Found problems: 85335
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 $(11,0)$).
2011 AMC 12/AHSME, 5
Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What percent of the birds that were not swans were geese?
$ \textbf{(A)}\ 20 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 60
$
2000 Croatia National Olympiad, Problem 2
The incircle of a triangle $ABC$ touches $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Find the angles of $\triangle A_1B_1C_1$ in terms of the angles of $\triangle ABC$.
1969 Putnam, B4
Show that any curve of unit length can be covered by a closed rectangle of area $1 \slash 4$.
2019 Jozsef Wildt International Math Competition, W. 24
If $a$, $b$, $c > 0$, prove that$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{a+b}{a+b+2c}+\frac{b+c}{2a+b+c}+\frac{c+a}{a+2b+c}$$
MBMT Guts Rounds, 2015.16
Your math teacher asks you to rationalize the denominator of the expression $\frac{a}{b + \sqrt{c}}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. You find that $\frac{a}{b + \sqrt{c}}$ is equal to $\frac{30 - 5\sqrt{14}}{11}$. Compute the triple $(a,b,c)$.