Found problems: 15925
V Soros Olympiad 1998 - 99 (Russia), 9.3
On the coordinate plane, draw a set of points $M(x;y)$, whose coordinates satisfy the equation $$\sqrt{(x - 1)^2+ y^2} +\sqrt{x^2 + (y -1)^2} = \sqrt2.$$
2005 Nordic, 2
Let $a,b,c$ be positive real numbers. Prove that \[\frac{2a^2}{b+c} + \frac{2b^2}{c+a} + \frac{2c^2}{a+b} \geq a+b+c\](this is, of course, a joke!)
[b]EDITED with exponent 2 over c[/b]
Russian TST 2022, P1
Let $a{}$ and $b{}$ be positive integers. Prove that for any real number $x{}$ \[\sum_{j=0}^a\binom{a}{j}\big(2\cos((2j-a)x)\big)^b=\sum_{j=0}^b\binom{b}{j}\big(2\cos((2j-b)x)\big)^a.\]
2023 VN Math Olympiad For High School Students, Problem 2
a) Given a prime number $p$ and $2$ polynomials$$P(x)=a_nx^n+...+a_1x+a_0; Q(x)=b_mx^m+...+b_1x+b_0.$$
We know that the product $P(x)Q(x)$ is a polynomial whose coefficents are all divisible by $p.$
Prove that: at least $1$ in $2$ polynomials $P(x),Q(x)$ has all coefficents are all divisible by $p.$
b) Prove that the product of $2$ original polynomials is a original polynomial.
2015 Thailand Mathematical Olympiad, 2
Let $a, b, c$ be positive reals with $abc = 1$. Prove the inequality
$$\frac{a^5}{a^3 + 1}+\frac{b^5}{b^3 + 1}+\frac{c^5}{c^3 + 1} \ge \frac32$$
and determine all values of a, b, c for which equality is attained
2006 Estonia National Olympiad, 2
Find the smallest possible distance of points $ P$ and $ Q$ on a $ xy$-plane, if $ P$ lies on the line $ y \equal{} x$ and $ Q$ lies on the curve $ y \equal{} 2^x$.
1996 Hungary-Israel Binational, 4
$ a_1, a_2, \cdots, a_n$ is a sequence of real numbers, and $ b_1, b_2, \cdots, b_n$ are real numbers that satisfy the condition $ 1 \ge b_1 \ge b_2 \ge \cdots \ge b_n \ge 0$. Prove that there exists a natural number $ k \le n$ that satisifes $ |a_1b_1 \plus{} a_2b_2 \plus{} \cdots \plus{} a_nb_n| \le |a_1 \plus{} a_2 \plus{} \cdots \plus{} a_k|$
2018 Middle European Mathematical Olympiad, 4
(a) Prove that for every positive integer $m$ there exists an integer $n\ge m$ such that
$$\left \lfloor \frac{n}{1} \right \rfloor \cdot \left \lfloor \frac{n}{2} \right \rfloor \cdots \left \lfloor \frac{n}{m} \right \rfloor =\binom{n}{m} \\\\\\\\\\\\\\\ (*)$$
(b) Denote by $p(m)$ the smallest integer $n \geq m$ such that the equation $ (*)$ holds. Prove that
$p(2018) = p(2019).$
Remark: For a real number $x,$ we denote by $\left \lfloor x \right \rfloor$ the largest integer not larger than $x.$
2018 Baltic Way, 1
A finite collection of positive real numbers (not necessarily distinct) is [i]balanced [/i]if each number is less than the sum of the others. Find all $m \ge 3$ such that every balanced finite collection of $m$ numbers can be split into three parts with the property that the sum of the numbers in each part is less than the sum of the numbers in the two other parts.
1999 Vietnam Team Selection Test, 1
Let a sequence of positive reals $\{u_n\}^{\infty}_{n=1}$ be given. For every positive integer $n$, let $k_n$ be the least positive integer satisfying:
\[\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.\]
Show that the sequence $\left\{\frac{k_{n+1}}{k_n}\right\}$ has finite limit if and only if $\{u_n\}$ does.
2000 JBMO ShortLists, 14
Let $m$ and $n$ be positive integers with $m\le 2000$ and $k=3-\frac{m}{n}$. Find the smallest positive value of $k$.
1992 Poland - First Round, 4
Determine all functions $f: R \longrightarrow R$ such that
$f(x+y)-f(x-y)=f(x)*f(y)$ for $x,y \in R$
1949-56 Chisinau City MO, 48
Calculate $\sin^3 a + \cos^3 a$ if you know that $\sin a+ \cos a = m$.
2015 Paraguayan Mathematical Olympiad, Problem 1
Alexa wrote the first $16$ numbers of a sequence:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\]
Then she continued following the same pattern, until she had $2015$ numbers in total.
What was the last number she wrote?
2022-IMOC, A4
Let the set of all bijective functions taking positive integers to positive integers be $\mathcal B.$ Find all functions $\mathbf F:\mathcal B\to \mathbb R$ such that $$(\mathbf F(p)+\mathbf F(q))^2=\mathbf F(p \circ p)+\mathbf F(p\circ q)+\mathbf F(q\circ p)+\mathbf F(q\circ q)$$ for all $p,q \in \mathcal B.$
[i]Proposed by ckliao914[/i]
1947 Moscow Mathematical Olympiad, 123
Find the remainder after division of the polynomial $x+x^3 +x^9 +x^{27} +x^{81} +x^{243}$ by $x-1$.
2011 All-Russian Olympiad, 2
In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in [i]his[/i] notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?
1988 Greece Junior Math Olympiad, 1
i) Simplify $\left(a-\frac{4ab}{a+b}+b\right): \left(\frac{a}{a+b}-\frac{b}{b-a}-\frac{2ab}{a^2-b^2}\right)$
ii) Simplify $\frac{2x^2-(3a+b)x+a^2+ab}{2x^2-(a+3b)x+ab+b^2}$
2012 France Team Selection Test, 2
Determine all non-constant polynomials $X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0$ with integer coefficients for which the roots are exactly the numbers $a_0,a_1,\ldots ,a_{n-1}$ (with multiplicity).
2018 CCA Math Bonanza, I13
$P\left(x\right)$ is a polynomial of degree at most $6$ such that such that $P\left(1\right)$, $P\left(2\right)$, $P\left(3\right)$, $P\left(4\right)$, $P\left(5\right)$, $P\left(6\right)$, and $P\left(7\right)$ are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ in some order. What is the maximum possible value of $P\left(8\right)$?
[i]2018 CCA Math Bonanza Individual Round #13[/i]
2024 239 Open Mathematical Olympiad, 3
a) (version for grades 10-11)
Let $P$ be a point lying in the interior of a triangle. Show that the product of the distances from $P$ to the sides of the triangle is at least $8$ times less than the product of the distances from $P$ to the tangents to the circumcircle at the vertices of the triangle.
b) (version for grades 8-9)
Is it true that for any triangle there exists a point $P$ for which equality in the inequality from a) holds?
2008 IMO Shortlist, 4
For an integer $ m$, denote by $ t(m)$ the unique number in $ \{1, 2, 3\}$ such that $ m \plus{} t(m)$ is a multiple of $ 3$. A function $ f: \mathbb{Z}\to\mathbb{Z}$ satisfies $ f( \minus{} 1) \equal{} 0$, $ f(0) \equal{} 1$, $ f(1) \equal{} \minus{} 1$ and $ f\left(2^{n} \plus{} m\right) \equal{} f\left(2^n \minus{} t(m)\right) \minus{} f(m)$ for all integers $ m$, $ n\ge 0$ with $ 2^n > m$. Prove that $ f(3p)\ge 0$ holds for all integers $ p\ge 0$.
[i]Proposed by Gerhard Woeginger, Austria[/i]
2007 Romania National Olympiad, 1
Let $a, b, c, d \in \mathbb{N^{*}}$ such that the equation \[x^{2}-(a^{2}+b^{2}+c^{2}+d^{2}+1)x+ab+bc+cd+da=0 \] has an integer solution. Prove that the other solution is integer too and both solutions are perfect squares.
2013 IFYM, Sozopol, 2
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?
[i]Proposed by Mahan Malihi[/i]
2021 European Mathematical Cup, 1
We say that a quadruple of nonnegative real numbers $(a,b,c,d)$ is [i]balanced [/i]if $$a+b+c+d=a^2+b^2+c^2+d^2.$$
Find all positive real numbers $x$ such that $$(x-a)(x-b)(x-c)(x-d)\geq 0$$ for every balanced quadruple $(a,b,c,d)$. \\ \\
(Ivan Novak)