Olympiad problems

2014 Balkan MO Shortlist, A5

$\boxed{A5}$Let $n\in{N},n>2$,and suppose $a_1,a_2,...,a_{2n}$ is a permutation of the numbers $1,2,...,2n$ such that $a_1<a_3<...<a_{2n-1}$ and $a_2>a_4>...>a_{2n}.$Prove that \[(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2>n^3\]

2000 Harvard-MIT Mathematics Tournament, 1

Tags: algebra

If $a = 2b + c$, $b = 2c + d$, $2c = d + a -1$, $d = a - c$, what is $b$?

CVM 2020, Problem 2+

Tags: algebra

Find all the real solutions to $$n=\sum_{i=1}^n x_i=\sum_{1\le i<j\le n} x_ix_j$$ [i]Proposed by Carlos Dominguez, Valle[/i]

2022 Caucasus Mathematical Olympiad, 2

Tags: algebra , construction

Prove that infinitely many positive integers can be represented as $(a-1)/b + (b-1)/c + (c-1)/a$, where $a$, $b$ and $c$ are pairwise distinct positive integers greater than 1.

1967 IMO Shortlist, 4

Tags: linear algebra , system of equations , arithmetic sequence , algebra , matrix

In what case does the system of equations $\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$ have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.

2022 ITAMO, 3

Tags: number theory , algebra

In a mathematical competition $n=10\,000$ contestants participate. During the final party, in sequence, the first one takes $1/n$ of the cake, the second one takes $2/n$ of the remaining cake, the third one takes $3/n$ of the cake that remains after the first and the second contestant, and so on until the last one, who takes all of the remaining cake. Determine which competitor takes the largest piece of cake.

1988 Austrian-Polish Competition, 2

Tags: algebra , inequalities , sum

If $a_1 \le a_2 \le .. \le a_n$ are natural numbers ($n \ge 2$), show that the inequality $$\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0$$ holds for all $n$-tuples $(x_1,...,x_n) \ne (0,..., 0)$ of real numbers if and only if $a_2 \ge 2$.

2012 Spain Mathematical Olympiad, 2

Tags: function , algebra

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[(x-2)f(y)+f(y+2f(x))=f(x+yf(x))\] for all $x,y\in\mathbb{R}$.

1989 IMO Longlists, 77

Tags: algebra

Let $ a, b, c, r,$ and $ s$ be real numbers. Show that if $ r$ is a root of $ ax^2\plus{}bx\plus{}c \equal{} 0$ and s is a root of $ \minus{}ax^2\plus{}bx\plus{}c \equal{} 0,$ then \[ \frac{a}{2} x^2 \plus{} bx \plus{} c \equal{} 0\] has a root between $ r$ and $ s.$

1998 Brazil Team Selection Test, Problem 3

Tags: number theory , algebra , combinatorics

Show that it is possible to color the points of $\mathbb Q\times\mathbb Q$ in two colors in such a way that any two points having distance $1$ have distinct colors.

2015 Auckland Mathematical Olympiad, 3

Tags: algebra

Several pounamu stones weigh altogether $10$ tons and none of them weigh more than $1$ tonne. A truck can carry a load which weight is at most $3$ tons. What is the smallest number of trucks such that bringing all stones from the quarry will be guaranteed?

2005 Slovenia Team Selection Test, 2

Tags: algebra , functional , functional equation

Find all functions $f : R^+ \to R^+$ such that $x^2(f(x)+ f(y)) = (x+y)f (f(x)y)$ for any $x,y > 0$.

2020-21 IOQM India, 5

Tags: inequalities , absolute value , algebra

Find the number of integer solutions to $||x| - 2020| < 5$.

1986 Iran MO (2nd round), 2

Tags: limit , function , calculus , derivative , algebra

[b](a)[/b] Sketch the diagram of the function $f$ if \[f(x)=4x(1-|x|) , \quad |x| \leq 1.\] [b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$ [b](c)[/b] Let $g$ be a function such that \[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\] Is the function $g$ continuous in the point $x=0 \ ?$ [b](d)[/b] Sketch the diagram of $g.$

2009 Junior Balkan Team Selection Tests - Romania, 1

Tags: algebra , inequalities

Let $a, b, c$ be positive real number such that $a + b + c \ge \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$ . Prove that $ \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\ge \frac{1}{ab}+ \frac{1}{bc}+ \frac{1}{ca}$ .

1956 Moscow Mathematical Olympiad, 337

Tags: number theory , algebra

* Assume that the number of a tree’s leaves is a multiple of $15$. Neglecting the shade of the trunk and branches prove that one can rip off the tree $7/15$ of its leaves so that not less than $8/15$ of its shade remains.

1985 IMO Longlists, 12

Tags: trigonometry , algebra

Find the maximum value of \[\sin^2 \theta_1+\sin^2 \theta_2+\cdots+\sin^2 \theta_n\] subject to the restrictions $0 \leq \theta_i , \theta_1+\theta_2+\cdots+\theta_n=\pi.$

2020 LMT Fall, B16

Tags: algebra

Let $f$ be a function $R \to R$ that satisfies the following equation: $$f (x)^2 + f (y)^2 = f (x^2 + y^2)+ f (0)$$ If there are $n$ possibilities for the function, find the sum of all values of $n \cdot f (12)$

2006 Cezar Ivănescu, 2

Tags: albegra , fractional part , number theory , algebra

[b]a)[/b] Prove that $ \{ a \} +\{ 1/a \} <3/2, $ for any positive real number $ a. $ [b]b)[/b] Give an example of a number $ b $ satisfying $ \{ b \} +\{ 1/b \} =1. $ [i]{} means fractional part[/i]

2016 Chile National Olympiad, 5

Tags: system of equations , algebra

Determine all triples $(x, y, z)$ of nonnegative real numbers that verify the following system of equations: $$x^2 - y = (z -1)^2 $$ $$y^2 - z = (x -1)^2$$ $$z^2 - x = (y - 1)^2$$

2004 Poland - Second Round, 1

Tags: algebra

Positive real numbers $a,b,c,d$ satisfy the equalities \[a^3+b^3+c^3=3d^3\\ b^4+c^4+d^4=3a^4\\ c^5+d^5+a^5=3b^5. \] Prove that $a=b=c=d$.

1996 Estonia Team Selection Test, 1

Tags: polynomial , algebra

Prove that the polynomial $P_n(x)=1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}$ has no real zeros if $n$ is even and has exatly one real zero if $n$ is odd

1994 Tournament Of Towns, (435) 1

Tags: algebra

Coefficients $p$ and $q$ of the equation $x^2+px+q = 0$ are changed and the new ones differ from the old ones by $0.001$ or less. Can the greater root of the new equation differ from that of the old one by $1000$ or more?

III Soros Olympiad 1996 - 97 (Russia), 11.8

Tags: algebra , trigonometry , system of equations

Solve the system of equations: $$ 2(3-2\cos y)^2+2(4-2\sin y)^2=2(3-x)^2+32=(x-2\cos y)^2+4\sin^2y$$

1986 Vietnam National Olympiad, 3

Tags: polynomial , logarithm , algebra

Suppose $ M(y)$ is a polynomial of degree $ n$ such that $ M(y) \equal{} 2^y$ for $ y \equal{} 1, 2, \ldots, n \plus{} 1$. Compute $ M(n \plus{} 2)$.