Found problems: 85335
2020 BMT Fall, 13
Compute the expected sum of elements in a subset of $\{1, 2, 3, . . . , 2020\}$ (including the empty set) chosen uniformly at random.
2017 Pan-African Shortlist, G3
Let $ABCDE$ be a regular pentagon, and $F$ some point on the arc $AB$ of the circumcircle of $ABCDE$. Show that
\[
\frac{FD}{FE + FC} = \frac{FB + FA}{FD} = \frac{-1 + \sqrt{5}}{2},
\]
and that $FD + FB + FA = FE + FC$.
2003 China Team Selection Test, 1
Let $S$ be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant $r$, such that there exists one way to colour all the points in $S$ with three colous so that the distance between any two points with same colour is less than $r$.
2013 Tuymaada Olympiad, 6
Quadratic trinomials with positive leading coefficients are arranged in the squares of a $6 \times 6$ table. Their $108$ coefficients are all integers from $-60$ to $47$ (each number is used once). Prove that at least in one column the sum of all trinomials has a real root.
[i]K. Kokhas & F. Petrov[/i]
2014 USAMO, 6
Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\gcd(a+i, b+j)>1$ for all $i, j\in\{0, 1, \ldots n\}$, then\[\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.\]
2011 Estonia Team Selection Test, 3
Does there exist an operation $*$ on the set of all integers such that the following conditions hold simultaneously:
$(1)$ for all integers $x,y,z$, $(x*y)*z=x*(y*z)$;
$(2)$ for all integers $x$ and $y$, $x*x*y=y*x*x=y$?
1999 Yugoslav Team Selection Test, Problem 4
For a natural number $d$, $M_d$ denotes the set of natural numbers which are not representable as the sum of at least two consecutive terms of an arithmetic progression with the common difference d whose terms are integers. Prove that each $c\in M_3$ can be written in the form $c=ab$, where $a\in M_1$ and $b\in M_2\setminus\{2\}$.
2016 Online Math Open Problems, 19
Let $S$ be the set of all polynomials $Q(x,y,z)$ with coefficients in $\{0,1\}$ such that there exists a homogeneous polynomial $P(x,y,z)$ of degree $2016$ with integer coefficients and a polynomial $R(x,y,z)$ with integer coefficients so that \[P(x,y,z) Q(x,y,z) = P(yz,zx,xy)+2R(x,y,z)\] and $P(1,1,1)$ is odd. Determine the size of $S$.
Note: A homogeneous polynomial of degree $d$ consists solely of terms of degree $d$.
[i]Proposed by Vincent Huang[/i]
1966 IMO Shortlist, 19
Construct a triangle given the radii of the excircles.
1949 Moscow Mathematical Olympiad, 167
The midpoints of alternative sides of a convex hexagon are connected by line segments. Prove that the intersection points of the medians of the two triangles obtained coincide.
1985 Tournament Of Towns, (084) T5
Every member of a given sequence, beginning with the second , is equal to the sum of the preceding one and the sum of its digits . The first member equals $1$ . Is there, among the members of this sequence, a number equal to $123456$ ?
(S. Fomin , Leningrad)
2001 Greece National Olympiad, 2
Prove that there are no positive integers $a,b$ such that $(15a +b)(a +15b)$ is a power of $3.$
2003 Romania National Olympiad, 2
An integer $ n$, $ n\ge2$ is called [i]friendly[/i] if there exists a family $ A_1,A_2,\ldots,A_n$ of subsets of the set $ \{1,2,\ldots,n\}$ such that:
(1) $ i\not\in A_i$ for every $ i\equal{}\overline{1,n}$;
(2) $ i\in A_j$ if and only if $ j\not\in A_i$, for every distinct $ i,j\in\{1,2,\ldots,n\}$;
(3) $ A_i\cap A_j$ is non-empty, for every $ i,j\in\{1,2,\ldots,n\}$.
Prove that:
(a) 7 is a friendly number;
(b) $ n$ is friendly if and only if $ n\ge7$.
[i]Valentin Vornicu[/i]
2020 HMIC, 1
Sir Alex is coaching a soccer team of $n$ players of distinct heights. He wants to line them up so that for each player $P$, the total number of players that are either to the left of $P$ and taller than $P$ or to the right of $P$ and shorter than $P$ is even. In terms of $n$, how many possible orders are there?
[i]Michael Ren[/i]
2020 Jozsef Wildt International Math Competition, W26
Let $P_n$ denote the $n$-th Pell number defined by $P_{n+1}=2P_n+P_{n-1}$, $P_0=0$, $P_1=1$. Furthermore, let $T_n$ denote the $n$-th triangular number, that is
$T_n=\binom{n+1}2$. Show that
$$\sum_{n=0}^\infty4T_n\cdot\frac{P_n}{3^{n+2}}=P_3+P_4$$
[i]Proposed by Ángel Plaza[/i]
2025 Kyiv City MO Round 2, Problem 4
Inside a convex quadrilateral \( ABCD \), a point \( P \) is chosen such that
\[
\angle PAD = \angle PAB = \angle PBC = \angle PCB = \angle PDA = 30^\circ.
\]
Prove that \( \angle CDP = 30^\circ \).
[i]Proposed by Vadym Solomka[/i]
1955 AMC 12/AHSME, 11
The negation of the statement "No slow learners attend this school" is:
$ \textbf{(A)}\ \text{All slow learners attend this school} \\
\textbf{(B)}\ \text{All slow learners do not attend this school} \\
\textbf{(C)}\ \text{Some slow learners attend this school} \\
\textbf{(D)}\ \text{Some slow learners do not attend this school} \\
\textbf{(E)}\ \text{No slow learners do not attend this school}$
2012 Online Math Open Problems, 41
Find the remainder when
\[ \sum_{i=2}^{63} \frac{i^{2011}-i}{i^2-1}. \]
is divided by 2016.
[i]Author: Alex Zhu[/i]
2023 Indonesia TST, 2
Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that
\[2^{j-i}x_ix_j>2^{s-3}.\]
2011 Princeton University Math Competition, A3 / B5
Two points are chosen uniformly at random on the sides of a square with side length 1. If $p$ is the probability that the distance between them is greater than $1$, what is $\lfloor 100p \rfloor$? (Note: $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
2021 Kyiv City MO Round 1, 10.3
Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. Let point $C$ be such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ at points $X$ and $Y$, respectively. Prove that $CX = CY$.
[i]Proposed by Oleksii Masalitin[/i]
2019 Taiwan TST Round 3, 1
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.
[i]Proposed by Mongolia[/i]
2023 USA TSTST, 3
Find all positive integers $n$ for which it is possible to color some cells of an infinite grid of unit squares red, such that each rectangle consisting of exactly $n$ cells (and whose edges lie along the lines of the grid) contains an odd number of red cells.
[i]Proposed by Merlijn Staps[/i]
2017 NIMO Problems, 6
Define $f(x) = x^2 - 45x + 21$. Find the sum of all positive integers $n$ with the following property: there is exactly one integer $i$ in the set $\{1, 2, \ldots, n\}$ such that $n$ divides $f(i)$.
[i]Proposed by Sharvil Kesarwani[/i]
2000 Estonia National Olympiad, 5
At a given plane with $2,000$ lines, all those with an odd number of different points of intersection with intersecting lines.
a) Can there be an odd number of red lines if in the plane given there are no parallel lines?
b) Can there be an odd number of red lines if none of any 3 given lines intersect at one point?