Found problems: 15925
1977 IMO Longlists, 28
Let $n$ be an integer greater than $1$. Define
\[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\]
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
2017 Middle European Mathematical Olympiad, 2
Determine the smallest possible real constant $C$ such that the inequality
$$|x^3 + y^3 + z^3 + 1| \leq C|x^5 + y^5 + z^5 + 1|$$
holds for all real numbers $x, y, z$ satisfying $x + y + z = -1$.
1992 Mexico National Olympiad, 5
$x, y, z$ are positive reals with sum $3$. Show that $$6 < \sqrt{2x+3} + \sqrt{2y+3} + \sqrt{2z+3}\le 3\sqrt5$$
2017 CMIMC Algebra, 6
Suppose $P$ is a quintic polynomial with real coefficients with $P(0)=2$ and $P(1)=3$ such that $|z|=1$ whenever $z$ is a complex number satisfying $P(z) = 0$. What is the smallest possible value of $P(2)$ over all such polynomials $P$?
2023 Bundeswettbewerb Mathematik, 2
Determine all triples $(x, y, z)$ of integers that satisfy the equation $x^2+ y^2+ z^2 - xy - yz - zx = 3$
2020 LIMIT Category 1, 8
Find the greatest integer which doesn't exceed $\frac{3^{100}+2^{100}}{3^{96}+2^{96}}$
(A)$81$
(B)$80$
(C)$79$
(D)$82$
1983 USAMO, 2
Prove that the roots of\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\] cannot all be real if $2a^2 < 5b$.
2017 South East Mathematical Olympiad, 1
Let $x_i \in \{0, 1\} (i = 1, 2, \cdots, n)$. If the function $f = f(x_1, x_2, \cdots, x_n)$ only equals $0$ or $1$, then define $f$ as an "$n$-variable Boolean function" and denote
$$D_n (f) = \{ (x_1, x_2, \cdots, x_n) | f(x_1, x_2, \cdots, x_n) = 0 \}$$.
$(1)$ Determine the number of $n$-variable Boolean functions;
$(2)$ Let $g$ be a $10$-variable Boolean function satisfying
$$g(x_1, x_2, \cdots, x_{10}) \equiv 1 + x_1 + x_1 x_2 + x_1 x_2 x_3 + \cdots + x_1 x_2\cdots x_{10} \pmod{2}$$
Evaluate the size of the set $D_{10} (g)$ and $\sum\limits_{(x_1, x_2, \cdots, x_{10}) \in D_{10} (g)} (x_1 + x_2 + x_3 + \cdots + x_{10})$.
1997 Baltic Way, 7
Let $P$ and $Q$ be polynomials with integer coefficients. Suppose that the integers $a$ and $a+1997$ are roots of $P$, and that $Q(1998)=2000$. Prove that the equation $Q(P(x))=1$ has no integer solutions.
2014 PUMaC Algebra B, 2
$f$ is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. $f$ satisfies the condition that $f(f(n))+f(n)=2n+3$ for all nonnegative integers $n$. Find $f(2014)$.
1973 Poland - Second Round, 3
Let $ f:\mathbb{R} \to \mathbb{R} $ be an increasing function satisfying the following conditions:
1. $ f(x+1) = f(x) + 1 $ for each $ x \in \mathbb{R} $,
2. there exists an integer p such that $ f(f(f(O))) = p $. Prove that for every real number $ x $
$$ \lim_{n\to \infty} \frac{x_n}{n} = \frac{p}{3}.$$
where $ x_1 = x $ and $ x_n =f(x_{n-1}) $ for $ n = 2, 3, \ldots $.
IV Soros Olympiad 1997 - 98 (Russia), 10.2
Solve the equation $$\sqrt[3]{x^3+6x^2-6x-1}=\sqrt{x^2+4x+1}$$
1988 IMO Longlists, 27
Assuming that the roots of $x^3 + p \cdot x^2 + q \cdot x + r = 0$ are real and positive, find a relation between $p,q$ and $r$ which gives a necessary condition for the roots to be exactly the cosines of the three angles of a triangle.
1949 Kurschak Competition, 1
Prove that $\sin x + \frac12 \sin 2x + \frac13 \sin 3x > 0$ for $0 < x < 180^o$.
2009 District Olympiad, 1
Let $A,B,C\in \mathcal{M}_3(\mathbb{R})$ such that $\det A=\det B=\det C$ and $\det(A+iB)=\det(C+iA)$. Prove that $\det (A+B)=\det (C+A)$.
2025 Poland - Second Round, 6
Let $1\le k\le n$. Suppose that the sequence $a_1, a_2, \ldots, a_n$ satisfies $0\le a_1 \le a_2 \le \ldots \le a_k$ and $0 \le a_n \le a_{n-1} \le \ldots \le a_k$. The sequence $b_1, b_2, \ldots, b_n$ is the nondecreasing permutation of $a_1, a_2, \ldots, a_n$. Prove that
\[\sum_{i=1}^n \sum_{j=1}^n (j-i)^2a_ia_j \le \sum_{i=1}^n \sum_{j=1}^n (j-i)^2b_ib_j \]
1995 IMO Shortlist, 1
Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k \minus{} 7$ where $ n$ is a positive integer.
1993 Iran MO (3rd Round), 6
Let $x_1, x_2, \ldots, x_{12}$ be twelve real numbers such that for each $1 \leq i \leq 12$, we have $|x_i| \geq 1$. Let $I=[a,b]$ be an interval such that $b-a \leq 2$. Prove that number of the numbers of the form $t= \sum_{i=1}^{12} r_ix_i$, where $r_i=\pm 1$, which lie inside the interval $I$, is less than $1000$.
2015 Thailand TSTST, 1
Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$,
\[f(f(x)-y^{2})=f(x)^{2}-2f(x)y^{2}+f(f(y)).\]
1988 Balkan MO, 2
Find all polynomials of two variables $P(x,y)$ which satisfy
\[P(a,b) P(c,d) = P (ac+bd, ad+bc), \forall a,b,c,d \in \mathbb{R}.\]
1987 Nordic, 4
Let $a, b$, and $c$ be positive real numbers. Prove: $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\le \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2}$ .
1995 French Mathematical Olympiad, Problem 2
Study the convergence of a sequence defined by $u_0\ge0$ and $u_{n+1}=\sqrt{u_n}+\frac1{n+1}$ for all $n\in\mathbb N_0$.
1954 Poland - Second Round, 4
Give the conditions under which the equation $$ \sqrt{x - a} + \sqrt{x - b} = \sqrt{x - c }$$ has roots, assuming that the numbers $ a $, $ b $, $ c $ are pairs of differences
1975 USAMO, 3
If $ P(x)$ denotes a polynomial of degree $ n$ such that $ P(k)\equal{}\frac{k}{k\plus{}1}$ for $ k\equal{}0,1,2,\ldots,n$, determine $ P(n\plus{}1)$.
1992 Czech And Slovak Olympiad IIIA, 4
Solve the equation $\cos 12x = 5\sin 3x+9\ tan ^2x+\ cot ^2x$