Found problems: 85335
2015 Iran MO (3rd round), 4
Let $ABC$ be a triangle with incenter $I$. Let $K$ be the midpoint of $AI$ and $BI\cap \odot(\triangle ABC)=M,CI\cap \odot(\triangle ABC)=N$. points $P,Q$ lie on $AM,AN$ respectively such that $\angle ABK=\angle PBC,\angle ACK=\angle QCB$. Prove that $P,Q,I$ are collinear.
2012 Iran MO (3rd Round), 2
Consider a set of $n$ points in plane. Prove that the number of isosceles triangles having their vertices among these $n$ points is $\mathcal O (n^{\frac{7}{3}})$. Find a configuration of $n$ points in plane such that the number of equilateral triangles with vertices among these $n$ points is $\Omega (n^2)$.
2022 International Zhautykov Olympiad, 5
A polynomial $f(x)$ with real coefficients of degree greater than $1$ is given. Prove that there are infinitely many positive integers which cannot be represented in the form \[f(n+1)+f(n+2)+\cdots+f(n+k)\]
where $n$ and $k$ are positive integers.
2009 Harvard-MIT Mathematics Tournament, 7
A line in the plane is called [i]strange[/i] if it passes through $(a,0)$ and $(0,10-a)$ for some $a$ in the interval $[0,10]$. A point in the plane is called [i]charming[/i] if it lies in the first quadrant and also lies [b]below[/b] some strange line. What is the area of the set of all charming points?
2004 Irish Math Olympiad, 2
Each of the players in a tennis tournament played one match against each of
the others. If every player won at least one match, show that there is a group
A; B; C of three players for which A beat B, B beat C and C beat A.
1970 AMC 12/AHSME, 14
Consider $x^2+px+q=0$ where $p$ and $q$ are positive numbers. If the roots of this equation differ by $1$, then $p$ equals
$\textbf{(A) }\sqrt{4q+1}\qquad\textbf{(B) }q-1\qquad\textbf{(C) }-\sqrt{4q+1}\qquad\textbf{(D) }q+1\qquad \textbf{(E) }\sqrt{4q-1}$
2018 China Team Selection Test, 2
Let $G$ be a simple graph with 100 vertices such that for each vertice $u$, there exists a vertice $v \in N \left ( u \right )$ and $ N \left ( u \right ) \cap N \left ( v \right ) = \o $. Try to find the maximal possible number of edges in $G$. The $ N \left ( . \right )$ refers to the neighborhood.
2019 Ukraine Team Selection Test, 3
Given an acute triangle $ABC$ . It's altitudes $AA_1 , BB_1$ and $CC_1$ intersect at a point $H$ , the orthocenter of $\vartriangle ABC$. Let the lines $B_1C_1$ and $AA_1$ intersect at a point $K$, point $M$ be the midpoint of the segment $AH$. Prove that the circumscribed circle of $\vartriangle MKB_1$ touches the circumscribed circle of $\vartriangle ABC$ if and only if $BA1 = 3A1C$.
(Bondarenko Mykhailo)
1986 IMO Longlists, 79
Let $AA_1,BB_1, CC_1$ be the altitudes in an acute-angled triangle $ABC$, $K$ and $M$ are points on the line segments $A_1C_1$ and $B_1C_1$ respectively. Prove that if the angles $MAK$ and $CAA_1$ are equal, then the angle $C_1KM$ is bisected by $AK.$
1952 Miklós Schweitzer, 8
For which values of $ z$ does the series
$ \sum_{n\equal{}1}^{\infty}c_1c_2\cdots c_n z^n$
converge, provided that $ c_k>0$ and
$ \sum_{k\equal{}1}^{\infty} \frac{c_k}{k}<\infty$ ?
2023 Indonesia TST, A
Let $a_1, a_2, a_3, a_4, a_5$ be non-negative real numbers satisfied
\[\sum_{k = 1}^{5} a_k = 20 \ \ \ \ \text{and} \ \ \ \ \sum_{k=1}^{5} a_k^2 = 100\]
Find the minimum and maximum of $\text{max} \{a_1, a_2, a_3, a_4, a_5\}$
1981 National High School Mathematics League, 6
In Cartesian coordinates, two areas $M,N$ are defined below:
$M:y\geq0,y\leq x,y\leq 2-x$;
$N:t\leq x\leq t+1$.
$t$ is a real number that $t\in[0,1]$.
Then the area of $M\cap N$ is
$\text{(A)}-t^2+t+\frac{1}{2}\qquad\text{(B)}-2t^2+2t\qquad\text{(C)}1-2t^2\qquad\text{(D)}\frac{1}{2}(t-2)^2$
1995 All-Russian Olympiad, 1
A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$.
[i]S. Tokarev[/i]
2010 LMT, 9
Let $ABC$ and $BCD$ be equilateral triangles, such that $AB=1,$ and $A \neq D.$ Find the area of triangle $ABD.$
1999 IMO Shortlist, 4
Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write
\[\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , \]
where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by \[ f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}\]
a) Prove that $S$ is infinite.
b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$
2023 ELMO Shortlist, C3
Find all pairs of positive integers \((a,b)\) with the following property: there exists an integer \(N\) such that for any integers \(m\ge N\) and \(n\ge N\), every \(m\times n\) grid of unit squares may be partitioned into \(a\times b\) rectangles and fewer than \(ab\) unit squares.
[i]Proposed by Holden Mui[/i]
2018 MIG, 3
Solve for $x$ if $4x + 1 = 37$.
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }10$
1987 AMC 8, 17
Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says:
"Bret is next to Carl."
"Abby is between Bret and Carl."
However each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2?
$\text{(A)}\ \text{Abby} \qquad \text{(B)}\ \text{Bret} \qquad \text{(C)}\ \text{Carl} \qquad \text{(D)}\ \text{Dana} \qquad \text{(E)}\ \text{There is not enough information to be sure.}$
2016 South East Mathematical Olympiad, 5
Let $n$ is positive integer, $D_n$ is a set of all positive divisor of $n$ and $f(n)=\sum_{d\in D_n}{\frac{1}{1+d}}$
Prove that for all positive integer $m$, $\sum_{i=1}^{m}{f(i)} <m$
TNO 2008 Senior, 4
Prove that the diagonals of a convex quadrilateral are perpendicular if and only if the sum of the squares of one pair of opposite sides is equal to the sum of the squares of the other pair.
2009 Sharygin Geometry Olympiad, 5
Rhombus $CKLN$ is inscribed into triangle $ABC$ in such way that point $L$ lies on side $AB$, point $N$ lies on side $AC$, point $K$ lies on side $BC$. $O_1, O_2$ and $O$ are the circumcenters of triangles $ACL, BCL$ and $ABC$ respectively. Let $P$ be the common point of circles $ANL$ and $BKL$, distinct from $L$. Prove that points $O_1, O_2, O$ and $P$ are concyclic.
(D.Prokopenko)
1961 AMC 12/AHSME, 11
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of triangle $APR$ is
${{ \textbf{(A)}\ 42\qquad\textbf{(B)}\ 40.5 \qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 39\frac{7}{8} }\qquad\textbf{(E)}\ \text{not determined by the given information} } $
2022 Latvia Baltic Way TST, P8
Call the intersection of two segments [i]almost perfect[/i] if for each of the segments the distance between the midpoint of the segment and the intersection is at least $2022$ times smaller than the length of the segment. Prove that there exists a closed broken line of segments such that every segment intersects at least one other segment, and every intersection of segments is [i]almost perfect[/i].
2005 Taiwan TST Round 3, 1
Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[
n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality.
[i]Proposed by Finbarr Holland, Ireland[/i]
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)