Found problems: 15925
2021 USEMO, 5
Given a polynomial $p(x)$ with real coefficients, we denote by $S(p)$ the sum of the squares of its coefficients. For example $S(20x+ 21)=20^2+21^2=841$.
Prove that if $f(x)$, $g(x)$, and $h(x)$ are polynomials with real coefficients satisfying the indentity $f(x) \cdot g(x)=h(x)^ 2$, then $$S(f) \cdot S(g) \ge S(h)^2$$
[i]Proposed by Bhavya Tiwari[/i]
1972 Bundeswettbewerb Mathematik, 4
Which natural numbers cannot be presented in that way: $[n+\sqrt{n}+\frac{1}{2}]$, $n\in\mathbb{N}$
$[y]$ is the greatest integer function.
2022 Austrian Junior Regional Competition, 1
Show that for all real numbers $x$ and $y$ with $x > -1$ and $y > -1$ and $x + y = 1$ the inequality
$$\frac{x}{y + 1} +\frac{y}{x + 1} \ge \frac23$$
holds. When does equality apply?
[i](Walther Janous)[/i]
2018 Hanoi Open Mathematics Competitions, 9
Each of the thirty squares in the diagram below contains a number $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ of which each number is used exactly three times. The sum of three numbers in three squares on each of the thirteen line segments is equal to $S$. [img]https://cdn.artofproblemsolving.com/attachments/8/0/3e056ebc252aee9ade1f45fd337cc6a2f84302.png[/img]
2014 PUMaC Individual Finals A, 2
Given $a,b,c \in\mathbb{R}^+$, and that $a^2+b^2+c^2=3$. Prove that
\[ \frac{1}{a^3+2}+\frac{1}{b^3+2}+\frac{1}{c^3+2}\ge 1 \]
2006 Cuba MO, 1
Determine all monic polynomials $P(x)$ of degree $3$ with coefficients integers, which are divisible by $x-1$, when divided by $ x-5$ leave the same remainder as when divided by$ x+5$ and have a root between $2$ and $3$.
1986 India National Olympiad, 2
Solve
\[ \left\{ \begin{array}{l}
\log_2 x\plus{}\log_4 y\plus{}\log_4 z\equal{}2 \\
\log_3 y\plus{}\log_9 z\plus{}\log_9 x\equal{}2 \\
\log_4 z\plus{}\log_{16} x\plus{}\log_{16} y\equal{}2 \\
\end{array} \right.\]
2020 USA IMO Team Selection Test, 1
Choose positive integers $b_1, b_2, \dotsc$ satisfying
\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]
and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?
[i]Carl Schildkraut and Milan Haiman[/i]
2023 Girls in Mathematics Tournament, 1
Define $(a_n)$ a sequence, where $a_1= 12, a_2= 24$ and for $n\geq 3$, we have: $$a_n= a_{n-2}+14$$
a) Is $2023$ in the sequence?
b) Show that there are no perfect squares in the sequence.
2022 IFYM, Sozopol, 4
Let $n$ be a natural number. To prove that the value of the expression
$$\prod^n_{i=0}\frac{x^{n-1}_i}{\prod_{j \ne i}(x_i - x_j)}$$
does not depend on the choice of the different real numbers $x_0, x_1, ... , x_n$.
1991 China National Olympiad, 2
Given $I=[0,1]$ and $G=\{(x,y)|x,y \in I\}$, find all functions $f:G\rightarrow I$, such that $\forall x,y,z \in I$ we have:
i. $f(f(x,y),z)=f(x,f(y,z))$;
ii. $f(x,1)=x, f(1,y)=y$;
iii. $f(zx,zy)=z^kf(x,y)$.
($k$ is a positive real number irrelevant to $x,y,z$.)
2004 Greece JBMO TST, 4
Let $a,b$ be positive real numbers such that $b^3+b\le a-a^3$. Prove that:
i) $b<a<1$
ii) $a^2+b^2<1$
2013 Indonesia MO, 5
Let $P$ be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics $P_1, P_2, P_3$ such that:
- $P(x) = P_1(x) + P_2(x) + P_3(x)$
- $P_1, P_2, P_3$ have positive leading coefficients and zero discriminants (and hence each has a double root)
- The roots of $P_1, P_2, P_3$ are different
2019 Iran MO (3rd Round), 1
Let $A_1,A_2, \dots A_k$ be points on the unit circle.Prove that:
$\sum\limits_{1\le i<j \le k} d(A_i,A_j)^2 \le k^2 $
Where $d(A_i,A_j)$ denotes the distance between $A_i,A_j$.
2013 National Olympiad First Round, 27
For how many pairs $(a,b)$ from $(1,2)$, $(3,5)$, $(5,7)$, $(7,11)$, the polynomial $P(x)=x^5+ax^4+bx^3+bx^2+ax+1$ has exactly one real root?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ 0
$
2021 Princeton University Math Competition, A2 / B4
For a bijective function $g : R \to R$, we say that a function $f : R \to R$ is its superinverse if it satisfies the following identity $(f \circ g)(x) = g^{-1}(x)$, where $g^{-1}$ is the inverse of $g$. Given $g(x) = x^3 + 9x^2 + 27x + 81$ and $f$ is its superinverse, find $|f(-289)|$.
2013 Thailand Mathematical Olympiad, 8
Let $p(x) = x^{2013} + a_{2012}x^{2012} + a_{2011}x^{2011} +...+ a_1x + a_0$ be a polynomial with real coefficients with roots $- b_{1006}, - b_{1005}, ... , -b_1, 0, b_1, ... , b_{1005}, b_{1006}$, where $b_1, b_2, ... , b_{1006}$ are positive reals with product $1$. Show that $a_3a_{2011} \le 1012036$
2021 Iran MO (3rd Round), 1
Positive real numbers $a, b, c$ and $d$ are given such that $a+b+c+d = 4$ prove that
$$\frac{ab}{a^2-\frac{4}{3}a+\frac{4}{3}} + \frac{bc}{b^2-\frac{4}{3}b+ \frac{4}{3}} + \frac{cd}{c^2-\frac{4}{3}c+ \frac{4}{3}} + \frac{da}{d^2-\frac{4}{3}d+ \frac{4}{3}}\leq 4.$$
2022 BMT, 2
The equation
$$4^x -5 \cdot 2^{x+1} +16 = 0$$
has two integer solutions for $x.$ Find their sum.
2018 Azerbaijan Junior NMO, 5
For a positive integer $n$, define $f(n)=n+P(n)$ and $g(n)=n\cdot S(n)$, where $P(n)$ and $S(n)$ denote the product and sum of the digits of $n$, respectively. Find all solutions to $f(n)=g(n)$
2019 Baltic Way, 5
The $2m$ numbers
$$1\cdot 2, 2\cdot 3, 3\cdot 4,\hdots,2m(2m+1)$$
are written on a blackboard, where $m\geq 2$ is an integer. A [i]move[/i] consists of choosing three numbers $a, b, c$, erasing them from the board and writing the single number
$$\frac{abc}{ab+bc+ca}$$
After $m-1$ such moves, only two numbers will remain on the blackboard. Supposing one of these is $\tfrac{4}{3}$, show that the other is larger than $4$.
2016 Moldova Team Selection Test, 9
Let $\alpha \in \left( 0, \dfrac{\pi}{2}\right)$.Find the minimum value of the expression
$$ P = (1+\cos\alpha)\left(1+\frac{1}{\sin \alpha} \right)+(1+\sin \alpha)\left(1+\frac{1}{\cos \alpha} \right) .$$
2022 VN Math Olympiad For High School Students, Problem 5
Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $p$ is an odd prime such that $p \equiv \pm 1(\bmod 5)$. Prove that:
a) ${5^{\frac{{p - 1}}{2}}} \equiv 1(\bmod p).$
b) ${F_{p - 1}} \equiv 0(\bmod p).$
c) $k(p)|p-1.$
II Soros Olympiad 1995 - 96 (Russia), 11.4
Draw on the coordinate plane a set of points $M(a, b)$ such that the equation $x^4+ax+b=0$ has a unique root satisfying the condition $0 \le x \le 1$.
2001 Brazil Team Selection Test, Problem 4
Prove that for all integers $n\ge3$ there exists a set $A_n=\{a_1,a_2,\ldots,a_n\}$ of $n$ distinct natural numbers such that, for each $i=1,2,\ldots,n$,
$$\prod_{\small{\begin{matrix}1\le k\le n\\k\ne i\end{matrix}}}a_k\equiv1\pmod{a_i}.$$