Found problems: 85335
1978 Miklós Schweitzer, 5
Suppose that $ R(z)= \sum_{n=-\infty}^{\infty} a_nz^n$ converges in a neighborhood of the unit circle $ \{ z : \;|z|=1\ \}$ in the complex plane, and $ R(z)=P(z) / Q(z)$ is a rational function in this neighborhood, where $ P$ and $ Q$ are polynomials of degree at most $ k$. Prove that there is a constant $ c$ independent of $ k$ such that \[ \sum_{n=-\infty} ^{\infty} |a_n| \leq ck^2 \max_{|z|=1} |R(z)|.\]
[i]H. S. Shapiro, G. Somorjai[/i]
2021 Princeton University Math Competition, B2
Let $p$ be an odd prime. Prove that for every integer $k$, there exist integers $a, b$ such that $p|a^2 + b^2 - k$.
1988 AMC 12/AHSME, 3
Four rectangular paper strips of length $10$ and width $1$ are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered?
[asy]
size(120);
draw((0,0)--(1,0)--(1,4)--(0,4)--(0,0)--(0,1)--(-1,1)--(-1,2)--(0,2)--(0,4)--(-1,4)--(-1,5)--(1,5)--(1,6)--(0,6)--(0,5)--(3,5)--(3,6)--(4,6)--(4,2)--(5,2)--(5,1)--(1,1)--(3,1)--(3,0)--(4,0)--(4,1));
draw((1,4)--(3,4)--(3,2)--(1,2)--(4,2)--(3,2)--(3,6)--(4,6)--(4,5)--(5,5)--(5,4)--(4,4));[/asy]
$ \textbf{(A)}\ 36 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 44 \qquad \textbf{(D)}\ 98 \qquad \textbf{(E)}\ 100 $
2000 Estonia National Olympiad, 4
On the side $AC$ of the triangle $ABC$, choose any point $D$ different from the vertices $A$ and C. Let $O_1$ and $O_2$ be circumcenters the triangles $ABD$ and $CBD$, respectively. Prove that the triangles $O_1DO_2$ and $ABC$ are similar.
2022 Durer Math Competition Finals, 1
To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.
2025 CMIMC Algebra/NT, 5
Consider all positive multiples of $77$ less than $1,000,000.$ What is the sum of all the odd digits that show up?
1993 Vietnam Team Selection Test, 1
Let $H$, $I$, $O$ be the orthocenter, incenter and circumcenter of a triangle. Show that $2 \cdot IO \geq IH$. When does the equality hold ?
2005 Romania National Olympiad, 4
Let $x_1,x_2,\ldots,x_n$ be positive reals. Prove that
\[ \frac 1{1+x_1} + \frac 1{1+x_1+x_2} + \cdots + \frac 1{1+x_1+\cdots + x_n} < \sqrt { \frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n}} . \]
[i]Bogdan Enescu[/i]
2005 Slovenia National Olympiad, Problem 2
For which prime numbers $p$ and $q$ is $(p+1)^q$ a perfect square?
2023 ELMO Shortlist, C8
Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An [i]operation[/i] consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors.
Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations.
[i]Proposed by Anthony Wang[/i]
1950 Miklós Schweitzer, 8
A coastal battery sights an enemy cruiser lying one kilometer off the coast and opens fire on it at the rate of one round per minute. After the first shot, the cruiser begins to move away at a speed of $ 60$ kilometers an hour. Let the probability of a hit be $ 0.75x^{ \minus{} 2}$, where $ x$ denotes the distance (in kilometers) between the cruiser and the coast ($ x\geq 1$), and suppose that the battery goes on firing till the cruiser either sinks or disappears. Further, let the probability of the cruiser sinking after $ n$ hits be $ 1 \minus{} \frac {1}{4^n}$ ($ n \equal{} 0,1,...$). Show that the probability of the cruiser escaping is $ \frac {2\sqrt {2}}{3\pi}$
2022 Stanford Mathematics Tournament, 4
Let $ABC$ be a triangle with $\angle A=\tfrac{135}{2}^\circ$ and $\overline{BC}=15$. Square $WXYZ$ is drawn inside $ABC$ such that $W$ is on $AB$, $X$ is on $AC$, $Z$ is on $BC$, and triangle $ZBW$ is similar to triangle $ABC$, but $WZ$ is not parallel to $AC$. Over all possible triangles $ABC$, find the maximum area of $WXYZ$.
LMT Guts Rounds, 2020 F17
In a regular square room of side length $2\sqrt{2}$ ft, two cats that can see $2$ feet ahead of them are randomly placed into the four corners such that they do not share the same corner. If the probability that they don't see the mouse, also placed randomly into the room can be expressed as $\frac{a-b\pi}{c},$ where $a,b,c$ are positive integers with a greatest common factor of $1,$ then find $a+b+c.$
[i]Proposed by Ada Tsui[/i]
2021 AIME Problems, 12
Let $A_1A_2A_3...A_{12}$ be a dodecagon (12-gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2003 Singapore Team Selection Test, 2
Let $M$ be a point on the diameter $AB$ of a semicircle $\Gamma$. The perpendicular at $M$ meets the semicircle $\Gamma$ at $P$. A circle inside $\Gamma$. touches $\Gamma$. and is tangent to $PM$ at $Q$ and $AM$ at $R$. Prove that $P B = RB$.
1992 Tournament Of Towns, (354) 3
Consider the sequence $a(n)$ defined by the following conditions:$$a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...$$ How many perfect squares no greater in value than $1000 000$ will be found among the first terms of the sequence? ( (Note: $[x]$ means the integer part of $x$, that is the greatest integer not greater than $x$.)
(A Andjans)
2006 Estonia National Olympiad, 2
In a right triangle, the length of one side is a prime and the lengths of the other
side and the hypotenuse are integral. The ratio of the triangle perimeter and the incircle diameter is also an integer. Find all possible side lengths of the triangle.
2017 Estonia Team Selection Test, 11
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
2007 Harvard-MIT Mathematics Tournament, 7
A student at Harvard named Kevin
Was counting his stones by $11$
He messed up $n$ times
And instead counted $9$s
And wound up at $2007$.
How many values of $n$ could make this limerick true?
2016 Iran Team Selection Test, 3
Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.
2000 Hong kong National Olympiad, 1
Let $O$ be the circumcentre of a triangle $ABC$ with $AB > AC > BC$. Let $D$ be a point on the minor arc $BC$ of the circumcircle and let $E$ and $F$ be points on $AD$ such that $AB \perp OE$ and $AC \perp OF$ . The lines $BE$ and $CF$ meet at $P$. Prove that if $PB=PC+PO$, then $\angle BAC = 30^{\circ}$.
2020 Costa Rica - Final Round, 2
Consider a square $ABCD$. Let $M$ be the midpoint of segment $AB$, $\Gamma_1$ be the circle tangent to $\overline{AD}$, $\overline{AM}$ and $\overline{MC}$ with radius $r > 0$ and let $\Gamma_2$ be the circle tangent to $\overline{AD}$, $\overline{DC}$ and $\overline{MC}$ with radius $R > 0$. Prove that $R =\frac{2r}{r+1}$.
2010 Germany Team Selection Test, 2
Determine all $n \in \mathbb{Z}^+$ such that a regular hexagon (i.e. all sides equal length, all interior angles same size) can be partitioned in finitely many $n-$gons such that they can be composed into $n$ congruent regular hexagons in a non-overlapping way upon certain rotations and translations.
2015 Bosnia Herzegovina Team Selection Test, 1
Determine the minimum value of the expression
$$\frac {a+1}{a(a+2)}+ \frac {b+1}{b(b+2)}+\frac {c+1}{c(c+2)}$$
for positive real numbers $a,b,c$ such that $a+b+c \leq 3$.
2024 Euler Olympiad, Round 1, 8
Let $P$ be a point inside a square $ABCD,$ such that $\angle BPC = 135^\circ $ and the area of triangle $ADP$ is twice as much as the area of triangle $PCD.$ Find $\frac {AP}{DP}.$
[i]Proposed by Andria Gvaramia, Georgia [/i]