Found problems: 85335
1991 National High School Mathematics League, 1
The number of regular triangles that three apexes are among eight vertex of a cube is
$\text{(A)}4\qquad\text{(B)}8\qquad\text{(C)}12\qquad\text{(D)}24$
1996 Rioplatense Mathematical Olympiad, Level 3, 3
The real numbers $x, y, z$, distinct in pairs satisfy $$\begin{cases} x^2=2 + y \\ y^2=2 + z \\ z^2=2 + x.\end{cases}$$
Find the possible values of $x^2 + y^2 + z^2$.
2011 South africa National Olympiad, 2
Suppose that $x$ and $y$ are real numbers that satisfy the system of equations
$2^x-2^y=1$
$4^x-4^y=\frac{5}{3}$
Determine $x-y$
2016 Junior Regional Olympiad - FBH, 4
Let $C$ and $D$ be points inside angle $\angle AOB$ such that $5\angle COD = 4\angle AOC$ and $3\angle COD = 2\angle DOB$. If $\angle AOB = 105^{\circ}$, find $\angle COD$
1957 Polish MO Finals, 6
A cube is given with base $ ABCD $, where $ AB = a $ cm. Calculate the distance of the line $ BC $ from the line passing through the point $ A $ and the center $ S $ of the face opposite the base.
2023 Brazil EGMO TST -wrong source, 2
Determine all the integers solutions $(x,y)$ of the following equation
$$\frac{x^2-4}{2x-1}+\frac{y^2-4}{2y-1}=x+y$$
1995 Baltic Way, 16
In the triangle $ABC$, let $\ell$ be the bisector of the external angle at $C$. The line through the midpoint $O$ of $AB$ parallel to $\ell$ meets $AC$ at $E$. Determine $|CE|$, if $|AC|=7$ and $|CB|=4$.
2018 Morocco TST., 4
Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.
2019 PUMaC Individual Finals A, B, B2
Let $G = (V, E)$ be a simple connected graph. Show that there exists a subset of edges $F \subseteq E$ such that every vertex in $H = (V, F)$ has odd degree if and only if $|V |$ is even.
Note: A connected graph is a graph such that any two vertices have a sequence of edges connecting one to the other.
Note: A simple graph has no loops (edges of the form $(v, v)$) or duplicate edges.
2003 Austria Beginners' Competition, 4
Prove that every rectangle circumscribed by a square is itself a square.
(A rectangle is circumscribed by a square if there is exactly one corner point of the square on each side of the rectangle.)
2010 CHKMO, 1
Given that $ \{a_n\}$ is a sequence in which all the terms are integers, and $ a_2$ is odd. For any natural number $ n$, $ n(a_{n \plus{} 1} \minus{} a_n \plus{} 3) \equal{} a_{n \plus{} 1} \plus{} a_n \plus{} 3$. Furthermore, $ a_{2009}$ is divisible by $ 2010$. Find the smallest integer $ n > 1$ such that $ a_n$ is divisible by $ 2010$.
P.S.: I saw EVEN instead of ODD. Got only half of the points.
VI Soros Olympiad 1999 - 2000 (Russia), 9.4
Is there a function $f(x)$, which satisfies both of the following conditions:
a) if $x \ne y$, then $f(x)\ne f(y)$
b) for all real $x$, holds the inequality $f(x^2-1998x)-f^2(2x-1999)\ge \frac14$?
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.