Found problems: 85335
2021 BMT, 9
Rakesh is flipping a fair coin repeatedly. If $T$ denotes the event where the coin lands on tails and $H$ denotes the event where the coin lands on heads, what is the probability Rakesh flips the sequence $HHH$ before the sequence $THH$?
2015 Baltic Way, 17
Find all positive integers $n$ for which $n^{n-1} - 1$ is divisible by $2^{2015}$, but not by $2^{2016}$.
1972 All Soviet Union Mathematical Olympiad, 163
The triangle table is constructed according to the rule: You put the natural number $a>1$ in the upper row, and then you write under the number $k$ from the left side $k^2$, and from the right side -- $(k+1)$. For example, if $a = 2$, you get the table on the picture. Prove that all the numbers on each particular line are different.
2
/ \
/ \
4 3
/ \ / \
16 5 9 4
/ \ / \ /\ / \
2021 South Africa National Olympiad, 5
Determine all polynomials $a(x)$, $b(x)$, $c(x)$, $d(x)$ with real coefficients satisfying the simultaneous equations
\begin{align*}
b(x) c(x) + a(x) d(x) & = 0 \\
a(x) c(x) + (1 - x^2) b(x) d(x) & = x + 1.
\end{align*}
2018 Bosnia and Herzegovina Team Selection Test, 4
Every square of $1000 \times 1000$ board is colored black or white. It is known that exists one square $10 \times 10$ such that all squares inside it are black and one square $10 \times 10$ such that all squares inside are white. For every square $K$ $10 \times 10$ we define its power $m(K)$ as an absolute value of difference between number of white and black squares $1 \times 1$ in square $K$. Let $T$ be a square $10 \times 10$ which has minimum power among all squares $10 \times 10$ in this board. Determine maximal possible value of $m(T)$
2023 All-Russian Olympiad, 5
Find the largest natural number $n$ for which the product of the numbers $n, n+1, n+2, \ldots, n+20$ is divisible by the square of one of them.
1992 Tournament Of Towns, (337) 5
$100$ silver coins ordered by weight and $101$ gold coins also ordered by weight are given. All coins have different weights. You are given a balance to compare weights of any two coins. How can you find the “middle” coin (that occupies the $101$-st place in weight among all $201$ coins) using the minimal number of weighings? Find this number and prove that a smaller number of weighings would be insufficient.
(A. Andjans, Riga)
2008 Indonesia TST, 2
Find all functions $f : R \to R$ that satisfies the condition $$f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy$$ for all real numbers $x, y$.
2016 NIMO Problems, 6
Let $ABCD$ be an isosceles trapezoid with $AD\parallel BC$ and $BC>AD$ such that the distance between the incenters of $\triangle ABC$ and $\triangle DBC$ is $16$. If the perimeters of $ABCD$ and $ABC$ are $120$ and $114$ respectively, then the area of $ABCD$ can be written as $m\sqrt n,$ where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $100m+n$.
[i]Proposed by David Altizio and Evan Chen[/i]
2021 Yasinsky Geometry Olympiad, 4
In triangle $ABC$, the point $H$ is the orthocenter. A circle centered at point $H$ and with radius $AH$ intersects the lines $AB$ and $AC$ at points $E$ and $D$, respectively. The point $X$ is the symmetric of the point $A$ with respect to the line $BC$ . Prove that $XH$ is the bisector of the angle $DXE$.
(Matthew of Kursk)
2020 SIME, 10
Consider all $2^{20}$ paths of length $20$ units on the coordinate plane starting from point $(0, 0)$ going only up or right, each one unit at a time. Each such path has a unique [i]bubble space[/i], which is the region of points on the coordinate plane at most one unit away from some point on the path. The average area enclosed by the bubble space of each path, over all $2^{20}$ paths, can be written as $\tfrac{m + n\pi}{p}$ where $m, n, p$ are positive integers and $\gcd(m, n, p) = 1$. Find $m + n + p$.
BIMO 2020, 1
Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$
2008 All-Russian Olympiad, 1
Numbers $ a,b,c$ are such that the equation $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots.Prove that if $ \minus{} 2\leq a \plus{} b \plus{} c\leq 0$,then at least one of these roots belongs to the segment $ [0,2]$
1964 AMC 12/AHSME, 34
If $n$ is a multiple of $4$, the sum $s=1+2i+3i^2+ ... +(n+1)i^{n}$, where $i=\sqrt{-1}$, equals:
$ \textbf{(A)}\ 1+i\qquad\textbf{(B)}\ \frac{1}{2}(n+2) \qquad\textbf{(C)}\ \frac{1}{2}(n+2-ni) \qquad$
$ \textbf{(D)}\ \frac{1}{2}[(n+1)(1-i)+2]\qquad\textbf{(E)}\ \frac{1}{8}(n^2+8-4ni) $
2017 NIMO Problems, 5
In triangle $ABC$, $AB=12$, $BC=17$, and $AC=25$. Distinct points $M$ and $N$ lie on the circumcircle of $ABC$ such that $BM=CM$ and $BN=CN$. If $AM + AN = \tfrac{a\sqrt{b}}{c}$, where $a, b, c$ are positive integers such that $\gcd(a, c) = 1$ and $b$ is not divisible by the square of a prime, compute $100a+10b+c$.
[i]Proposed by Michael Tang[/i]
2018-IMOC, A3
Find all functions $f:\mathbb R\to\mathbb R$ such that for reals $x,y$,
$$f(xf(y)+y)=yf(x)+f(y).$$
2013 Tuymaada Olympiad, 3
The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours).
Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected.
[i]V. Dolnikov[/i]
[b]EDIT.[/b] It is confirmed by the official solution that the graph is tacitly assumed to be [b]finite[/b].
2024 Junior Balkan Team Selection Tests - Moldova, 6
In the isosceles triangle $ABC$, with $AB=BC$, points $X$ and $Y$ are the midpoints of the sides $AB$ and $AC$, respectively. Point $Z$ is the foot of the perpendicular from $B$ to $CX$. Prove that the circumcenter of the triangle $XYZ$ is of the line $AC$.
2021 MMATHS, 2
Define the [i]digital reduction[/i] of a two-digit positive integer $\underline{AB}$ to be the quantity $\underline{AB} - A - B$. Find the greatest common divisor of the digital reductions of all the two-digit positive integers. (For example, the digital reduction of $62$ is $62 - 6 - 2 = 54.$)
[i]Proposed by Andrew Wu[/i]
2022 Kyiv City MO Round 2, Problem 2
Monica and Bogdan are playing a game, depending on given integers $n, k$. First, Monica writes some $k$ positive numbers. Bogdan wins, if he is able to find $n$ points on the plane with the following property: for any number $m$ written by Monica, there are some two points chosen by Bogdan with distance exactly $m$ between them. Otherwise, Monica wins.
Determine who has a winning strategy depending on $n, k$.
[i](Proposed by Fedir Yudin)[/i]
2023 Brazil Team Selection Test, 4
Find all positive integers $n$ with the following property: There are only a finite number of positive multiples of $n$ that have exactly $n$ positive divisors.
2020 China Northern MO, BP4
In $\triangle ABC$, $\angle BAC = 60^{\circ}$, point $D$ lies on side $BC$, $O_1$ and $O_2$ are the centers of the circumcircles of $\triangle ABD$ and $\triangle ACD$, respectively. Lines $BO_1$ and $CO_2$ intersect at point $P$. If $I$ is the incenter of $\triangle ABC$ and $H$ is the orthocenter of $\triangle PBC$, then prove that the four points $B,C,I,H$ are on the same circle.
2019 Ramnicean Hope, 3
Calculate $ \lfloor \log_3 5 +\log_5 7 +\log_7 3 \rfloor .$
[i]Petre Rău[/i]
LMT Guts Rounds, 20
Three vertices of a parallelogram are $(2,-4),(-2,8),$ and $(12,7.)$ Determine the sum of the three possible x-coordinates of the fourth vertex.
2014 Purple Comet Problems, 3
The diagram below shows a rectangle with side lengths $36$ and $48$. Each of the sides is trisected and edges are added between the trisection points as shown. Then the shaded corner regions are removed, leaving the octagon which is not shaded in the diagram. Find the perimeter of this octagon.
[asy]
size(4cm);
dotfactor=3.5;
pair A,B,C,D,E,F,G,H,W,X,Y,Z;
A=(0,12);
B=(0,24);
C=(16,36);
D=(32,36);
E=(48,24);
F=(48,12);
G=(32,0);
H=(16,0);
W=origin;
X=(0,36);
Y=(48,36);
Z=(48,0);
filldraw(W--A--H--cycle^^B--X--C--cycle^^D--Y--E--cycle^^F--Z--G--cycle,rgb(.76,.76,.76));
draw(W--X--Y--Z--cycle,linewidth(1.2));
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
[/asy]