Olympiad problems

2018 South East Mathematical Olympiad, 5

Tags: algebra

Let $\{a_n\}$ be a nonnegative real sequence. Define $$X_k = \sum_{i=1}^{2^k}a_i, Y_k = \sum_{i=1}^{2^k}\left\lfloor \frac{2^k}{i}\right\rfloor a_i, k=0,1,2,...$$ Prove that $X_n\le Y_n - \sum_{i=0}^{n-1} Y_i \le \sum_{i=0}^n X_i$ for all positive integer $n$. Here $\lfloor\alpha\rfloor$ denotes the largest integer that does not exceed $\alpha$.

2012 VJIMC, Problem 3

Tags: inequalities

Determine the smallest real number $C$ such that the inequality $$\frac x{\sqrt{yz}}\cdot\frac1{x+1}+\frac y{\sqrt{zx}}\cdot\frac1{y+1}+\frac z{\sqrt{xy}}\cdot\frac1{x+1}\le C$$holds for all positive real numbers $x,y$ and $z$ with $\frac1{x+1}+\frac1{y+1}+\frac1{z+1}=1$.

2009 AMC 8, 14

Tags:

Austin and Temple are $ 50$ miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging $ 60$ miles per hour. Leaving the car with her daughter, Bonnie rod a bus back to Austin along the same route and averaged $ 40$ miles per hour on the return trip. What was the average speed for the round trip, in miles per hour? $ \textbf{(A)}\ 46 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 54$

1953 Kurschak Competition, 2

Tags: number theory , Perfect Square , divides

$n$ and $d$ are positive integers such that $d$ divides $2n^2$. Prove that $n^2 + d$ cannot be a square.

2005 ITAMO, 1

Tags: number theory proposed , number theory

Determine all $n \geq 3$ for which there are $n$ positive integers $a_1, \cdots , a_n$ any two of which have a common divisor greater than $1$, but any three of which are coprime. Assuming that, moreover, the numbers $a_i$ are less than $5000$, find the greatest possible $n$.

2012 AMC 10, 6

Tags: AMC

In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x>y>0$, Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her values. Which of the following statements is necessarily correct? $ \textbf{(A)}\ \text{Her estimate is larger than }x-y\\ \textbf{(B)}\ \text{Her estimate is smaller than }x-y\\ \textbf{(C)}\ \text{Her estimate equals }x-y\\ \textbf{(D)}\ \text{Her estimate equals }y-x\\ \textbf{(E)}\ \text{Her estimate is }0 $

2007 VJIMC, Problem 4

Tags: inequalities , function

Let $f:[0,1]\to[0,\infty)$ be an arbitrary function satisfying $$\frac{f(x)+f(y)}2\le f\left(\frac{x+y}2\right)+1$$ for all pairs $x,y\in[0,1]$. Prove that for all $0\le u<v<w\le1$, $$\frac{w-v}{w-u}f(u)+\frac{v-u}{w-u}f(w)\le f(v)+2.$$

2004 AMC 12/AHSME, 21

Tags: trigonometry , search

If $ \displaystyle \sum_{n \equal{} 0}^{\infty} \cos^{2n} \theta \equal{} 5$, what is the value of $ \cos{2\theta}$? $ \textbf{(A)}\ \frac15 \qquad \textbf{(B)}\ \frac25 \qquad \textbf{(C)}\ \frac {\sqrt5}{5}\qquad \textbf{(D)}\ \frac35 \qquad \textbf{(E)}\ \frac45$

2024 Poland - Second Round, 6

Tags: factorial , number theory

Given is a prime number $p$. Prove that the number $$p \cdot (p^2 \cdot \frac{p^{p-1}-1}{p-1})!$$ is divisible by $$\prod_{i=1}^{p}(p^i)!.$$

2019 Jozsef Wildt International Math Competition, W. 9

Tags: limit , Sequences

Let $\alpha > 0$ be a real number. Compute the limit of the sequence $\{x_n\}_{n\geq 1}$ defined by $$x_n=\begin{cases} \sum \limits_{k=1}^n \sinh \left(\frac{k}{n^2}\right),& \text{when}\ n>\frac{1}{\alpha}\\ 0,& \text{when}\ n\leq \frac{1}{\alpha}\end{cases}$$

2007 Alexandru Myller, 4

Tags: algebra , finite sequence , induction

Let be a number $ n\ge 2, $ a binary funcion $ b:\mathbb{Z}\rightarrow\mathbb{Z}_2, $ and $ \frac{n^3+5n}{6} $ consecutive integers. Show that among these consecutive integers there are $ n $ of them, namely, $ b_1,b_2,\ldots ,b_n, $ that have the properties: $ \text{(i)} b\left( b_1\right) =b\left( b_2\right) =\cdots =b\left( b_n\right) $ $ \text{(ii)} 1\le b_2-b_1\le b_3-b_2\le \cdots\le b_n-b_{n-1} $

1994 Irish Math Olympiad, 4

Tags: vector , linear algebra , matrix , modular arithmetic , search , linear algebra unsolved

Consider all $ m \times n$ matrices whose all entries are $ 0$ or $ 1$. Find the number of such matrices for which the number of $ 1$-s in each row and in each column is even.

1990 Irish Math Olympiad, 5

Tags: geometry

Let $ABC$ be a right-angled triangle with right-angle at $A$. Let $X$ be the foot of the perpendicular from $A$ to $BC$, and $Y$ the mid-point of $XC$. Let $AB$ be extended to $D$ so that $|AB|=|BD|$. Prove that $DX$ is perpendicular to $AY$.

1966 AMC 12/AHSME, 19

Tags: geometry , 3D geometry , sphere , arithmetic sequence , algebra , linear equation

Let $s_1$ be the sum of the first $n$ terms of the arithmetic sequence $8,12,\cdots$ and let $s_2$ be the sum of the first $n$ terms of the arithmetic sequence $17,19\cdots$. Assume $n\ne 0$. Then $s_1=s_2$ for: $\text{(A)} \ \text{no value of n} \qquad \text{(B)} \ \text{one value of n} \qquad \text{(C)} \ \text{two values of n}$ $\text{(D)} \ \text{four values of n} \qquad \text{(E)} \ \text{more than four values of n}$

1953 AMC 12/AHSME, 40

Tags:

The negation of the statement "all men are honest," is: $ \textbf{(A)}\ \text{no men are honest} \qquad\textbf{(B)}\ \text{all men are dishonest} \\ \textbf{(C)}\ \text{some men are dishonest} \qquad\textbf{(D)}\ \text{no men are dishonest} \\ \textbf{(E)}\ \text{some men are honest}$

1981 Romania Team Selection Tests, 1.

Tags: algebra , polynomial , TST

Consider the polynomial $P(X)=X^{p-1}+X^{p-2}+\ldots+X+1$, where $p>2$ is a prime number. Show that if $n$ is an even number, then the polynomial \[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\] is divisible by $X^2+1$. [i]Mircea Becheanu[/i]

Kvant 2024, M2821

Tags: combinatorics

Peter and Basil take turns drawing roads on a plane, Peter starts. The road is either horizontal or a vertical line along which one can drive in only one direction (that direction is determined when the road is drawn). Can Basil always act in such a way that after each of his moves one could drive according to the rules between any two constructed crossroads, regardless of Peter's actions? Alexandr Perepechko

2015 IFYM, Sozopol, 2

Tags: function , induction , number theory proposed , number theory

Find all functions $f$ from positive integers to themselves such that: 1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$ 2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.

2012 Germany Team Selection Test, 1

Tags: algebra , polynomial , pigeonhole principle , number theory , IMO Shortlist , combinatorics

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

2001 Moldova National Olympiad, Problem 7

Tags: geometry , ratio

A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e. lines connecting the midpoints of two sides) in the same ratio. Determine this ratio.

2014-2015 SDML (Middle School), 12

Tags: function , real analysis , 2014-15 SDML Round 1 , SDML High School , SDML Middle School

Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$. $\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$

2012 ELMO Shortlist, 10

Tags: geometry , conics , circumcircle , projective geometry , algebra proposed , algebra

Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic. [i]David Yang.[/i]

1993 National High School Mathematics League, 2

Tags: function

$f(x)=a\sin x+b\sqrt[3]{x}+4$. If $f(\lg\log_{3}10)=5$, then the value of $f(\lg\lg 3)$ is $\text{(A)}-5\qquad\text{(B)}-3\qquad\text{(C)}3\qquad\text{(D)}$ not sure

2010 Oral Moscow Geometry Olympiad, 1

Tags: geometry , combinatorial geometry , combinatorics , Cyclic , polygon , convex polygon

Convex $n$-gon $P$, where $n> 3$, is cut into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is cyclic?

2019 Korea USCM, 5

Tags: Sequences , series , exponential , logarithms , college contests

A sequence $\{a_n\}_{n\geq 1}$ is defined by a recurrence relation $$a_1 = 1,\quad a_{n+1} = \log \frac{e^{a_n}-1}{a_n}$$ And a sequence $\{b_n\}_{n\geq 1}$ is defined as $b_n = \prod\limits_{i=1}^n a_i$. Evaluate an infinite series $\sum\limits_{n=1}^\infty b_n$.