Found problems: 85335
2016 Romanian Master of Mathematics, 4
Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $x^{2016}+y> 1-\frac{1}{100}$
1987 IMO Shortlist, 9
Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ?
[i]Proposed by Hungary.[/i]
[hide="Remark"]I'm not sure I'm posting this in a right Forum.[/hide]
2014 USAMTS Problems, 3b:
A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in [i]almost-order[/i] with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters.
(b) How many different ways are there to line up $20$ people in [i]almost-order[/i] if their heights are $120, 125, 130,$ $135,$ $140,$ $145,$ $150,$ $155,$ $160,$ $164, 165, 170, 175, 180, 185, 190, 195, 200, 205$, and $210$ centimeters? (Note that there is someone of height $164$ centimeters.)
2013 Taiwan TST Round 1, 3
Find all $g:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in R$,
\[(4x+g(x)^2)g(y)=4g(\frac{y}{2}g(x))+4xyg(x)\]
2007 Turkey Junior National Olympiad, 2
In a qualification group with $15$ volleyball teams, each team plays with all the other teams exactly once. Since there is no tie in volleyball, there is a winner in every match. After all matches played, a team would be qualified if its total number of losses is not exceeding $N$. If there are at least $7$ teams qualified, find the possible least value of $N$.
2017 Saudi Arabia BMO TST, 1
Let $a, b, c$ be positive real numbers. Prove that
$$\frac{a(b^2 + c^2)}{(b + c)(a^2 + bc)} + \frac{b(c^2 + a^2)}{(c + a)(b^2 + ca)} + \frac{c(a^2 + b^2)}{(a + b)(c^2 + ab)} \ge \frac32$$
2012 International Zhautykov Olympiad, 2
Equilateral triangles $ACB'$ and $BDC'$ are drawn on the diagonals of a convex quadrilateral $ABCD$ so that $B$ and $B'$ are on the same side of $AC$, and $C$ and $C'$ are on the same sides of $BD$. Find $\angle BAD + \angle CDA$ if $B'C' = AB+CD$.
MathLinks Contest 7th, 4.2
Find the number of finite sequences $ \{a_1,a_2,\ldots,a_{2n\plus{}1}\}$, formed with nonnegative integers, for which $ a_1\equal{}a_{2n\plus{}1}\equal{}0$ and $ |a_k \minus{}a_{k\plus{}1}|\equal{}1$, for all $ k\in\{1,2,\ldots,2n\}$.
2014 AMC 10, 15
In rectangle $ABCD$, $DC = 2CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD$?
[asy]
draw((0, 0)--(0, 1)--(2, 1)--(2, 0)--cycle);
draw((0, 0)--(sqrt(3)/3, 1));
draw((0, 0)--(sqrt(3), 1));
label("A", (0, 1), N);
label("B", (2, 1), N);
label("C", (2, 0), S);
label("D", (0, 0), S);
label("E", (sqrt(3)/3, 1), N);
label("F", (sqrt(3), 1), N);
[/asy]
${ \textbf{(A)}\ \ \frac{\sqrt{3}}{6}\qquad\textbf{(B)}\ \frac{\sqrt{6}}{8}\qquad\textbf{(C)}\ \frac{3\sqrt{3}}{16}\qquad\textbf{(D)}}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{4}$
2018 Istmo Centroamericano MO, 3
Determine all sequences of integers $a_1, a_2,. . .,$ such that:
(i) $1 \le a_i \le n$ for all $1 \le i \le n$.
(ii) $| a_i - a_j| = | i - j |$ for any $1 \le i, j \le n$
2006 Bulgaria National Olympiad, 2
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a function that satisfies for all $x>y>0$
\[f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}\]
a) Prove that $f(2x)=4f(x)$ for all $x>0$;
b) Find all such functions.
[i]Nikolai Nikolov, Oleg Mushkarov [/i]
Putnam 1939, B7
Do either $(1)$ or $(2)$:
$(1)$ Let $ai = \sum_{n=0}^{\infty} \dfrac{x^{3n+i}}{(3n+i)!}$ Prove that $a_0^3 + a_1^3 + a_2^3 - 3 a_0a_1a_2 = 1.$
$(2)$ Let $O$ be the origin, $\lambda$ a positive real number, $C$ be the conic $ax^2 + by^2 + cx + dy + e = 0,$ and $C\lambda$ the conic $ax^2 + by^2 + \lambda cx + \lambda dy + \lambda 2e = 0.$ Given a point $P$ and a non-zero real number $k,$ define the transformation $D(P,k)$ as follows. Take coordinates $(x',y')$ with $P$ as the origin. Then $D(P,k)$ takes $(x',y')$ to $(kx',ky').$ Show that $D(O,\lambda)$ and $D(A,-\lambda)$ both take $C$ into $C\lambda,$ where $A$ is the point $(\dfrac{-c \lambda} {(a(1 + \lambda))}, \dfrac{-d \lambda} {(b(1 + \lambda))}) $. Comment on the case $\lambda = 1.$
2009 All-Russian Olympiad Regional Round, 10.6
Circle $\omega$ inscribed in triangle $ABC$ touches sides $BC$, $CA$, $AB$ at points $A_1$, $B_1$ and $C_1$ respectively. On the extension of segment $AA_1$, point $A$ is taken as point D such that $AD= AC_1$. Lines $DB_1$ and $DC_1$ intersect a second time circle $\omega$ at points $B_2$ and $C_2$. Prove that $B_2C_2$ is the diameter of circle of $\omega$.
2018 Taiwan TST Round 1, 2
Assume $ a,b,c $ are arbitrary reals such that $ a+b+c = 0 $. Show that $$ \frac{33a^2-a}{33a^2+1}+\frac{33b^2-b}{33b^2+1}+\frac{33c^2-c}{33c^2+1} \ge 0 $$
2011 Today's Calculation Of Integral, 684
On the $xy$ plane, find the area of the figure bounded by the graphs of $y=x$ and $y=\left|\ \frac34 x^2-3\ \right |-2$.
[i]2011 Kyoto University entrance exam/Science, Problem 3[/i]
2015 Online Math Open Problems, 27
For integers $0 \le m,n \le 64$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor m/2^k \right\rfloor$ and $\left\lfloor n/2^k \right\rfloor$ are both odd integers. Consider a $65 \times 65$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 65$) is \[ (-1)^{\alpha(i-1, j-1)}. \] Compute the remainder when $\det M$ is divided by $1000$.
[i] Proposed by Evan Chen [/i]
2010 Germany Team Selection Test, 3
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i]Proposed by North Korea[/i]
2015 HMIC, 2
Let $m,n$ be positive integers with $m \ge n$. Let $S$ be the set of pairs $(a,b)$ of relatively prime positive integers such that $a,b \le m$ and $a+b > m$.
For each pair $(a,b)\in S$, consider the nonnegative integer solution $(u,v)$ to the equation $au - bv = n$ chosen with $v \ge 0$ minimal, and let $I(a,b)$ denote the (open) interval $(v/a, u/b)$.
Prove that $I(a,b) \subseteq (0,1)$ for every $(a,b)\in S$, and that any fixed irrational number $\alpha\in(0,1)$ lies in $I(a,b)$ for exactly $n$ distinct pairs $(a,b)\in S$.
[i]Victor Wang, inspired by 2013 ISL N7[/i]
2002 Irish Math Olympiad, 2
$ (a)$ A group of people attends a party. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a common acquaintance in the group. What is the maximum possible number of people present?
$ (b)$ If, in addition, the group contains three mutual acquaintances, what is the maximum possible number of people?
2005 Poland - Second Round, 2
A rhombus $ABCD$ with $\angle BAD=60^{\circ}$ is given. Points $E$ on side $AB$ and $F$ on side $AD$ are such that $\angle ECF=\angle ABD$. Lines $CE$ and $CF$ respectively meet line $BD$ at $P$ and $Q$. Prove that $\frac{PQ}{EF}=\frac{AB}{BD}$.
2013 Stanford Mathematics Tournament, 8
The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, it intersects the graph $y=f(x)$ at precisely one point, which is $\frac{1}{\sqrt{m}}$ units from the origin. Let $a$ be the unique real number for which $f$ takes on its maximum value at $x=a$ (you may assume that such an $a$ exists). Find $\int_{0}^{a}f(x) \, dx$.
PEN C Problems, 4
Let $M$ be an integer, and let $p$ be a prime with $p>25$. Show that the set $\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}$ contains a quadratic non-residue to modulus $p$.
2003 Gheorghe Vranceanu, 2
Let be a real number $ a $ and a function $ f:[a,\infty )\longrightarrow\mathbb{R} $ that is continuous at $ a. $ Prove that $ f $ is primitivable on $ (a,\infty ) $ if and only if $ f $ is primitivable on $ [a,\infty ) . $
1955 Moscow Mathematical Olympiad, 311
Find all numbers $a$ such that
(1) all numbers $[a], [2a], . . . , [Na]$ are distinct and
(2) all numbers $\left[ \frac{1}{a}\right], \left[ \frac{2}{a}\right], ..., \left[ \frac{M}{a}\right]$ are distinct.
2009 Today's Calculation Of Integral, 432
Define the function $ f(t)\equal{}\int_0^1 (|e^x\minus{}t|\plus{}|e^{2x}\minus{}t|)dx$. Find the minimum value of $ f(t)$ for $ 1\leq t\leq e$.