Olympiad problems

2016 ASDAN Math Tournament, 4

Tags:

The radius $r$ of a circle is increasing at a rate of $2$ meters per minute. Find the rate of change, in $\text{meters}^2/\text{minute}$, of the area when $r$ is $6$ meters.

2020 Brazil National Olympiad, 6

Tags: combinatorics

Let $k$ be a positive integer. Arnaldo and Bernaldo play a game in a table $2020\times 2020$, initially all the cells are empty. In each round a player chooses a empty cell and put one red token or one blue token, Arnaldo wins if in some moment, there are $k$ consecutive cells in the same row or column with tokens of same color, if all the cells have a token and there aren't $k$ consecutive cells(row or column) with same color, then Bernaldo wins. If the players play alternately and Arnaldo goes first, determine for which values of $k$, Arnaldo has the winning strategy.

1990 India Regional Mathematical Olympiad, 4

Tags: modular arithmetic , number theory , algebra , binomial theorem

Find the remainder when $2^{1990}$ is divided by $1990.$

Russian TST 2019, P3

Tags: number theory

Prove that there are infinitely many positive integers $m$ such that the number of odd distinct prime factor of $m(m+3)$ is a multiple of $3$.

1984 Putnam, B4

Tags: function , integration , calculus

Find, with proof, all real-valued functions $y=g(x)$ defined and continuous on $[0,\infty)$, positive on $(0,\infty)$, such that for all $x>0$ the $y$-coordinate of the centroid of the region $$R_x=\{(s,t)\mid0\le s\le x,\enspace0\le t\le g(s)\}$$is the same as the average value of $g$ on $[0,x]$.

2014 Contests, 3

Tags: geometry , 3d geometry , tetrahedron , sphere

A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.

KoMaL A Problems 2018/2019, A. 750

Tags: circles , geometry

Let $k_1,k_2,\ldots,k_5$ be five circles in the lane such that $k_1$ and $k_2$ are externally tangent to each other at point $T,$ $k_3$ and $k_4$ are exetrnally tangent to both $k_1$ and $k_2,$ $k_5$ is externally tangent to $k_3$ and $k_4$ at points $U$ and $V,$ respectively, and $k_5$ intersects $k_1$ at $P$ and $Q,$ like shown in the figure. Prove that \[\frac{PU}{QU}\cdot\frac{PV}{QV}=\frac{PT^2}{QT^2}.\]

1994 AMC 8, 2

Tags:

$\dfrac{1}{10}+\dfrac{2}{10}+\dfrac{3}{10}+\dfrac{4}{10}+\dfrac{5}{10}+\dfrac{6}{10}+\dfrac{7}{10}+\dfrac{8}{10}+\dfrac{9}{10}+\dfrac{55}{10}=$ $\text{(A)}\ 4\dfrac{1}{2} \qquad \text{(B)}\ 6.4 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11$

2021 Yasinsky Geometry Olympiad, 3

Tags: geometry , concyclic

Prove that in triangle $ABC$, the foot of the altitude $AH$, the point of tangency of the inscribed circle with side $BC$ and projections of point $A$ on the bisectors $\angle B$ and $\angle C$ of the triangle lie on one circle. (Dmitry Prokopenko)

2016 Chile TST IMO, 4

Tags: algebra , polynomial

Let $ f $ and $ g $ be two nonzero polynomials with integer coefficients such that $ \deg(f) > \deg(g) $. Suppose that for infinitely many prime numbers $ p $, the polynomial $ pf + g $ has a rational root. Prove that $ f $ has a rational root. Clarification: A rational root of a polynomial $ f $ is a number $ q \in \mathbb{Q} $ such that $ f(q) = 0 $.