Olympiad problems

2022 Iran MO (3rd Round), 1

Tags: polynomial , algebra

We call polynomial $S(x)\in\mathbb{R}[x]$ sadeh whenever it's divisible by $x$ but not divisible by $x^2$. For the polynomial $P(x)\in\mathbb{R}[x]$ we know that there exists a sadeh polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists sadeh polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{1401}$.

1967 German National Olympiad, 3

Tags: algebra , inequalities

Prove the following theorem: If $n > 2$ is a natural number, $a_1, ..., a_n$ are positive real numbers and becomes $\sum_{i=1}^n a_i = s$, then the following holds $$\sum_{i=1}^n \frac{a_i}{s - a_i} \ge \frac{n}{n - 1}$$

III Soros Olympiad 1996 - 97 (Russia), 10.1

Tags: trigonometry , algebra

Find the smallest natural number $n$ for which the equality $\sin n^o= \sin (1997n)^o$ holds.

2016 Hanoi Open Mathematics Competitions, 6

Tags: interval , number theory , minimum , algebra

Determine the smallest positive number $a$ such that the number of all integers belonging to $(a, 2016a]$ is $2016$.

1977 Germany Team Selection Test, 1

Tags: rearrangement inequality , inequality , permutation , algebra

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

1964 AMC 12/AHSME, 37

Tags: special factorizations , difference of squares , algebra

Given two positive number $a$, $b$ such that $a<b$. Let A.M. be their arithmetic mean and let G.M. be their positive geometric mean. Then A.M. minus G.M. is always less than: $\textbf{(A) }\dfrac{(b+a)^2}{ab}\qquad\textbf{(B) }\dfrac{(b+a)^2}{8b}\qquad\textbf{(C) }\dfrac{(b-a)^2}{ab}$ $\textbf{(D) }\dfrac{(b-a)^2}{8a}\qquad \textbf{(E) }\dfrac{(b-a)^2}{8b}$

1969 Polish MO Finals, 4

Tags: algebra , inequalities

Show that if natural numbers $a,b, p,q,r,s$ satisfy the conditions $$qr- ps = 1 \,\,\,\,\, and \,\,\,\,\, \frac{p}{q}<\frac{a}{b}<\frac{r}{s},$$ then $b \ge q+s.$

2006 Miklós Schweitzer, 7

Tags: periodic , real analysis , algebra

Suppose that the function $f: Z \to Z$ can be written in the form $f = g_1+...+g_k$ , where $g_1,. . . , g_k: Z \to R$ are real-valued periodic functions, with period $a_1,...,a_k$. Does it follow that f can be written in the form $f = h_1 +. . + h_k$ , where $h_1,. . . , h_k: Z \to Z$ are periodic functions with integer values, also with period $a_1,...,a_k$?

1994 All-Russian Olympiad, 1

Tags: polynomial , quadratic , algebra

Let be given three quadratic polynomials: $P_1(x) = x^2 + p_1x+q_1, P_2(x) = x^2+ p_2x+q_2, P_3(x) = x^2 + p_3x+q_3$. Prove that the equation $|P_1(x)|+|P_2(x)| = |P_3(x)|$ has at most eight real roots.

2010 Romania National Olympiad, 1

Tags: quadratic , vieta , function , algebra

Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that \[\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.\] Prove that $(a_n)_{n\ge0}$ is a geometric sequence. [i]Lucian Dragomir[/i]

2025 Kyiv City MO Round 1, Problem 1

Tags: bruteforce , number theory , algebra

How many three-digit numbers are there, which do not have a zero in their decimal representation and whose sum of digits is $7$?

2015 Indonesia MO, 4

Tags: function , algebra , functional equation

Let function pair $f,g : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies \[ f(g(x)y + f(x)) = (y+2015)f(x) \] for every $x,y \in \mathbb{R^+} $ a. Prove that $f(x) = 2015g(x)$ for every $x \in \mathbb{R^+}$ b. Give an example of function pair $(f,g)$ that satisfies the statement above and $f(x), g(x) \geq 1$ for every $x \in \mathbb{R^+}$

2005 Tournament of Towns, 1

Tags: analytic geometry , algebra , polynomial

On the graph of a polynomial with integral coefficients are two points with integral coordinates. Prove that if the distance between these two points is integral, then the segment connecting them is parallel to the $x$-axis. [i](4 points)[/i]

1976 IMO Longlists, 43

Tags: polynomial , algebra

Prove that if for a polynomial $P(x, y)$, we have \[P(x - 1, y - 2x + 1) = P(x, y),\] then there exists a polynomial $\Phi(x)$ with $P(x, y) = \Phi(y - x^2).$

2010 Romanian Master of Mathematics, 6

Tags: polynomial , algebra , number theory , pigeonhole principle , relatively prime , induction

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$. Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set. [i]Dan Schwarz, Romania[/i]

2021 Princeton University Math Competition, A1 / B3

Tags: algebra

Compute the sum of all real numbers x which satisfy the following equation $$\frac {8^x - 19 \cdot 4^x}{16 - 25 \cdot 2^x}= 2$$

2013 Cuba MO, 5

Tags: polynomial , algebra

Let the real numbers be $a, b, c, d$ with $a \ge b$ and $c \ge d$. Prove that the equation $$(x + a) (x + d) + (x + b) (x + c) = 0$$ has real roots.

1999 Yugoslav Team Selection Test, Problem 3

Tags: sequence , algebra

Consider the set $A_n=\{x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n\}$ of $2n$ variables. How many permutations of set $A_n$ are there for which it is possible to assign real values from the interval $(0,1)$ to the $2n$ variables so that: (i) $x_i+y_i=1$ for each $i$; (ii) $x_1<x_2<\ldots<x_n$; (iii) the $2n$ terms of the permutation form a strictly increasing sequence?

2024 Thailand TST, 1

Tags: polynomial , algebra , number theory

Determine all polynomials $P$ with integer coefficients for which there exists an integer $a_n$ such that $P(a_n)=n^n$ for all positive integers $n$.

1974 Swedish Mathematical Competition, 5

Tags: algebra , system , system of equations

Find the smallest positive real $t$ such that \[\left\{ \begin{array}{l} x_1 + x_3 = 2t x_2 \\ x_2 + x_4 = 2t x_3 \\ x_3 + x_5=2t x_4 \\ \end{array} \right. \] has a solution $x_1$, $x_2$, $x_3$, $x_4$, $x_5$ in non-negative reals, not all zero.

2005 Spain Mathematical Olympiad, 1

Tags: algebra , number theory

Let $a$ and $b$ be integers. Demonstrate that the equation $$(x-a)(x-b)(x-3) +1 = 0$$ has an integer solution.

2018 China Team Selection Test, 4

Tags: function , fe , functional equation , algebra

Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.

2007 AMC 12/AHSME, 21

Tags: function , algebra , quadratic , vieta , quadratic formula

The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function $ f(x) \equal{} ax^{2} \plus{} bx \plus{} c$ are equal. Their common value must also be which of the following? $ \textbf{(A)}\ \text{the coefficient of }x^{2}\qquad \textbf{(B)}\ \text{the coefficient of }x$ $ \textbf{(C)}\ \text{the y \minus{} intercept of the graph of }y \equal{} f(x)$ $ \textbf{(D)}\ \text{one of the x \minus{} intercepts of the graph of }y \equal{} f(x)$ $ \textbf{(E)}\ \text{the mean of the x \minus{} intercepts of the graph of }y \equal{} f(x)$

2005 Bundeswettbewerb Mathematik, 2

Tags: algebra

Let be $x$ a rational number. Prove: There are only finitely many triples $(a,b,c)$ of integers with $a<0$ and $b^2-4ac=5$ such that $ax^2+bx+c$ is positive.

1996 May Olympiad, 3

Tags: algebra

$A$ and $B$ are two cylindrical containers that contain water. The height of the water at$ A$ is $1000$ cm and at $B$, $350$ cm. Using a pump, water is transferred from $A$ to $B$. It is noted that, in container $A$, the height of the water decreases $4$ cm per minute and in $B$ it increases $9$ cm per minute. After how much time, since the pump was started, will the heights at $A$ and $B$ be the same?