Found problems: 85335
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]
1998 Vietnam National Olympiad, 1
Does there exist an infinite sequence $\{x_{n}\}$ of reals satisfying the following conditions
i)$|x_{n}|\leq 0,666$ for all $n=1,2,...$
ii)$|x_{m}-x_{n}|\geq \frac{1}{n(n+1)}+\frac{1}{m(m+1)}$ for all $m\not = n$?
1998 Gauss, 11
Kalyn cut rectangle R from a sheet of paper and then cut figure S from R. All the cuts were made
parallel to the sides of the original rectangle. In comparing R to S
(A) the area and perimeter both decrease
(B) the area decreases and the perimeter increases
(C) the area and perimeter both increase
(D) the area increases and the perimeter decreases
(E) the area decreases and the perimeter stays the same
2012-2013 SDML (Middle School), 10
Two of the diagonals of a regular pentagon are selected at random. What is the probability that the two selected diagonals intersect inside the pentagon?
$\text{(A) }\frac{2}{5}\qquad\text{(B) }\frac{1}{5}\qquad\text{(C) }\frac{7}{10}\qquad\text{(D) }\frac{3}{5}\qquad\text{(E) }\frac{1}{2}$
1998 Romania National Olympiad, 3
Suppose $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function for which the inequality $f'(x) \leq f'(x+\frac{1}{n})$ holds for every $x\in\mathbb{R}$ and every $n\in\mathbb{N}$.Prove that f is continiously differentiable
2018 AIME Problems, 2
The number \(n\) can be written in base \(14\) as \(\underline{a}\) \(\underline{b}\) \(\underline{c}\), can be written in base \(15\) as \(\underline{a}\) \(\underline{c}\) \(\underline{b}\), and can be written in base \(6\) as \(\underline{a}\) \(\underline{c}\) \(\underline{a}\) \(\underline{c}\), where \(a > 0\). Find the base-\(10\) representation of \(n\).
2023 BMT, 8
A circle intersects equilateral triangle $\vartriangle XY Z$ at $A,$ $B$, $C$, $D$, $E$, and $F$ such that points $X$, $A$, $B$, $Y$ , $C$, $D$, $Z$, $E$, and $F$ lie on the equilateral triangle in that order. If $AC^2 +CE^2 +EA^2 = 1900$ and $BD^2 + DF^2 + FB^2 = 2092$, compute the positive difference between the areas of triangles $\vartriangle ACE$ and $\vartriangle BDF$.
1998 Tournament Of Towns, 2
$ABCD$ is a parallelogram. A point $M$ is found on the side $AB$ or its extension such that $\angle MAD = \angle AMO$ where $O$ is the intersection point of the diagonals of the parallelogram. Prove that $MD = MG$.
(M Smurov)
2013 India IMO Training Camp, 3
For a positive integer $n$, a cubic polynomial $p(x)$ is said to be [i]$n$-good[/i] if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.
2017 NMTC Junior, 4
a) $a,b,c,d$ are positive reals such that $abcd=1$. Prove that \[\sum_{cyc} \frac{1+ab}{1+a}\geq 4.\]
(b)In a scalene triangle $ABC$, $\angle BAC =120^{\circ}$. The bisectors of angles $A,B,C$ meets the opposite sides in $P,Q,R$ respectively. Prove that the circle on $QR$ as diameter passes through the point $P$.
Russian TST 2020, P1
Determine all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ satisfying $xf(xf(2y))=y+xyf(x)$ for all $x,y>0$.