Found problems: 85335
2013 Bangladesh Mathematical Olympiad, 10
Higher Secondary P10
$X$ is a set of $n$ elements. $P_m(X)$ is the set of all $m$ element subsets (i.e. subsets that contain exactly $m$ elements) of $X$. Suppose $P_m(X)$ has $k$ elements. Prove that the elements of $P_m(X)$ can be ordered in a sequence $A_1, A_2,...A_i,...A_k$ such that it satisfies the two conditions:
(A) each element of $P_m(X)$ occurs exactly once in the sequence,
(B) for any $i$ such that $0<i<k$, the size of the set $A_i \cap A_{i+1}$ is $m-1$.
2001 USAMO, 1
Each of eight boxes contains six balls. Each ball has been colored with one of $n$ colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer $n$ for which this is possible.
1990 IMO Shortlist, 20
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.
2007 Purple Comet Problems, 13
Find the circumradius of the triangle with side lengths $104$, $112$, and $120$.
2001 Hungary-Israel Binational, 2
Points $A, B, C, D$ lie on a line $l$, in that order. Find the locus of points $P$ in the plane for which $\angle{APB}=\angle{CPD}$.
PEN E Problems, 32
Let $n \ge 5$ be an integer. Show that $n$ is prime if and only if $n_{i} n_{j} \neq n_{p} n_{q}$ for every partition of $n$ into $4$ integers, $n=n_{1}+n_{2}+n_{3}+n_{4}$, and for each permutation $(i, j, p, q)$ of $(1, 2, 3, 4)$.
2016 Tournament Of Towns, 5
In convex hexagonal pyramid 11 edges are equal to 1.Find all possible values of 12th edge.
2010 Romania National Olympiad, 2
Prove that there is a similarity between a triangle $ABC$ and the triangle having as sides the medians of the triangle $ABC$ if and only if the squares of the lengths of the sides of triangle $ABC$ form an arithmetic sequence.
[i]Marian Teler & Marin Ionescu[/i]
2005 Today's Calculation Of Integral, 53
Find the maximum value of the following integral.
\[\int_0^{\infty} e^{-x}\sin tx\ dx\]
2016 CCA Math Bonanza, I12
Let $f$ be a function from the set $X = \{1,2, \dots, 10\}$ to itself. Call a partition ($S$, $T$) of $X$ $f-balanced$ if for all $s \in S$ we have $f(s) \in T$ and for all $t \in T$ we have $f(t) \in S$. (A partition $(S, T)$ is a pair of subsets $S$ and $T$ of $X$ such that $S\cap T = \emptyset$ and $S \cup T = X$. Note that $(S, T)$ and $(T, S)$ are considered the same partition).
Let $g(f)$ be the number of $f-balanced$ partitions, and let $m$ equal the maximum value of $g$ over all functions $f$ from $X$ to itself. If there are $k$ functions satisfying $g(f) = m$, determine $m+k$.
[i]2016 CCA Math Bonanza Individual #12[/i]
2009 Tournament Of Towns, 7
Initially a number $6$ is written on a blackboard. At $n$-th step an integer $k$ on the blackboard is replaced by $k+gcd(k,n)$. Prove that at each step the number on the blackboard increases either by $1$ or by a prime number.
MBMT Team Rounds, 2015 E15
In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. If the area of triangle $DOC$ is $4$ and the area of triangle $AOB$ is $36$, compute the minimum possible value of the area of $ABCD$.
2013 Chile TST Ibero, 2
Let $a \in \mathbb{N}$ such that $a + n^2$ can be expressed as the sum of two squares for all $n \in \mathbb{N}$. Prove that $a$ is the square of a natural number.
1970 IMO Longlists, 30
Let $u_1, u_2, \ldots, u_n, v_1, v_2, \ldots, v_n$ be real numbers. Prove that
\[1+ \sum_{i=1}^n (u_i+v_i)^2 \leq \frac 43 \Biggr( 1+ \sum_{i=1}^n u_i^2 \Biggl) \Biggr( 1+ \sum_{i=1}^n v_i^2 \Biggl) .\]
2014 Irish Math Olympiad, 2
Prove that for $N>1$ that $(N^{2})^{2014} - (N^{11})^{106}$ is divisible by $N^6 + N^3 +1$
Is this just a proof by induction or is there a more elegant method? I don't think calculating $N = 2$ was expected.
2015 Saudi Arabia Pre-TST, 3.1
Let $ABC$ be a triangle, $I$ its incenter, and $D$ a point on the arc $BC$ of the circumcircle of $ABC$ not containing $A$. The bisector of the angle $\angle ADB$ intesects the segment $AB$ at $E$. The bisector of the angle $\angle CDA$ intesects the segment $AC$ at $F$. Prove that the points $E, F,I$ are collinear.
(Malik Talbi)
2019 Online Math Open Problems, 3
Let $k$ be a positive real number. Suppose that the set of real numbers $x$ such that $x^2+k|x| \leq 2019$ is an interval of length $6$. Compute $k$.
[i]Proposed by Luke Robitaille[/i]
2012 Indonesia TST, 3
Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.
1979 Polish MO Finals, 1
Let be given a set $\{r_1,r_2,...,r_k\}$ of natural numbers that give distinct remainders when divided by a natural number $m$. Prove that if $k > m/2$, then for every integer $n$ there exist indices $i$ and $j$ (not necessarily distinct) such that $r_i +r_j -n$ is divisible by $m$.
2018 AMC 12/AHSME, 10
A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list?
$\textbf{(A)}\ 202\qquad\textbf{(B)}\ 223\qquad\textbf{(C)}\ 224\qquad\textbf{(D)}\ 225\qquad\textbf{(E)}\ 234$
2015 Estonia Team Selection Test, 12
Call an $n$-tuple $(a_1, . . . , a_n)$ [i]occasionally periodic [/i] if there exist a nonnegative integer $i$ and a positive integer $p$ satisfying $i + 2p \le n$ and $a_{i+j} = a_{i+p+j}$ for every $j = 1, 2, . . . , p$. Let $k$ be a positive integer. Find the least positive integer $n$ for which there exists an $n$-tuple $(a_1, . . . , a_n)$ with elements from set $\{1, 2, . . . , k\}$, which is not occasionally periodic but whose arbitrary extension $(a_1, . . . , a_n, a_{n+1})$ is occasionally periodic for any $a_{n+1} \in \{1, 2, . . . , k\}$.
2019 New Zealand MO, 1
A positive integer is called sparkly if it has exactly 9 digits, and for any n between 1 and 9 (inclusive), the nth digit is a positive multiple of n. How many positive integers are sparkly?
2008 Croatia Team Selection Test, 4
Let $ S$ be the set of all odd positive integers less than $ 30m$ which are not multiples of $ 5$, where $ m$ is a given positive integer. Find the smallest positive integer $ k$ such that each $ k$-element subset of $ S$ contains two distinct numbers, one of which divides the other.
2021-IMOC, C4
There is a city with many houses, where the houses are connected by some two-way roads. It is known that for any two houses $A,B$, there is exactly one house $C$ such that both $A,B$ are connected to $C$. Show that for any two houses not connected directly by a road, they have the same number of roads adjacent to them.
[i]ST[/i]
2019 Bangladesh Mathematical Olympiad, 9
Let $ABCD$ is a convex quadrilateral.The internal angle bisectors of $\angle {BAC}$ and $\angle {BDC}$ meets at $P$.$\angle {APB}=\angle {CPD}$.Prove that $AB+BD=AC+CD$.