Found problems: 15925
2012 India IMO Training Camp, 2
Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that
\[\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}\]
1986 China Team Selection Test, 2
Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent:
[b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$
[b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.
1999 Czech and Slovak Match, 4
Find all positive integers $k$ for which the following assertion holds:
If $F(x)$ is polynomial with integer coefficients ehich satisfies $F(c) \leq k$ for all $c \in \{0,1, \cdots,k+1 \}$, then \[F(0)= F(1) = \cdots =F(k+1).\]
2006 Swedish Mathematical Competition, 3
A cubic polynomial $f$ with a positive leading coefficient has three different positive zeros. Show that $f'(a)+ f'(b)+ f'(c) > 0$.
1988 Bulgaria National Olympiad, Problem 6
Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.
2007 Grigore Moisil Intercounty, 2
Prove that if all roots of a monic cubic polynomial have modulus $ 1, $ then, the two middle coefficients have the same modulus.
1985 Traian Lălescu, 1.4
Let $ a $ be a non-negative real number distinct from $ 1. $
[b]a)[/b] For which positive values $ x $ the equation
$$ \left\lfloor\log_a x\right\rfloor +\left\lfloor \frac{1}{3} +\log_a x\right\rfloor =\left\lfloor 2\cdot\log_a x\right\rfloor $$
is true?
[b]b)[/b] Solve $ \left\lfloor\log_3 x\right\rfloor +\left\lfloor \frac{1}{3} +\log_3 x\right\rfloor =3. $
1975 IMO, 6
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
2015 Peru IMO TST, 9
Let $A$ be a finite set of functions $f: \Bbb{R}\to \Bbb{R.}$ It is known that: [list] [*] If $f, g\in A$ then $f (g (x)) \in A.$ [*] For all $f \in A$ there exists $g \in A$ such that $f (f (x) + y) = 2x + g (g (y) - x),$ for all $x, y\in \Bbb{R}.$ [/list] Let $i:\Bbb{R}\to \Bbb{R}$ be the identity function, ie, $i (x) = x$ for all $x\in \Bbb{R}.$ Prove that $i \in A.$
1954 Moscow Mathematical Olympiad, 285
The absolute values of all roots of the quadratic equation $x^2+Ax+B = 0$ and $x^2+Cx+D = 0$ are less then $1$. Prove that so are absolute values of the roots of the quadratic equation $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.
2010 Junior Balkan Team Selection Tests - Romania, 2
Let $n$ be an integer, $n \ge 2$. For each number $k = 1, 2, ....., n,$ denote by $a _ k$ the number of multiples of $k$ in the set $\{1, 2,. .., n \}$ and let $x _ k = \frac {1} {1} + \frac {1} {2} + \frac {1} {3} _... + \frac {1} {a _ k}$ .
Show that: $$\frac {x _ 1 + x _ 2 + ... + x _ n} {n} \le \frac {1} {1 ^ 2} + \frac {1} {2 ^ 2} + ... + \frac {1} {n ^ 2} $$.
2012 HMNT, 4
Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012$.
2019 Silk Road, 4
The sequence $ \{a_n \} $ is defined as follows: $ a_0 = 1 $ and $ {a_n} = \sum \limits_ {k = 1} ^ {[\sqrt n]} {{a_ {n - {k ^ 2 }}}} $ for $ n \ge 1. $
Prove that among $ a_1, a_2, \ldots, a_ {10 ^ 6} $ there are at least $500$ even numbers.
(Here, $ [x] $ is the largest integer not exceeding $ x $.)
2024 Azerbaijan National Mathematical Olympiad, 2
Find all the real number triples $(x, y, z)$ satisfying the following system of inequalities under the condition $0 < x, y, z < \sqrt{2}$:
$$y\sqrt{4-x^2y^2}\ge \frac{2}{\sqrt{xz}}$$
$$x\sqrt{4-x^2z^2}\ge \frac{2}{\sqrt{yz}}$$
$$z\sqrt{4-y^2z^2}\ge \frac{2}{\sqrt{xy}}$$.
2010 Iran MO (3rd Round), 2
prove the third sylow theorem: suppose that $G$ is a group and $|G|=p^em$ which $p$ is a prime number and $(p,m)=1$. suppose that $a$ is the number of $p$-sylow subgroups of $G$ ($H<G$ that $|H|=p^e$). prove that $a|m$ and $p|a-1$.(Hint: you can use this: every two $p$-sylow subgroups are conjugate.)(20 points)
1994 Tournament Of Towns, (439) 5
The periods of two periodic sequences are coprime (i.e. relatively prime) numbers $m$ and $n$.. What is the maximal length of initial sections of the two sequences which can coincide? (The period $p$ of a sequence $a_1$,$a_2$, $...$ is the minimal $p$ such that $a_n = a_{n+p}$ for all $n$.)
(AY Belov)
2021 Harvard-MIT Mathematics Tournament., 3
Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$, compute the smallest possible value of $|P(0)|.$
2024 Caucasus Mathematical Olympiad, 7
The positive numbers $a_1, a_2, \ldots , a_{2024}$ are placed on a circle clockwise in this order. Let $A_i$ be the arithmetic mean of the number $a_i$ and one or several following it clockwise. Prove that the largest of the numbers $A_1, A_2, \ldots , A_{2024}$ is not less than the arithmetic mean of all numbers $a_1, a_2, \ldots , a_{2024}$.
1981 Dutch Mathematical Olympiad, 1
$f(x) = [x] + [2x] + [3x] + [4x] + [5x] + [6x]$. What values does $f$ take?
2019 Saudi Arabia JBMO TST, 2
Let $a, b, c$ be non-negative real numbers. Prove that $$a\sqrt{3a^2+6b^2}+b\sqrt{3b^2+6c^2}+c\sqrt{3c^2+6a^2}=>(a+b+c)^2$$
2023 German National Olympiad, 6
The equation $x^3-3x^2+1=0$ has three real solutions $x_1<x_2<x_3$. Show that for any positive integer $n$, the number $\left\lceil x_3^n\right\rceil$ is a multiple of $3$.
2000 Harvard-MIT Mathematics Tournament, 2
The temperatures $f^o F$ and $c^o C$ are equal when $f = \frac95 c + 32$. What temperature is the same in both $^o F$ and $^o C$?
2013 Hitotsubashi University Entrance Examination, 5
Throw a die $n$ times, let $a_k$ be a number shown on the die in the $k$-th place. Define $s_n$ by $s_n=\sum_{k=1}^n 10^{n-k}a_k$.
(1) Find the probability such that $s_n$ is divisible by 4.
(2) Find the probability such that $s_n$ is divisible by 6.
(3) Find the probability such that $s_n$ is divisible by 7.
Last Edited
Thanks, jmerry & JBL
2018 CMIMC Algebra, 5
Suppose $a$, $b$, and $c$ are nonzero real numbers such that \[bc+\frac1a = ca+\frac2b = ab+\frac7c = \frac1{a+b+c}.\] Find $a+b+c$.
1999 Estonia National Olympiad, 4
$32$ stones, with pairwise different weights, and lever scales without weights are given. How to determine by $35$ scaling, which stone is the heaviest and which is the second by weight?