Found problems: 85335
2012-2013 SDML (High School), 2
If five boys and three girls are randomly divided into two four-person teams, what is the probability that all three girls will end up on the same team?
$\text{(A) }\frac{1}{7}\qquad\text{(B) }\frac{2}{7}\qquad\text{(C) }\frac{1}{10}\qquad\text{(D) }\frac{1}{14}\qquad\text{(E) }\frac{1}{28}$
1980 Poland - Second Round, 3
There is a sphere $ K $ in space and points $ A, B $ outside the sphere such that the segment $ AB $ intersects the interior of the sphere. Prove that the set of points $ P $ for which the segments $ AP $ and $ BP $ are tangent to the sphere $ K $ is contained in a certain plane.
2013 Mexico National Olympiad, 2
Let $ABCD$ be a parallelogram with the angle at $A$ obtuse. Let $P$ be a point on segment $BD$. The circle with center $P$ passing through $A$ cuts line $AD$ at $A$ and $Y$ and cuts line $AB$ at $A$ and $X$. Line $AP$ intersects $BC$ at $Q$ and $CD$ at $R$. Prove $\angle XPY = \angle XQY + \angle XRY$.
1989 IMO Longlists, 48
A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle, i.e. both the incircle and circumcircle exists. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals.
2017 Romania National Olympiad, 2
Let be two natural numbers $ n\ge 2, k, $ and $ k\quad n\times n $ symmetric real matrices $ A_1,A_2,\ldots ,A_k. $ Then, the following relations are equivalent:
$ 1)\quad \left| \sum_{i=1}^k A_i^2 \right| =0 $
$ 2)\quad \left| \sum_{i=1}^k A_iB_i \right| =0,\quad\forall B_1,B_2,\ldots ,B_k\in \mathcal{M}_n\left( \mathbb{R} \right) $
$ || $ [i]denotes the determinant.[/i]
2020 CMIMC Algebra & Number Theory, 4
For all real numbers $x$, let $P(x)=16x^3 - 21x$. What is the sum of all possible values of $\tan^2\theta$, given that $\theta$ is an angle satisfying \[P(\sin\theta) = P(\cos\theta)?\]
2023 Indonesia MO, 4
Determine whether or not there exists a natural number $N$ which satisfies the following three criteria:
1. $N$ is divisible by $2^{2023}$, but not by $2^{2024}$,
2. $N$ only has three different digits, and none of them are zero,
3. Exactly 99.9% of the digits of $N$ are odd.
Bangladesh Mathematical Olympiad 2020 Final, #8
Let $ABC$ be a triangle where$\angle$[b]B=55[/b] and $\angle$ [b]C = 65[/b]. [b]D[/b] is the mid-point of [b]BC[/b]. Circumcircle of [b]ACD[/b] and[b] ABD[/b] cuts [b]AB[/b] and[b] AC[/b] at point [b]F[/b] and [b]E[/b] respectively. Center of circumcircle of [b]AEF[/b] is[b] O[/b]. $\angle$[b]FDO[/b] = ?
2005 Colombia Team Selection Test, 6
$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win?
[i]Proposed by A. Slinko & S. Marshall, New Zealand[/i]
2000 Harvard-MIT Mathematics Tournament, 9
How many hexagons are in the figure below with vertices on the given vertices?
(Note that a hexagon need not be convex, and edges may cross!)
[img]https://cdn.artofproblemsolving.com/attachments/1/9/437add8a9225760e7059b8dc2d481d562a7da2.png[/img]
2023 Austrian MO Regional Competition, 4
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$
holds.
[i](Walther Janous)[/i]
2016 SDMO (High School), 2
Let $a$, $b$, $c$, $d$ be four integers. Prove that $$\left(b-a\right)\left(c-a\right)\left(d-a\right)\left(d-c\right)\left(d-b\right)\left(c-b\right)$$ is divisible by $12$.
1939 Moscow Mathematical Olympiad, 053
What is the greatest number of parts that $5$ spheres can divide the space into?
1984 Czech And Slovak Olympiad IIIA, 1
A cube $A_1A_2A_3A_4A_5A_6A_7A_8$ is given in space. We will mark its center with the letter $S$ (intersection of solid diagonals). Find all natural numbers $k$ for which there exists a plane not containing the point $S$ and intersecting just $k$ of the rays $SA_1, SA_2, .. SA_8$
2025 Abelkonkurransen Finale, 3b
An acute angled triangle \(ABC\) has circumcenter \(O\). The lines \(AO\) and \(BC\) intersect at \(D\), while \(BO\) and \(AC\) intersect at \(E\) and \(CO\) and \(AB\) intersect at \(F\). Show that if the triangles \(ABC\) and \(DEF\) are similar(with vertices in that order), than \(ABC\) is equilateral.
2022 Kyiv City MO Round 2, Problem 3
In triangle $ABC$ the median $BM$ is equal to half of the side $BC$. Show that $\angle ABM = \angle BCA + \angle BAC$.
[i](Proposed by Anton Trygub)[/i]
2007 Ukraine Team Selection Test, 10
Find all positive integers $ n$ such that acute-angled $ \triangle ABC$ with $ \angle BAC<\frac{\pi}{4}$ could be divided into $ n$ quadrilateral. Every quadrilateral is inscribed in circle and radiuses of circles are in geometric progression.
[hide] be carefull ! :lol: [/hide]
1946 Putnam, A4
Let $g(x)$ be a function that has a continuous first derivative $g'(x)$. Suppose that $g(0)=0$ and $|g'(x)| \leq |g(x)|$ for all values of $x.$ Prove that $g(x)$ vanishes identically.
2022 Harvard-MIT Mathematics Tournament, 9
Let $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le3$, let $D_i $, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$.
Suppose that the areas of the triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, $D_1D_2D_3$, $E_1E_2E_3$, and $F_1F_2F_3$ are $2$, $3$, $4$, $20$, $21$, and $2020$, respectively. Compute the largest possible area of $G_1G_2G_3$.
2011 Akdeniz University MO, 5
For all $n \in {\mathbb Z^+}$ we define
$$I_n=\{\frac{0}{n},\frac{1}{n},\frac{2}{n},\dotsm,\frac{n-1}{n},\frac{n}{n},\frac{n+1}{n},\dotsm\}$$
infinite cluster. For whichever $x$ and $y$ real number, we say $\mid{x-y}\mid$ is between distance of the $x$ and $y$.
[b]a[/b]) For all $n$'s we find a number in $I_n$ such that, the between distance of the this number and $\sqrt 2$ is less than $\frac{1}{2n}$
[b]b[/b]) We find a $n$ such that, between distance of the a number in $I_n$ and $\sqrt 2$ is less than $\frac{1}{2011n}$
1982 IMO Longlists, 48
Given a finite sequence of complex numbers $c_1, c_2, \ldots , c_n$, show that there exists an integer $k$ ($1 \leq k \leq n$) such that for every finite sequence $a_1, a_2, \ldots, a_n$ of real numbers with $1 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$, the following inequality holds:
\[\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.\]
2017 IFYM, Sozopol, 8
The points with integer coordinates in a plane are painted in two colors – blue and red. Prove that there exist an infinite monochromatic subset that is symmetrical on some point.
2006 Bosnia and Herzegovina Team Selection Test, 6
Let $a_1$, $a_2$,...,$a_n$ be constant real numbers and $x$ be variable real number $x$. Let $f(x)=cos(a_1+x)+\frac{cos(a_2+x)}{2}+\frac{cos(a_3+x)}{2^2}+...+\frac{cos(a_n+x)}{2^{n-1}}$. If $f(x_1)=f(x_2)=0$, prove that $x_1-x_2=m\pi$, where $m$ is integer.
2022 Dutch Mathematical Olympiad, 1
A positive integer n is called [i]primary divisor [/i] if for every positive divisor $d$ of $n$ at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is divisor primary, because its positive divisors $1$, $2$, $4$, and $8$ each differ by $1$ from a prime number ($2$, $3$, $5$, and $7$, respectively), while $9$ is not divisor primary, because the divisor $9$ does not differ by $1$ from a prime number (both $8$ and $10$ are composite). Determine the largest primary divisor number.
2007 Indonesia MO, 3
Let $ a,b,c$ be positive real numbers which satisfy $ 5(a^2\plus{}b^2\plus{}c^2)<6(ab\plus{}bc\plus{}ca)$. Prove that these three inequalities hold: $ a\plus{}b>c$, $ b\plus{}c>a$, $ c\plus{}a>b$.