Found problems: 15925
1987 IMO Longlists, 15
Let $a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3$ be nine strictly positive real numbers. We set
\[S_1 = a_1b_2c_3, \quad S_2 = a_2b_3c_1, \quad S_3 = a_3b_1c_2;\]\[T_1 = a_1b_3c_2, \quad T_2 = a_2b_1c_3, \quad T_3 = a_3b_2c_1.\]
Suppose that the set $\{S1, S2, S3, T1, T2, T3\}$ has at most two elements.
Prove that
\[S_1 + S_2 + S_3 = T_1 + T_2 + T_3.\]
2008 China Girls Math Olympiad, 2
Let $ \varphi(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$ be a polynomial with real coefficients. Given that $ \varphi(x)$ has three positive real roots and that $ \varphi(0) < 0$, prove that
\[ 2b^3 \plus{} 9a^2d \minus{} 7abc \leq 0.
\]
2006 Iran MO (3rd Round), 4
$p(x)$ is a real polynomial that for each $x\geq 0$, $p(x)\geq 0$. Prove that there are real polynomials $A(x),B(x)$ that $p(x)=A(x)^{2}+xB(x)^{2}$
2006 Germany Team Selection Test, 2
Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.
1948 Moscow Mathematical Olympiad, 145
Without tables and such, prove that $\frac{1}{\log_2 \pi}+\frac{1}{\log_5 \pi} >2$
2019 New Zealand MO, 2
Find all real solutions to the equation $(x^2 + 3x + 1)^{x^2-x-6} = 1$.
1983 Putnam, B2
For positive integers $n$, let $C(n)$ be the number of representation of $n$ as a sum of nonincreasing powers of $2$, where no power can be used more than three times. For example, $C(8)=5$ since the representations of $8$ are:
$$8,4+4,4+2+2,4+2+1+1,\text{ and }2+2+2+1+1.$$Prove or disprove that there is a polynomial $P(x)$ such that $C(n)=\lfloor P(n)\rfloor$ for all positive integers $n$.
2010 IMO Shortlist, 8
Given six positive numbers $a,b,c,d,e,f$ such that $a < b < c < d < e < f.$ Let $a+c+e=S$ and $b+d+f=T.$ Prove that
\[2ST > \sqrt{3(S+T)\left(S(bd + bf + df) + T(ac + ae + ce) \right)}.\]
[i]Proposed by Sung Yun Kim, South Korea[/i]
1996 All-Russian Olympiad, 7
Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coeficients in $M$, all of whose roots are real and belong to $M$?
[i]E. Malinnikova[/i]
1991 Putnam, A3
Find all real polynomials $ p(x)$ of degree $ n \ge 2$ for which there exist real numbers $ r_1 < r_2 < ... < r_n$ such that
(i) $ p(r_i) \equal{} 0, 1 \le i \le n$, and
(ii) $ p' \left( \frac {r_i \plus{} r_{i \plus{} 1}}{2} \right) \equal{} 0, 1 \le i \le n \minus{} 1$.
[b]Follow-up:[/b] In terms of $ n$, what is the maximum value of $ k$ for which $ k$ consecutive real roots of a polynomial $ p(x)$ of degree $ n$ can have this property? (By "consecutive" I mean we order the real roots of $ p(x)$ and ignore the complex roots.) In particular, is $ k \equal{} n \minus{} 1$ possible for $ n \ge 3$?
1965 AMC 12/AHSME, 13
Let $ n$ be the number of number-pairs $ (x,y)$ which satisfy $ 5y \minus{} 3x \equal{} 15$ and $ x^2 \plus{} y^2 \le 16$. Then $ n$ is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \text{more than two, but finite} \qquad \textbf{(E)}\ \text{greater than any finite number}$
2014 Putnam, 6
Let $n$ be a positive integer. What is the largest $k$ for which there exist $n\times n$ matrices $M_1,\dots,M_k$ and $N_1,\dots,N_k$ with real entries such that for all $i$ and $j,$ the matrix product $M_iN_j$ has a zero entry somewhere on its diagonal if and only if $i\ne j?$
2020 Kosovo National Mathematical Olympiad, 1
Compare the following two numbers: $2^{2^{2^{2^{2}}}}$ and $3^{3^{3^{3}}}$.
1983 Dutch Mathematical Olympiad, 3
Suppose that $ a,b,c,p$ are real numbers with $ a,b,c$ not all equal, such that: $ a\plus{}\frac{1}{b}\equal{}b\plus{}\frac{1}{c}\equal{}c\plus{}\frac{1}{a}\equal{}p.$ Determine all possible values of $ p$ and prove that $ abc\plus{}p\equal{}0$.
2003 Kazakhstan National Olympiad, 8
Determine all functions $f: \mathbb R \to \mathbb R$ with the property
\[f(f(x)+y)=2x+f(f(y)-x), \quad \forall x,y \in \mathbb R.\]
2006 China Second Round Olympiad, 3
Solve the system of equations in real numbers:
\[ \begin{cases} x-y+z-w=2 \\ x^2-y^2+z^2-w^2=6 \\ x^3-y^3+z^3-w^3=20 \\ x^4-y^4+z^4-w^4=66 \end{cases} \]
1997 Pre-Preparation Course Examination, 4
Let $f : \mathbb N \to \mathbb N$ be an injective function such that there exists a positive integer $k$ for which $f(n) \leq n^k$. Prove that there exist infinitely many primes $q$ such that the equation $f(x) \equiv 0 \pmod q$ has a solution in prime numbers.
2022 Kyiv City MO Round 1, Problem 3
You are given $n$ not necessarily distinct real numbers $a_1, a_2, \ldots, a_n$. Let's consider all $2^n-1$ ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to $1$?
For example, if $a = [-1, 2, 2]$, then we got $3$ once, $4$ once, $2$ twice, $-1$ once, $1$ twice, so the total number of ones here is $2$.
[i](Proposed by Anton Trygub)[/i]
2022 Poland - Second Round, 1
Find all real quadruples $(a,b,c,d)$ satisfying the system of equations
$$
\left\{ \begin{array}{ll}
ab+cd = 6 \\
ac + bd = 3 \\
ad + bc = 2 \\
a + b + c + d = 6.
\end{array} \right.
$$
2013 AMC 12/AHSME, 5
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $\$105$, Dorothy paid $\$125$, and Sammy paid $\$175$. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$?
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 $
2016 IFYM, Sozopol, 8
Prove that there exist infinitely many natural numbers $n$, for which there $\exists \, f:\{0,1…n-1\}\rightarrow \{0,1…n-1\}$, satisfying the following conditions:
1) $f(x)\neq x$;
2) $f(f(x))=x$;
3) $f(f(f(x+1)+1)+1)=x$ for $\forall x\in \{0,1…n-1\}$.
2024 5th Memorial "Aleksandar Blazhevski-Cane", P2
Let $x,y$ and $z$ be positive real numbers such that $xy+z^2=8$. Determine the smallest possible value of the expression $$\frac{x+y}{z}+\frac{y+z}{x^2}+\frac{z+x}{y^2}.$$
1996 USAMO, 1
Prove that the average of the numbers $n \sin n^{\circ} \; (n = 2,4,6,\ldots,180)$ is $\cot 1^{\circ}$.
2000 Iran MO (3rd Round), 1
Let $n$ be a positive integer. Suppose $S$ is a set of ordered $n-\mbox{tuples}$ of
nonnegative integers such that, whenever $(a_1,\dots,an)\in S$ and $b_i$ are nonnegative integers with$b_i\le a_i$, the $n-\text{tuple}$ $(b_1,\dots,b_n)$ is also in $S$. If $h_m$
is the number of elements of $S$ with the sum of components equal to$m$,
prove that $h_m$ is a polynomial in $m$ for all sufficiently large$m$.
2000 APMO, 1
Compute the sum: $\sum_{i=0}^{101} \frac{x_i^3}{1-3x_i+3x_i^2}$ for $x_i=\frac{i}{101}$.