Found problems: 85335
MIPT Undergraduate Contest 2019, 1.1 & 2.1
In $\mathbb{R}^3$, let there be a cube $Q$ and a sequence of other cubes, all of which are homothetic to $Q$ with coefficients of homothety that are each smaller than $1$. Prove that if this sequence of homothetic cubes completely fills $Q$, the sum of their coefficients of homothety is not less than $4$.
1992 Tournament Of Towns, (346) 4
On the plane is give a broken line $ABCD$ in which $AB = BC = CD = 1$, and $AD$ is not equal to $1$. The positions of $B$ and $C$ are fixed but $A$ and $D$ change their positions in turn according to the following rule (preserving the distance rules given): the point $A$ is reflected with respect to the line $BD$, then $D$ is reflected with respect to the line $AC$ (in which $A$ occupies its new position), then $A$ is reflected with respect to the line $BD$ ($D$ occupying its new position), $D$ is reflected with respect to the line $AC$, and so on. Prove that after several steps $A$ and $D$ coincide with their initial positions.
(M Kontzewich)
2006 International Zhautykov Olympiad, 1
Solve in positive integers the equation
\[ n \equal{} \varphi(n) \plus{} 402 ,
\]
where $ \varphi(n)$ is the number of positive integers less than $ n$ having no common prime factors with $ n$.
1961 IMO Shortlist, 4
Consider triangle $P_1P_2P_3$ and a point $p$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers \[ \dfrac{P_1P}{PQ_1}, \dfrac{P_2P}{PQ_2}, \dfrac{P_3P}{PQ_3} \]
at least one is $\leq 2$ and at least one is $\geq 2$
2019 Belarusian National Olympiad, 9.8
Andrey and Sasha play the game, making moves alternate. On his turn, Andrey marks on the plane an arbitrary point that has not yet been marked. After that, Sasha colors this point in one of two colors: white and black. Sasha wins if after his move it is impossible to draw a line such that all white points lie in one half-plane, while all black points lie in another half-plane with respect to this line.
[b]a)[/b] Prove that Andrey can make moves in such a way that Sasha will never win.
[b]b)[/b] Suppose that Andrey can mark only integer points on the Cartesian plane. Can Sasha guarantee himself a win regardless of Andrey's moves?
[i](N. Naradzetski)[/i]
1974 Miklós Schweitzer, 8
Prove that there exists a topological space $ T$ containing the real line as a subset, such that the Lebesgue-measurable functions, and only those, extend continuously over $ T$. Show that the real line cannot be an everywhere-dense subset of such a space $ T$.
[i]A. Csaszar[/i]
1990 AMC 8, 1
What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $ 4,5,6,7,8,9 $ in one of the six boxes in this addition problem?
[asy]
unitsize(12);
draw((0,0)--(10,0)); draw((-1.5,1.5)--(-1.5,2.5)); draw((-1,2)--(-2,2));
draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); draw((1,4)--(3,4)--(3,6)--(1,6)--cycle);
draw((4,1)--(6,1)--(6,3)--(4,3)--cycle); draw((4,4)--(6,4)--(6,6)--(4,6)--cycle);
draw((7,1)--(9,1)--(9,3)--(7,3)--cycle); draw((7,4)--(9,4)--(9,6)--(7,6)--cycle);[/asy]
$ \text{(A)}\ 947\qquad\text{(B)}\ 1037\qquad\text{(C)}\ 1047\qquad\text{(D)}\ 1056\qquad\text{(E)}\ 1245 $
1998 Turkey Junior National Olympiad, 1
Let $F$, $D$, and $E$ be points on the sides $[AB]$, $[BC]$, and $[CA]$ of $\triangle ABC$, respectively, such that $\triangle DEF$ is an isosceles right triangle with hypotenuse $[EF]$. The altitude of $\triangle ABC$ passing through $A$ is $10$ cm. If $|BC|=30$ cm, and $EF \parallel BC$, calculate the perimeter of $\triangle DEF$.
1977 IMO Longlists, 19
Given any integer $m>1$ prove that there exist infinitely many positive integers $n$ such that the last $m$ digits of $5^n$ are a sequence $a_m,a_{m-1},\ldots ,a_1=5\ (0\le a_j<10)$ in which each digit except the last is of opposite parity to its successor (i.e., if $a_i$ is even, then $a_{i-1}$ is odd, and if $a_i$ is odd, then $a_{i-1}$ is even).
2024 VJIMC, 3
Let $n$ be a positive integer and let $G$ be a simple undirected graph on $n$ vertices. Let $d_i$ be the
degree of its $i$-th vertex, $i = 1, \dots , n$. Denote $\Delta=\max d_i$. Prove that if
\[\sum_{i=1}^n d_i^2>n\Delta(n-\Delta),\]
then $G$ contains a triangle.
1996 Putnam, 2
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be circles whose centers are $10$ units apart, and whose radii are $1$ and $3$. Find, with proof, the locus of all points $M$ for which there exists points $X\in \mathcal{C}_1,Y\in \mathcal{C}_2$ such that $M$ is the midpoint of $XY$.
2021 AIME Problems, 6
Segments $\overline{AB}, \overline{AC},$ and $\overline{AD}$ are edges of a cube and $\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\sqrt{10}$, $CP=60\sqrt{5}$, $DP=120\sqrt{2}$, and $GP=36\sqrt{7}$. Find $AP.$
2016 Iran MO (3rd Round), 2
Let $P$ be a polynomial with integer coefficients. We say $P$ is [i]good [/i] if there exist infinitely many prime numbers $q$ such that the set $$X=\left\{P(n) \mod q : \quad n\in \mathbb N\right\}$$ has at least $\frac{q+1}{2}$ members.
Prove that the polynomial $x^3+x$ is good.
2010 Costa Rica - Final Round, 1
Consider points $D,E$ and $F$ on sides $BC,AC$ and $AB$, respectively, of a triangle $ABC$, such that $AD, BE$ and $CF$ concurr at a point $G$. The parallel through $G$ to $BC$ cuts $DF$ and $DE$ at $H$ and $I$, respectively. Show that triangles $AHG$ and $AIG$ have the same areas.
2012 Bogdan Stan, 4
Let be a group of order $ 2002 $ having the property that the application $ x\mapsto x^4 $ is and endomorphism of it.
Show that this group is cyclic.
2000 Saint Petersburg Mathematical Olympiad, 9.6
Excircle of $ABC$ is tangent to the side $BC$ at point $K$ and is tangent to the extension of $AB$ at point $L$. Another excircle is tangent to extensions of sides $AB$ and $BC$ at points $M$ and $N$. Lines $KL$ and $MN$ intersect at point $X$. Prove that $CX$ is the bisector of angle $ACN$.
[I]Proposed by S. Berlov[/i]
1995 IMO, 3
Determine all integers $ n > 3$ for which there exist $ n$ points $ A_{1},\cdots ,A_{n}$ in the plane, no three collinear, and real numbers $ r_{1},\cdots ,r_{n}$ such that for $ 1\leq i < j < k\leq n$, the area of $ \triangle A_{i}A_{j}A_{k}$ is $ r_{i} \plus{} r_{j} \plus{} r_{k}$.
1980 Spain Mathematical Olympiad, 1
Among the triangles that have a side of length $5$ m and the angle opposite of $30^o$, determine the one with maximum area, calculating the value of the other two angles and area of triangle.
2014 Miklós Schweitzer, 5
Let $ \alpha $ be a non-real algebraic integer of degree two, and let $ \mathbb{P} $ be the set of irreducible elements of the ring $ \mathbb{Z}[ \alpha ] $. Prove that
\[ \sum_{p\in \mathbb{P}}^{{}}\frac{1}{|p|^{2}}=\infty \]
2010 ELMO Problems, 1
Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$, there is a circle passing through all four or a line passing through some three.
[i]Carl Lian.[/i]
2022 Durer Math Competition (First Round), 5
Let $a_1 \le a_2 \le ... \le a_n$ be real numbers for which $$\sum_{i=1}^{n} a_i^{2k+1} = 0$$ holds for all integers $0 \le k < n$. Show that in this case, $a_i = -a_{n+1-i}$ holds for all $1 \le i \le n$.
2014 Iran Team Selection Test, 2
find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.
2005 Federal Competition For Advanced Students, Part 2, 1
The function $f : (0,...2005) \rightarrow N$ has the properties that $f(2x+1)=f(2x)$, $f(3x+1)=f(3x)$ and $f(5x+1)=f(5x)$ with $x \in (0,1,2,...,2005)$. How many different values can the function assume?
2004 Putnam, A6
Suppose that $f(x,y)$ is a continuous real-valued function on the unit square $0\le x\le1,0\le y\le1.$ Show that
$\int_0^1\left(\int_0^1f(x,y)dx\right)^2dy + \int_0^1\left(\int_0^1f(x,y)dy\right)^2dx$
$\le\left(\int_0^1\int_0^1f(x,y)dxdy\right)^2 + \int_0^1\int_0^1\left[f(x,y)\right]^2dxdy.$
1991 Putnam, A1
The rectangle with vertices $(0,0)$, $(0,3)$, $(2,0)$ and $(2,3)$ is rotated clockwise through a right angle about the point $(2,0)$, then about $(5,0)$, then about $(7,0$), and finally about $(10,0)$. The net effect is to translate it a distance $10$ along the $x$-axis. The point initially at $(1,1)$ traces out a curve. Find the area under this curve (in other words, the area of the region bounded by the curve, the $x$-axis and the lines parallel to the $y$-axis through $(1,0)$ and $(11,0)$).