Found problems: 348
2009 Putnam, A6
Let $ f: [0,1]^2\to\mathbb{R}$ be a continuous function on the closed unit square such that $ \frac{\partial f}{\partial x}$ and $ \frac{\partial f}{\partial y}$ exist and are continuous on the interior of $ (0,1)^2.$ Let $ a\equal{}\int_0^1f(0,y)\,dy,\ b\equal{}\int_0^1f(1,y)\,dy,\ c\equal{}\int_0^1f(x,0)\,dx$ and $ d\equal{}\int_0^1f(x,1)\,dx.$ Prove or disprove: There must be a point $ (x_0,y_0)$ in $ (0,1)^2$ such that
$ \frac{\partial f}{\partial x}(x_0,y_0)\equal{}b\minus{}a$ and $ \frac{\partial f}{\partial y}(x_0,y_0)\equal{}d\minus{}c.$
2007 Putnam, 5
Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$
\[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\]
($ \lfloor a\rfloor$ means the largest integer $ \le a.$)
2011 Morocco National Olympiad, 1
Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.
2004 France Team Selection Test, 1
Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that
$a_1 + ... + a_n = b_1 + ... + b_n = 1$.
Find the minimal value of
$ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.
2012 Centers of Excellency of Suceava, 4
Let be two real numbers $ a<b $ and a differentiable function $ f:[a,b]\longrightarrow\mathbb{R} $ that has a bounded derivative.
Show that if $ \frac{f(b)-f(a)}{b-a} $ is equal to the global supremum or infimum of $ f', $ then $ f $ is polynomial with degree $ 1. $
[i]Cătălin Țigăeru[/i]
2010 Contests, 522
Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.
2013 Today's Calculation Of Integral, 867
Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$
2006 Harvard-MIT Mathematics Tournament, 1
A nonzero polynomial $f(x)$ with real coefficients has the property that $f(x)=f^\prime(x)f^{\prime\prime}(x)$. What is the leading coefficient of $f(x)$?
2006 Stanford Mathematics Tournament, 2
Find the minimum value of $ 2x^2\plus{}2y^2\plus{}5z^2\minus{}2xy\minus{}4yz\minus{}4x\minus{}2z\plus{}15$ for real numbers $ x$, $ y$, $ z$.
Today's calculation of integrals, 859
In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$
Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$
1989 Balkan MO, 2
Let $\overline{a_{n}a_{n-1}\ldots a_{1}a_{0}}$ be the decimal representation of a prime positive integer such that $n>1$ and $a_{n}>1$. Prove that the polynomial $P(x)=a_{n}x^{n}+\ldots +a_{1}x+a_{0}$ cannot be written as a product of two non-constant integer polynomials.
2014 Taiwan TST Round 2, 1
Let $a_i > 0$ for $i=1,2,\dots,n$ and suppose $a_1 + a_2 + \dots + a_n = 1$. Prove that for any positive integer $k$,
\[ \left( a_1^k + \frac{1}{a_1^k} \right) \left( a_2^k + \frac{1}{a_2^k} \right) \dots \left( a_n^k + \frac{1}{a_n^k} \right) \ge \left( n^k + \frac{1}{n^k} \right)^n. \]
2007 German National Olympiad, 6
For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$
1991 Arnold's Trivium, 1
Sketch the graph of the derivative and the graph of the integral of a function given by a free-hand graph.
2013 F = Ma, 16
A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way.
How much time passes until the cloud collapses fully? (The constant $0.5427$ is actually $\sqrt{\frac{3 \pi}{32}}$.)
$\textbf{(A) } \frac{0.5427}{r_0^2 \sqrt{G \rho_0}}\\ \\
\textbf{(B) } \frac{0.5427}{r_0 \sqrt{G \rho_0}}\\ \\
\textbf{(C) } \frac{0.5427}{\sqrt{r_0} \sqrt{G \rho_0}}\\ \\
\textbf{(D) } \frac{0.5427}{\sqrt{G \rho_0}}\\ \\
\textbf{(E) } \frac{0.5427}{\sqrt{G \rho_0}}r_0$
2005 Korea National Olympiad, 4
Find all $f: \mathbb R \to\mathbb R$ such that for all real numbers $x$, $f(x) \geq 0$ and for all real numbers $x$ and $y$, \[ f(x+y)+f(x-y)-2f(x)-2y^2=0. \]
2011 Romania National Olympiad, 2
[color=darkred]Let $u:[a,b]\to\mathbb{R}$ be a continuous function that has finite left-side derivative $u_l^{\prime}(x)$ in any point $x\in (a,b]$ . Prove that the function $u$ is monotonously increasing if and only if $u_l^{\prime}(x)\ge 0$ , for any $x\in (a,b]$ .[/color]
2008 China Western Mathematical Olympiad, 4
Let P be an interior point of a regular n-gon $ A_1 A_2 ...A_n$, the lines $ A_i P$ meet the regular n-gon at another point $ B_i$, where $ i\equal{}1,2,...,n$. Prove that sums of all $ PA_i\geq$ sum of all $ PB_i$.
2022 JHMT HS, 2
Suppose that $f$ is a differentiable function such that $f(0) = 20$ and $|f'(x)| \leq 4$ for all real numbers $x$. Let $a$ and $b$ be real numbers such that [i]every[/i] such function $f$ satisfies $a \leq f(22) \leq b$. Find the smallest possible value of $|a| + |b|$.
2024 All-Russian Olympiad Regional Round, 11.7
Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ in points $A$, $B$. The intersection of tangents to $G_2$ in points $A$, $B$ is point $C$. If $C \in G_1$, find all possible values of $p$.
2009 Harvard-MIT Mathematics Tournament, 10
Let $a$ and $b$ be real numbers satisfying $a>b>0$. Evaluate \[\int_0^{2\pi}\dfrac{1}{a+b\cos(\theta)}d\theta.\] Express your answer in terms of $a$ and $b$.
1992 Brazil National Olympiad, 1
The equation $x^3+px+q=0$ has three distinct real roots. Show that $p<0$
1999 Putnam, 2
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P^{\prime\prime}(x)$, where $Q(x)$ is a quadratic polynomial and $P^{\prime\prime}(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.
2006 Swedish Mathematical Competition, 3
A cubic polynomial $f$ with a positive leading coefficient has three different positive zeros. Show that $f'(a)+ f'(b)+ f'(c) > 0$.
Gheorghe Țițeica 2024, P2
Find all monotonic and twice differentiable functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f''+4f+3f^2+8f^3=0.$$