Olympiad problems

2006 Sharygin Geometry Olympiad, 5

Tags: paper , folding , square , combinatorics , combinatorial geometry , geometry

a) Fold a $10 \times 10$ square from a $1 \times 118$ rectangular strip. b) Fold a $10 \times 10$ square from a $1 \times (100+9\sqrt3)$ rectangular strip (approximately $1\times 115.58$). The strip can be bent, but not torn.

2019 CHMMC (Fall), 8

Tags: algebra

Consider an infinite sequence of reals $x_1, x_2, x_3, ...$ such that $x_1 = 1$, $x_2 =\frac{2\sqrt3}{3}$ and with the recursive relationship $$n^2 (x_n - x_{n-1} - x_{n-2}) - n(3x_n + 2x_{n-1} + x_{n-2}) + (x_nx_{n-1}x_{n-2} + 2x_n) = 0.$$ Find $x_{2019}$.

2020 Online Math Open Problems, 7

Tags: Online Math Open

On a $9\times 9$ square lake composed of unit squares, there is a $2\times 4$ rectangular iceberg also composed of unit squares (it could be in either orientation; that is, it could be $4\times 2$ as well). The sides of the iceberg are parallel to the sides of the lake. Also, the iceberg is invisible. Lily is trying to sink the iceberg by firing missiles through the lake. Each missile fires through a row or column, destroying anything that lies in its row or column. In particular, if Lily hits the iceberg with any missile, she succeeds. Lily has bought $n$ missiles and will fire all $n$ of them at once. Let $N$ be the smallest possible value of $n$ such that Lily can guarantee that she hits the iceberg. Let $M$ be the number of ways for Lily to fire $N$ missiles and guarantee that she hits the iceberg. Compute $100M+N$. [i]Proposed by Brandon Wang[/i]

1956 AMC 12/AHSME, 39

Tags:

The hypotenuse $ c$ and one arm $ a$ of a right triangle are consecutive integers. The square of the second arm is: $ \textbf{(A)}\ ca \qquad\textbf{(B)}\ \frac {c}{a} \qquad\textbf{(C)}\ c \plus{} a \qquad\textbf{(D)}\ c \minus{} a \qquad\textbf{(E)}\ \text{none of these}$

2014 India Regional Mathematical Olympiad, 2

Tags: inequalities , inequalities unsolved

let $x,y$ be positive real numbers. prove that $ 4x^4+4y^3+5x^2+y+1\geq 12xy $

LMT Team Rounds 2021+, B6

Tags: combinatorics

Maisy is at the origin of the coordinate plane. On her first step, she moves $1$ unit up. On her second step, she moves $ 1$ unit to the right. On her third step, she moves $2$ units up. On her fourth step, she moves $2$ units to the right. She repeats this pattern with each odd-numbered step being $ 1$ unit more than the previous step. Given that the point that Maisy lands on after her $21$st step can be written in the form $(x, y)$, find the value of $x + y$. Proposed by Audrey Chun

1971 Bulgaria National Olympiad, Problem 1

Tags: number theory

A natural number is called [i]triangular[/i] if it may be presented in the form $\frac{n(n+1)}2$. Find all values of $a$ $(1\le a\le9)$ for which there exist a triangular number all digit of which are equal to $a$.

2024 Portugal MO, 1

Tags: number theory

A number is called cool if the sum of its digits is multiple of $17$ and the sum of digits of its successor is multiple of $17$. What is the smallest cool number?

2017 IFYM, Sozopol, 3

Tags: number theory

A row of $2n$ real numbers is called [i]“Sozopolian”[/i], if for each $m$, such that $1\leq m\leq 2n$, the sum of the first $m$ members of the row is an integer or the sum of the last $m$ members of the row is an integer. What’s the least number of integers that a [i]Sozopolian[/i] row can have, if the number of its members is: a) 2016; b) 2017?

2008 Peru MO (ONEM), 2

Tags: trigonometry , algebra , inequalities

Let $a$ and $b$ be real numbers for which the following is true: $acscx + b cot x \ge 1$, for all $0 <x < \pi$ Find the least value of $a^2 + b$.

2018 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorics , Coloring

In $n$ transparent boxes there are red balls and blue balls. One needs to choose $50$ boxes such that, together, they contain at least half of the red balls and at least half of the blue balls. Is such a choice possible irrespective on the number of balls and on the way they are distributed in the boxes, if: a) $n = 100$ b) $n = 99$?

2010 QEDMO 7th, 9

Tags: number theory , divisible

Let $p$ be an odd prime number and $c$ an integer for which $2c -1$ is divisible by $p$. Prove that $$(-1)^{\frac{p+1}{2}}+\sum_{n=0}^{\frac{p-1}{2}} {2n \choose n}c^n$$ is divisible by $p$.

2008 Sharygin Geometry Olympiad, 3

Tags: inequalities , geometry , inradius , trigonometry , geometry unsolved

(R.Pirkuliev) Prove the inequality \[ \frac1{\sqrt {2\sin A}} \plus{} \frac1{\sqrt {2\sin B}} \plus{} \frac1{\sqrt {2\sin C}}\leq\sqrt {\frac {p}{r}}, \] where $ p$ and $ r$ are the semiperimeter and the inradius of triangle $ ABC$.

2025 Spain Mathematical Olympiad, 4

Tags: geometry , Spain

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$, satisfying $AB<AC$. The tangent line at $A$ to the circumcicle of $ABC$ intersects $BC$ in $T$. Let $X$ be the midpoint of $AH$. Prove that $\angle ATX=\angle OTB$.

1950 AMC 12/AHSME, 24

Tags:

The equation $ x\plus{}\sqrt{x\minus{}2}\equal{}4$ has: $\textbf{(A)}\ \text{2 real roots} \qquad \textbf{(B)}\ \text{1 real and 1 imaginary root} \qquad \textbf{(C)}\ \text{2 imaginary roots} \qquad \textbf{(D)}\ \text{No roots} \qquad \textbf{(E)}\ \text{1 real root}$

2012 Philippine MO, 3

Tags: trigonometry , inequalities , inequalities unsolved

If $ab>0$ and $\displaystyle 0<x<\frac{\pi}{2}$, prove that \[ \left ( 1+\frac{a^2}{\sin x} \right ) \left ( 1+\frac{b^2}{\cos x} \right ) \geq \frac{(1+\sqrt{2}ab)^2 \sin 2x}{2}. \]

PEN H Problems, 66

Tags: inequalities , Diophantine Equations

Let $b$ be a positive integer. Determine all $2002$-tuples of non-negative integers $(a_{1}, a_{2}, \cdots, a_{2002})$ satisfying \[\sum^{2002}_{j=1}{a_{j}}^{a_{j}}=2002{b}^{b}.\]

2008 ISI B.Math Entrance Exam, 10

Tags: number theory , greatest common divisor

If $p$ is a prime number and $a>1$ is a natural number , then show that the greatest common divisor of the two numbers $a-1$ and $\frac{a^p-1}{a-1}$ is either $1$ or $p$ .

2009 Indonesia MO, 2

Tags: inequalities , function , floor function , number theory unsolved , number theory

For any real $ x$, let $ \lfloor x\rfloor$ be the largest integer that is not more than $ x$. Given a sequence of positive integers $ a_1,a_2,a_3,\ldots$ such that $ a_1>1$ and \[ \left\lfloor\frac{a_1\plus{}1}{a_2}\right\rfloor\equal{}\left\lfloor\frac{a_2\plus{}1}{a_3}\right\rfloor\equal{}\left\lfloor\frac{a_3\plus{}1}{a_4}\right\rfloor\equal{}\cdots\] Prove that \[ \left\lfloor\frac{a_n\plus{}1}{a_{n\plus{}1}}\right\rfloor\leq1\] holds for every positive integer $ n$.

2018 AMC 12/AHSME, 14

Tags: AMC , AMC 10 , 2018 AMC 10B , 2018 AMC , 2018 AMC 12B , AMC 12

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age? $\textbf{(A) } 7 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11 $

2023 AMC 10, 24

Tags: 2023 amc 10A , 2023 AMC

Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is $\frac{3}{7}$ unit. What is the area of the region inside the frame not occupied by the blocks? [asy] unitsize(1cm); draw(scale(3)*polygon(6)); filldraw(shift(dir(0)*2+dir(120)*0.4)*polygon(6), lightgray); filldraw(shift(dir(60)*2+dir(180)*0.4)*polygon(6), lightgray); filldraw(shift(dir(120)*2+dir(240)*0.4)*polygon(6), lightgray); filldraw(shift(dir(180)*2+dir(300)*0.4)*polygon(6), lightgray); filldraw(shift(dir(240)*2+dir(360)*0.4)*polygon(6), lightgray); filldraw(shift(dir(300)*2+dir(420)*0.4)*polygon(6), lightgray); [/asy] $\textbf{(A)}~\frac{13 \sqrt{3}}{3}\qquad\textbf{(B)}~\frac{216 \sqrt{3}}{49}\qquad\textbf{(C)}~\frac{9 \sqrt{3}}{2} \qquad\textbf{(D)}~ \frac{14 \sqrt{3}}{3}\qquad\textbf{(E)}~\frac{243 \sqrt{3}}{49}$

2011 Brazil Team Selection Test, 2

Tags: geometry , geometric transformation , combinatorics proposed , combinatorics

Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$.

PEN A Problems, 75

Tags: modular arithmetic , Divisibility Theory

Find all triples $(a,b,c)$ of positive integers such that $2^{c}-1$ divides $2^{a}+2^{b}+1$.

Kvant 2022, M2689

Tags: combinatorics , Kvant

There are 1000 gentlemen listed in the register of a city with numbers from 1 to 1000. Any 720 of them form a club. The mayor wants to impose a tax on each club, which is paid by all club members in equal shares (the tax is an arbitrary non-negative real number). At the same time, the total tax paid by a gentleman should not exceed his number in the register. What is the largest tax the mayor can collect? [i]Proposed by I. Bogdanov[/i]

2024 South Africa National Olympiad, 3

Tags: combinatorics

Each of the lattice points $(x,y)$ (where $x$ and $y$ are integers) in the plane can be coloured black or white. A single strike by an $L$-shaped punch changes the colour of the four lattice points $(a,b)$, $(a+1,b)$, $(a,b+1)$ and $(a,b+2)$. All lattice points are initially coloured white. Prove that after any number of strikes, the number of black lattice points will be either zero or greater than or equal to four.