Found problems: 85335
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.
1998 AIME Problems, 14
An $m\times n\times p$ rectangular box has half the volume of an $(m+2)\times(n+2)\times(p+2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$ What is the largest possible value of $p$?
2014 India Regional Mathematical Olympiad, 2
let $x,y$ be positive real numbers.
prove that
$ 4x^4+4y^3+5x^2+y+1\geq 12xy $
2013 Romania Team Selection Test, 4
Let $n$ be an integer greater than 1. The set $S$ of all diagonals of a $ \left( 4n-1\right) $-gon is partitioned into $k$ sets, $S_{1},S_{2},\ldots ,S_{k},$ so that, for every pair of distinct indices $i$ and $j,$ some diagonal in $S_{i}$ crosses some diagonal in $S_{j};$ that is, the two diagonals share an interior point. Determine the largest possible value of $k $ in terms of $n.$
1981 Putnam, A4
A point $P$ moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it
meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection.
Let $N( t)$ be the number of starting directions from a fixed interior point $P_0$ for which $P$ escapes within $t$ units of time. Find the least constant $a$ for which constants $b$ and $c$ exist such that
$$N(t) \leq at^2 +bt+c$$
for all $t>0$ and all initial points $P_0 .$
2014 Greece National Olympiad, 3
For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once).
Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S_1}{S_2}=\frac{39}{64}.$
2023 Taiwan Mathematics Olympiad, 3
Let $O$ be the center of circle $\Gamma$, and $A$, $B$ be two points on $\Gamma$ so that $O, A$ and $B$ are not collinear. Let $M$ be the midpoint of $AB$. Let $P$ and $Q$ be points on $OA$ and $OB$, respectively, so that $P \neq A$ and $P, M, Q$ are collinear. Let $X$ be the intersection of the line passing through $P$ and parallel to $AB$ and the line passing through $Q$ and parallel to $OM$. Let $Y$ be the intersection of the line passing through $X$ and parallel to $OA$ and the line passing through $B$ and orthogonal to $OX$. Prove that: if $X$ is on $\Gamma$, then $Y$ is also on $\Gamma$.
[i]
Proposed by usjl[/i]
2014 Belarus Team Selection Test, 2
Let $x,y,z$ be pairwise distinct real numbers such that $x^2-1/y = y^2 -1/z = z^2 -1/x$.
Given $z^2 -1/x = a$, prove that $(x + y + z)xyz= -a^2$.
(I. Voronovich)
2006 Iran MO (3rd Round), 1
Let $A$ be a family of subsets of $\{1,2,\ldots,n\}$ such that no member of $A$ is contained in another. Sperner’s Theorem states that $|A|\leq{n\choose{\lfloor\frac{n}{2}\rfloor}}$. Find all the families for which the equality holds.
2013-2014 SDML (Middle School), 8
On a windless day, a pigeon can fly from Albatrocity to Finchester and back in $3$ hours and $45$ minutes. However, when there is a $10$ mile per hour win blowing from Albatrocity to Finchester, it takes the pigeon $4$ hours to make the round trip. How many miles is it from Albatrocity to Finchester?
2024 China Western Mathematical Olympiad, 1
For positive integer $n$, note $S_n=1^{2024}+2^{2024}+ \cdots +n^{2024}$.
Prove that there exists infinitely many positive integers $n$, such that $S_n$ isn’t divisible by $1865$ but $S_{n+1}$ is divisible by $1865$
1990 AMC 8, 11
The numbers on the faces of this cube are consecutive whole numbers. The sums of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3));
draw((3,3)--(5,5));
label("$15$",(1.5,1.2),N); label("$11$",(4,2.3),N); label("$14$",(2.5,3.7),N);[/asy]
$ \text{(A)}\ 75\qquad\text{(B)}\ 76\qquad\text{(C)}\ 78\qquad\text{(D)}\ 80\qquad\text{(E)}\ 81 $
2007 Harvard-MIT Mathematics Tournament, 13
Determine the largest integer $n$ such that $7^{2048}-1$ is divisible by $2^n$.