Found problems: 85335
2022 Tuymaada Olympiad, 7
A $1 \times 5n$ rectangle is partitioned into tiles, each of the tile being either a separate $1 \times 1$ square or a broken domino consisting of two such squares separated by four squares (not belonging to the domino). Prove that the number of such partitions is a perfect fifth power.
[i](K. Kokhas)[/i]
2023 Israel National Olympiad, P2
The non-negative integers $x,y$ satisfy $\sqrt{x}+\sqrt{x+60}=\sqrt{y}$. Find the largest possible value for $x$.
2019 Ukraine Team Selection Test, 1
In a triangle $ABC$, $\angle ABC= 60^o$, point $I$ is the incenter. Let the points $P$ and $T$ on the sides $AB$ and $BC$ respectively such that $PI \parallel BC$ and $TI \parallel AB$ , and points $P_1$ and $T_1$ on the sides $AB$ and $BC$ respectively such that $AP_1 = BP$ and $CT_1 = BT$. Prove that point $I$ lies on segment $P_1T_1$.
(Anton Trygub)
1986 ITAMO, 6
Show that for any positive integer $n$ there exists an integer $m > 1$ such that $(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1}$.
2008 Irish Math Olympiad, 5
Suppose that $ x, y$ and $ z$ are positive real numbers such that $ xyz \ge 1$.
(a) Prove that
$ 27 \le (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2$,
with equality if and only if $ x \equal{} y \equal{} z \equal{} 1$.
(b) Prove that
$ (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2$ $ \le 3(x \plus{} y \plus{} z)^2$,
with equality if and only if $ x \equal{} y \equal{} z \equal{} 1$.
1984 IMO Shortlist, 4
Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that:
\[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\]
where $ [x]$ denotes the greatest integer not exceeding $ x$.
1996 India National Olympiad, 6
There is a $2n \times 2n$ array (matrix) consisting of $0's$ and $1's$ and there are exactly $3n$ zeroes. Show that it is possible to remove all the zeroes by deleting some $n$ rows and some $n$ columns.
2017 F = ma, 3
A ball of radius R and mass m is magically put inside a thin shell of the same mass and radius 2R. The system is at rest on a horizontal frictionless surface initially. When the ball is, again magically, released inside the shell, it sloshes around in the shell and eventually stops at the bottom of the shell. How far does the shell move from its initial contact point with the surface?
$\textbf{(A)}R\qquad
\textbf{(B)}\frac{R}{2}\qquad
\textbf{(C)}\frac{R}{4}\qquad
\textbf{(D)}\frac{3R}{8}\qquad
\textbf{(E)}\frac{R}{8}$
2024 HMNT, 5
Let $ABCD$ be a trapezoid with $AB \parallel CD, AB=20, CD=24,$ and area $880.$ Compute the area of the triangle formed by the midpoints of $AB, AC,$ and $BD.$
MOAA Team Rounds, 2023.8
Two consecutive positive integers $n$ and $n+1$ have the property that they both have $6$ divisors but a different number of distinct prime factors. Find the sum of the possible values of $n$.
[i]Proposed by Harry Kim[/i]
2018 District Olympiad, 2
Show that the number
\[\sqrt[n]{\sqrt{2019} + \sqrt{2018}} + \sqrt[n]{\sqrt{2019} - \sqrt{2018}}\]
is irrational for any $n\ge 2$.
2007 Alexandru Myller, 3
Let $ ABC $ be a right angle in $ A, $ and $ M $ be the mid of $ BC. $ On the perpendicular of $ AM $ through $ A $ choose a point $ D $ so that $ DM $ meets $ AB $ at a point, namely $ P. $ Let $ E $ be the projection of $ D $ on $ BC. $ Show that $ \angle BPM =\angle EAC. $
2018 CMIMC Number Theory, 7
For each $q\in\mathbb Q$, let $\pi(q)$ denote the period of the repeating base-$16$ expansion of $q$, with the convention of $\pi(q)=0$ if $q$ has a terminating base-$16$ expansion. Find the maximum value among \[\pi\left(\frac11\right),~\pi\left(\frac12\right),~\dots,~\pi\left(\frac1{70}\right).\]
1992 Brazil National Olympiad, 8
In a chess tournament each player plays every other player once. A player gets 1 point for a win, 0.5 point for a draw and 0 for a loss. Both men and women played in the tournament and each player scored the same total of points against women as against men. Show that the total number of players must be a square.
MathLinks Contest 5th, 2.1
For what positive integers $k$ there exists a function $f : N \to N$ such that for all $n \in N$ we have
$\underbrace{\hbox{f(f(... f(n)....))}}_{\hbox{k times}} = f(n) + 2$ ?
2015 Oral Moscow Geometry Olympiad, 5
On the $BE$ side of a regular $ABE$ triangle, a $BCDE$ rhombus is built outside it. The segments $AC$ and $BD$ intersect at point $F$. Prove that $AF <BD$.
1982 AMC 12/AHSME, 20
The number of pairs of positive integers $(x,y)$ which satisfy the equation $x^2+y^2=x^3$ is
$\textbf {(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf {(C) } 2 \qquad \textbf {(D) } \text{not finite} \qquad \textbf {(E) } \text{none of these}$
1988 Irish Math Olympiad, 2
2. Let $x_1, . . . , x_n$ be $n$ integers, and let $p$ be a positive integer, with $p < n$. Put
$$S_1 = x_1 + x_2 + . . . + x_p$$
$$T_1 = x_{p+1} + x_{p+2} + . . . + x_n$$
$$S_2 = x_2 + x_3 + . . . + x_{p+1}$$
$$T_2 = x_{p+2} + x_{p+3} + . . . + x_n + x_1$$
$$...$$
$$S_n=x_n+x_1+...+x_{p-1}$$
$$T_n=x_p+x_{p+1}+...+x_{n-1}$$
For $a = 0, 1, 2, 3$, and $b = 0, 1, 2, 3$, let $m(a, b)$ be the number of numbers $i$, $1 \leq i \leq n$, such that $S_i$ leaves remainder $a$ on division by $4$ and $T_i$ leaves remainder $b$ on division by $4$. Show that $m(1, 3)$ and $m(3, 1)$ leave the same remainder when divided by $4$ if, and only if, $m(2, 2)$ is even.
2008 Iran Team Selection Test, 4
Let $ P_1,P_2,P_3,P_4$ be points on the unit sphere. Prove that $ \sum_{i\neq j}\frac1{|P_i\minus{}P_j|}$ takes its minimum value if and only if these four points are vertices of a regular pyramid.
2003 All-Russian Olympiad Regional Round, 9.7
Prove that of any six four-digit numbers, mutual prime in total, you can always choose five numbers that are also relatively prime in total.
[hide=original wording]Докажите, что из любых шести четырехзначных чисел, взаимно простых в совокупности, всегда можно выбратьпя ть чисел, также взаимно простых в совокупности.[/hide]
2023 Dutch BxMO TST, 5
Find all pairs of prime numbers $(p,q)$ for which
\[2^p = 2^{q-2} + q!.\]
1989 AIME Problems, 1
Compute $\sqrt{(31)(30)(29)(28)+1}$.
2005 Taiwan TST Round 1, 2
Does there exist an positive integer $n$, so that for any positive integer $m<1002$, there exists an integer $k$ so that \[\displaystyle \frac{m}{1002} < \frac{k}{n} < \frac {m+1}{1003}\] holds? If $n$ does not exist, prove it; if $n$ exists, determine the minimum value of it.
I know this problem was easy, but it still appeared on our TST, and so I posted it here.
1986 National High School Mathematics League, 9
$f(x)=\frac{4^x}{4^x+2}$, then $f(\frac{1}{1001})+f(\frac{2}{1001})+\cdots+f(\frac{1000}{1001})=$________.
2008 Putnam, B3
What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?