Found problems: 85335
2022 MMATHS, 4
How many ways are there to choose three digits $A,B,C$ with $1 \le A \le 9$ and $0 \le B,C \le 9$ such that $\overline{ABC}_b$ is even for all choices of base $b$ with $b \ge 10$?
1995 Tournament Of Towns, (449) 5
Four equal right-angled triangles are given. We are allowed to cut any triangle into two new ones along the altitude dropped on to the hypotenuse. This operation may be repeated with any of the triangles from the new set. Prove that after any number of such operations there will be congruent triangles among those obtained.
(AV Shapovalov)
2015 Sharygin Geometry Olympiad, 4
A fixed triangle $ABC$ is given. Point $P$ moves on its circumcircle so that segments $BC$ and $AP$ intersect. Line $AP$ divides triangle $BPC$ into two triangles with incenters $I_1$ and $I_2$. Line $I_1I_2$ meets $BC$ at point $Z$. Prove that all lines $ZP$ pass through a fixed point.
(R. Krutovsky, A. Yakubov)
1986 ITAMO, 7
On a long enough highway, a passenger in a bus observes the traffic. He notes that, during an hour, the bus going with a constant velocity overpasses $a$ cars and gets overpassed by $b$ cars, while $c$ cars pass in the opposite direction. Assuming that the traffic is the same in both directions, is it possible to determine the number of cars that pass along the highway per hour? (You may assume that the velocity of a car can take only two values.)
2013 Balkan MO, 3
Let $S$ be the set of positive real numbers. Find all functions $f\colon S^3 \to S$ such that, for all positive real numbers $x$, $y$, $z$ and $k$, the following three conditions are satisfied:
(a) $xf(x,y,z) = zf(z,y,x)$,
(b) $f(x, ky, k^2z) = kf(x,y,z)$,
(c) $f(1, k, k+1) = k+1$.
([i]United Kingdom[/i])
2002 Iran Team Selection Test, 3
A "[i]2-line[/i]" is the area between two parallel lines. Length of "2-line" is distance of two parallel lines. We have covered unit circle with some "2-lines". Prove sum of lengths of "2-lines" is at least 2.
2011 Math Prize for Girls Olympiad, 2
Let $\triangle ABC$ be an equilateral triangle. If $0 < r < 1$, let $D_r$ be the point on $\overline{AB}$ such that $AD_r = r \cdot AB$, let $E_r$ be the point on $\overline{BC}$ such that $BE_r = r \cdot BC$, and let $P_r$ be the point where $\overline{AE_r}$ and $\overline{CD_r}$ intersect. Prove that the set of points $P_r$ (over all $0 < r < 1$) lie on a circle.
2010 Hong kong National Olympiad, 2
Let $n$ be a positive integer. Find the number of sequences $x_{1},x_{2},\ldots x_{2n-1},x_{2n}$, where $x_{i}\in\{-1,1\}$ for each $i$, satisfying the following condition: for any integer $k$ and $m$ such that $1\le k\le m\le n$ then the following inequality holds \[\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2\]
2018 Irish Math Olympiad, 9
The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$ for $n \ge 1$.
Prove that there exists at most one $n$ for which $a_n$ is the cube of an integer.
2023 AMC 10, 16
In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
$\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66$
1986 AIME Problems, 1
What is the sum of the solutions to the equation $\sqrt[4]x =\displaystyle \frac{12}{7-\sqrt[4]x}$?
2016 Auckland Mathematical Olympiad, 1
It is known that in a set of five coins three are genuine (and have the same weight) while two coins are fakes, each of which has a different weight from a genuine coin. What is the smallest number of weighings on a scale with two cups that is needed to locate one genuine coin?
2010 Indonesia TST, 4
How many natural numbers $(a,b,n)$ with $ gcd(a,b)=1$ and $ n>1 $ such that the equation \[ x^{an} +y^{bn} = 2^{2010} \] has natural numbers solution $ (x,y) $
1970 IMO, 3
Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.
1995 National High School Mathematics League, 2
Complex numbers of apexes of 20-regular polygon inscribed to unit circle refer to are $Z_1,Z_2,\cdots,Z_{20}$ on complex plane. Then the number of points in $Z_1^{1995},Z_2^{1995},\cdots,Z_{20}^{1995}$ refer to is
$\text{(A)}4\qquad\text{(B)}5\qquad\text{(C)}10\qquad\text{(D)}20$
1955 Moscow Mathematical Olympiad, 316
Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.
1965 Swedish Mathematical Competition, 5
Let $S$ be the set of all real polynomials $f(x) = ax^3 + bx^2 + cx + d$ such that $|f(x)| \le 1$ for all $ -1 \le x \le 1$. Show that the set of possible $|a|$ for $f$ in $S$ is bounded above and find the smallest upper bound.
2020-2021 OMMC, 4
In 3-dimensional space, two spheres centered at points $O_1$ and $O_2$ with radii $13$ and $20$ respectively intersect in a circle. Points $A, B, C$ lie on that circle, and lines $O_1A$ and $O_1B$ intersect sphere $O_2$ at points $D$ and $E$ respectively. Given that $O_1O_2 = AC = BC = 21,$ $DE$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are positive integers. Find $a+b+c$.
2021 MOAA, 5
Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
Denmark (Mohr) - geometry, 2002.1
An interior point in a rectangle is connected by line segments to the midpoints of its four sides. Thus four domains (polygons) with the areas $a, b, c$ and $d$ appear (see the figure). Prove that $a + c = b + d$.
[img]https://1.bp.blogspot.com/-BipDNHELjJI/XzcCa68P3HI/AAAAAAAAMXY/H2Iqya9VItMLXrRqsdyxHLTXCAZ02nEtgCLcBGAsYHQ/s0/2002%2BMohr%2Bp1.png[/img]
2005 China Northern MO, 5
Let $x, y, z$ be positive real numbers such that $x^2 + xy + y^2 = \frac{25}{4}$, $y^2 + yz + z^2 = 36$, and $z^2 + zx + x^2 = \frac{169}{4}$. Find the value of $xy + yz + zx$.
Indonesia Regional MO OSP SMA - geometry, 2019.1
Given cube $ ABCD.EFGH $ with $ AB = 4 $ and $ P $ midpoint of the side $ EFGH $. If $ M $ is the midpoint of $ PH $, find the length of segment $ AM $.
2017 China Northern MO, 4
Positive intenger $n\geq3$. $a_1,a_2,\cdots,a_n$ are $n$ positive intengers that are pairwise coprime, satisfying that there exists $k_1,k_2,\cdots,k_n\in\{-1,1\}, \sum_{i=1}^{n}k_ia_i=0$. Are there positive intengers $b_1,b_2,\cdots,b_n$, for any $k\in\mathbb{Z}_+$, $b_1+ka_1,b_2+ka_2,\cdots,b_n+ka_n$ are pairwise coprime?
1955 Moscow Mathematical Olympiad, 288
We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.
1987 Traian Lălescu, 2.2
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} ,f(x)=\left\{\begin{matrix} \sin x , & x\not\in\mathbb{Q} \\ 0, & x\in\mathbb{Q}\end{matrix}\right. . $
[b]a)[/b] Determine the maximum length of an interval $ I\subset\mathbb{R} $ such that $ f|_I $ is discontinuous everywhere, yet has the intermediate value property.
[b]b)[/b] Study the convergence of the sequence $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ defined by $ x_0\in (0,\pi /2),x_{n+1}=f\left( x_n\right),\forall n\ge 0. $