Found problems: 85335
2002 Flanders Junior Olympiad, 1
Prove that for all $a,b,c \in \mathbb{R}^+_0$ we have \[\frac{a}{bc}+\frac{b}{ac}+\frac{c}{ab} \ge \frac2a+\frac2b-\frac2c\] and determine when equality occurs.
2019 Baltic Way, 3
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
$$f(xf(y)-y^2)=(y+1)f(x-y)$$
holds for all $x,y\in\mathbb{R}$.
1996 AMC 12/AHSME, 8
If $3 = k \cdot 2^r$ and $15 = k \cdot 4^r$, then $r =$
$\text{(A)}\ - \log_2 5 \qquad \text{(B)}\ \log_5 2 \qquad \text{(C)}\ \log_{10} 5 \qquad \text{(D)}\ \log_2 5 \qquad \text{(E)}\ \displaystyle \frac{5}{2}$
2010 Today's Calculation Of Integral, 607
On the coordinate plane, Let $C$ be the graph of $y=(\ln x)^2\ (x>0)$ and for $\alpha >0$, denote $L(\alpha)$ be the tangent line of $C$ at the point $(\alpha ,\ (\ln \alpha)^2).$
(1) Draw the graph.
(2) Let $n(\alpha)$ be the number of the intersection points of $C$ and $L(\alpha)$. Find $n(\alpha)$.
(3) For $0<\alpha <1$, let $S(\alpha)$ be the area of the region bounded by $C,\ L(\alpha)$ and the $x$-axis. Find $S(\alpha)$.
2010 Tokyo Institute of Technology entrance exam, Second Exam.
2019 USEMO, 6
Let $ABC$ be an acute scalene triangle with circumcenter $O$ and altitudes $\overline{AD}$, $\overline{BE}$, $\overline{CF}$. Let $X$, $Y$, $Z$ be the midpoints of $\overline{AD}$, $\overline{BE}$, $\overline{CF}$. Lines $AD$ and $YZ$ intersect at $P$, lines $BE$ and $ZX$ intersect at $Q$, and lines $CF$ and $XY$ intersect at $R$.
Suppose that lines $YZ$ and $BC$ intersect at $A'$, and lines $QR$ and $EF$ intersect at $D'$. Prove that the perpendiculars from $A$, $B$, $C$, $O$, to the lines $QR$, $RP$, $PQ$, $A'D'$, respectively, are concurrent.
[i]Ankan Bhattacharya[/i]
2014 Brazil National Olympiad, 3
Let $N$ be an integer, $N>2$. Arnold and Bernold play the following game: there are initially $N$ tokens on a pile. Arnold plays first and removes $k$ tokens from the pile, $1\le k < N$. Then Bernold removes $m$ tokens from the pile, $1\le m\le 2k$ and so on, that is, each player, on its turn, removes a number of tokens from the pile that is between $1$ and twice the number of tokens his opponent took last. The player that removes the last token wins.
For each value of $N$, find which player has a winning strategy and describe it.
2015 Cuba MO, 3
Determine the smallest integer of the form $\frac{ \overline{AB}}{B}$ .where $A$ and $B$ are three-digit positive integers and $\overline{AB}$ denotes the six-digit number that is form by writing the numbers $A$ and $B$ consecutively.
1996 May Olympiad, 2
Considering the three-digit natural numbers, how many of them, when adding two of their digits, are double of their remainder? Justify your answer.
2019 Durer Math Competition Finals, 16
Triangle $ABC$ has side lengths $13$, $14$ and $15$. Let $k, k_A,k_B,k_C$ be four circles of radius $ r$ inside the triangle such that $k_A$ is tangent to sides $AB$ and $AC$, $k_B$ is tangent to sides $BA$ and $BC$, $k_C$ is tangent to sides $CA$ and $CB$, and $k$ is externally tangent to circles $k_A$, $k_B$ and $k_C$. Let $r = m/n$ where $m$ and $n$ are coprime. Find $m + n$.
2005 JBMO Shortlist, 3
Let $ABCDEF$ be a regular hexagon and $M\in (DE)$, $N\in(CD)$ such that $m (\widehat {AMN}) = 90^\circ$ and $AN = CM \sqrt {2}$. Find the value of $\frac{DM}{ME}$.
1987 AMC 12/AHSME, 13
A long piece of paper $5$ cm wide is made into a roll for cash registers by wrapping it $600$ times around a cardboard tube of diameter $2$ cm, forming a roll $10$ cm in diameter. Approximate the length of the paper in meters. (Pretend the paper forms $600$ concentric circles with diameters evenly spaced from $2$ cm to $10$ cm.)
$ \textbf{(A)}\ 36\pi \qquad\textbf{(B)}\ 45\pi \qquad\textbf{(C)}\ 60\pi \qquad\textbf{(D)}\ 72\pi \qquad\textbf{(E)}\ 90\pi $
1987 Romania Team Selection Test, 9
Prove that for all real numbers $\alpha_1,\alpha_2,\ldots,\alpha_n$ we have \[ \sum_{i=1}^n \sum_{j=1}^n ij \cos {(\alpha_i - \alpha_j )} \geq 0. \]
[i]Octavian Stanasila[/i]
2015 Olympic Revenge, 2
Given $v = (a,b,c,d) \in \mathbb{N}^4$, let $\Delta^{1} (v) = (|a-b|,|b-c|,|c-d|,|d-a|)$ and $\Delta^{k} (v) = \Delta(\Delta^{k-1} (v))$ for $k > 1$. Define $f(v) = \min\{k \in \mathbb{N} : \Delta^k (v) = (0,0,0,0)\}$ and $\max(v) = \max\{a,b,c,d\}.$ Show that $f(v) < 1000\log \max(v)$ for all sufficiently large $v$ and $f(v) > 0.001 \log \max (v)$ for infinitely many $v$.
2015 Saudi Arabia JBMO TST, 3
A natural number is called $nice$ if it doesn't contain 0 and if we add the product of its digit to the number, we obtain number with the same product of its digits. Prove that there is a nice 2015-digit number.
2015 Korea Junior Math Olympiad, 2
For a positive integer $m$, prove that the number of pairs of positive integers $(x,y)$ which satisfies the following two conditions is even or $0$.
(i): $x^2-3y^2+2=16m$
(ii): $2y \le x-1$
1994 IMC, 6
Let $f\in C^2[0,N]$ and $|f'(x)|<1$, $f''(x)>0$ for every $x\in [0, N]$. Let $0\leq m_0\ <m_1 < \cdots < m_k\leq N$ be integers such that $n_i=f(m_i)$ are also integers for $i=0,1,\ldots, k$. Denote $b_i=n_i-n_{i-1}$ and $a_i=m_i-m_{i-1}$ for $i=1,2,\ldots, k$.
a) Prove that
$$-1<\frac{b_1}{a_1}<\frac{b_2}{a_2}<\cdots < \frac{b_k}{a_k}<1$$
b) Prove that for every choice of $A>1$ there are no more than $N / A$ indices $j$ such that $a_j>A$.
c) Prove that $k\leq 3N^{2/3}$ (i.e. there are no more than $3N^{2/3}$ integer points on the curve $y=f(x)$, $x\in [0,N]$).
1971 IMO Longlists, 30
Prove that the system of equations
\[2yz+x-y-z=a,\\ 2xz-x+y-z=a,\\ 2xy-x-y+z=a, \]
$a$ being a parameter, cannot have five distinct solutions. For what values of $a$ does this system have four distinct integer solutions?
TNO 2008 Senior, 4
Prove that the diagonals of a convex quadrilateral are perpendicular if and only if the sum of the squares of one pair of opposite sides is equal to the sum of the squares of the other pair.
2003 IMO Shortlist, 6
Each pair of opposite sides of a convex hexagon has the following property: the distance between their midpoints is equal to $\dfrac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.
2024 Junior Balkan Team Selection Tests - Romania, P4
Let $n\geqslant 2$ be an integer and $A{}$ a set of $n$ points in the plane. Find all integers $1\leqslant k\leqslant n-1$ with the following property: any two circles $C_1$ and $C_2$ in the plane such that $A\cap\text{Int}(C_1)\neq A\cap\text{Int}(C_2)$ and $|A\cap\text{Int}(C_1)|=|A\cap\text{Int}(C_2)|=k$ have at least one common point.
[i]Cristi Săvescu[/i]
2022 Peru MO (ONEM), 3
Let $R$ be the set of real numbers and $f : R \to R$ be a function that satisfies:
$$f(xy) + y + f(x + f(y)) = (y + 1)f(x),$$ for all real numbers $x, y$.
a) Determine the value of $f(0)$.
b) Prove that $f(x) = 2-x$ for every real number $x$.
2016 Iran Team Selection Test, 4
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.
2014 Singapore Senior Math Olympiad, 15
Let $x,y$ be real numbers such that $y=|x-1|$. What is the smallest value of $(x-1)^2+(y-2)^2$?
2000 Moldova National Olympiad, Problem 2
Prove that if real numbers $a,b,c,d$ satisfy $a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2$, then they also satisfy $a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4$.
2017 HMNT, 9
Let $A, B, C, D$ be points chosen on a circle, in that order. Line $BD$ is reflected over lines $AB$ and $DA$ to obtain lines $\ell_1$ and $\ell_2$ respectively. If lines $\ell_1$, $\ell_2$, and $AC$ meet at a common point and if $AB = 4$, $BC = 3$, $CD = 2$, compute the length $DA$.