Found problems: 85335
2010 LMT, 14
Al and Bob are joined by Carl and D’Angelo, and they decide to play a team game of Rock Paper Scissors. A game is called [i]perfect[/i] if some two of the four play the same thing, and the other two also play the same thing, but something different. For example, an example of a perfect game would be Al and Bob playing rock, and Carl and D’Angelo playing scissors, but if all four play paper, we do not have a perfect game. What is the probability of a perfect game?
2015 Thailand TSTST, 1
Let $A$ and $B$ be nonempty sets and let $f : A \to B$. Prove that the following statements are equivalent:
$\text{(i) }$ $f$ is surjective.
$\text{(ii)} $ For every set $C$ and and every functions $g, h : B \to C$, if $g\circ f = h \circ f$ then $g = h$.
2018 Hong Kong TST, 4
In triangle $ABC$ with incentre $I$, let $M_A,M_B$ and $M_C$ by the midpoints of $BC, CA$ and $AB$ respectively, and $H_A,H_B$ and $H_C$ be the feet of the altitudes from $A,B$ and $C$ to the respective sides. Denote by $\ell_b$ the line being tangent tot he circumcircle of triangle $ABC$ and passing through $B$, and denote by $\ell_b'$ the reflection of $\ell_b$ in $BI$. Let $P_B$ by the intersection of $M_AM_C$ and $\ell_b$, and let $Q_B$ be the intersection of $H_AH_C$ and $\ell_b'$. Defined $\ell_c,\ell_c',P_C,Q_C$ analogously. If $R$ is the intersection of $P_BQ_B$ and $P_CQ_C$, prove that $RB=RC$.
2011 Pre-Preparation Course Examination, 3
Calculate number of the hamiltonian cycles of the graph below: (15 points)
2014 Singapore Junior Math Olympiad, 3
In the triangle $ABC$, the bisector of $\angle A$ intersects the bisection of $\angle B$ at the point $I, D$ is the foot of the perpendicular from $I$ onto $BC$. Prove that the bisector of $\angle BIC$ is perpendicular to the bisector $\angle AID$.
2005 Austrian-Polish Competition, 3
Let $a_0, a_1, a_2, ... , a_n$ be real numbers, which fulfill the following two conditions:
a) $0 = a_0 \leq a_1 \leq a_2 \leq ... \leq a_n$.
b) For all $0 \leq i < j \leq n$ holds: $a_j - a_i \leq j-i$.
Prove that
$$\left( \displaystyle \sum_{i=0}^n a_i \right)^2 \geq \sum_{i=0}^n a_i^3.$$
1993 India National Olympiad, 3
If $a,b,c,d \in \mathbb{R}_{+}$ and $a+b +c +d =1$, show that \[ ab +bc +cd \leq \dfrac{1}{4}. \]
2014 Contests, 1
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
[i](A. Golovanov)[/i]
2008 Greece Team Selection Test, 1
Find all possible values of $a\in \mathbb{R}$ and $n\in \mathbb{N^*}$ such that $f(x)=(x-1)^n+(x-2)^{2n+1}+(1-x^2)^{2n+1}+a$
is divisible by $\phi (x)=x^2-x+1$
2014 BAMO, 3
Suppose that for two real numbers $x$ and $y$ the following equality is true:
$$(x+ \sqrt{1+ x^2})(y+\sqrt{1+y^2})=1$$
Find (with proof) the value of $x+y$.
2006 Mathematics for Its Sake, 1
[b]a)[/b] Show that there are $ 4 $ equidistant parallel planes that passes through the vertices of the same tetrahedron.
[b]b)[/b] How many such $ \text{4-tuplets} $ of planes does exist, in function of the tetrahedron?
2010 Purple Comet Problems, 16
Half the volume of a 12 foot high cone-shaped pile is grade A ore while the other half is grade B ore. The pile is worth \$62. One-third of the volume of a similarly shaped 18 foot pile is grade A ore while the other two-thirds is grade B ore. The second pile is worth \$162. Two-thirds of the volume of a similarly shaped 24 foot pile is grade A ore while the other one-third is grade B ore. What is the value in dollars (\$) of the 24 foot pile?
1998 Tournament Of Towns, 5
The sum of the length, width, and height of a rectangular parallelepiped will be called its size. Can it happen that one rectangular parallelepiped contains another one of greater size?
(A Shen)
2012 Pre - Vietnam Mathematical Olympiad, 1
Let $n \geq 2$ be a positive integer. Suppose there exist non-negative integers ${n_1},{n_2},\ldots,{n_k}$ such that $2^n - 1 \mid \sum_{i = 1}^k {{2^{{n_i}}}}$. Prove that $k \ge n$.
2018 Estonia Team Selection Test, 12
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$.
Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.
2000 All-Russian Olympiad Regional Round, 9.5
In a $99\times 101$ table , cubes of natural numbers, as shown in figure . Prove that the sum of all numbers in the table are divisible by $200$.
[img]https://cdn.artofproblemsolving.com/attachments/3/e/dd3d38ca00a36037055acaaa0c2812ae635dcb.png[/img]
2011 Tournament of Towns, 4
Each diagonal of a convex quadrilateral divides it into two isosceles triangles. The two diagonals of the same quadrilateral divide it into four isosceles triangles. Must this quadrilateral be a square?
2014 NIMO Problems, 5
Let $r$, $s$, $t$ be the roots of the polynomial $x^3+2x^2+x-7$. Then \[ \left(1+\frac{1}{(r+2)^2}\right)\left(1+\frac{1}{(s+2)^2}\right)\left(1+\frac{1}{(t+2)^2}\right)=\frac{m}{n} \] for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
[i]Proposed by Justin Stevens[/i]
2006 Baltic Way, 17
Determine all positive integers $n$ such that $3^{n}+1$ is divisible by $n^{2}$.
2006 Sharygin Geometry Olympiad, 9.1
Given a circle of radius $K$. Two other circles, the sum of the radii of which are also equal to $K$, tangent to the circle from the inside. Prove that the line connecting the points of tangency passes through one of the common points of these circles.
2019 EGMO, 1
Find all triples $(a, b, c)$ of real numbers such that $ab + bc + ca = 1$ and
$$a^2b + c = b^2c + a = c^2a + b.$$
1987 IMO Shortlist, 23
Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$,
\[[r^m] \equiv -1 \pmod k .\]
[i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients.
[i]Proposed by Yugoslavia.[/i]
2018 Hong Kong TST, 3
On a rectangular board with $m$ rows and $n$ columns, where $m\leq n$, some squares are coloured black in a way that no two rows are alike. Find the biggest integer $k$ such that for every possible colouring to start with one can always color $k$ columns entirely red in such a way that no two rows are still alike.
2003 Cuba MO, 4
Let $f : N \to N$ such that $f(p) = 1$ for all p prime and $f(ab) =bf(a) + af(b)$ for all $a, b \in N$. Prove that if $n = p^{a_1}_1 p^{a_1}_2... p^{a_1}_k$ is the canonical distribution of $n$ and $p_i$ does not divide $a_i$ ($i = 1, 2, ..., k$) then $\frac{n}{gcd(n,f(n))}$ is square free (not divisible by a square greater than $1$).
Kvant 2023, M2734
Real numbers are placed at the vertices of an $n{}$-gon. On each side, we write the sum of the numbers on its endpoints. For which $n{}$ is it possible that the numbers on the sides form a permutation of $1, 2, 3,\ldots , n$?
[i]From the folklore[/i]