Found problems: 15925
2012 European Mathematical Cup, 4
Olja writes down $n$ positive integers $a_1, a_2, \ldots, a_n$ smaller than $p_n$ where $p_n$ denotes the $n$-th prime number. Oleg can choose two (not necessarily different) numbers $x$ and $y$ and replace one of them with their product $xy$. If there are two equal numbers Oleg wins. Can Oleg guarantee a win?
[i]Proposed by Matko Ljulj.[/i]
1960 Czech and Slovak Olympiad III A, 4
Determine the (real) domain of a function $$y=\sqrt{1-\frac{x}{4}|x|+\sqrt{1-\frac{x}{2}|x|\,}\,}-\sqrt{1-\frac{x}{4}|x|-\sqrt{1-\frac{x}{2}|x|\,}\,}$$ and draw its graph.
2004 Swedish Mathematical Competition, 4
If $0 < v <\frac{\pi}{2}$ and $\tan v = 2v$, decide whether $sinv < \frac{20}{21}$.
2015 Balkan MO Shortlist, A5
Let $m, n$ be positive integers and $a, b$ positive real numbers different from $1$ such thath $m > n$ and
$$\frac{a^{m+1}-1}{a^m-1} = \frac{b^{n+1}-1}{b^n-1} = c$$. Prove that $a^m c^n > b^n c^{m}$
(Turkey)
V Soros Olympiad 1998 - 99 (Russia), 9.3
On the coordinate plane, draw a set of points $M(x;y)$, whose coordinates satisfy the equation $$\sqrt{(x - 1)^2+ y^2} +\sqrt{x^2 + (y -1)^2} = \sqrt2.$$
2020 Princeton University Math Competition, 15
Suppose that f is a function $f : R_{\ge 0} \to R$ so that for all $x, y \in R_{\ge 0}$ (nonnegative reals) we have that $$f(x)+f(y) = f(x+y+xy)+f(x)f(y).$$ Given that $f\left(\frac{3}{5} \right) = \frac12$ and$ f(1) = 3$, determine
$$\lfloor \log_2 (-f(10^{2021} - 1)) \rfloor.$$
2017 Junior Balkan Team Selection Tests - Romania, 4
Let $a, b, c, d$ be non-negative real numbers satisfying $a + b + c + d = 3$. Prove that
$$\frac{a}{1 + 2b^3} + \frac{b}{1 + 2c^3} +\frac{c}{1 + 2d^3} +\frac{d}{1 + 2a^3} \ge \frac{a^2 + b^2 + c^2 + d^2}{3}$$
When does the equality hold?
2022 CMIMC, 1.5
Grant is standing at the beginning of a hallway with infinitely many lockers, numbered, $1, 2, 3, \ldots$ All of the lockers are initially closed. Initially, he has some set $S = \{1, 2, 3, \ldots\}$.
Every step, for each element $s$ of $S$, Grant goes through the hallway and opens each locker divisible by $s$ that is closed, and closes each locker divisible by $s$ that is open. Once he does this for all $s$, he then replaces $S$ with the set of labels of the currently open lockers, and then closes every door again.
After $2022$ steps, $S$ has $n$ integers that divide ${10}^{2022}$. Find $n$.
[i]Proposed by Oliver Hayman[/i]
2010 Morocco TST, 2
Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$
1998 Belarus Team Selection Test, 2
Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.
2016 Saudi Arabia GMO TST, 1
Let $S = x + y +z$ where $x, y, z$ are three nonzero real numbers satisfying the following system of inequalities:
$$xyz > 1$$
$$x + y + z >\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$
Prove that $S$ can take on any real values when $x, y, z$ vary
1964 Miklós Schweitzer, 2
Let $ p$ be a prime and let \[ l_k(x,y)\equal{}a_kx\plus{}b_ky \;(k\equal{}1,2,...,p^2)\ .\] be homogeneous linear polynomials with integral coefficients. Suppose that for every pair $ (\xi,\eta)$ of integers, not both divisible by $ p$, the values $ l_k(\xi,\eta), \;1\leq k\leq p^2 $, represent every residue class $ \textrm{mod} \;p$ exactly $ p$ times. Prove that the set of pairs $ \{(a_k,b_k): 1\leq k \leq p^2 \}$ is identical $ \textrm{mod} \;p$ with the set $ \{(m,n): 0\leq m,n \leq p\minus{}1 \}.$
2020 Bangladesh Mathematical Olympiad National, Problem 2
How many integers $n$ are there subject to the constraint that $1 \leq n \leq 2020$ and $n^n$ is a perfect square?
1982 Bulgaria National Olympiad, Problem 5
Find all values of parameters $a,b$ for which the polynomial
$$x^4+(2a+1)x^3+(a-1)^2x^2+bx+4$$can be written as a product of two monic quadratic polynomials $\Phi(x)$ and $\Psi(x)$, such that the equation $\Psi(x)=0$ has two distinct roots $\alpha,\beta$ which satisfy $\Phi(\alpha)=\beta$ and $\Phi(\beta)=\alpha$.
2017 Bulgaria EGMO TST, 1
Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions
\[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\]
for all $x \in \mathbb{Q^+}.$
2013 Balkan MO Shortlist, A1
Positive real numbers $a, b,c$ satisfy $ab + bc+ ca = 3$. Prove the inequality $$\frac{1}{4+(a+b)^2}+\frac{1}{4+(b+c)^2}+\frac{1}{4+(c+a)^2}\le \frac{3}{8}$$
2010 Ukraine Team Selection Test, 3
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
2022 Princeton University Math Competition, 3
Provided that $\{a_i\}^{28}_{i=1}$ are the $28$ distinct roots of $29x^{28} + 28x^{27} + ... + 2x + 1 = 0$, then the absolute value of $\sum^{28}_{i=1}\frac{1}{(1-a_i)^2}$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$.
2021 Federal Competition For Advanced Students, P2, 4
Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$.
(Walther Janous)
1953 AMC 12/AHSME, 43
If the price of an article is increased by percent $ p$, then the decrease in percent of sales must not exceed $ d$ in order to yield the same income. The value of $ d$ is:
$ \textbf{(A)}\ \frac{1}{1\plus{}p} \qquad\textbf{(B)}\ \frac{1}{1\minus{}p} \qquad\textbf{(C)}\ \frac{p}{1\plus{}p} \qquad\textbf{(D)}\ \frac{p}{p\minus{}1} \qquad\textbf{(E)}\ \frac{1\minus{}p}{1\plus{}p}$
2010 Saudi Arabia IMO TST, 2
a) Prove that for each positive integer $n$ there is a unique positive integer $a_n$ such that $$(1 + \sqrt5)^n =\sqrt{a_n} + \sqrt{a_n+4^n} . $$
b) Prove that $a_{2010}$ is divisible by $5\cdot 4^{2009}$ and find the quotient
2007 Bulgaria National Olympiad, 3
Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.
2008 National Olympiad First Round, 3
Let $P(x) = 1-x+x^2-x^3+\dots+x^{18}-x^{19}$ and $Q(x)=P(x-1)$. What is the coefficient of $x^2$ in polynomial $Q$?
$
\textbf{(A)}\ 840
\qquad\textbf{(B)}\ 816
\qquad\textbf{(C)}\ 969
\qquad\textbf{(D)}\ 1020
\qquad\textbf{(E)}\ 1140
$
2007 District Olympiad, 1
Let be three real numbers $ a,b,c, $ all in the interval $ (0,\infty ) $ or all in the interval $ (0,1). $ Prove the following inequality:
$$ \sum_{\text{cyc}}\log_a bc\ge 4\cdot\sum_{\text{cyc}} \log_{ab} c . $$
Brazil L2 Finals (OBM) - geometry, 2015.3
Let $ABC$ be a triangle and $n$ a positive integer. Consider on the side $BC$ the points $A_1, A_2, ..., A_{2^n-1}$ that divide the side into $2^n$ equal parts, that is, $BA_1=A_1A_2=...=A_{2^n-2}A_{2^n-1}=A_{2^n-1}C$. Set the points $B_1, B_2, ..., B_{2^n-1}$ and $C_1, C_2, ..., C_{2^n-1}$ on the sides $CA$ and $AB$, respectively, analogously. Draw the line segments $AA_1, AA_2, ..., AA_{2^n-1}$, $BB_1, BB_2, ..., BB_{2^n-1}$ and $CC_1, CC_2, ..., CC_{2^n-1}$. Find, in terms of $n$, the number of regions into which the triangle is divided.