Olympiad problems

2006 Stanford Mathematics Tournament, 24

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The number 555,555,555,555 factors into eight distinct prime factors, each with a multiplicity of 1. What are the three largest prime factors of 555,555,555,555?

2011 Morocco TST, 2

Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$ [i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]

2010 Moldova National Olympiad, 9.1

Tags: algebra , cauchy inequality

$a$,$b$,$c$ are real. What is the highest value of $a+b+c$ if $a^2+4b^2+9c^2-2a-12b+6c+2=0$

2021 AMC 12/AHSME Spring, 1

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How many integer values satisfy $|x|<3\pi$? $\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20$

2004 China Team Selection Test, 1

Tags: geometric transformation , geometry

Points $D,E,F$ are on the sides $BC, CA$ and $AB$, respectively which satisfy $EF || BC$, $D_1$ is a point on $BC,$ Make $D_1E_1 || D_E, D_1F_1 || DF$ which intersect $AC$ and $AB$ at $E_1$ and $F_1$, respectively. Make $\bigtriangleup PBC \sim \bigtriangleup DEF$ such that $P$ and $A$ are on the same side of $BC.$ Prove that $E, E_1F_1, PD_1$ are concurrent. [color=red][Edit by Darij: See my post #4 below for a [b]possible correction[/b] of this problem. However, I am not sure that it is in fact the problem given at the TST... Does anyone have a reliable translation?][/color]

PEN M Problems, 8

Tags: algebra , polynomial , recursive sequences

The Bernoulli sequence $\{B_{n}\}_{n \ge 0}$ is defined by \[B_{0}=1, \; B_{n}=-\frac{1}{n+1}\sum^{n}_{k=0}{{n+1}\choose k}B_{k}\;\; (n \ge 1)\] Show that for all $n \in \mathbb{N}$, \[(-1)^{n}B_{n}-\sum \frac{1}{p},\] is an integer where the summation is done over all primes $p$ such that $p| 2k-1$.

2010 Philippine MO, 2

Tags: geometry , cyclic quadrilateral , perimeter

On a cyclic quadrilateral $ABCD$, there is a point $P$ on side $AD$ such that the triangle $CDP$ and the quadrilateral $ABCP$ have equal perimeters and equal areas. Prove that two sides of $ABCD$ have equal lengths.

2007 National Olympiad First Round, 31

Tags: geometry , perimeter

A square-shaped field is divided into $n$ rectangular farms whose sides are parallel to the sides of the field. What is the greatest value of $n$, if the sum of the perimeters of the farms is equal to $100$ times of the perimeter of the field? $ \textbf{(A)}\ 10000 \qquad\textbf{(B)}\ 20000 \qquad\textbf{(C)}\ 50000 \qquad\textbf{(D)}\ 100000 \qquad\textbf{(E)}\ 200000 $

2024 OMpD, 4

Tags: function , real analysis

Let $ n $ be a positive integer. Determine the largest possible value of $ k $ with the following property: there exists a bijective function $ \phi: [0, 1] \to [0, 1]^k $ and a constant $ C > 0 $ such that, for all $ x, y \in [0, 1] $, \[ \| \phi(x) - \phi(y) \| \leq C \| x - y \|^k. \] Note: $ \| \cdot \| $ denotes the Euclidean norm, that is, $ \| (a_1, \ldots, a_n) \| = \sqrt{a_1^2 + \cdots + a_n^2} $.

2019 Nigerian Senior MO Round 3, 2

Tags: triangle inequality , inequalities , algebra

Let $abc$ be real numbers satisfying $ab+bc+ca=1$. Show that $\frac{|a-b|}{|1+c^2|}$ + $\frac{|b-c|}{|1+a^2|}$ $>=$ $\frac{|c-a|}{|1+b^2|}$

2017 ASDAN Math Tournament, 4

Tags: algebra test

What is the maximum possible value for the sum of the squares of the roots of $x^4+ax^3+bx^2+cx+d$ where $a$, $b$, $c$, and $d$ are $2$, $0$, $1$, and $7$ in some order?

1996 All-Russian Olympiad, 5

Tags: ratio , euler , arithmetic sequence , number theory

Show that in the arithmetic progression with first term 1 and ratio 729, there are infinitely many powers of 10. [i]L. Kuptsov[/i]

1986 Canada National Olympiad, 4

Tags: number theory

For all positive integers $n$ and $k$, define $F(n,k) = \sum_{r = 1}^n r^{2k - 1}$. Prove that $F(n,1)$ divides $F(n,k)$.

2019 Swedish Mathematical Competition, 6

Tags: sequence , perfect square , number theory

Is there an infinite sequence of positive integers $\{a_n\}_{n = 1}^{\infty}$ which contains each positive integer exactly once and is such that the number $a_n + a_{n + 1} $ is a perfect square for each $n$?

1993 AMC 8, 16

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$\dfrac{1}{1+\dfrac{1}{2+\dfrac{1}{3}}} =$ $\text{(A)}\ \dfrac{1}{6} \qquad \text{(B)}\ \dfrac{3}{10} \qquad \text{(C)}\ \dfrac{7}{10} \qquad \text{(D)}\ \dfrac{5}{6} \qquad \text{(E)}\ \dfrac{10}{3}$

2013 India PRMO, 8

Tags: geometry , trapezium , trapezoid

Let $AD$ and $BC$ be the parallel sides of a trapezium $ABCD$. Let $P$ and $Q$ be the midpoints of the diagonals $AC$ and $BD$. If $AD = 16$ and $BC = 20$, what is the length of $PQ$?

1976 Miklós Schweitzer, 11

Tags: probability , geometry , geometric transformation , reflection , probability and stats , logarithm

Let $ \xi_1,\xi_2,...$ be independent, identically distributed random variables with distribution \[ P(\xi_1=-1)=P(\xi_1=1)=\frac 12 .\] Write $ S_n=\xi_1+\xi_2+...+\xi_n \;(n=1,2,...),\ \;S_0=0\ ,$ and \[ T_n= \frac{1}{\sqrt{n}} \max _{ 0 \leq k \leq n}S_k .\] Prove that $ \liminf_{n \rightarrow \infty} (\log n)T_n=0$ with probability one. [i]P. Revesz[/i]

2010 All-Russian Olympiad, 4

Tags: rhombus , geometry

In a acute triangle $ABC$, the median, $AM$, is longer than side $AB$. Prove that you can cut triangle $ABC$ into $3$ parts out of which you can construct a rhombus.

2018-IMOC, G5

Tags: geometry , incenter , circumcenter , orthocenter , parallel , euler line

Suppose $I,O,H$ are incenter, circumcenter, orthocenter of $\vartriangle ABC$ respectively. Let $D = AI \cap BC$,$E = BI \cap CA$, $F = CI \cap AB$ and $X$ be the orthocenter of $\vartriangle DEF$. Prove that $IX \parallel OH$.

2012 IFYM, Sozopol, 7

Tags: midpoint , geometry , orthocenter

Let $\Delta ABC$ be a triangle with orthocenter $H$ and midpoints $M_a,M_b$, and $M_c$ of $BC$, $AC$, and $AB$ respectively. A circle with center $H$ intersects the lines $M_bM_a$, $M_bM_c$, and $M_cM_a$ in points $U_1,U_2,V_1,V_2,W_1,W_2$ respectively. Prove that $CU_1=CU_2=AV_1=AV_2=BW_1=BW_2$.

2019 AMC 12/AHSME, 6

Tags: ellipse , conic

In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units? $\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$

2008 Grigore Moisil Intercounty, 2

Tags: algebra

Prove that the equation $ z^3\plus{}z^2\plus{}z\minus{}6\equal{}0$ doesn't have roots $ x$ with $ |x|\equal{}2$.

1970 IMO Longlists, 17

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

In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?

2020 Paraguay Mathematical Olympiad, 2

Tags: algebra , recurrence relation

Laura is putting together the following list: $a_0, a_1, a_2, a_3, a_4, ..., a_n$, where $a_0 = 3$ and $a_1 = 4$. She knows that the following equality holds for any value of $n$ integer greater than or equal to $1$: $$a_n^2-2a_{n-1}a_{n+1} =(-2)^n.$$Laura calculates the value of $a_4$. What value does it get?

2020 Purple Comet Problems, 19

Tags: number theory

Find the least prime number greater than $1000$ that divides $2^{1010} \cdot 23^{2020} + 1$.