Olympiad problems

2014 Putnam, 1

A [i]base[/i] 10 [i]over-expansion[/i] of a positive integer $N$ is an expression of the form $N=d_k10^k+d_{k-1}10^{k-1}+\cdots+d_0 10^0$ with $d_k\ne 0$ and $d_i\in\{0,1,2,\dots,10\}$ for all $i.$ For instance, the integer $N=10$ has two base 10 over-expansions: $10=10\cdot 10^0$ and the usual base 10 expansion $10=1\cdot 10^1+0\cdot 10^0.$ Which positive integers have a unique base 10 over-expansion?

2003 China Western Mathematical Olympiad, 3

Tags: inequalities

The non-negative numbers $ x_1, x_2, \ldots, x_5$ satisfy $ \sum_{i \equal{} 1}^5 \frac {1}{1 \plus{} x_i} \equal{} 1$. Prove that $ \sum_{i \equal{} 1}^5 \frac {x_i}{4 \plus{} x_i^2} \leq 1$.

CIME I 2018, 2

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An underground line has $26$ stops, including the first and the final one, and all the stops are numbered from $1$ to $26$ according to their order. Inside the train, for each pair $(x,y)$ with $1\leq x < y \leq 26$ there is exactly one passenger that goes from the $x$-th stop to the $y$-th one. If every passenger wants to take a seat during his journey, find the minimum number of seats that must be available on the train. [i]Proposed by [b]FedeX333X[/b][/i]

2007 Grigore Moisil Intercounty, 2

Tags: algebra , polynomial , complex number

Prove that if all roots of a monic cubic polynomial have modulus $ 1, $ then, the two middle coefficients have the same modulus.

1979 IMO Longlists, 33

Tags: trigonometry , inequality , geometric inequality , trigonometric identities , inequalities

Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

2018-2019 Winter SDPC, 2

Tags: number theory

Call a number [i]precious[/i] if it is the sum of two distinct powers of two. Find all precious numbers $n$ such that $n^2$ is also precious.

2016 PUMaC Geometry A, 7

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ and let $AC$ and $BD$ intersect at $X$. Let the line through $A$ parallel to $BD$ intersect line $CD$ at $E$ and $\omega$ at $Y \ne A$. If $AB = 10, AD = 24, XA = 17$, and $XB = 21$, then the area of $\vartriangle DEY$ can be written in simplest form as $\frac{m}{n}$ . Find $m + n$.

1987 Polish MO Finals, 1

Tags: point , combinatorial geometry , combinatorics , square , distance , geometry

There are $n \ge 2$ points in a square side $1$. Show that one can label the points $P_1, P_2, ... , P_n$ such that $\sum_{i=1}^n |P_{i-1} - P_i|^2 \le 4$, where we use cyclic subscripts, so that $P_0$ means $P_n$.

2017 Online Math Open Problems, 2

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The numbers $a,b,c,d$ are $1,2,2,3$ in some order. What is the greatest possible value of $a^{b^{c^d}}$? [i]Proposed by Yannick Yao and James Lin[/i]

2023 Iran MO (3rd Round), 2

Tags: geometry

In triangle $\triangle ABC$ , $M$ is the midpoint of arc $(BAC)$ and $N$ is the antipode of $A$ in $(ABC)$. The line through $B$ perpendicular to $AM$ , intersects $AM , (ABC)$ at $D,P$ respectively and a line through $D$ perpendicular to $AC$ , intersects $BC,AC$ at $F,E$ respectively. Prove that $PE,MF,ND$ are concurrent.

2021 BMT, 12

Tags: algebra

Let $a$, $b$, and $c$ be the solutions of the equation $$x^3 - 3 \cdot 2021^2x = 2 \cdot 20213.$$ Compute $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$

1985 ITAMO, 7

Tags:

Assume that $a$, $b$, $c$, and $d$ are positive integers such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$. Determine $d - b$.

Revenge EL(S)MO 2024, 5

Tags: hyperbola , ellipse , geometry , conic

In triangle $ABC$ let the $A$-foot be $E$ and the $B$-excenter be $L$. Suppose the incircle of $ABC$ is tangent to $AC$ at $I$. Construct a hyperbola $\mathcal H$ through $A$ with $B$ and $C$ as the foci such that $A$ lies on the branch of the $\mathcal H$ closer to $C$. Construct an ellipse $\mathcal E$ passing through $I$ with $B$ and $C$ as the foci. Suppose $\mathcal E$ meets $\overline{AB}$ again at point $H$. Let $\overline{CH}$ and $\overline{BI}$ intersect the $C$-branch of $\mathcal H$ at points $M$ and $O$ respectively. Prove $E$, $L$, $M$, $O$ are concyclic. Proposed by [i]Alex Wang[/i]

2010 Postal Coaching, 4

Tags: relatively prime , number theory

For each $n\in \mathbb{N}$, let $S(n)$ be the sum of all numbers in the set $\{ 1, 2, 3, \cdots , n \}$ which are relatively prime to $n$. $(a)$ Show that $2 \cdot S(n)$ is not a perfect square for any $n$. $(b)$ Given positive integers $m, n$, with odd $n$, show that the equation $2 \cdot S(x) = y^n$ has at least one solution $(x, y)$ among positive integers such that $m|x$.

2007 India IMO Training Camp, 3

Tags: function , ratio , algebra

Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation \[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\] For all $x,y\in\mathbb R.$

1969 IMO, 3

Tags: geometry , 3d geometry , tetrahedron , triangle inequality

For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.

2021 Middle European Mathematical Olympiad, 7

Tags: number theory , diophantine equation , easy number theorem

Find all pairs $(n, p)$ of positive integers such that $p$ is prime and \[ 1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdot + p^2). \]

1951 AMC 12/AHSME, 49

Tags: pythagorean theorem , geometry

The medians of a right triangle which are drawn from the vertices of the acute angles are $ 5$ and $ \sqrt {40}$. The value of the hypotenuse is: $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 2\sqrt {40} \qquad\textbf{(C)}\ \sqrt {13} \qquad\textbf{(D)}\ 2\sqrt {13} \qquad\textbf{(E)}\ \text{none of these}$

1995 AIME Problems, 13

Tags: floor function , inequalities

Let $f(n)$ be the integer closest to $\sqrt[4]{n}.$ Find $\sum_{k=1}^{1995}\frac 1{f(k)}.$

1955 Moscow Mathematical Olympiad, 311

Tags: number theory , floor function

Find all numbers $a$ such that (1) all numbers $[a], [2a], . . . , [Na]$ are distinct and (2) all numbers $\left[ \frac{1}{a}\right], \left[ \frac{2}{a}\right], ..., \left[ \frac{M}{a}\right]$ are distinct.

2016 German National Olympiad, 6

Tags: cyclic function , function , inequality , maximum , algebra

Let \[ f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3 \] be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$. Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.

2017 Online Math Open Problems, 23

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Call a nonempty set $V$ of nonzero integers \emph{victorious} if there exists a polynomial $P(x)$ with integer coefficients such that $P(0)=330$ and that $P(v)=2|v|$ holds for all elements $v\in V$. Find the number of victorious sets. [i]Proposed by Yannick Yao[/i]

2004 Germany Team Selection Test, 2

Tags: geometry , fixed point

Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.

2003 Austria Beginners' Competition, 3

Tags: number theory , consecutive , divide , divisible

a) Show that the product of $5$ consecutive even integers is divisible by $15$. b) Determine the largest integer $D$ such that the product of $5$ consecutive even integers is always divisible by $D$.

2010 Junior Balkan Team Selection Tests - Moldova, 5

Tags: inequalities

For any strictly positive numbers $a$ and $b$ , prove the inequality $$\frac{a}{a+b} \cdot \frac{a+2b}{a+3b} < \sqrt{ \frac{a}{a+4b}}.$$