Olympiad problems

1971 Czech and Slovak Olympiad III A, 1

Let $a,b,c$ real numbers. Show that there are non-negative $x,y,z,xyz\neq0$ such that \begin{align*} cy-bz &\ge 0, \\ az-cx &\ge 0, \\ bx-ay &\ge 0. \end{align*}

2012 Korea Junior Math Olympiad, 7

Tags: maximum , positive real , sum , algebra , inequalities

If all $x_k$ ($k = 1, 2, 3, 4, 5)$ are positive reals, and $\{a_1,a_2, a_3, a_4, a_5\} = \{1, 2,3 , 4, 5\}$, find the maximum of $$\frac{(\sqrt{s_1x_1} +\sqrt{s_2x_2}+\sqrt{s_3x_3}+\sqrt{s_4x_4}+\sqrt{s_5x_5})^2}{a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 + a_5x_5}$$ ($s_k = a_1 + a_2 +... + a_k$)

2018 Malaysia National Olympiad, A4

Tags: algebra

Given points $A, B, C, D, E$, and $F$ on a line (not necessarily in that order) with $AB = 2$, $BC = 6$, $CD = 8$, $DE = 10$, $EF = 20$, and $FA = 22$. Find the distance between the two furthest points on the line.

2009 All-Russian Olympiad Regional Round, 10.1

Tags: polynomial , algebra

Square trinomial $f(x)$ is such that the polynomial $(f(x)) ^3- f(x)$ has exactly three real roots. Find the ordinate of the vertex of the graph of this trinomial.

2003 Iran MO (3rd Round), 23

Tags: polynomial , algebra

Find all homogeneous linear recursive sequences such that there is a $ T$ such that $ a_n\equal{}a_{n\plus{}T}$ for each $ n$.

2009 USAMTS Problems, 2

Tags: quadratic , algebra , quadratic formula

Let $a, b, c, d$ be four real numbers such that \begin{align*}a + b + c + d &= 8, \\ ab + ac + ad + bc + bd + cd &= 12.\end{align*} Find the greatest possible value of $d$.

1996 All-Russian Olympiad Regional Round, 9.1

Tags: algebra , trinomial

Find all pairs of square trinomials $x^2 + ax + b$, $ x^2 + cx + d$ such that $a$ and $b$ are the roots of the second trinomial, $c$ and $d$ are the roots of the first.

2022 VIASM Summer Challenge, Problem 2

Tags: algebra , inequalities

Let $S$ be the set of real numbers $k$ with the following property: for all set of real numbers $(a,b,c)$ satisfying $ab+bc+ca=1$, we always have the inequality:$$\frac{a}{{\sqrt {{a^2} + ab + {b^2} + k} }} + \frac{b}{{\sqrt {{b^2} + bc + {c^2} + k} }} + \frac{c}{{\sqrt {{c^2} + ca + {a^2} + k} }} \ge \sqrt {\frac{3}{{k + 1}}} .$$ a) Assume that $k\in S$. Prove that: $k\ge 2$. b) Prove that: $2\in S$.

2008 Czech and Slovak Olympiad III A, 1

Tags: algebra

Find all pairs of real numbers $(x,y)$ satisfying: \[x+y^2=y^3,\]\[y+x^2=x^3.\]

2006 Moldova National Olympiad, 12.2

Tags: limit , calculus , integration , algebra , floor function , function

Let $a, b, n \in \mathbb{N}$, with $a, b \geq 2.$ Also, let $I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx $ and $I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.$ Find $\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.$

2016 Indonesia TST, 2

Tags: invariant , symmetry , quadratic , algebra , polynomial , vieta , number theory

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

2006 South africa National Olympiad, 6

Tags: function , floor function , algebra

Consider the function $f$ defined by \[f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right )\] for all positive integers $n$. (Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.) Prove that (a) $f(n+1)>f(n)$ for infinitely many $n$. (b) $f(n+1)<f(n)$ for infinitely many $n$.

2021 Cyprus JBMO TST, 2

Tags: algebra , cube roots

Let $x,y$ be real numbers with $x \geqslant \sqrt{2021}$ such that \[ \sqrt[3]{x+\sqrt{2021}}+\sqrt[3]{x-\sqrt{2021}} = \sqrt[3]{y}\] Determine the set of all possible values of $y/x$.

2021 Vietnam National Olympiad, 1

Tags: algebra

Let $(x_n)$ define by $x_1\in \left(0;\dfrac{1}{2}\right)$ and $x_{n+1}=3x_n^2-2nx_n^3$ for all $n\ge 1$. a) Prove that $(x_n)$ convergence to $0$. b) For each $n\ge 1$, let $y_n=x_1+2x_2+\cdots+n x_n$. Prove that $(y_n)$ has a limit.

2015 Costa Rica - Final Round, 5

Tags: inequalities , algebra

Let $a,b \in R^+$ with $ab = 1$, prove that $$\frac{1}{a^3 + 3b}+\frac{1}{b^3 + 3a}\le \frac12.$$

2009 Croatia Team Selection Test, 1

Tags: polynomial , absolute value , algebra

Determine the lowest positive integer n such that following statement is true: If polynomial with integer coefficients gets value 2 for n different integers, then it can't take value 4 for any integer.

1973 All Soviet Union Mathematical Olympiad, 187

Tags: inequalities , algebra

Prove that for every positive $x_1, x_2, x_3, x_4, x_5$ holds inequality: $$(x_1 + x_2 + x_3 + x_4 + x_5)^2 \ge 4(x_1x_2 + x_3x_4 + x_5x_1 + x_2x_3 + x_4x_5)$$

2021 Argentina National Olympiad, 6

Tags: algebra

Decide if it is possible to choose $330$ points in the plane so that among all the distances that are formed between two of them there are at least $1700$ that are equal.

2023 USA IMOTST, 1

Tags: floor function , algebra , function

Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \] [i]Carl Schildkraut[/i]

2016 Puerto Rico Team Selection Test, 6

Tags: composition , function , algebra , recurrence

$N$ denotes the set of all natural numbers. Define a function $T: N \to N$ such that $T (2k) = k$ and $T (2k + 1) = 2k + 2$. We write $T^2 (n) = T (T (n))$ and in general $T^k (n) = T^{k-1} (T (n))$ for all $k> 1$. (a) Prove that for every $n \in N$, there exists $k$ such that $T^k (n) = 1$. (b) For $k \in N$, $c_k$ denotes the number of elements in the set $\{n: T^k (n) = 1\}$. Prove that $c_{k + 2} = c_{k + 1} + c_k$, for $1 \le k$.

Russian TST 2017, P2

Tags: inequalities , algebra

Let $a_1, a_2,...,a_n$ be positive real numbers, prove that $$\sum {\frac{a_{i+1}}{a_i}} \ge \sum{\sqrt{\frac{a_{i+1}^2+1}{a_i^2+1}}}$$ $a_{n+1}=a_1$

2019 Saudi Arabia Pre-TST + Training Tests, 5.3

Tags: algebra , min , inequalities

Let $x, y, z, a,b, c$ are pairwise different integers from the set $\{1,2,3, 4,5,6\}$. Find the smallest possible value for expression $xyz + abc - ax - by - cz$.

2014 Putnam, 4

Tags: algebra , polynomial , calculus , inequalities

Show that for each positive integer $n,$ all the roots of the polynomial \[\sum_{k=0}^n 2^{k(n-k)}x^k\] are real numbers.

2014 BMT Spring, 4

Tags: polynomial , algebra

The function $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$.

2011 QEDMO 10th, 5

Tags: polynomial , algebra

A polynomial $f (x)$ with real coefficients is called [i]completely reducible[/i] if it is a product of at least two non-constant polynomials whose coefficientsare all nonnegative real numbers. Show: If $f (x^{2011})$ is completely reducible, then $f(x)$ is also.