Olympiad problems

1991 Greece Junior Math Olympiad, 4

Tags: algebra

Let $x+y=a$ and $xy=b$. Calculate exression $ x^4+y^4$ in terms of $a$ and $b$.

2016 Benelux, 2

Tags: number theory

Let $n$ be a positive integer. Suppose that its positive divisors can be partitioned into pairs (i.e. can be split in groups of two) in such a way that the sum of each pair is a prime number. Prove that these prime numbers are distinct and that none of these are a divisor of $n.$

1983 USAMO, 2

Tags: USA(J)MO , USAMO , calculus , derivative , quadratics , algebra , marvio

Prove that the roots of\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\] cannot all be real if $2a^2 < 5b$.

2023 Korea - Final Round, 2

Tags: algebra , function , FKMO

Function $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies the following condition. (Condition) For each positive real number $x$, there exists a positive real number $y$ such that $(x + f(y))(y + f(x)) \leq 4$, and the number of $y$ is finite. Prove $f(x) > f(y)$ for any positive real numbers $x < y$. ($\mathbb{R^+}$ is a set for all positive real numbers.)

1990 IMO Shortlist, 6

Tags: combinatorics , game , game strategy , algorithm , IMO , IMO 1990

Given an initial integer $ n_0 > 1$, two players, $ {\mathcal A}$ and $ {\mathcal B}$, choose integers $ n_1$, $ n_2$, $ n_3$, $ \ldots$ alternately according to the following rules : [b]I.)[/b] Knowing $ n_{2k}$, $ {\mathcal A}$ chooses any integer $ n_{2k \plus{} 1}$ such that \[ n_{2k} \leq n_{2k \plus{} 1} \leq n_{2k}^2. \] [b]II.)[/b] Knowing $ n_{2k \plus{} 1}$, $ {\mathcal B}$ chooses any integer $ n_{2k \plus{} 2}$ such that \[ \frac {n_{2k \plus{} 1}}{n_{2k \plus{} 2}} \] is a prime raised to a positive integer power. Player $ {\mathcal A}$ wins the game by choosing the number 1990; player $ {\mathcal B}$ wins by choosing the number 1. For which $ n_0$ does : [b]a.)[/b] $ {\mathcal A}$ have a winning strategy? [b]b.)[/b] $ {\mathcal B}$ have a winning strategy? [b]c.)[/b] Neither player have a winning strategy?

1973 AMC 12/AHSME, 6

Tags:

If 554 is the base $ b$ representation of the square of the number whose base $ b$ representation is 24, then $ b$, when written in base 10, equals $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$

2013 South East Mathematical Olympiad, 7

Tags: combinatorics unsolved , combinatorics

Given a $3\times 3$ grid, we call the remainder of the grid an “[i]angle[/i]” when a $2\times 2$ grid is cut out from the grid. Now we place some [i]angles[/i] on a $10\times 10$ grid such that the borders of those [i]angles[/i] must lie on the grid lines or its borders, moreover there is no overlap among the [i]angles[/i]. Determine the maximal value of $k$, such that no matter how we place $k$ [i]angles[/i] on the grid, we can always place another [i]angle[/i] on the grid.

2021 Putnam, A2

Tags: Putnam , Putnam 2021

For every positive real number $x$, let \[ g(x)=\lim_{r\to 0} ((x+1)^{r+1}-x^{r+1})^{\frac{1}{r}}. \] Find $\lim_{x\to \infty}\frac{g(x)}{x}$. [hide=Solution] By the Binomial Theorem one obtains\\ $\lim_{x \to \infty} \lim_{r \to 0} \left((1+r)+\frac{(1+r)r}{2}\cdot x^{-1}+\frac{(1+r)r(r-1)}{6} \cdot x^{-2}+\dots \right)^{\frac{1}{r}}$\\ $=\lim_{r \to 0}(1+r)^{\frac{1}{r}}=\boxed{e}$ [/hide]

1988 AIME Problems, 9

Tags: geometry , 3D geometry , modular arithmetic , AMC , AIME

Find the smallest positive integer whose cube ends in 888.

2006 Hungary-Israel Binational, 1

Tags: quadratics , geometry , power of a point , geometry unsolved

A point $ P$ inside a circle is such that there are three chords of the same length passing through $ P$. Prove that $ P$ is the center of the circle.

2001 National Olympiad First Round, 26

Tags:

Berk tries to guess the two-digit number that Ayca picks. After each guess, Ayca gives a hint indicating the number of digits which match the number picked. If Berk can guarantee to guess Ayca's number in $n$ guesses, what is the smallest possible value of $n$? $ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 20 $

2005 China Team Selection Test, 3

Tags: inequalities , algebra , polynomial , Computer solutions , 4-variable inequality , Symmetric inequality

Let $a,b,c,d >0$ and $abcd=1$. Prove that: \[ \frac{1}{(1+a)^2}+\frac{1}{(1+b)^2}+\frac{1}{(1+c)^2}+\frac{1}{(1+d)^2} \geq 1 \]

1955 AMC 12/AHSME, 21

Tags: geometry , area of a triangle

Represent the hypotenuse of a right triangle by $ c$ and the area by $ A$. The atltidue on the hypotenuse is: $ \textbf{(A)}\ \frac{A}{c} \qquad \textbf{(B)}\ \frac{2A}{c} \qquad \textbf{(C)}\ \frac{A}{2c} \qquad \textbf{(D)}\ \frac{A^2}{c} \qquad \textbf{(E)}\ \frac{A}{c^2}$

2001 India Regional Mathematical Olympiad, 3

Tags: floor function , calculus , integration

Find the number of positive integers $x$ such that \[ \left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] . \]

2011 ELMO Shortlist, 5

Tags: inequalities , geometry , inequalities proposed

Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that \[\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,\]where $k=2+\sqrt{3}$. [i]Victor Wang.[/i]

2019 Israel National Olympiad, 4

Tags: algebra , number theory

In the beginning, the number 1 is written on the board 9999 times. We are allowed to perform the following actions: [list] [*] Erase four numbers of the form $x,x,y,y$, and instead write the two numbers $x+y,x-y$. (The order or location of the erased numbers does not matter) [*] Erase the number 0 from the board, if it's there. [/list] Is it possible to reach a state where: [list=a] [*] Only one number remains on the board? [*] At most three numbers remain on the board? [/list]

2005 iTest, 2

Tags: algebra

$f(0) = 0$ $f(1) = 1$ $f(2) = 3$ $f(3) = 5$ $f(4) = 9$ $f(5) = 11$ $f(6) = 29$ $f(11) = 31$ $f(20) = ? $

2016 Harvard-MIT Mathematics Tournament, 11

Tags:

Define $\phi^!(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when \[ \sum_{\substack{2 \le n \le 50 \\ \gcd(n,50)=1}} \phi^!(n) \] is divided by $50$.

STEMS 2021 Math Cat C, Q1

Tags: function , algebra , combinatorics

Let $M>1$ be a natural number. Tom and Jerry play a game. Jerry wins if he can produce a function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying [list] [*]$f(M) \ne M$ [/*] [*] $f(k)<2k$ for all $k \in \mathbb{N}$[/*] [*] $f^{f(n)}(n)=n$ for all $n \in \mathbb{N}$. For each $\ell>0$ we define $f^{\ell}(n)=f\left(f^{\ell-1}(n)\right)$ and $f^0(n)=n$[/*] [/list] Tom wins otherwise. Prove that for infinitely many $M$, Tom wins, and for infinitely many $M$, Jerry wins. [i]Proposed by Anant Mudgal[/i]

1986 Traian Lălescu, 1.4

Tags: function , real analysis , Darboux

Let $ f:(0,1)\longrightarrow \mathbb{R} $ be a bounded function having the property of Darboux. Then: [b]a)[/b] There exists $ g:[0,1)\longrightarrow\mathbb{R} $ with Darboux’s property such that $ g\bigg|_{(0,1)} =f\bigg|_{(0,1)} . $ [b]b)[/b] The function above is uniquely determined if and only if $ f $ has limit at $ 0. $

1955 AMC 12/AHSME, 15

Tags: ratio , geometry

The ratio of the areas of two concentric circles is $ 1: 3$. If the radius of the smaller is $ r$, then the difference between the radii is best approximated by: $ \textbf{(A)}\ 0.41r \qquad \textbf{(B)}\ 0.73 \qquad \textbf{(C)}\ 0.75 \qquad \textbf{(D)}\ 0.73r \qquad \textbf{(E)}\ 0.75r$

2010 Math Prize for Girls Olympiad, 2

Tags:

Prove that for every positive integer $n$, there exist integers $a$ and $b$ such that $4a^2 + 9b^2 - 1$ is divisible by $n$.

2020 Durer Math Competition Finals, 6

Tags: combinatorics , game , game strategy

(Game) At the beginning of the game the organisers place $4$ piles of paper disks onto the table. The player who is in turn takes away a pile, then divides one of the remaining piles into two nonempty piles. Whoever is unable to move, loses. [i]Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.[/i]

2010 South africa National Olympiad, 5

Tags: geometry unsolved , geometry

(a) A set of lines is drawn in the plane in such a way that they create more than 2010 intersections at a particular angle $\alpha$. Determine the smallest number of lines for which this is possible. (b) Determine the smallest number of lines for which it is possible to obtain exactly 2010 such intersections.

1943 Eotvos Mathematical Competition, 2

Tags: geometry , perimeter , geometric inequality

Let $P$ be any point inside an acute triangle. Let $D$ and $d$ be respectively the maximum and minimum distances from $P$ to any point on the perimeter of the triangle. (a) Prove that $D \ge 2d$. (b) Determine when equality holds