AC$ are called $E$ and $F$, respectively. Let $M$ be the circumcentre of $BEF$. Prove that $AC$ and $ BM$ are parallel.

1951 Putnam, B7

Tags: volume , sphere , four dimensions

Find the volume of the four-dimensional hypersphere $x^2 +y^2 +z^2 +t^2 =r^2$ and the hypervolume of its interior $x^2 +y^2 +z^2 +t^2 <r^2$

2007 Balkan MO Shortlist, A7

Tags: polynomial , floor function , induction , number theory , algebra

Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which \[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\] is a rational number.

2021 CCA Math Bonanza, I1

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Compute the number of positive integer divisors of $2121$ with a units digit of $1$. [i]2021 CCA Math Bonanza Individual Round #1[/i]

2009 Poland - Second Round, 1

Tags: induction , function , inequalities

Let $a_1\ge a_2\ge \ldots \ge a_n>0$ be $n$ reals. Prove the inequality \[a_1a_2\ldots a_{n-1}+(2a_2-a_1)(2a_3-a_2)\ldots (2a_n-a_{n-1})\ge 2a_2a_3\ldots a_n\]

2024 Regional Olympiad of Mexico West, 5

Tags: number theory , recurrence relation

Consider a sequence of positive integers $a_1,a_2,a_3,...$ such that $a_1>1$ and $$a_{n+1}=\frac{a_n}{p}+p,$$ where $p$ is the greatest prime factor of $a_n$. Prove that for any choice of $a_1$, the sequence $a_1,a_2,a_3,...$ has an infinite terms that are equal between them.

2013 NIMO Summer Contest, 3

Tags: probability

Jacob and Aaron are playing a game in which Aaron is trying to guess the outcome of an unfair coin which shows heads $\tfrac{2}{3}$ of the time. Aaron randomly guesses ``heads'' $\tfrac{2}{3}$ of the time, and guesses ``tails'' the other $\tfrac{1}{3}$ of the time. If the probability that Aaron guesses correctly is $p$, compute $9000p$. [i]Proposed by Aaron Lin[/i]

2010 Contests, 1

Tags: induction , modular arithmetic , combinatorics

For a finite non empty set of primes $P$, let $m(P)$ denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of $P$. (i) Show that $|P|\le m(P)$, with equality if and only if $\min(P)>|P|$. (ii) Show that $m(P)<(|P|+1)(2^{|P|}-1)$. (The number $|P|$ is the size of set $P$) [i]Dan Schwarz, Romania[/i]

1987 Greece Junior Math Olympiad, 2

Tags: algebra

Solve $(x-4)(x-5)(x-6)(x-7)=1680$

2024 CMI B.Sc. Entrance Exam, 4

Tags: inequalities , algebra

(a) For non negetive $a,b,c, r$ prove that \[a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r (c-a)(c-b) \geq 0 \] (b) Find an inequality for non negative $a,b,c$ with $a^4+b^4+c^4 + abc(a+b+c)$ on the greater side. (c) Prove that if $abc = 1$ for non negative $a,b,c$, $a^4+b^4+c^4+a^3+b^3+c^3+a+b+c \geq \frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}+3$

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , divide

Let $p$ be a prime number, $p> 5$. Determine the non-zero natural numbers $x$ with the property that $5p + x$ divides $5p ^ n + x ^ n$, whatever $n \in N ^ {*} $.

2011 May Olympiad, 2

Tags: perfect cube , perfect square , number theory , digit

Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be.