Found problems: 15925
2015 India IMO Training Camp, 2
Let $A$ be a finite set of pairs of real numbers such that for any pairs $(a,b)$ in $A$ we have $a>0$. Let $X_0=(x_0, y_0)$ be a pair of real numbers(not necessarily from $A$). We define $X_{j+1}=(x_{j+1}, y_{j+1})$ for all $j\ge 0$ as follows: for all $(a,b)\in A$, if $ax_j+by_j>0$ we let $X_{j+1}=X_j$; otherwise we choose a pair $(a,b)$ in $A$ for which $ax_j+by_j\le 0$ and set $X_{j+1}=(x_j+a, y_j+b)$. Show that there exists an integer $N\ge 0$ such that $X_{N+1}=X_N$.
2009 Stanford Mathematics Tournament, 9
Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x=\frac{y^2}{2}$
2021 Peru PAGMO TST, P3
Find all the quaterns $(x,y,z,w)$ of real numbers (not necessarily distinct) that solve the following system of equations:
$$x+y=z^2+w^2+6zw$$
$$x+z=y^2+w^2+6yw$$
$$x+w=y^2+z^2+6yz$$
$$y+z=x^2+w^2+6xw$$
$$y+w=x^2+z^2+6xz$$
$$z+w=x^2+y^2+6xy$$
2018 MOAA, 8
Suppose that k and x are positive integers such that $$\frac{k}{2}=\left( \sqrt{1 +\frac{\sqrt3}{2}}\right)^x+\left( \sqrt{1 -\frac{\sqrt3}{2}}\right)^x.$$
Find the sum of all possible values of $k$
2024 ELMO Shortlist, A6
Let $\mathbb R^+$ denote the set of positive real numbers. Find all functions $f:\mathbb R^+\to\mathbb R$ and $g:\mathbb R^+\to\mathbb R$ such that for all $x,y\in\mathbb R^+$, $g(x)-g(y)=(x-y)f(xy)$.
[i]Linus Tang[/i]
2018 Istmo Centroamericano MO, 3
Determine all sequences of integers $a_1, a_2,. . .,$ such that:
(i) $1 \le a_i \le n$ for all $1 \le i \le n$.
(ii) $| a_i - a_j| = | i - j |$ for any $1 \le i, j \le n$
2023 Romania National Olympiad, 3
Determine all positive integers $n$ for which the number
\[
N = \frac{1}{n \cdot (n + 1)}
\]
can be represented as a finite decimal fraction.
1999 Romania National Olympiad, 4
a) Prove that if $x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n$ are positive real numbers satisfying the conditions [list=i]
[*] $x_1y_1<x_2y_2<\ldots<x_ny_n$;
[*] $x_1+x_2+\ldots+x_k \ge y_1+y_2+ \ldots +y_k,$ for $k=\overline{1,n},$[/list] then $$\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n} \le \frac{1}{y_1}+\frac{1}{y_2}+\ldots+\frac{1}{y_n}.$$
b) Let $A=\{a_1,a_2,\ldots,a_n\}$ be a set of positive integers with the property that for any distinct subsets $B$ and $C$ of $A$ we have $\sum_{x \in B} x \neq \sum_{x \in C} x.$ Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_n}<2.$$
PEN Q Problems, 7
Let $f(x)=x^{n}+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.
2012 Moldova Team Selection Test, 1
Prove that polynomial $x^8+98x^4+1$ can be factorized in $Z[X]$.
2010 Kazakhstan National Olympiad, 4
Let $x$- minimal root of equation $x^2-4x+2=0$.
Find two first digits of number $ \{x+x^2+....+x^{20} \}$ after $0$, where $\{a\}$- fractional part of $a$.
1995 Nordic, 3
Let $n \ge 2$ and let $x_1, x_2, ..., x_n$ be real numbers satisfying $x_1 +x_2 +...+x_n \ge 0$ and $x_1^2+x_2^2+...+x_n^2=1$. Let $M = max \{x_1, x_2,... , x_n\}$. Show that $M \ge \frac{1}{\sqrt{n(n-1)}}$ (1) .When does equality hold in (1)?
2010 Stanford Mathematics Tournament, 10
Compute the base 10 value of $14641_{99}$
2022 Purple Comet Problems, 23
There are prime numbers $a$, $b$, and $c$ such that the system of equations
$$a \cdot x - 3 \cdot y + 6 \cdot z = 8$$
$$b \cdot x + 3\frac12 \cdot y + 2\frac13 \cdot z = -28$$
$$c \cdot x - 5\frac12 \cdot y + 18\frac13 \cdot z = 0$$
has infinitely many solutions for $(x, y, z)$. Find the product $a \cdot b \cdot c$.
2022 Kyiv City MO Round 1, Problem 2
For any reals $x, y$, show the following inequality:
$$\sqrt{(x+4)^2 + (y+2)^2} + \sqrt{(x-5)^2 + (y+4)^2} \le \sqrt{(x-2)^2 + (y-6)^2} + \sqrt{(x-5)^2 + (y-6)^2} + 20$$
[i](Proposed by Bogdan Rublov)[/i]
1985 IMO Longlists, 40
Each of the numbers $x_1, x_2, \dots, x_n$ equals $1$ or $-1$ and
\[\sum_{i=1}^n x_i x_{i+1} x_{i+2} x_{i+3} =0.\]
where $x_{n+i}=x_i $ for all $i$. Prove that $4\mid n$.
2020 Dutch IMO TST, 1
Given are real numbers $a_1, a_2,..., a_{2020}$, not necessarily different.
For every $n \ge 2020$, define $a_{n + 1}$ as the smallest real zero of the polynomial $$P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n$$, if it exists. Assume that $a_{n + 1}$ exists for all $n \ge 2020$.
Prove that $a_{n + 1} \le a_n$ for all $n \ge 2021$.
1979 Kurschak Competition, 2
$f$ is a real-valued function defined on the reals such that $f(x) \le x$ and $f(x + y) \le f(x) + f(y)$ for all $x, y$. Prove that $f(x) = x$ for all $x$.
2019 Czech-Polish-Slovak Junior Match, 4
Determine all possible values of the expression $xy+yz+zx$ with real numbers $x, y, z$ satisfying the conditions $x^2-yz = y^2-zx = z^2-xy = 2$.
2009 APMO, 2
Let $ a_1$, $ a_2$, $ a_3$, $ a_4$, $ a_5$ be real numbers satisfying the following equations:
$ \frac{a_1}{k^2\plus{}1}\plus{}\frac{a_2}{k^2\plus{}2}\plus{}\frac{a_3}{k^2\plus{}3}\plus{}\frac{a_4}{k^2\plus{}4}\plus{}\frac{a_5}{k^2\plus{}5} \equal{} \frac{1}{k^2}$ for $ k \equal{} 1, 2, 3, 4, 5$
Find the value of $ \frac{a_1}{37}\plus{}\frac{a_2}{38}\plus{}\frac{a_3}{39}\plus{}\frac{a_4}{40}\plus{}\frac{a_5}{41}$ (Express the value in a single fraction.)
2018 Romania National Olympiad, 3
Let $a, b, c \ge 0$ so that $ab + bc + ca = 3$. Prove that:
$$\frac{a}{a^2+7}+\frac{b}{b^2+7}+\frac{c}{c^2+7}\le \frac38$$
2023 All-Russian Olympiad Regional Round, 10.10
Prove that for all positive reals $x, y, z$, the inequality $(x-y)\sqrt{3x^2+y^2}+(y-z)\sqrt{3y^2+z^2}+(z-x)\sqrt{3z^2+x^2} \geq 0$ is satisfied.
2015 Azerbaijan JBMO TST, 1
With the conditions $a,b,c\in\mathbb{R^+}$ and $a+b+c=1$, prove that \[\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\geq\frac{69}{4}\]
1998 All-Russian Olympiad, 1
Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.
2018 China Team Selection Test, 3
Prove that there exists a constant $C>0$ such that
$$H(a_1)+H(a_2)+\cdots+H(a_m)\leq C\sqrt{\sum_{i=1}^{m}i a_i}$$
holds for arbitrary positive integer $m$ and any $m$ positive integer $a_1,a_2,\cdots,a_m$, where $$H(n)=\sum_{k=1}^{n}\frac{1}{k}.$$