Found problems: 85335
1962 AMC 12/AHSME, 25
Given square $ ABCD$ with side $ 8$ feet. A circle is drawn through vertices $ A$ and $ D$ and tangent to side $ BC.$ The radius of the circle, in feet, is:
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 4 \sqrt{2} \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 5 \sqrt{2} \qquad
\textbf{(E)}\ 6$
1984 All Soviet Union Mathematical Olympiad, 383
The teacher wrote on a blackboard: $$x^2 + 10x + 20$$ Then all the pupils in the class came up in turn and either decreased or increased by $1$ either the free coefficient or the coefficient at $x$, but not both. Finally they have obtained: $$x^2 + 20x + 10$$ Is it true that some time during the process there was written the square polynomial with the integer roots?
2023 Stanford Mathematics Tournament, 7
Triangle $ABC$ has $AC = 5$. $D$ and $E$ are on side $BC$ such that $AD$ and $AE$ trisect $\angle BAC$, with $D$ closer to $B$ and $DE =\frac32$, $EC =\frac52$ . From $B$ and $E$, altitudes $BF$ and $EG$ are drawn onto side $AC$. Compute $\frac{CF}{CG}-\frac{AF}{AG}$ .
2010 Contests, 3
Given $a_1\ge 1$ and $a_{k+1}\ge a_k+1$ for all $k\ge 1,2,\dots,n$, show that $a_1^3+a_2^3+\dots+a_n^3\ge (a_1+a_2+\dots+a_n)^2$
2024 Junior Balkan Team Selection Tests - Romania, P3
Determine all positive integers $a,b,c,d,e,f$ satisfying the following condition: for any two of them, $x{}$ and $y{},$ two of the remaining numbers, $z{}$ and $t{},$ satisfy $x/y=z/t.$
[i]Cristi Săvescu[/i]
2007 AIME Problems, 6
An integer is called [i]parity-monotonic[/i] if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ is $a_{i}$ is even. How many four-digit parity-monotonic integers are there?
2019 Greece National Olympiad, 4
Given a $n\times m$ grid we play the following game . Initially we place $M$ tokens in each of $M$ empty cells and at the end of the game we need to fill the whole grid with tokens.For that purpose we are allowed to make the following move:If an empty cell shares a common side with at least two other cells that contain a token then we can place a token in this cell.Find the minimum value of $M$ in terms of $m,n$ that enables us to win the game.
2003 France Team Selection Test, 2
A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.
2010 Harvard-MIT Mathematics Tournament, 7
Let $a_1$, $a_2$, and $a_3$ be nonzero complex numbers with non-negative real and imaginary parts. Find the minimum possible value of \[\dfrac{|a_1+a_2+a_3|}{\sqrt[3]{|a_1a_2a_3|}}.\]
1996 Bundeswettbewerb Mathematik, 4
Find all natural numbers $n$ for which $n2^{n-1} +1$ is a perfect square.
2008 Hong Kong TST, 4
Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$.
(a) Show that all such lines $ AB$ are concurrent.
(b) Find the locus of midpoints of all such segments $ AB$.
2013 ISI Entrance Examination, 5
Let $AD$ be a diameter of a circle of radius $r,$ and let $B,C$ be points on the circle such that $AB=BC=\frac r2$ and $A\neq C.$ Find the ratio $\frac{CD}{r}.$
2020 LIMIT Category 1, 12
$q$ is the smallest rational number having the following properties:
(i) $q>\frac{31}{17}$
(ii) when $q$ is written in its reduced form $\frac{a}{b}$, then $b<17$
As in part (ii) above, find $a+b$.
1997 Akdeniz University MO, 2
If $x$ and $y$ are positive reals, prove that
$$x^2\sqrt{\frac{x}{y}}+y^2\sqrt{\frac{y}{x}} \geq x^2+y^2$$
2014 Harvard-MIT Mathematics Tournament, 7
Find the maximum possible number of diagonals of equal length in a convex hexagon.
Kvant 2020, M2626
An infinite number of participants gathered for the Olympiad, who were registered under the numbers $1, 2,\ldots$. It turns out that for every $n = 1, 2,\ldots$ a participant with number $n{}$ has at least $n{}$ friends among the remaining participants (note: friendship is mutual). There is a hotel with an infinite number of double rooms. Prove that the participants can be accommodated in double rooms so that there is a couple of friends in each room.
[i]Proposed by V. Bragin, P. Kozhevnikov[/i]
2021 Saint Petersburg Mathematical Olympiad, 4
Stierlitz wants to send an encryption to the Center, which is a code containing $100$ characters, each a "dot" or a "dash". The instruction he received from the Center the day before about conspiracy reads:
i) when transmitting encryption over the radio, exactly $49$ characters should be replaced with their opposites;
ii) the location of the "wrong" characters is decided by the transmitting side and the Center is not informed of it.
Prove that Stierlitz can send $10$ encryptions, each time choosing some $49$ characters to flip, such that when the Center receives these $10$ ciphers, it may unambiguously restore the original code.
1999 All-Russian Olympiad, 3
A triangle $ABC$ is inscribed in a circle $S$. Let $A_0$ and $C_0$ be the midpoints of the arcs $BC$ and $AB$ on $S$, not containing the opposite vertex, respectively. The circle $S_1$ centered at $A_0$ is tangent to $BC$, and the circle $S_2$ centered at $C_0$ is tangent to $AB$. Prove that the incenter $I$ of $\triangle ABC$ lies on a common tangent to $S_1$ and $S_2$.
2016 Saudi Arabia Pre-TST, 2.2
Given four numbers $x, y, z, t$, let $(a, b, c, d)$ be a permutation of $(x, y, z, t)$ and set $x_1 =|a- b|$, $y_1 = |b-c|$, $z_1 = |c-d|$, and $t_1 = |d -a|$. From $x_1, y_1, z_1, t_1$, form in the same fashion the numbers $x_2, y_2, z_2, t_2$, and so on. It is known that $x_n = x, y_n = y, z_n = z, t_n = t$ for some $n$. Find all possible values of $(x, y, z, t)$.
2016 Korea Summer Program Practice Test, 8
There are distinct points $A_1, A_2, \dots, A_{2n}$ with no three collinear. Prove that one can relabel the points with the labels $B_1, \dots, B_{2n}$ so that for each $1 \le i < j \le n$ the segments $B_{2i-1} B_{2i}$ and $B_{2j-1} B_{2j}$ do not intersect and the following inequality holds.
\[ B_1 B_2 + B_3 B_4 + \dots + B_{2n-1} B_{2n} \ge \frac{2}{\pi} (A_1 A_2 + A_3 A_4 + \dots + A_{2n-1} A_{2n}) \]
1968 AMC 12/AHSME, 6
Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$, and let side $BC$ be extended through $C$, to meet in point $E$. Let $S$ represent the degree-sum of angles $CDE$ and $DCE$, and let $S'$ represent the degree-sum of angles $BAD$ and $ABC$. If $r=S/S'$, then:
$\textbf{(A)}\ r=1\text{ sometimes, }r>1\text{ sometimes} \qquad\\
\textbf{(B)}\ r=1\text{ sometimes, }r<1\text{ sometimes} \qquad\\
\textbf{(C)}\ 0<r<1\qquad
\textbf{(D)}\ r>1 \qquad
\textbf{(E)}\ r=1 $
2021 Peru IMO TST, P3
Suppose the function $f:[1,+\infty)\to[1,+\infty)$ satisfies the following two conditions:
(i) $f(f(x))=x^2$ for any $x\geq 1$;
(ii) $f(x)\leq x^2+2021x$ for any $x\geq 1$.
1. Prove that $x<f(x)<x^2$ for any $x\geq 1$.
2. Prove that there exists a function $f$ satisfies the above two conditions and the following one:
(iii) There are no real constants $c$ and $A$, such that $0<c<1$, and $\frac{f(x)}{x^2}<c$ for any $x>A$.
2009 AMC 12/AHSME, 25
The first two terms of a sequence are $ a_1 \equal{} 1$ and $ a_2 \equal{} \frac {1}{\sqrt3}$. For $ n\ge1$,
\[ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} a_{n \plus{} 1}}{1 \minus{} a_na_{n \plus{} 1}}.
\]What is $ |a_{2009}|$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2 \minus{} \sqrt3\qquad \textbf{(C)}\ \frac {1}{\sqrt3}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 2 \plus{} \sqrt3$
2006 Belarusian National Olympiad, 2
Find all triples $(x, y,z)$ such that $x, y, z \in (0,1)$ and $$\left(x+\frac{1}{2x}-1\right) \left(y+\frac{1}{2y}-1\right) \left(z+\frac{1}{2z}-1\right) = \left(1-\frac{xy}{z}\right)\left(1-\frac{yz}{x}\right)\left(1-\frac{zx}{y}\right)$$
(D. Bazylev)
2001 Federal Competition For Advanced Students, Part 2, 1
Prove that $\frac{1}{25} \sum_{k=0}^{2001} \left[ \frac{2^k}{25}\right]$ is a positive integer.