Found problems: 85335
2013 Indonesia MO, 4
Suppose $p > 3$ is a prime number and
\[S = \sum_{2 \le i < j < k \le p-1} ijk\]
Prove that $S+1$ is divisible by $p$.
2023 Math Prize for Girls Olympiad, 4
Let $O=(0,0)$ be the origin of the $xy$-plane. We say a lattice triangle $ABC$ is [i]marine[/i] if it has centroid $O$ and area $\tfrac{3}{2}$.
Let $P$ be any point in the plane which is not a lattice point. Prove that $P$ lies in the interior of some marine triangle if and only if the line segment $\overline{OP}$ does not pass through any lattice points besides $O$.
(A [i]lattice point[/i] is a point whose $x$-coordinate and $y$-coordinate are both integers. A [i]lattice triangle[/i] is a triangle whose vertices are lattice points.)
2009 Tournament Of Towns, 1
Each of $10$ identical jars contains some milk, up to $10$ percent of its capacity. At any time, we can tell the precise amount of milk in each jar. In a move, we may pour out an exact amount of milk from one jar into each of the other $9$ jars, the same amount in each case. Prove that we can have the same amount of milk in each jar after at most $10$ moves.
[i](4 points)[/i]
2001 All-Russian Olympiad Regional Round, 10.6
Given triangle $ABC$. Point $B_1$ is marked on line $AC$ so that $AB = AB_1$, while $B_1$ and $C$ are on the same side of $A$. Through points $C$, $B_1$ and the foot of the bisector of angle $A$ of triangle $ABC$, a circle $\omega$ is drawn, intersecting for second time the circle circumscribed around triangle $ABC$, at point $Q$. Prove that the tangent drawn to $\omega$ at point $Q$ is parallel to $AC$.
2002 Belarusian National Olympiad, 5
Prove that there exist infinitely many positive integers which cannot be presented in the form $x_1^3+x_2^5+x_3^7+x_4^9+x_5^{11}$ where $x_1,x_2,x_3,x_4,x_5$ are positive integers.
(V. Bernik)
MathLinks Contest 6th, 4.1
Let $F$ be a family of n subsets of a set $K$ with $5$ elements, such that any two subsets in $F$ have a common element. Find the minimal value of $n$ such that no matter how we choose $F$ with the properties above, there exists exactly one element of $K$ which belongs to all the sets in $F$.
2011 Dutch BxMO TST, 4
Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$.
Let $b$ be the greatest positive integer such that $2^b \le n$. Prove that $a \le b + 3$.
2016 AMC 12/AHSME, 25
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n \ge 2$. What is the smallest positive integer $k$ such that the product $a_1a_2 \cdots a_k$ is an integer?
$\textbf{(A)}\ 17 \qquad
\textbf{(B)}\ 18 \qquad
\textbf{(C)}\ 19 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 21$
2017 Purple Comet Problems, 17
Let $a_0$, $a_1$, ..., $a_6$ be real numbers such that $a_0 + a_1 + ... + a_6 = 1$ and
$$a_0 + a_2 + a_3 + a_4 + a_5 + a_6 =\frac{1}{2}$$
$$a_0 + a_1 + a_3 + a_4 + a_5 + a_6 = \frac{2}{3}$$
$$a_0 + a_1 + a_2 + a_4 + a_5 + a_6 =\frac{7}{8}$$
$$a_0 + a_1 + a_2 + a_3 + a_5 + a_6 =\frac{29}{30}$$
$$a_0 + a_1 + a_2 + a_3 + a_4 + a_6 =\frac{143}{144}$$
$$a_0 + a_1 + a_2 + a_3 + a_4 + a_5 =\frac{839}{840}$$
The value of $a_0$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2006 Austrian-Polish Competition, 4
A positive integer $d$ is called [i]nice[/i] iff for all positive integers $x,y$ hold: $d$ divides $(x+y)^{5}-x^{5}-y^{5}$ iff $d$ divides $(x+y)^{7}-x^{7}-y^{7}$ .
a) Is 29 nice?
b) Is 2006 nice?
c) Prove that infinitely many nice numbers exist.
1991 Greece National Olympiad, 3
Prove that exists triangle that can be partitions in $2050$ congruent triangles.
2005 Germany Team Selection Test, 2
Let n be a positive integer, and let $a_1$, $a_2$, ..., $a_n$, $b_1$, $b_2$, ..., $b_n$ be positive real numbers such that $a_1\geq a_2\geq ...\geq a_n$ and $b_1\geq a_1$, $b_1b_2\geq a_1a_2$, $b_1b_2b_3\geq a_1a_2a_3$, ..., $b_1b_2...b_n\geq a_1a_2...a_n$.
Prove that $b_1+b_2+...+b_n\geq a_1+a_2+...+a_n$.
2011 Morocco National Olympiad, 4
Two circles $C_{1}$ and $C_{2}$ intersect in $A$ and $B$. A line passing through $B$ intersects $C_{1}$ in $C$ and $C_{2}$ in $D$. Another line passing through $B$ intersects $C_{1}$ in $E$ and $C_{2}$ in $F$, $(CF)$ intersects $C_{1}$ and $C_{2}$ in $P$ and $Q$ respectively. Make sure that in your diagram, $B, E, C, A, P \in C_{1}$ and $B, D, F, A, Q \in C_{2}$ in this order. Let $M$ and $N$ be the middles of the arcs $BP$ and $BQ$ respectively. Prove that if $CD=EF$, then the points $C,F,M,N$ are cocylic in this order.
1981 Romania Team Selection Tests, 2.
Consider a tetrahedron $OABC$ with $ABC$ equilateral. Let $S$ be the area of the triangle of sides $OA$, $OB$ and $OC$. Show that $V\leqslant \dfrac12 RS$ where $R$ is the circumradius and $V$ is the volume of the tetrahedron.
[i]Stere Ianuș[/i]
2019 Tournament Of Towns, 3
An integer $1$ is written on the blackboard. We are allowed to perform the following operations:to change the number $x$ to $3x+1$ of to $[\frac{x}{2}]$. Prove that we can get all positive integers using this operations.
2018 Dutch IMO TST, 4
Let $A$ be a set of functions $f : R\to R$.
For all $f_1, f_2 \in A$ there exists a $f_3 \in A$ such that $f_1(f_2(y) - x)+ 2x = f_3(x + y)$ for all $x, y \in R$.
Prove that for all $f \in A$, we have $f(x - f(x))= 0$ for all $x \in R$.
2014 Saudi Arabia BMO TST, 1
A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.
1983 IMO Longlists, 73
Let $ABC$ be a nonequilateral triangle. Prove that there exist two points $P$ and $Q$ in the plane of the triangle, one in the interior and one in the exterior of the circumcircle of $ABC$, such that the orthogonal projections of any of these two points on the sides of the triangle are vertices of an equilateral triangle.
2013 Saudi Arabia BMO TST, 2
Define Fibonacci sequence $\{F\}_{n=0}^{\infty}$ as $F_0 = 0, F_1 = 1$ and $F_{n+1} = F_n +F_{n-1}$ for every integer $n > 1$. Determine all quadruples $(a, b, c,n)$ of positive integers with a $< b < c$ such that each of $a, b,c,a + n, b + n,c + 2n$ is a term of the Fibonacci sequence.
V Soros Olympiad 1998 - 99 (Russia), 11.3
Find the area of the figure on the coordinate plane bounded by the straight lines $x = 0$, $x = 2$ and the graphs of the functions $y =\sqrt{x^3+ 1}$ and $y = - \sqrt[3]{x^2+ 2x}$.
1987 China Team Selection Test, 1
a.) For all positive integer $k$ find the smallest positive integer $f(k)$ such that $5$ sets $s_1,s_2, \ldots , s_5$ exist satisfying:
[b]i.[/b] each has $k$ elements;
[b]ii.[/b] $s_i$ and $s_{i+1}$ are disjoint for $i=1,2,...,5$ ($s_6=s_1$)
[b]iii.[/b] the union of the $5$ sets has exactly $f(k)$ elements.
b.) Generalisation: Consider $n \geq 3$ sets instead of $5$.
2001 AMC 12/AHSME, 13
The parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ and vertex $ (h,k)$ is reflected about the line $ y \equal{} k$. This results in the parabola with equation $ y \equal{} dx^2 \plus{} ex \plus{} f$. Which of the following equals $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f$?
$ \textbf{(A)} \ 2b \qquad \textbf{(B)} \ 2c \qquad \textbf{(C)} \ 2a \plus{} 2b \qquad \textbf{(D)} \ 2h \qquad \textbf{(E)} \ 2k$
1941 Putnam, A3
A circle of radius $a$ rolls in the plane along the $x$-axis. Show that the envelope of a diameter is a cycloid, coinciding with the cycloid traced out by a point on the circumference of a circle of diameter $a$, likewise rolling in the plane along the $x$-axis.
2016 Purple Comet Problems, 11
Find the number of three-digit positive integers which have three distinct digits where the sum of the digits is an even number such as 925 and 824.
2021 Harvard-MIT Mathematics Tournament., 5
Let $n$ be the product of the first $10$ primes, and let
$$S=\sum_{xy\mid n} \varphi(x) \cdot y,$$
where $\varphi(x)$ denotes the number of positive integers less than or equal to $x$ that are relatively prime to $x$, and the sum is taken over ordered pairs $(x, y)$ of positive integers for which $xy$ divides $n$. Compute $\tfrac{S}{n}.$