Found problems: 85335
2020 Taiwan APMO Preliminary, P1
Let $\triangle ABC$ satisfies $\cos A:\cos B:\cos C=1:1:2$, then $\sin A=\sqrt[s]{t}$($s\in\mathbb{N},t\in\mathbb{Q^+}$ and $t$ is an irreducible fraction). Find $s+t$.
Estonia Open Senior - geometry, 2004.1.3
a) Does there exist a convex quadrangle $ABCD$ satisfying the following conditions
(1) $ABCD$ is not cyclic;
(2) the sides $AB, BC, CD$ and $DA$ have pairwise different lengths;
(3) the circumradii of the triangles $ABC, ADC, BAD$ and $BCD$ are equal?
b) Does there exist such a non-convex quadrangle?
1998 Finnish National High School Mathematics Competition, 5
$15\times 36$-checkerboard is covered with square tiles. There are two kinds of tiles, with side $7$ or $5.$
Tiles are supposed to cover whole squares of the board and be non-overlapping.
What is the maximum number of squares to be covered?
1981 IMO Shortlist, 1
[b]a.)[/b] For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers?
[b]b.)[/b] For which $n>2$ is there exactly one set having this property?
2019 CCA Math Bonanza, I2
Square $1$ is drawn with side length $4$. Square $2$ is then drawn inside of Square $1$, with its vertices at the midpoints of the sides of Square $1$. Given Square $n$ for a positive integer $n$, we draw Square $n+1$ with vertices at the midpoints of the sides of Square $n$. For any positive integer $n$, we draw Circle $n$ through the four vertices of Square $n$. What is the area of Circle $7$?
[i]2019 CCA Math Bonanza Individual Round #2[/i]
1994 Canada National Olympiad, 5
Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$, and let $H$ be any interior point on $AD$. Lines $BH,CH$, when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$.
2010 Today's Calculation Of Integral, 604
Let $r$ be a positive integer. Determine the value of $a$ for which the limit value $\lim_{n\to\infty} \frac{\sum_{k=1}^n k^r}{n^a} $ has a non zero finite value, then find the limit value.
1956 Tokyo Institute of Technology entrance exam
2018 Kazakhstan National Olympiad, 3
Is there exist a function $f:\mathbb {N}\to \mathbb {N}$ with for $\forall m,n \in \mathbb {N}$ $$f\left(mf\left(n\right)\right)=f\left(m\right)f\left(m+n\right)+n ?$$
2011 AMC 12/AHSME, 19
A lattice point in an $xy$-coordinate system is any point $(x,y)$ where both $x$ and $y$ are integers. The graph of $y=mx+2$ passes through no lattice point with $0<x \le 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$?
$ \textbf{(A)}\ \frac{51}{101} \qquad
\textbf{(B)}\ \frac{50}{99} \qquad
\textbf{(C)}\ \frac{51}{100} \qquad
\textbf{(D)}\ \frac{52}{101} \qquad
\textbf{(E)}\ \frac{13}{25} $
1997 Vietnam Team Selection Test, 1
The function $ f : \mathbb{N} \to \mathbb{Z}$ is defined by $ f(0) \equal{} 2$, $ f(1) \equal{} 503$ and $ f(n \plus{} 2) \equal{} 503f(n \plus{} 1) \minus{} 1996f(n)$ for all $ n \in\mathbb{N}$. Let $ s_1$, $ s_2$, $ \ldots$, $ s_k$ be arbitrary integers not smaller than $ k$, and let $ p(s_i)$ be an arbitrary prime divisor of $ f\left(2^{s_i}\right)$, ($ i \equal{} 1, 2, \ldots, k$). Prove that, for any positive integer $ t$ ($ t\le k$), we have $ 2^t \Big | \sum_{i \equal{} 1}^kp(s_i)$ if and only if $ 2^t | k$.
1995 Irish Math Olympiad, 2
Determine all integers $ a$ for which the equation $ x^2\plus{}axy\plus{}y^2\equal{}1$ has infinitely many distinct integer solutions $ x,y$.
1952 Moscow Mathematical Olympiad, 221
Prove that if for any positive $p$ all roots of the equation $ax^2 + bx + c + p = 0$ are real and positive then $a = 0$.
2013 CIIM, Problem 4
Let $a_1,b_1,c_1,a_2,b_2,c_2$ be positive real number and $F,G:(0,\infty)\to(0,\infty)$ be to differentiable and positive functions that satisfy the identities: $$\frac{x}{F} = 1 + a_1x+ b_1y + c_1G$$ $$\frac{y}{G} = 1 + a_2x+ b_2y + c_2F.$$
Prove that if $0 < x_1 \leq x_2$ and $0 < y_2 \leq y_1$, then $F(x_1,x_2) \leq F(x_2,y_2)$ and $G(x_1,y_1) \geq G(x_2,y_2).$
1974 IMO Longlists, 36
Consider the binomial coefficients $\binom{n}{k}=\frac{n!}{k!(n-k)!}\ (k=1,2,\ldots n-1)$. Determine all positive integers $n$ for which $\binom{n}{1},\binom{n}{2},\ldots ,\binom{n}{n-1}$ are all even numbers.
2005 India Regional Mathematical Olympiad, 3
If $a,b,c$ are positive three real numbers such that $| a-b | \geq c , | b-c | \geq a, | c-a | \geq b$ . Prove that one of $a,b,c$ is equal to the sum of the other two.
2012 JBMO ShortLists, 3
Let $a$ , $b$ , $c$ be positive real numbers such that $a+b+c=a^2+b^2+c^2$ . Prove that :
\[\frac{a^2}{a^2+ab}+\frac{b^2}{b^2+bc}+\frac{c^2}{c^2+ca} \geq \frac{a+b+c}{2}\]
2019 Czech-Austrian-Polish-Slovak Match, 1
Let $\omega$ be a circle. Points $A,B,C,X,D,Y$ lie on $\omega$ in this order such that $BD$ is its diameter and $DX=DY=DP$ , where $P$ is the intersection of $AC$ and $BD$. Denote by $E,F$ the intersections of line $XP$ with lines $AB,BC$, respectively. Prove that points $B,E,F,Y$ lie on a single circle.
2018 Middle European Mathematical Olympiad, 1
Let $Q^+$ denote the set of all positive rational number and let $\alpha\in Q^+.$ Determine all functions $f:Q^+ \to (\alpha,+\infty )$ satisfying $$f(\frac{ x+y}{\alpha}) =\frac{ f(x)+f(y)}{\alpha}$$
for all $x,y\in Q^+ .$
1999 Mongolian Mathematical Olympiad, Problem 6
Show that there exists a positive integer $n$ such that the decimal representations of $3^n$ and $7^n$ both start with the digits $10$.
2018 Moscow Mathematical Olympiad, 3
Are there such natural $n$, that exist polynomial of degree $n$ and with $n$ different real roots, and
a) $P(x)P(x+1)=P(x^2)$
b) $P(x)P(x+1)=P(x^2+1)$
2011 Harvard-MIT Mathematics Tournament, 3
Evaluate $\displaystyle \int_1^\infty \left(\frac{\ln x}{x}\right)^{2011} dx$.
2012 China Western Mathematical Olympiad, 3
Let $A$ be a set of $n$ elements and $A_1, A_2, ... A_k$ subsets of $A$ such that for any $2$ distinct subsets $A_i, A_j$ either they are disjoint or one contains the other. Find the maximum value of $k$
2009 Princeton University Math Competition, 6
Find the smallest positive $\alpha$ (in degrees) for which all the numbers \[\cos{\alpha},\cos{2\alpha},\ldots,\cos{2^n\alpha},\ldots\] are negative.
2012 ELMO Shortlist, 1
Let $n\ge2$ be a positive integer. Given a sequence $\left(s_i\right)$ of $n$ distinct real numbers, define the "class" of the sequence to be the sequence $\left(a_1,a_2,\ldots,a_{n-1}\right)$, where $a_i$ is $1$ if $s_{i+1} > s_i$ and $-1$ otherwise.
Find the smallest integer $m$ such that there exists a sequence $\left(w_i\right)$ of length $m$ such that for every possible class of a sequence of length $n$, there is a subsequence of $\left(w_i\right)$ that has that class.
[i]David Yang.[/i]
2018 ASDAN Math Tournament, 2
Given that $\sec x+\tan x=2018$, compute $\csc x+\cot x$.