Found problems: 85335
2019 Latvia Baltic Way TST, 13
Let $s(k)$ denotes sum of digits of positive integer $k$. Prove that there are infinitely many positive integers $n$, which are not divisible by $10$ and satisfies:
$$s(n^2) < s(n) - 5$$
2018 Serbia Team Selection Test, 5
Let $H $ be the orthocenter of $ABC $ ,$AB\neq AC $ ,and let $F $ be a point on circumcircle of $ABC $ such that $\angle AFH=90^{\circ} $.$K $ is the symmetric point of $H $ wrt $B $.Let $P $ be a point such that $\angle PHB=\angle PBC=90^{\circ} $,and $Q $ is the foot of $B $ to $CP $.Prove that $HQ $ is tangent to tge circumcircle of $FHK $.
1992 IMO Longlists, 45
Let $n$ be a positive integer. Prove that the number of ways to express $n$ as a sum of distinct positive integers (up to order) and the number of ways to express $n$ as a sum of odd positive integers (up to order) are the same.
1959 Polish MO Finals, 5
In the plane of the triangle $ ABC $ a straight line moves which intersects the sides $ AC $ and $ BC $ in such points $ D $ and $ E $ that $ AD = BE $. Find the locus of the midpoint $ M $ of the segment $ DE $.
2022 Denmark MO - Mohr Contest, 3
The square $ABCD$ has side length $1$. The point $E$ lies on the side $CD$. The line through $A$ and $E$ intersects the line through $B$ and $C$ at the point $F$. Prove that $$\frac{1}{|AE|^2}+\frac{1}{|AF|^2}= 1.$$
[img]https://cdn.artofproblemsolving.com/attachments/5/8/4e803eb7748f7a72783065717044cfc06f565f.png[/img]
2014 Singapore MO Open, 5
Determine the largest odd positive integer $n$ such that every odd integer $k$ with $1<k<n$ and $\gcd(k, n)=1$ is a prime.
2004 Tuymaada Olympiad, 3
An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\circ.$
$B_{1}$ is the foot of perpendicular from $B$ to $CO,$ $B_{2}$ is the foot of perpendicular from $B_{1}$ to $AC.$
Similarly, $C_{1}$ is the foot of perpendicular from $C$ to $BO,$ $C_{2}$ is the foot of perpendicular from $C_{1}$ to $AB.$
The lines $B_{1}B_{2}$ and $C_{1}C_{2}$ intersect at $A_{3}.$ The points $B_{3}$ and $C_{3}$ are defined in the same way.
Find the circumradius of triangle $A_{3}B_{3}C_{3}.$
[i]Proposed by F.Bakharev, F.Petrov[/i]
2015 Online Math Open Problems, 13
Let $ABC$ be a scalene triangle whose side lengths are positive integers. It is called [i]stable[/i] if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?
[i]Proposed by Evan Chen[/i]
2016 Kyrgyzstan National Olympiad, 6
Given three pairwise tangent equal circles $\Omega_i (i=1,2,3)$ with radius $r$.
The circle $\Gamma $ touches the three circles internally (circumscribed about 3 circles).The three equal circles $\omega_i (i=1,2,3)$ with radius $x$ touches $\Omega_i$ and $\Omega_{i+1}$ externally ($\Omega_4= \Omega_1$) and touches $\Gamma$ internally.Find $x$ in terms of $r$
1982 Miklós Schweitzer, 8
Show that for any natural number $ n$ and any real number $ d > 3^n / (3^n\minus{}1)$, one can find a covering of the unit square with $ n$ homothetic triangles with area of the union less than $ d$.
2000 Moldova National Olympiad, Problem 6
Assuming that real numbers $x$ and $y$ satisfy $y\left(1+x^2\right)=x\left(\sqrt{1-4y^2}-1\right)$, find the maximum value of $xy$.
2024 Brazil EGMO TST, 4
The infinite sequence \( a_1, a_2, \ldots \) is defined by \( a_1 = 1 \) and, for each \( n \geq 1 \), the number \( a_{n+1} \) is the smallest positive integer greater than \( a_n \) that has the following property: for each \( k \in \{1, 2, \ldots, n\} \), the number \( a_{n+1} + a_k \) is not a perfect square. Prove that, for all \( n \), it holds that \( a_n \leq (n - 1)^2 + 1 \).
2006 Cezar Ivănescu, 1
[b]a)[/b] $ \lim_{n\to\infty } \frac{1}{n^2}\sum_{i=0}^n\sqrt{\binom{n+i}{2}} $
[b]b)[/b] $ \lim_{n\to\infty } \frac{a^{H_n}}{1+n} ,\quad a>0 $
2004 All-Russian Olympiad Regional Round, 10.8
Given natural numbers $p < k < n$. On an endless checkered plane some cells are marked so that in any rectangle $(k + 1) \times n$ ($n$ cells horizontally, $k + 1$ vertically) marked exactly $p$ cells. Prove that there is a $k \times (n + 1)$ rectangle ($n + 1$ cell horizontally, $k$ - vertically), in which no less than $p + 1$ cells.
2008 Germany Team Selection Test, 2
Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$.
[i]Author: Christopher Bradley, United Kingdom [/i]
PEN I Problems, 18
Do there exist irrational numbers $a, b>1$ and $\lfloor a^{m}\rfloor \not=\lfloor b^{n}\rfloor $ for any positive integers $m$ and $n$?
1997 All-Russian Olympiad, 2
A class consists of 33 students. Each student is asked how many other students in the class have his first name, and how many have his last name. It turns out that each number from 0 to 10 occurs among the answers. Show that there are two students in the class with the same first and last name.
[i]A. Shapovalov[/i]
2015 District Olympiad, 2
At a math contest there were $ 50 $ participants, where they were given $ 3 $ problems each to solve. The results have shown that every candidate has solved correctly at least one problem, and that a total of $ 100 $ problems have been evaluated by the jury as correct.
Show that there were, at most, $ 25 $ winners who got the maximum score.
2021 Indonesia MO, 5
Let $P(x) = x^2 + rx + s$ be a polynomial with real coefficients. Suppose $P(x)$ has two distinct real roots, both of which are less than $-1$ and the difference between the two is less than $2$. Prove that $P(P(x)) > 0$ for all real $x$.
1996 IMO Shortlist, 7
Let $ABC$ be an acute triangle with circumcenter $O$ and circumradius $R$. $AO$ meets the circumcircle of $BOC$ at $A'$, $BO$ meets the circumcircle of $COA$ at $B'$ and $CO$ meets the circumcircle of $AOB$ at $C'$. Prove that \[OA'\cdot OB'\cdot OC'\geq 8R^{3}.\] Sorry if this has been posted before since this is a very classical problem, but I failed to find it with the search-function.
2015 Purple Comet Problems, 17
How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements differ by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.
1940 Moscow Mathematical Olympiad, 061
Given two lines on a plane, find the locus of all points with the difference between the distance to one line and the distance to the other equal to the length of a given segment.
2018 India IMO Training Camp, 2
Let $A,B,C$ be three points in that order on a line $\ell$ in the plane, and suppose $AB>BC$. Draw semicircles $\Gamma_1$ and $\Gamma_2$ respectively with $AB$ and $BC$ as diameters, both on the same side of $\ell$. Let the common tangent to $\Gamma_1$ and $\Gamma_2$ touch them respectively at $P$ and $Q$, $P\ne Q$. Let $D$ and $E$ be points on the segment $PQ$ such that the semicircle $\Gamma_3$ with $DE$ as diameter touches $\Gamma_2$ in $S$ and $\Gamma_1$ in $T$.
[list=1][*]Prove that $A,C,S,T$ are concyclic.
[*]Prove that $A,C,D,E$ are concyclic.[/list]
Durer Math Competition CD 1st Round - geometry, 2012.C5
In a triangle, the line between the center of the inscribed circle and the center of gravity is parallel to one of the sides. Prove that the sidelengths form an arithmetic sequence.
1985 Tournament Of Towns, (086) 2
The integer part $I (A)$ of a number $A$ is the greatest integer which is not greater than $A$ , while the fractional part $F(A)$ is defined as $A - I(A)$ .
(a) Give an example of a positive number $A$ such that $F(A) + F( 1/A) = 1$ .
(b) Can such an $A$ be a rational number?
(I. Varge, Romania)