Found problems: 85335
PEN H Problems, 41
Suppose that $A=1,2,$ or $3$. Let $a$ and $b$ be relatively prime integers such that $a^{2}+Ab^2 =s^3$ for some integer $s$. Then, there are integers $u$ and $v$ such that $s=u^2 +Av^2$, $a =u^3 - 3Avu^2$, and $b=3u^{2}v -Av^3$.
1957 AMC 12/AHSME, 41
Given the system of equations
\[ ax \plus{} (a \minus{} 1)y \equal{} 1 \\
(a \plus{} 1)x \minus{} ay \equal{} 1.
\]
For which one of the following values of $ a$ is there no solution $ x$ and $ y$?
$ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ \minus{} 1\qquad \textbf{(D)}\ \frac {\pm \sqrt {2}}{2}\qquad \textbf{(E)}\ \pm\sqrt {2}$
2019 Auckland Mathematical Olympiad, 3
Let $x$ be the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$, where $a$, $b$, $c$, $d$ are non-negative integers. Prove that $x$ is odd.
2017 Online Math Open Problems, 18
Let $a,b,c$ be real nonzero numbers such that $a+b+c=12$ and \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}=1.\] Compute the largest possible value of $abc-\left(a+2b-3c\right)$.
[i]Proposed by Tristan Shin[/i]
2012 Kazakhstan National Olympiad, 2
Given an inscribed quadrilateral $ABCD$, which marked the midpoints of the points $M, N, P, Q$ in this order. Let diagonals $AC$ and $BD$ intersect at point $O$. Prove that the triangle $OMN, ONP, OPQ, OQM$ have the same radius of the circles
1995 Iran MO (2nd round), 3
In a quadrilateral $ABCD$ let $A', B', C'$ and $D'$ be the circumcenters of the triangles $BCD, CDA, DAB$ and $ABC$, respectively. Denote by $S(X, YZ)$ the plane which passes through the point $X$ and is perpendicular to the line $YZ.$ Prove that if $A', B', C'$ and $D'$ don't lie in a plane, then four planes $S(A, C'D'), S(B, A'D'), S(C, A'B')$ and $S(D, B'C')$ pass through a common point.
2023 Grand Duchy of Lithuania, 3
The midpoints of the sides $BC$, $CA$ and $AB$ of triangle $ABC$ are $M$, $N$ and $P$ respectively . $G$ is the intersection point of the medians. The circumscribed circle around $BGP$ intersects the line $MP$ at the point $K$ (different than $P$).The circle circumscribed around $CGN$ intersects the line $MN$ at point $L$ (different than $N$). Prove that $\angle BAK = \angle CAL$.
1994 Turkey Team Selection Test, 1
Let $P,Q,R$ be points on the sides of $\triangle ABC$ such that $P \in [AB],Q\in[BC],R\in[CA]$ and
$\frac{|AP|}{|AB|} = \frac {|BQ|}{|BC|} =\frac{|CR|}{|CA|} =k < \frac 12$
If $G$ is the centroid of $\triangle ABC$, find the ratio $\frac{Area(\triangle PQG)}{Area(\triangle PQR)}$ .
1996 Moldova Team Selection Test, 10
Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$.
[b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$.
[b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.
2013 Czech-Polish-Slovak Junior Match, 6
There is a square $ABCD$ in the plane with $|AB|=a$. Determine the smallest possible radius value of three equal circles to cover a given square.
the 16th XMO, 4
Given an integer $n$ ,For a sequence of $X$ with the number of $n$ and $Y$ with the number of $100n$ , we call it a [b]spring [/b] . We have two following rules
$\blacksquare$ Choose four adjacent character , if it is $YXXY$ , than it can be changed into $XYYX$
$\blacksquare $ Choose. four adjacent character , if it is $XYYX $ , than it can be changed into $YXXY$
If [b]spring [/b] $A$ can become $B$ using the rules , than we call they are [b][color=#3D85C6]similar [/color][/b]
Thy to find the maximum of $C$ such that there exists $C$ distinct [b]springs[/b] and they are [b][color=#3D85C6]similar[/color][/b]
2022 Putnam, A1
Determine all ordered pairs of real numbers $(a,b)$ such that the line $y=ax+b$ intersects the curve $y=\ln(1+x^2)$ in exactly one point.
1997 Singapore Senior Math Olympiad, 1
Let $x_1,x_2,x_3,x_4, x_5,x_6$ be positive real numbers. Show that
$$\left( \frac{x_2}{x_1} \right)^5+\left( \frac{x_4}{x_2} \right)^5+\left( \frac{x_6}{x_3} \right)^5+\left( \frac{x_1}{x_4} \right)^5+\left( \frac{x_3}{x_5} \right)^5+\left( \frac{x_5}{x_6} \right)^5 \ge \frac{x_1}{x_2}+\frac{x_2}{x_4}+\frac{x_3}{x_6}+\frac{x_4}{x_1}+\frac{x_5}{x_3}+\frac{x_6}{x_5}$$
2012 National Olympiad First Round, 19
What is the sum of real roots of the equation $x^4-7x^3+14x^2-14x+4=0$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
2024 Ukraine National Mathematical Olympiad, Problem 3
$2024$ positive real numbers with sum $1$ are arranged on a circle. It is known that any two adjacent numbers differ at least in $2$ times. For each pair of adjacent numbers, the smaller one was subtracted from the larger one, and then all these differences were added together. What is the smallest possible value of this resulting sum?
[i]Proposed by Oleksiy Masalitin[/i]
1985 IMO Longlists, 81
Given the side $a$ and the corresponding altitude $h_a$ of a triangle $ABC$, find a relation between $a$ and $h_a$ such that it is possible to construct, with straightedge and compass, triangle $ABC$ such that the altitudes of $ABC$ form a right triangle admitting $h_a$ as hypotenuse.
Kvant 2024, M2808
Some participants of the tournament are friends with each other, and everyone has at least one friend. Each participant of the tournament was given a T-shirt with the number of his friends at the tournament written on it. Prove that at least one participant has the arithmetic mean of the numbers written on his friends' T-shirts, not less than the arithmetic mean of the numbers on all T-shirts.
[i] From Czech-Slovak Olympiad 1991 [/i]
2007 Croatia Team Selection Test, 6
$\displaystyle 2n$ students $\displaystyle (n \geq 5)$ participated at table tennis contest, which took $\displaystyle 4$ days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this:
- there is only one winner;
- there are $\displaystyle 3$ students on the second place;
- no student lost all $\displaystyle 4$ matches.
How many students won only a single match and how many won exactly $\displaystyle 2$ matches? (In the above conditions)
1966 IMO Longlists, 54
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$
2017 Taiwan TST Round 2, 6
Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$.
[list=a]
[*] Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$.
[*] Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$.
[/list]
2014 Stanford Mathematics Tournament, 1
The coordinates of three vertices of a parallelogram are $A(1, 1)$, $B(2, 4)$, and $C(-5, 1)$. Compute the area of the parallelogram.
1995 AMC 12/AHSME, 23
The sides of a triangle have lengths $11$,$15$, and $k$, where $k$ is an integer. For how many values of $k$ is the triangle obtuse?
$\textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 13 \qquad
\textbf{(E)}\ 14$
2005 China Girls Math Olympiad, 2
Find all ordered triples $ (x, y, z)$ of real numbers such that
\[ 5 \left(x \plus{} \frac{1}{x} \right) \equal{} 12 \left(y \plus{} \frac{1}{y} \right) \equal{} 13 \left(z \plus{} \frac{1}{z} \right),\]
and \[ xy \plus{} yz \plus{} zy \equal{} 1.\]
2023 Assam Mathematics Olympiad, 16
$n$ is a positive integer such that the product of all its positive divisors is $n^3$. Find all such $n$ less than $100$.
2001 Bosnia and Herzegovina Team Selection Test, 1
On circle there are points $A$, $B$ and $C$ such that they divide circle in ratio $3:5:7$. Find angles of triangle $ABC$