Found problems: 348
1991 Baltic Way, 8
Let $a, b, c, d, e$ be distinct real numbers. Prove that the equation
\[(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e)\]
\[+(x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e)\]
\[+(x - b)(x - c)(x - d)(x - e) = 0\]
has four distinct real solutions.
2010 Today's Calculation Of Integral, 635
Suppose that a function $f(x)$ defined in $-1<x<1$ satisfies the following properties (i) , (ii), (iii).
(i) $f'(x)$ is continuous.
(ii) When $-1<x<0,\ f'(x)<0,\ f'(0)=0$, when $0<x<1,\ f'(x)>0$.
(iii) $f(0)=-1$
Let $F(x)=\int_0^x \sqrt{1+\{f'(t)\}^2}dt\ (-1<x<1)$. If $F(\sin \theta)=c\theta\ (c :\text{constant})$ holds for $-\frac{\pi}{2}<\theta <\frac{\pi}{2}$, then find $f(x)$.
[i]1975 Waseda University entrance exam/Science and Technology[/i]
1970 IMO Longlists, 47
Given a polynomial
\[P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),\]
where $a, b, c \neq 0$, prove that $P(x)$ is divisible by
\[Q(x) = abx^2 + (a^2 + b^2)x + ab\]
and conclude that $P(x_0)$ is divisible by $(a + b)^3$ for $x_0 = (a + b + 1)^n, n \in \mathbb N$.
2023 IMC, 1
Find all functions $f: \mathbb{R} \to \mathbb{R}$ that have a continuous second derivative and for which the equality $f(7x+1)=49f(x)$ holds for all $x \in \mathbb{R}$.
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$.
2003 SNSB Admission, 5
Let be an holomorphic function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ having the property that $ |f(z)|\le e^{|\text{Im}(z)|} , $ for all complex numbers $ z. $
Prove that the restriction of any of its derivatives (of any order) to the real numbers is everywhere dominated by $ 1. $
2000 Romania National Olympiad, 2
For any partition $ P $ of $ [0,1] $ , consider the set
$$ \mathcal{A}(P)=\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f’\bigg|_{[0,1]}\right.\wedge\int_0^1 |f(x)|dx =1\wedge \left( y\in P\implies f (y ) =0\right)\right\} . $$
Prove that there exists a partition $ P_0 $ of $ [0,1] $ such that
$$ g\in \mathcal{A}\left( P_0\right)\implies \sup_{x\in [0,1]} \big| g’(x)\big| >4\cdot \# P. $$
Here, $ \# D $ denotes the natural number $ d $ such that $ 0=x_0<x_1<\cdots <x_d=1 $ is a partition $ D $ of $
[0,1] . $
ICMC 6, 2
Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f'(x) > f(x)>0$ for all real numbers $x$. Show that $f(8) > 2022f(0)$.
[i]Proposed by Ethan Tan[/i]
2025 Bulgarian Spring Mathematical Competition, 12.1
In terms of the real numbers $a$ and $b$ determine the minimum value of $$ \sqrt{(x+a)^2+1}+\sqrt{(x+1-a)^2+1}+\sqrt{(x+b)^2+1}+\sqrt{(x+1-b)^2+1}$$ as well as all values of $x$ which attain it.
2006 All-Russian Olympiad, 1
Prove that $\sin\sqrt{x}<\sqrt{\sin x}$ for every real $x$ such that $0<x<\frac{\pi}{2}$.
2014 Harvard-MIT Mathematics Tournament, 10
Fix a positive real number $c>1$ and positive integer $n$. Initially, a blackboard contains the numbers $1,c,\ldots, c^{n-1}$. Every minute, Bob chooses two numbers $a,b$ on the board and replaces them with $ca+c^2b$. Prove that after $n-1$ minutes, the blackboard contains a single number no less than \[\left(\dfrac{c^{n/L}-1}{c^{1/L}-1}\right)^L,\] where $\phi=\tfrac{1+\sqrt 5}2$ and $L=1+\log_\phi(c)$.
2014 NIMO Problems, 6
Let $N=10^6$. For which integer $a$ with $0 \leq a \leq N-1$ is the value of \[\binom{N}{a+1}-\binom{N}{a}\] maximized?
[i]Proposed by Lewis Chen[/i]
2009 ISI B.Stat Entrance Exam, 4
A sequence is called an [i]arithmetic progression of the first order[/i] if the differences of the successive terms are constant. It is called an [i]arithmetic progression of the second order[/i] if the differences of the successive terms form an arithmetic progression of the first order. In general, for $k\geq 2$, a sequence is called an [i]arithmetic progression of the $k$-th order[/i] if the differences of the successive terms form an arithmetic progression of the $(k-1)$-th order.
The numbers
\[4,6,13,27,50,84\]
are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the $n$-th term of this progression.
2011 China Team Selection Test, 2
Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.
2010 Contests, 522
Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.
2010 Today's Calculation Of Integral, 561
Evaluate
\[ \int_{\minus{}1}^1 \frac{1\plus{}2x^2\plus{}3x^4\plus{}4x^6\plus{}5x^8\plus{}6x^{10}\plus{}7x^{12}}{\sqrt{(1\plus{}x^2)(1\plus{}x^4)(1\plus{}x^6)}}dx.\]
1999 Tuymaada Olympiad, 2
Can the graphs of a polynomial of degree 20 and the function $\displaystyle y={1\over x^{40}}$ have exactly 30 points of intersection?
[i]Proposed by K. Kokhas[/i]
2009 Stanford Mathematics Tournament, 9
Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x=\frac{y^2}{2}$
2012 Math Prize For Girls Problems, 19
Define $L(x) = x - \frac{x^2}{2}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by
\[
a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),
\]
where there are $n$ iterations of $L$. For example,
\[
a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr).
\]
As $n$ approaches infinity, what value does $n a_n$ approach?
2008 ISI B.Stat Entrance Exam, 3
Study the derivatives of the function
\[y=\sqrt{x^3-4x}\]
and sketch its graph on the real line.
2005 Today's Calculation Of Integral, 50
Let $a,b$ be real numbers such that $a<b$.
Evaluate
\[\lim_{b\rightarrow a} \frac{\displaystyle\int_a^b \ln |1+(x-a)(b-x)|dx}{(b-a)^3}\].
1991 Arnold's Trivium, 34
Investigate the singular points on the curve $y=x^3$ in the projective plane.
1964 Miklós Schweitzer, 8
Let $ F$ be a closed set in the $ n$-dimensional Euclidean space. Construct a function that is $ 0$ on $ F$, positive outside $ F$ , and whose partial derivatives all exist.
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]
2000 Putnam, 3
Let $f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt)$, where each $a_j$ is areal and $a_N$ is not equal to $0$.
Let $N_k$ denote the number of zeroes (including multiplicites) of $\dfrac{d^k f}{dt^k}$. Prove that \[ N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N. \] [color=green][Only zeroes in [0, 1) should be counted.][/color]