Found problems: 85335
1997 Canadian Open Math Challenge, 5
Two cubes have their faces painted either red or blue. The 1st cube has five red faces and one blue face. When the two cubes are rolled simultaneously, the probability that the two top faces show the same color is $\frac{1}{2}$. How many red faces are there on the second cube?
1995 Turkey Team Selection Test, 2
Let $n\in\mathbb{N}$ be given. Prove that the following two conditions are equivalent:
$\quad(\text{i})\: n|a^n-a$ for any positive integer $a$;
$\quad(\text{ii})\:$ For any prime divisor $p$ of $n$, $p^2 \nmid n$ and $p-1|n-1$.
2012 Olympic Revenge, 2
We define $(x_1, x_2, \ldots , x_n) \Delta (y_1, y_2, \ldots , y_n) = \left( \sum_{i=1}^{n}x_iy_{2-i}, \sum_{i=1}^{n}x_iy_{3-i}, \ldots , \sum_{i=1}^{n}x_iy_{n+1-i} \right)$, where the indices are taken modulo $n$.
Besides this, if $v$ is a vector, we define $v^k = v$, if $k=1$, or $v^k = v \Delta v^{k-1}$, otherwise.
Prove that, if $(x_1, x_2, \ldots , x_n)^k = (0, 0, \ldots , 0)$, for some natural number $k$, then $x_1 = x_2 = \ldots = x_n = 0$.
2021 USAJMO, 2
Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that \[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\] Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.
2002 District Olympiad, 2
a) Let $x$ be a real number such that $x^2+x$ and $x^3+2x$ are rational numbers. Show that $x$ is a rational number.
b) Show that there exist irrational numbers $x$ such that $x^2+x$and $x^3-2x$ are rational.
2013 Purple Comet Problems, 12
How many four-digit positive integers have no adjacent equal even digits? For example, count numbers such as $1164$ and $2035$ but not $6447$ or $5866$.
2018 Yasinsky Geometry Olympiad, 5
In the trapezium $ABCD$ ($AD // BC$), the point $M$ lies on the side of $CD$, with $CM:MD=2:3$, $AB=AD$, $BC:AD=1:3$. Prove that $BD \perp AM$.
2013 AMC 12/AHSME, 25
Let $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $?
${ \textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{(D}} \ 431 \qquad \textbf{(E)} \ 441 $
2011 IMO, 4
Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.
[i]Proposed by Morteza Saghafian, Iran[/i]
2008 Gheorghe Vranceanu, 3
If the circumradius of any three consecutive vertices of a convex polygon is at most $ 1, $ show that the discs of radius $ 1 $ centered at each vertex cover the polygon and its interior.
2006 Belarusian National Olympiad, 3
A finite set $V \in Z^2$ of vectors with integer coordinates is chosen on the plane. Each of them is painted one of the $n$ colors. The color is [i]suitable[/i] for the vector if this vector may be presented as' a linear combination (with integer coefficients) of the vectors from $V$ of this color. It is known,that for any vector from $Z^2$ there exist a suitable color. Find all $n$ such that there must exist a color which is suitable for any vector from $Z^2$ .
(V. Lebed)
2009 Korea Junior Math Olympiad, 5
Acute triangle $\triangle ABC$ satis es $AB < AC$. Let the circumcircle of this triangle be $O$, and the midpoint of $BC,CA,AB$ be $D,E,F$. Let $P$ be the intersection of the circle with $AB$ as its diameter and line $DF$, which is in the same side of $C$ with respect to $AB$. Let $Q$ be the intersection of the circle with $AC$ as its diameter and the line $DE$, which is in the same side of $B$ with respect to $AC$. Let $PQ \cap BC = R$, and let the line passing through $R$ and perpendicular to $BC$ meet $AO$ at $X$. Prove that $AX = XR$.
2012 Kosovo National Mathematical Olympiad, 3
Prove that for any integer $n\geq 2$ it holds that
$\dbinom {2n}{n}>\frac {4^n}{2n}$.
2016 Romania Team Selection Tests, 3
Given a positive integer $n$, show that for no set of integers modulo $n$, whose size exceeds $1+\sqrt{n+4}$, is it possible that the pairwise sums of unordered pairs be all distinct.
2022 Bulgaria EGMO TST, 6
Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold:
(a) the union of any two white subsets is white;
(b) the union of any two black subsets is black;
(c) there are exactly $N$ white subsets.
2003 Spain Mathematical Olympiad, Problem 2
Does there exist such a finite set of real numbers ${M}$ that has at least two distinct elements and has the property that for two numbers, ${a}$, ${b}$, belonging to ${M}$, the number ${2a - b^2}$ is also an element in ${M}$?
Swiss NMO - geometry, 2008.1
Let $ABC$ be a triangle with $\angle BAC \ne 45^o$ and $\angle ABC \ne 135^o$. Let $P$ be the point on the line $AB$ with $\angle CPB = 45^o$. Let $O_1$ and $O_2$ be the centers of the circumcircles of the triangles $ACP$ and $BCP$ respectively. Show that the area of the square $CO_1P O_2$ is equal to the area of the triangle $ABC$.
2024 Taiwan TST Round 1, N
Given a prime number $p$, a set is said to be $p$-good if the set contains exactly three elements $a, b, c$ and $a + b \equiv c \pmod{p}$.
Find all prime number $p$ such that $\{ 1, 2, \cdots, p-1 \}$ can be partitioned into several $p$-good sets.
[i]Proposed by capoouo[/i]
2010 Contests, 3
Let $ABCD$ be a convex quadrilateral. $AC$ and $BD$ meet at $P$, with $\angle APD=60^{\circ}$. Let $E,F,G$, and $H$ be the midpoints of $AB,BC,CD$ and $DA$ respectively. Find the greatest positive real number $k$ for which
\[EG+3HF\ge kd+(1-k)s \]
where $s$ is the semi-perimeter of the quadrilateral $ABCD$ and $d$ is the sum of the lengths of its diagonals. When does the equality hold?
2019 Bosnia and Herzegovina Junior BMO TST, 4
$4.$ Let there be a variable positive integer whose last two digits are $3's$. Prove that this number is divisible by a prime greater than $7$.
1981 Spain Mathematical Olympiad, 1
Calculate the sum of $n$ addends
$$7 + 77 + 777 +...+ 7... 7.$$
2012 JHMT, 8
A red unit cube $ABCDEF GH$ (with $E$ below $A$, $F$ below $B$, etc.) is pushed into the corner of a room with vertex $E$ not visible, so that faces $ABF E$ and $ADHE$ are adjacent to the wall and face $EF GH$ is adjacent to the floor. A string of length $2$ is dipped in black paint, and one of its endpoints is attached to vertex $A$. How much surface area on the three visible faces of the cube can be painted black by sweeping the string over it?
1999 Turkey Team Selection Test, 3
Determine all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the set
\[\left \{ \frac{f(x)}{x}: x \neq 0 \textnormal{ and } x \in \mathbb{R}\right \}\]
is finite, and for all $x \in \mathbb{R}$
\[f(x-1-f(x)) = f(x) - x - 1\]
2024 Malaysian IMO Training Camp, 8
Given a triangle $ABC$, let $I$ be the incenter, and $J$ be the $A$-excenter. A line $\ell$ through $A$ perpendicular to $BC$ intersect the lines $BI$, $CI$, $BJ$, $CJ$ at $P$, $Q$, $R$, $S$ respectively. Suppose the angle bisector of $\angle BAC$ meet $BC$ at $K$, and $L$ is a point such that $AL$ is a diameter in $(ABC)$.
Prove that the line $KL$, $\ell$, and the line through the centers of circles $(IPQ)$ and $(JRS)$, are concurrent.
[i]Proposed by Chuah Jia Herng & Ivan Chan Kai Chin[/i]
2012 Bundeswettbewerb Mathematik, 2
Are there positive integers $a$ and $b$ such that both $a^2 + 4b$ and $b^2 + 4a$ are perfect squares?