Olympiad problems

2013 Lusophon Mathematical Olympiad, 5

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Find all the numbers of $5$ non-zero digits such that deleting consecutively the digit of the left, in each step, we obtain a divisor of the previous number.

1998 India National Olympiad, 2

Let $a$ and $b$ be two positive rational numbers such that $\sqrt[3] {a} + \sqrt[3]{b}$ is also a rational number. Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers.

2014 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: set , partition , number theory , inequalities

Let $n \in N$ such that $1 + 2 + ... + n$ is divisible by $3$. Integers $a_1\ge a_2\ge a_3\ge 2$ have sum $n$ and they satisfy $1 + 2 + ... + a_1\le \frac{1}{3}( 1 + 2 + ... + n ) $ and $1 + 2 + ... + (a_1+ a_2) \le \frac{2}{3}( 1 + 2 + ... + n )$. Prove that there is a partition of $\{ 1 , 2 , ... , n\}$ in three subsets $A_1, A_2, A_3$ with cardinals $| A_i| = a_i, i = 1 , 2 , 3$, and with equal sums of their elements .

2006 Stanford Mathematics Tournament, 15

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The odometer of a family car shows 15,951 miles. The driver noticed that this number is palindromic: it reads the same backward as forwards. "Curious," the driver said to himself, "it will be a long time before that happens again." Surprised, he saw his third palindromic odometer reading (not counting 15,951) exactly five hours later. How many miles per hour was the car traveling in those 5 hours (assuming speed was constant)?

2021 USEMO, 5

Tags: polynomial , function , algebra

Given a polynomial $p(x)$ with real coefficients, we denote by $S(p)$ the sum of the squares of its coefficients. For example $S(20x+ 21)=20^2+21^2=841$. Prove that if $f(x)$, $g(x)$, and $h(x)$ are polynomials with real coefficients satisfying the indentity $f(x) \cdot g(x)=h(x)^ 2$, then $$S(f) \cdot S(g) \ge S(h)^2$$ [i]Proposed by Bhavya Tiwari[/i]

2020 Harvard-MIT Mathematics Tournament, 4

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For positive integers $n$ and $k$, let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$. For example, $\mho(90, 3)=2$, since the only prime factors of $90$ that are at least $3$ are $3$ and $5$. Find the closest integer to \[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.\] [i]Proposed by Daniel Zhu.[/i]

2021 Miklós Schweitzer, 9

Tags: probability

For a given natural number $n$, two players randomly (uniformly distributed) select a common number $0 \le j \le n$, and then each of them independently randomly selects a subset of $\{1,2, \cdots, n \}$ with $j$ elements. Let $p_n$ be the probability that the same set was chosen. Prove that \[ \sum_{k=1}^{n} p_k = 2 \log{n} + 2 \gamma - 1 + o(1), \quad (n \to \infty),\] where $\gamma$ is the Euler constant.

2008 AMC 10, 23

Tags: rectangle , geometry

A rectangular floor measures $ a$ by $ b$ feet, where $ a$ and $ b$ are positive integers with $ b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $ 1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $ (a,b)$? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

2007 Paraguay Mathematical Olympiad, 4

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Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three [i]aligned[/i] numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$?

2004 Germany Team Selection Test, 2

Tags: menelaus , angle bisector , geometry

In a triangle $ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on the side $AC$. The lines $BE$ and $AD$ meet at a point $F$. Prove: If $\frac{BF}{FE}=\frac{BC}{AB}+1$, then the line $BE$ bisects the angle $ABC$.

2001 National High School Mathematics League, 9

Tags: 3d geometry , geometry

The length of edge of cube $ABCD-A_1B_1C_1D_1$ is $1$, then the distance between lines $A_1C_1$ and $BD_1$ is________.

1983 IMO Longlists, 42

Tags: geometry , ratio

Consider the square $ABCD$ in which a segment is drawn between each vertex and the midpoints of both opposite sides. Find the ratio of the area of the octagon determined by these segments and the area of the square $ABCD.$

2001 Vietnam National Olympiad, 2

Tags: number theory , relatively prime

Let $N = 6^{n}$, where $n$ is a positive integer, and let $M = a^{N}+b^{N}$, where $a$ and $b$ are relatively prime integers greater than $1. M$ has at least two odd divisors greater than $1$ are $p,q$. Find the residue of $p^{N}+q^{N}\mod 6\cdot 12^{n}$.

1988 IMO Shortlist, 21

Tags: partial order , sperner , combinatorics

Forty-nine students solve a set of 3 problems. The score for each problem is a whole number of points from 0 to 7. Prove that there exist two students $ A$ and $ B$ such that, for each problem, $ A$ will score at least as many points as $ B.$

2011 NIMO Summer Contest, 2

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The sum of three consecutive integers is $15$. Determine their product.

2007 Harvard-MIT Mathematics Tournament, 7

Tags: law of cosines , trig identities , trigonometry , geometry

Convex quadrilateral $ABCD$ has sides $AB=BC=7$, $CD=5$, and $AD=3$. Given additionally that $m\angle ABC=60^\circ$, find $BD$.

1996 Austrian-Polish Competition, 2

Tags: geometry

A convex hexagon $ ABCDEF$ satisfies the following conditions: 1) $ AB\parallel DE$, $ BC\parallel EF$, and $ CD\parallel FA$. 2) The distances between these pairs of parallel lines are the same. 3) $ \angle FAB \equal{} \angle CDE \equal{} 90^\circ$ Prove that the diagonals $ BE$ and $ CF$ of the hexagon intersect with angle $ 45$ degrees. $ \bullet$ Thank you dear [b]Babis Stergiou[/b] for your translation. :P

2002 India IMO Training Camp, 4

Tags: orthocenter , triangle , concurrency , circumcircle , geometry

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.

2018 HMNT, 7

Tags: number theory

Anders is solving a math problem, and he encounters the expression $\sqrt{15!}$. He attempts to simplify this radical as $a\sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible values of $ab$ can be expressed in the form $q\cdot 15!$ for some rational number $q$. Find $q$.

2005 Miklós Schweitzer, 3

Tags: number theory

Let $\alpha\leq 22$ be non-negative integer. For which $\alpha$ does the equation $$8x^{23}-5^{\alpha}y^{23}=1$$ have the most integer solutions (x,y)? What can we say about $\alpha\geq 23$? [hide=Note]I believe the eqn has solutions only when $\alpha=0$. taking modulo 47, $\alpha\equiv 9,17\pmod{23}$ or ($23|\alpha$ and $47|x$). taking modulo 139 and 277 eliminates the $\alpha\equiv 9,17\pmod{23}$ cases. 139=23*6+1 , 277=23*12+1[/hide]

LMT Speed Rounds, 2010.4

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Determine the largest positive integer that is a divisor of all three of $A=2^{2010}\times3^{2010}, B=3^{2010}\times5^{2010},$ and $C=5^{2010}\times2^{2010}.$

2000 District Olympiad (Hunedoara), 1

Tags: matrix , algebra , linear algebra

Solve in the set of $ 2\times 2 $ integer matrices the equation $$ X^2-4\cdot X+4\cdot\left(\begin{matrix}1\quad 0\\0\quad 1\end{matrix}\right) =\left(\begin{matrix}7\quad 8\\12\quad 31\end{matrix}\right) . $$

1966 Spain Mathematical Olympiad, 3

Tags: similarity , pentagon , geometry

Given a regular pentagon, consider the convex pentagon limited by its diagonals. You are asked to calculate: a) The similarity relation between the two convex pentagons. b) The relationship of their areas. c) The ratio of the homothety that transforms the first into the second.

2003 JBMO Shortlist, 3

Tags: geometry

Let $G$ be the centroid of triangle $ABC$, and $A'$ the symmetric of $A$ wrt $C$. Show that $G, B, C, A'$ are concyclic if and only if $GA \perp GC$.

PEN A Problems, 91

Tags: divisibility theory

Determine all pairs $(a, b)$ of positive integers such that $ab^2+b+7$ divides $a^2 b+a+b$.