Found problems: 85335
2017 Singapore Junior Math Olympiad, 3
Let $ABC$ be a triangle with $AB=AC$. Let $D$ be a point on $BC$, and $E$ a point on $AD$ such that $\angle BED=\angle BAC=2\angle CED$. Prove that $BD=2CD$.
2024 Azerbaijan Senior NMO, 1
Numbers from 1 to 100 are written on the board in ascending order to make the following large number: 12345678910111213...9899100. Then 100 digits of this number are deleted to get the largest possible number. Find the first 10 digits of the number after deletion.
2010 IMO Shortlist, 3
Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\]
[i]Proposed by Mariusz Skałba, Poland[/i]
Cono Sur Shortlist - geometry, 2021.G3
Let $ABCD$ be a parallelogram with vertices in order clockwise and let $E$ be the intersection of its diagonals. The circle of diameter $DE$ intersects the segment $AD$ at $L$ and $EC$ at $H$. The circumscribed circle of $LEB$ intersects the segment $BC$ at $O$. Prove that the lines $HD$ , $LE$ and $BC$ are concurrent if and only if $EO = EC$.
2014 National Olympiad First Round, 19
What is the largest possible value of $\dfrac{x^2+2x+6}{x^2+x+5}$ where $x$ is a positive real number?
$
\textbf{(A)}\ \dfrac{14}{11}
\qquad\textbf{(B)}\ \dfrac{9}{7}
\qquad\textbf{(C)}\ \dfrac{13}{10}
\qquad\textbf{(D)}\ \dfrac{4}{3}
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2022 Peru MO (ONEM), 2
Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and let $G$ be the point of the segment $AD$ such that $AG = 2GD$. Let $E$ and $F$ be points on the sides $AB$ and $AC$, respectively, such that$ G$ lies on the segment $EF$. Let $M$ and $N$ be points of the segments $AE$ and $AF$, respectively, such that $ME = EB$ and $NF = FC$.
a) Prove that the area of the quadrilateral $BMNC$ is equal to four times the area of the triangle $DEF$.
b) Prove that the quadrilaterals $MNFE$ and $AMDN$ have the same area.
2019 Teodor Topan, 2
Let $ I $ be a nondegenerate interval, and let $ F $ be a primitive of a function $ f:I\longrightarrow\mathbb{R} . $ Show that for any distinct $ a,b\in I, $ the tangents to the graph of $ F $ at the points $ (a,F(a)) ,(b,F(b)) $ are concurrent at a point whose abscisa is situated in the interval $ (a,b). $
[i]Nicolae Bourbăcuț[/i]
1985 All Soviet Union Mathematical Olympiad, 399
Given a straight line $\ell$ and the point $O$ out of the line. Prove that it is possible to move an arbitrary point $A$ in the same plane to the $O$ point, using only rotations around $O$ and symmetry with respect to the $\ell$.
1998 China Team Selection Test, 1
In acute-angled $\bigtriangleup ABC$, $H$ is the orthocenter, $O$ is the circumcenter and $I$ is the incenter. Given that $\angle C > \angle B > \angle A$, prove that $I$ lies within $\bigtriangleup BOH$.
2013 Greece Team Selection Test, 2
For the several values of the parameter $m\in \mathbb{N^{*}}$,find the pairs of integers $(a,b)$ that satisfy the relation
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{[a,m]+[b,m]}{(a+b)m}=\frac{10}{11}$,
and,moreover,on the Cartesian plane $Oxy$ the lie in the square $D=\{(x,y):1\leq x\leq 36,1\leq y\leq 36\}$.
[i][u]Note:[/u]$[k,l]$ denotes the least common multiple of the positive integers $k,l$.[/i]
2000 Moldova National Olympiad, Problem 5
Solve in real numbers the equation
$$\left(x^2-3x-2\right)^2-3\left(x^2-3x-2\right)-2-x=0.$$
2014 NIMO Problems, 6
10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence $11$ possible places) with a uniform $\tfrac{1}{11}$ probability of each location. Then, either the frontmost or the backmost student is removed from the row (each with a $\tfrac{1}{2}$ probability).
Suppose you are the eighth in the line from the front. The probability that you exit the row from the front rather than the back is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m+n$.
[i]Proposed by Lewis Chen[/i]
LMT Team Rounds 2021+, 15
In triangle $ABC$ with $AB = 26$, $BC = 28$, and $C A = 30$, let $M$ be the midpoint of $AB$ and let $N$ be the midpoint of $C A$. The circumcircle of triangle $BCM$ intersects $AC$ at $X\ne C$, and the circumcircle of triangle $BCN $intersects $AB$ at $Y\ne B$. Lines $MX$ and $NY$ intersect $BC$ at $P$ and $Q$, respectively. The area of quadrilateral $PQY X$ can be expressed as $\frac{p}{q}$ for positive integers $p$ and $q$ such that gcd$(p,q) = 1$. Find $q$.
2018 Romania National Olympiad, 4
In the rectangular parallelepiped $ABCDA'B'C'D'$ we denote by $M$ the center of the face $ABB'A'$. We denote by $M_1$ and $M_2$ the projections of $M$ on the lines $B'C$ and $AD'$ respectively. Prove that:
a) $MM_1 = MM_2$
b) if $(MM_1M_2) \cap (ABC) = d$, then $d \parallel AD$;
c) $\angle (MM_1M_2), (A B C)= 45^ o \Leftrightarrow \frac{BC}{AB}=\frac{BB'}{BC}+\frac{BC}{BB'}$.
2017 Irish Math Olympiad, 2
$5$ teams play in a soccer competition where each team plays one match against each of the other four teams. A winning team gains $5$ points and a losing team $0$ points. For a $0-0$ draw both teams gain $1$ point, and for other draws ($1-1,2-2,3-3,$etc.) both teams gain 2 points. At the end of the competition, we write down the total points for each team, and we find that they form 5 consecutive integers. What is the minimum number of goals scored?
2002 Estonia National Olympiad, 5
The teacher writes numbers $1$ at both ends of the blackboard. The first student adds a $2$ in the middle between them, each next student adds the sum of each two adjacent numbers already on the blackboard between them (hence there are numbers $1, 3, 2, 3, 1$ on the blackboard after the second student, $1, 4, 3, 5, 2, 5, 3, 4, 1$ after the third student etc.) Find the sum of all numbers on the blackboard after the $n$-th student.
2012 JBMO TST - Turkey, 2
Let $S=\{1,2,3,\ldots,2012\}.$ We want to partition $S$ into two disjoint sets such that both sets do not contain two different numbers whose sum is a power of $2.$ Find the number of such partitions.
1983 IMO Longlists, 64
The sum of all the face angles about all of the vertices except one of a given polyhedron is $5160$. Find the sum of all of the face angles of the polyhedron.
2007 Iran Team Selection Test, 3
Find all solutions of the following functional equation: \[f(x^{2}+y+f(y))=2y+f(x)^{2}. \]
2023 ITAMO, 3
Let $s(n)$ denote the sum of the digits of $n$.
a) Do there exist distinct positive integers $a, b$, such that $2023a+s(a)=2023b+s(b)$?
b) Do there exist distinct positive integers $a, b$, such that $a+2023s(a)=b+2023s(b)$?
2011 Croatia Team Selection Test, 1
Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality
\[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.\]
2006 China Second Round Olympiad, 7
Let $f(x)=\sin^4x-\sin x\cos x+cos^4 x$. Find the range of $f(x)$.
2015 ASDAN Math Tournament, 15
In a given acute triangle $\triangle ABC$ with the values of angles given (known as $a$, $b$, and $c$), the inscribed circle has points of tangency $D,E,F$ where $D$ is on $BC$, $E$ is on $AB$, and $F$ is on $AC$. Circle $\gamma$ has diameter $BC$, and intersects $\overline{EF}$ at points $X$ and $Y$. Find $\tfrac{XY}{BC}$ in terms of the angles $a$, $b$, and $c$.
2022 South East Mathematical Olympiad, 2
In acute triangle ABC AB>AC. H is the orthocenter. M is midpoint of BC and AD is the symmedian line. Prove that if $\angle ADH= \angle MAH$, EF bisects segment AD.
[img]https://s2.loli.net/2022/08/02/t9xzTV8IEv1qQRm.jpg[/img]
2021 MOAA, 8
Andrew chooses three (not necessarily distinct) integers $a$, $b$, and $c$ independently and uniformly at random from $\{1,2,3,4,5,6,7\}$. Let $p$ be the probability that $abc(a+b+c)$ is divisible by $4$. If $p$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$.
[i]Proposed by Andrew Wen[/i]