Olympiad problems

2002 China Team Selection Test, 3

The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$. Prove that $ \alpha\beta$ is not a perfect square.

1992 Irish Math Olympiad, 1

Tags: algebra

Let $n > 2$ be an integer and let $m = \sum k^3$, where the sum is taken over all integers $k$ with $1 \leq k < n$ that are relatively prime to $n$. Prove that $n$ divides $m$.

V Soros Olympiad 1998 - 99 (Russia), 11.6

Tags: geometry , rectangle

Cut the $10$ cm $x 20$ cm rectangle into two pieces with one straight cut so that they can be placed inside the $19.4$ cm diameter circle without intersecting.

2015 Federal Competition For Advanced Students, P2, 4

Tags: algebra , inequalities

Let $x,y,z$ be positive real numbers with $x+y+z \ge 3$. Prove that $\frac{1}{x+y+z^2} + \frac{1}{y+z+x^2} + \frac{1}{z+x+y^2} \le 1$ When does equality hold? (Karl Czakler)

2009 Stanford Mathematics Tournament, 15

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What is the largest integer $n$ for which $\frac{2008!}{31^n}$ is an integer?

1994 AMC 12/AHSME, 24

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A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$. The smallest value that the range (largest observation minus smallest) can assume for such a sample is $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 10 $

1996 Tournament Of Towns, (503) 6

Tags: combinatorics

At first all $2^n$ rows of a $2^n \times n$ table were filled with all different $n$-tuples of numbers $+1$ and $-1$. Then some of the numbers were replaced by Os. Prove that one can choose a (non-empty) set of rows such that: (a) the sum of all the numbers in all the chosen rows is $0$; (b) the sum of all the chosen rows equals the zero row, that is, the sum of numbers in each column of the chosen rows equals $0$. (G Kondakov, V Chernorutskii)

1994 Tournament Of Towns, (440) 6

Tags: digit , number theory

Let $c_n$ be the first digit of $2^n$ (in decimal representation). Prove that the number of different $13$-tuples $< c_k$,$...$, $c_{k+12}>$ is equal to $57$. (AY Belov,)

2006 Harvard-MIT Mathematics Tournament, 4

Tags: geometry , area of a triangle , heron's formula

Let $ABC$ be a triangle such that $AB=2$, $CA=3$, and $BC=4$. A semicircle with its diameter on $BC$ is tangent to $AB$ and $AC$. Compute the area of the semicircle.

2024 Philippine Math Olympiad, P6

Tags: floor function , number theory

The sequence $\{a_n\}_{n\ge 1}$ of real numbers is defined as follows: $$a_1=1, \quad \text{and}\quad a_{n+1}=\frac{1}{2\lfloor a_n \rfloor -a_n+1} \quad \text{for all} \quad n\ge 1$$ Find $a_{2024}$.

2020 CHMMC Winter (2020-21), 5

Tags: combinatorics , algebra

Suppose that a professor has $n \ge 4$ students. Let $P$ denote the set of all ordered pairs $(n, k)$ such that the number of ways for the professor to choose one pair of students equals the number of ways for the professor to choose $k > 1$ pairs of students. For each such ordered pair $(n, k) \in P$, consider the sum $n+k=s$. Find the sum of all $s$ over all ordered pairs $(n, k)$ in $P$. [i]If the same value of $s$ appears in multiple distinct elements $(n, k)$ in $P$, count this value multiple times.[/i]

2021 Simon Marais Mathematical Competition, A2

Tags: sequence , number theory , greatest common divisor

Define the sequence of integers $a_1, a_2, a_3, \ldots$ by $a_1 = 1$, and \[ a_{n+1} = \left(n+1-\gcd(a_n,n) \right) \times a_n \] for all integers $n \ge 1$. Prove that $\frac{a_{n+1}}{a_n}=n$ if and only if $n$ is prime or $n=1$. [i]Here $\gcd(s,t)$ denotes the greatest common divisor of $s$ and $t$.[/i]

2023 Iran MO (2nd Round), P2

Tags: number theory

2. Prove that for any $2\le n \in \mathbb{N}$ there exists positive integers $a_1,a_2,\cdots,a_n$ such that $\forall i\neq j: \text{gcd}(a_i,a_j) = 1$ and $\forall i: a_i \ge 1402$ and the given relation holds. $$[\frac{a_1}{a_2}]+[\frac{a_2}{a_3}]+\cdots+[\frac{a_n}{a_1}] = [\frac{a_2}{a_1}]+[\frac{a_3}{a_2}]+\cdots+[\frac{a_1}{a_n}]$$

2020 LMT Fall, 18

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Given that $\sqrt{x+2y}-\sqrt{x-2y}=2,$ compute the minimum value of $x+y.$ [i]Proposed by Alex Li[/i]

1994 AIME Problems, 9

Tags: probability , number theory , relatively prime , recursion

A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$

2014 Iran Team Selection Test, 4

Tags: function , algebra , functional equation

Find all functions $f:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ such that $x,y\in \mathbb{R}^{+},$ \[ f\left(\frac{y}{f(x+1)}\right)+f\left(\frac{x+1}{xf(y)}\right)=f(y) \]

2013 Hanoi Open Mathematics Competitions, 15

Tags: rational , algebra

Denote by $Q$ and $N^*$ the set of all rational and positive integer numbers, respectively. Suppose that $\frac{ax + b}{x} \in Q$ for every $x \in N^*$: Prove that there exist integers $A,B,C$ such that $\frac{ax + b}{x}= \frac{Ax + B}{Cx}$ for all $x \in N^* $

1987 Tournament Of Towns, (157) 1

Tags: area , perpendicular , square , geometry

From vertex $A$ in square $ABCD$ (of side length $1$ ) two lines are drawn , one intersecting side $BC$ and the other intersecting side $CD$. The angle between these lines is $\theta$. From vertices $B$ and $D$ we construct perpendiculars to each of these lines . Find the area of the quadrilateral whose vertices are the four feet of these perpendiculars.

1941 Putnam, B2

Tags: convergence , series

Find (i) $\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i^{2}}}$. (ii) $\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i}}$. (iii) $\lim_{n\to \infty} \sum_{i=1}^{n^{2}} \frac{1}{\sqrt{n^2 +i}}$.

1999 Polish MO Finals, 2

Tags: integration , function , triangle inequality , inequalities

Prove that for any $ 2n$ real numbers $ a_{1}$, $ a_{2}$, ..., $ a_{n}$, $ b_{1}$, $ b_{2}$, ..., $ b_{n}$, we have $ \sum_{i < j}{\left|a_{i}\minus{}a_{j}\right|}\plus{}\sum_{i < j}{\left|b_{i}\minus{}b_{j}\right|}\leq\sum_{i,j\in\left[1,n\right]}{\left|a_{i}\minus{}b_{j}\right|}$.

1997 AIME Problems, 9

Tags: floor function

Given a nonnegative real number $x,$ let $\langle x\rangle$ denote the fractional part of $x;$ that is, $\langle x\rangle=x-\lfloor x\rfloor,$ where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x.$ Suppose that $a$ is positive, $\langle a^{-1}\rangle=\langle a^2\rangle,$ and $2<a^2<3.$ Find the value of $a^{12}-144a^{-1}.$

1995 Israel Mathematical Olympiad, 5

Tags: inequalities , algebra

Let $n$ be an odd positive integer and let $x_1,x_2,...,x_n$ be n distinct real numbers that satisfy $|x_i -x_j| \le 1$ for $1 \le i < j \le n$. Prove that $$\sum_{i<j} |x_i -x_j| \le \left[\frac{n}{2} \right] \left(\left[\frac{n}{2} \right]-1 \right)$$

2013 Romanian Master of Mathematics, 5

Tags: floor function , induction , number theory

Given a positive integer $k\geq2$, set $a_1=1$ and, for every integer $n\geq 2$, let $a_n$ be the smallest solution of equation \[x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_i}}\right\rfloor\] that exceeds $a_{n-1}$. Prove that all primes are among the terms of the sequence $a_1,a_2,\ldots$

1969 IMO Longlists, 19

Tags: number theory , relatively prime , periodic sequence , divisibility

$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$

1973 Bundeswettbewerb Mathematik, 1

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A positive integer has 1000 digits (decimal system), all but at most one of them being the digit $5$. Show that this number isn't a perfect square.