Found problems: 85335
2014 Albania Round 2, 4
Solve the equation,$$ \sin (\pi \log x) + \cos (\pi \log x) = 1$$
2017 China Team Selection Test, 4
Given integer $d>1,m$,prove that there exists integer $k>l>0$, such that $$(2^{2^k}+d,2^{2^l}+d)>m.$$
2023 Thailand Mathematical Olympiad, 9
Prove that there exists an infinite sequence of positive integers $a_1,a_2,a_3,\dots$ such that for any positive integer $k$, $a_k^2+a_k+2023$ has at least $k$ distinct positive divisors.
2003 Baltic Way, 7
A subset of $X$ of $\{1,2,3, \ldots 10000 \}$ has the following property: If $a,b$ are distinct elements of $X$, then $ab\not\in X$. What is the maximal number of elements in $X$?
2012 Hanoi Open Mathematics Competitions, 11
[Help me] Suppose that the equation $x^3+px^2+qx+r = 0$ has 3 real roots $x_1; x_2; x_3$; where p; q; r are integer numbers. Put $S_n = x_1^n+x_2^n+x_3^n$ ; n = 1; 2; : : : Prove that $S_{2012}$ is an integer.
2002 Moldova National Olympiad, 1
Integers $ a_1,a_2,\ldots a_9$ satisfy the relations $ a_{k\plus{}1}\equal{}a_k^3\plus{}a_k^2\plus{}a_k\plus{}2$ for $ k\equal{}1,2,...,8$. Prove that among these numbers there exist three with a common divisor greater than $ 1$.
2021 CCA Math Bonanza, L5.2
Define the sequences $x_0,x_1,x_2,\ldots$ and $y_0,y_1,y_2,\ldots$ such that $x_0=1$, $y_0=2021$, and for all nonnegative integers $n$, we have $x_{n+1}=\sqrt{x_ny_n}$ and $y_{n+1}=\frac{x_n+y_n}{2}.$ There is some constant $X$ such that as $n$ grows large, $x_n-X$ and $y_n-X$ both approach $0$. Estimate $X$.
An estimate of $E$ earns $\max(0,2-0.02|A-E|)$ points, where $A$ is the actual answer.
[i]2021 CCA Math Bonanza Lightning Round #5.2[/i]
VI Soros Olympiad 1999 - 2000 (Russia), 11.4
Given isosceles triangle $ABC$ ($AB = AC$). A straight line $\ell$ is drawn through its vertex $B$ at a right angle with $AB$ . On the straight line $AC$, an arbitrary point $D$ is taken, different from the vertices, and a straight line is drawn through it at a right angle with $AC$, intersecting $\ell$ at the point $F$. Prove that the center of the circle circumscribed around the triangle $BCD$ lies on the circumscribed circle of triangle $ABD$.
2005 Germany Team Selection Test, 1
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
1991 Arnold's Trivium, 89
Calculate the sum of vector products $[[x, y], z] + [[y, z], x] + [[z, x], y]$
2007 Putnam, 4
Let $ n$ be a positive integer. Find the number of pairs $ P,Q$ of polynomials with real coefficients such that
\[ (P(X))^2\plus{}(Q(X))^2\equal{}X^{2n}\plus{}1\]
and $ \text{deg}P<\text{deg}{Q}.$
2019 Indonesia MO, 7
Determine all solutions of
\[ x + y^2 = p^m \]
\[ x^2 + y = p^n \]
For $x,y,m,n$ positive integers and $p$ being a prime.
2001 Romania National Olympiad, 4
Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$.
a) Show that:
\[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\]
b) Show that:
\[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]
2016 SGMO, Q1
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for any pair of naturals $m,n$,
$$\gcd(f(m),n) = \gcd(m,f(n)).$$
2014 Dutch IMO TST, 1
Determine all pairs $(a,b)$ of positive integers satisfying
\[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]
1999 Gauss, 3
Susan wants to place 35.5 kg of sugar in small bags. If each bag holds 0.5 kg, how many bags are needed?
$\textbf{(A)}\ 36 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 53 \qquad \textbf{(D)}\ 70 \qquad \textbf{(E)}\ 71$
2006 MOP Homework, 2
Points $P$ and $Q$ lies inside triangle $ABC$ such that $\angle ACP =\angle BCQ$ and $\angle CAP = \angle BAQ$. Denote by $D,E$, and $F$ the feet of perpendiculars from $P$ to lines $BC,CA$, and $AB$, respectively. Prove that if $\angle DEF = 90^o$, then $Q$ is the orthocenter of triangle $BDF$.
1996 All-Russian Olympiad Regional Round, 10.8
There are $1996$ points marked on a straight line at regular intervals. Petya colors half of them red and the rest blue. Then Vasya divides them into pairs ''red'' - ''blue'' so that the sum distances between points in pairs was maximum. Prove that this maximum does not depend on what coloring Petya made.
2012 Online Math Open Problems, 36
Let $s_n$ be the number of solutions to $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$, where $a_1,a_2,a_3$ and $a_4$ are elements of the set $\{2, 3, 5, 7\}$ and $b_1$ and $b_2$ are elements of the set $\{ 1, 2, 3, 4\}$. Find the number of $n$ for which $s_n$ is odd.
[i]Author: Alex Zhu[/i]
[hide="Clarification"]$s_n$ is the number of [i]ordered[/i] solutions $(a_1, a_2, a_3, a_4, b_1, b_2)$ to the equation, where each $a_i$ lies in $\{2, 3, 5, 7\}$ and each $b_i$ lies in $\{1, 2, 3, 4\}$. [/hide]
2001 All-Russian Olympiad, 2
The two polynomials $(x) =x^4+ax^3+bx^2+cx+d$ and $Q(x) = x^2+px+q$ take negative values on an interval $I$ of length greater than $2$, and nonnegative values outside of $I$. Prove that there exists $x_0 \in \mathbb R$ such that $P(x_0) < Q(x_0)$.
2016 IMAR Test, 2
Given a positive integer $n$, does there exist a planar polygon and a point in its plane such that every line through that point meets the boundary of the polygon at exactly $2n$ points?
1992 Irish Math Olympiad, 2
If $a_1$ is a positive integer, form the sequence $a_1,a_2,a_3,\dots$ by letting $a_2$ be the product of the digits of $a_1$, etc.. If $a_k$ consists of a single digit, for some $k\ge 1$, $a_k$ is called a [i]digital root[/i] of $a_1$. It is easy to check that every positive integer has a unique root. $($For example, if $a_1=24378$, then $a_2=1344$, $a_3=48$, $a_4=32$, $a_5=6$, and thus $6$ is the digital root of $24378.)$ Prove that the digital root of a positive integer $n$ equals $1$ if, and only if, all the digits of $n$ equal $1$.
2014 Estonia Team Selection Test, 1
In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.
2014 India Regional Mathematical Olympiad, 1
Three positive real numbers $a,b,c$ are such that $a^2+5b^2+4c^2-4ab-4bc=0$. Can $a,b,c$ be the lengths of te sides of a triangle? Justify your answer.
2004 German National Olympiad, 3
Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.