Found problems: 85335
2015 South Africa National Olympiad, 2
Determine all pairs of real numbers $a$ and $x$ that satisfy the simultaneous equations $$5x^3 + ax^2 + 8 = 0$$ and $$5x^3 + 8x^2 + a = 0.$$
2005 Abels Math Contest (Norwegian MO), 4a
Show that for all positive real numbers $a, b$ and $c$, the inequality $(a+b)(a+c)\ge 2\sqrt{abc(a+b+c)}$ is true.
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 9
We draw a circle with radius 5 on a gridded paper where the grid consists of squares with sides of length 1. The center of the circle is placed in the middle of one of the squares. All the squares through which the circle passes are colored. How many squares are colored? (The figure illustrates this for a smaller circle.)
[img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1995Number9.jpg[/img]
A. 24
B. 32
C. 40
D. 64
E. None of these
2019 China Team Selection Test, 5
Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds.
Swiss NMO - geometry, 2009.7
Points $A, M_1, M_2$ and $C$ are on a line in this order. Let $k_1$ the circle with center $M_1$ passing through $A$ and $k_2$ the circle with center $M_2$ passing through $C$. The two circles intersect at points $E$ and $F$. A common tangent of $k_1$ and $k_2$, touches $k_1$ at $B$ and $k_2$ at $D$. Show that the lines $AB, CD$ and $EF$ intersect at one point.
1996 Flanders Math Olympiad, 3
Consider the points $1,\frac12,\frac13,...$ on the real axis. Find the smallest value $k \in \mathbb{N}_0$ for which all points above can be covered with 5 [b]closed[/b] intervals of length $\frac1k$.
2010 All-Russian Olympiad, 3
Let $O$ be the circumcentre of the acute non-isosceles triangle $ABC$. Let $P$ and $Q$ be points on the altitude $AD$ such that $OP$ and $OQ$ are perpendicular to $AB$ and $AC$ respectively. Let $M$ be the midpoint of $BC$ and $S$ be the circumcentre of triangle $OPQ$. Prove that $\angle BAS =\angle CAM$.
2003 Greece JBMO TST, 3
Consider the set $M=\{1,2,3,...,2003\}$. How many subsets of $M$ with even number of elements exist?
1978 Vietnam National Olympiad, 5
A river has a right-angle bend. Except at the bend, its banks are parallel lines of distance $a$ apart. At the bend the river forms a square with the river flowing in across one side and out across an adjacent side. What is the longest boat of length $c$ and negligible width which can pass through the bend?
2003 Abels Math Contest (Norwegian MO), 3
Let $ABC$ be a triangle with $AC> BC$, and let $S$ be the circumscribed circle of the triangle. $AB$ divides $S$ into two arcs. Let $D$ be the midpoint of the arc containing $C$.
(a) Show that $\angle ACB +2 \cdot \angle ACD = 180^o$.
(b) Let $E$ be the foot of the altitude from $D$ on $AC$. Show that $BC +CE = AE$.
2018 Costa Rica - Final Round, N3
Let $a$ and $b$ be positive integers such that $2a^2 + a = 3b^2 + b$. Prove that $a-b$ is a perfect square.
2018 Tournament Of Towns, 2.
Aladdin has several gold ingots, and sometimes he asks the Genie to give him more. The Genie first adds a thousand ingots, but then takes half of the total number. Could it be possible that after asking the Genie for gold ten times, the number of Aladdin’s gold ingots increased, assuming that each time the Genie took half, he took an integer number of ingots? (5 points)
Alexandr Perepechko
LMT Team Rounds 2021+, A8
Isosceles $\triangle{ABC}$ has interior point $O$ such that $AO = \sqrt{52}$, $BO = 3$, and $CO = 5$. Given that $\angle{ABC}=120^{\circ}$, find the length $AB$.
[i]Proposed by Powell Zhang[/i]
2018 Brazil Undergrad MO, 5
Consider the set $A = \left\{\frac{j}{4}+\frac{100}{j}|j=1,2,3,..\right\} $ What is the smallest number that belongs to the $ A $ set?
1973 All Soviet Union Mathematical Olympiad, 183
$N$ men are not acquainted each other. You need to introduce some of them to some of them in such a way, that no three men will have equal number of acquaintances. Prove that it is possible for all $N$.
2022 Czech-Polish-Slovak Junior Match, 4
Find all triples $(a, b, c)$ of integers that satisfy the equations
$ a + b = c$ and $a^2 + b^3 = c^2$
2007 Postal Coaching, 1
Let $ABC$ be an isosceles triangle with $AC = BC$, and let $M$ be the midpoint of $AB$. Let $P$ be a point inside the triangle such that $\angle PAB =\angle PBC$. Prove that $\angle APM + \angle BPC = 180^o$.
1959 IMO, 4
Construct a right triangle with given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.
1977 Bundeswettbewerb Mathematik, 4
Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent.
[b]Note by Darij:[/b] A [i]cyclic quadrilateral [/i]is a quadrilateral inscribed in a circle.
2021 Novosibirsk Oral Olympiad in Geometry, 2
The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.
2011 Laurențiu Duican, 3
Let $ n\ge 2 $ be a perfect square and let be $ n $ natural numbers $ m_1,m_2,\ldots ,m_n. $ Prove that if the polynom
$$ X^2-\left( 1+ m_1^2+m_2^2+\cdots +m_n^2 \right) X+m_1m_2+m_2m_3+\cdots +m_{n-1}m_n +m_nm_1\in \mathbb{N} [X] $$
is reducible, then its two roots are perfect squares.
1996 May Olympiad, 1
A terrain ( $ABCD$ ) has a rectangular trapezoidal shape. The angle in $A$ measures $90^o$. $AB$ measures $30$ m, $AD$ measures $20$ m and $DC$ measures 45 m. This land must be divided into two areas of the same area, drawing a parallel to the $AD$ side . At what distance from $D$ do we have to draw the parallel?
[img]https://1.bp.blogspot.com/-DnyNY3x4XKE/XNYvRUrLVTI/AAAAAAAAKLE/gohd7_S9OeIi-CVUVw-iM63uXE5u-WmGwCK4BGAYYCw/s400/image002.gif[/img]
2010 China Team Selection Test, 3
Let $A$ be a finite set, and $A_1,A_2,\cdots, A_n$ are subsets of $A$ with the following conditions:
(1) $|A_1|=|A_2|=\cdots=|A_n|=k$, and $k>\frac{|A|}{2}$;
(2) for any $a,b\in A$, there exist $A_r,A_s,A_t\,(1\leq r<s<t\leq n)$ such that
$a,b\in A_r\cap A_s\cap A_t$;
(3) for any integer $i,j\, (1\leq i<j\leq n)$, $|A_i\cap A_j|\leq 3$.
Find all possible value(s) of $n$ when $k$ attains maximum among all possible systems $(A_1,A_2,\cdots, A_n,A)$.
2012 IFYM, Sozopol, 2
Let $p$ and $q=4p+1$ be prime numbers. Determine the least power $i$ of 2 for which $2^i\equiv 1\,(mod\, q)$.
2002 China Team Selection Test, 2
$ m$ and $ n$ are positive integers. In a $ 8 \times 8$ chessboard, $ (m,n)$ denotes the number of grids a Horse can jump in a chessboard ($ m$ horizontal $ n$ vertical or $ n$ horizontal $ m$ vertical ). If a $ (m,n) \textbf{Horse}$ starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a $ \textbf{H Route}$. For example, the $ (1,2) \textbf{Horse}$ has its $ \textbf{H Route}$. Find the smallest positive integer $ t$, such that from any grid of the chessboard, the $ (t,t\plus{}1) \textbf{Horse}$ does not has any $ \textbf{H Route}$.