Found problems: 85335
2006 Iran MO (3rd Round), 4
$f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is a bijective map, that Image of every $n-1$-dimensional affine space is a $n-1$-dimensional affine space.
1) Prove that Image of every line is a line.
2) Prove that $f$ is an affine map. (i.e. $f=goh$ that $g$ is a translation and $h$ is a linear map.)
1953 AMC 12/AHSME, 15
A circular piece of metal of maximum size is cut out of a square piece and then a square piece of maximum size is cut out of the circular piece. The total amount of metal wasted is:
$ \textbf{(A)}\ \frac{1}{4} \text{ the area of the original square}\\
\textbf{(B)}\ \frac{1}{2} \text{ the area of the original square}\\
\textbf{(C)}\ \frac{1}{2} \text{ the area of the circular piece}\\
\textbf{(D)}\ \frac{1}{4} \text{ the area of the circular piece}\\
\textbf{(E)}\ \text{none of these}$
2019 Romanian Master of Mathematics Shortlist, original P5
Two ants are moving along the edges of a convex polyhedron. The route of every ant ends in its starting point, so that one ant does not pass through the same point twice along its way. On every face $F$ of the polyhedron are written the number of edges of $F$ belonging to the route of the first ant and the number of edges of $F$ belonging to the route of the second ant. Is there a polyhedron and a pair of routes described as above, such that only one face contains a pair of distinct numbers?
[i]Proposed by Nikolai Beluhov[/i]
2017 ELMO Shortlist, 1
Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$
[i]Proposed by Michael Ren[/i]
2002 Junior Balkan Team Selection Tests - Romania, 4
Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$. Find the least number $k$ such that $s(M,N)\le k$, for all points $M,N$.
[i]Dinu Șerbănescu[/i]
1997 German National Olympiad, 1
Prove that there are no perfect squares $a,b,c$ such that $ab-bc = a$.
1998 Flanders Math Olympiad, 3
a magical $3\times3$ square is a $3\times3$ matrix containing all number from 1 to 9, and of which the sum of every row, every column, every diagonal, are all equal.
Determine all magical $3\times3$ square
2006 Putnam, A6
Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral.
2005 India Regional Mathematical Olympiad, 3
If $a,b,c$ are positive three real numbers such that $| a-b | \geq c , | b-c | \geq a, | c-a | \geq b$ . Prove that one of $a,b,c$ is equal to the sum of the other two.
2023 OMpD, 3
For each positive integer $x$, let $\varphi(x)$ be the number of integers $1 \leq k \leq x$ that do not have prime factors in common with $x$. Determine all positive integers $n$ such that there are distinct positive integers $a_1,a_2, \ldots, a_n$ so that the set: $$S = \{a_1, a_2, \ldots, a_n, \varphi(a_1), \varphi(a_2), \ldots, \varphi(a_n)\}$$ Have exactly $2n$ consecutive integers (in some order).
2020 LMT Fall, 13
Let set $S$ contain all positive integers that are one less than a perfect square. Find the sum of all powers of $2$ that can be expressed as the product of two (not necessarily distinct) members of $S.$
[i]Proposed by Alex Li[/i]
2014 Saudi Arabia Pre-TST, 4.1
Let $p$ be a prime number and $n \ge 2$ a positive integer, such that $p | (n^6 -1)$. Prove that $n > \sqrt{p}-1$.
2008 Silk Road, 3
Let $ G$ be a graph with $ 2n$ vertexes and $ 2n(n\minus{}1)$ edges.If we color some edge to red,then vertexes,which are connected by this edge,must be colored to red too. But not necessary that all edges from the red vertex are red.
Prove that it is possible to color some vertexes and edges in $ G$,such that all red vertexes has exactly $ n$ red edges.
2003 Hong kong National Olympiad, 1
Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$
PEN H Problems, 25
What is the smallest positive integer $t$ such that there exist integers $x_{1},x_{2}, \cdots, x_{t}$ with \[{x_{1}}^{3}+{x_{2}}^{3}+\cdots+{x_{t}}^{3}=2002^{2002}\;\;?\]
2019 Saint Petersburg Mathematical Olympiad, 7
Let $\omega$ and $O$ be respectively the circumcircle and the circumcenter of a triangle $ABC$. The line $AO$ intersects $\omega$ second time at $A'$. $M_B$ and $M_C$ are the midpoints of $AC$ and $AB$, respectively. The lines $A'M_B$ and $A'M_C$ intersect $\omega$ secondly at points $B'$ and $C$, and also intersect $BC$ at points $D_B$ and $D_C$, respectively. The circumcircles of $CD_BB'$ and $BD_CC'$ intersect at points $P$ and $Q$.
Prove that $O$, $P$, $Q$ are collinear.
[i] (М. Германсков)[/i]
[hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]
Geometry Mathley 2011-12, 10.1
Let $ABC$ be a triangle with two angles $B,C$ not having the same measure, $I$ be its incircle, $(O)$ its circumcircle. Circle $(O_b)$ touches $BA,BC$ and is internally tangent to $(O)$ at $B_1$. Circle $(O_c)$ touches $CA,CB$ and is internally tangent to $(O)$ at $C_1$. Let $S$ be the intersection of $BC$ and $B_1C_1$. Prove that $\angle AIS = 90^o$.
Nguyễn Minh Hà
2002 China Team Selection Test, 2
Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \cdots k_{2002}$ such that for any positive integer $ n \geq 2001$, one of $ k_12^n \plus{} 1, k_22^n \plus{} 1, \cdots, k_{2002}2^n \plus{} 1$ is prime?
2017 ELMO Shortlist, 1
Let $0<k<\frac{1}{2}$ be a real number and let $a_0, b_0$ be arbitrary real numbers in $(0,1)$. The sequences $(a_n)_{n\ge 0}$ and $(b_n)_{n\ge 0}$ are then defined recursively by
$$a_{n+1} = \dfrac{a_n+1}{2} \text{ and } b_{n+1} = b_n^k$$
for $n\ge 0$. Prove that $a_n<b_n$ for all sufficiently large $n$.
[i]Proposed by Michael Ma
2018 Purple Comet Problems, 23
Let $a, b$, and $c$ be integers simultaneously satisfying the equations $4abc + a + b + c = 2018$ and $ab + bc + ca = -507$. Find $|a| + |b|+ |c|$.
2006 Romania National Olympiad, 4
Let $a,b,c \in \left[ \frac 12, 1 \right]$. Prove that \[ 2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3 . \]
[i]selected by Mircea Lascu[/i]
1991 Putnam, A6
An $n$-sum of type $1$ is a finite sequence of positive integers $a_1,a_2,\ldots,a_r$, such that:
$(1)$ $a_1+a_2+\ldots+a_r=n$;
$(2)$ $a_1>a_2+a_3,a_2>a_3+a_4,\ldots, a_{r-2}>a_{r-1}+a_r$, and $a_{r-1}>a_r$. For example, there are five $7$-sums of type $1$, namely: $7$; $6,1$; $5,2$; $4,3$; $4,2,1$. An $n$-sum of type $2$ is a finite sequence of positive integers $b_1,b_2,\ldots,b_s$ such that:
$(1)$ $b_1+b_2+\ldots+b_s=n$;
$(2)$ $b_1\ge b_2\ge\ldots\ge b_s$;
$(3)$ each $b_i$ is in the sequence $1,2,4,\ldots,g_j,\ldots$ defined by $g_1=1$, $g_2=2$, $g_j=g_{j-1}+g_{j-2}+1$; and
$(4)$ if $b_1=g_k$, then $1,2,4,\ldots,g_k$ is a subsequence. For example, there are five $7$-sums of type $2$, namely: $4,2,1$; $2,2,2,1$; $2,2,1,1,1$; $2,1,1,1,1,1$; $1,1,1,1,1,1,1$. Prove that for $n\ge1$ the number of type $1$ and type $2$ $n$-sums is the same.
1966 IMO Longlists, 8
We are given a bag of sugar, a two-pan balance, and a weight of $1$ gram. How do we obtain $1$ kilogram of sugar in the smallest possible number of weighings?
1956 Czech and Slovak Olympiad III A, 4
Let a semicircle $AB$ be given and let $X$ be an inner point of the arc. Consider a point $Y$ on ray $XA$ such that $XY=XB$. Find the locus of all points $Y$ when $X$ moves on the arc $AB$ (excluding the endpoints).
2006 Kazakhstan National Olympiad, 3
The racing tournament has $12$ stages and $ n $ participants. After each stage, all participants, depending on the occupied place $ k $, receive points $ a_k $ (the numbers $ a_k $ are natural and $ a_1> a_2> \dots> a_n $). For what is the smallest $ n $ the tournament organizer can choose the numbers $ a_1 $, $ \dots $, $ a_n $ so that after the penultimate stage for any possible distribution of places at least two participants had a chance to take first place.