Olympiad problems

2017 Taiwan TST Round 3, 3

Tags: geometry

Given a $ \triangle ABC $ with circumcircle $ \Gamma. $ Let $ A' $ be the antipode of $ A $ in $ \Gamma $ and $ D $ be the point s.t. $ \triangle BCD $ is an equilateral triangle ($ A $ and $ D $ are on the opposite side of $ BC $). Let the perpendicular from $ A' $ to $ A'D $ cuts $ CA, $ $ AB $ at $ E, $ $ F, $ resp. and $ T $ be the point s.t. $ \triangle ETF $ is an isosceles triangle with base $ EF $ and base angle $ 30^{\circ} $ ($ A $ and $ T $ are on the opposite side of $ EF $). Prove that $ AT $ passes through the 9-point center of $ \triangle ABC. $ [i]Proposed by Telv Cohl[/i]

2017 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra , inequality

a) Find : $A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$ b) Prove that for any $(a,b,c) \in A$ next inequality hold : \begin{align*} \frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8} \end{align*}

2017 Saudi Arabia IMO TST, 1

Tags: combinatorics

In the garden of Wonderland, there are $2016$ apples, $2017$ bananas and $2018$ oranges.Two monkeys Adu and Bakar play the following game: alternatively each of them takes and eats one fruit of any kind except for the one that he took in previous turn (in the first turn, each of them can take a fruit of any kind). Who can not take a fruit is the loser. Which monkey has the winning strategy if Adu plays first?

2011 Saudi Arabia Pre-TST, 4.2

Tags: number theory , factorial

Find positive integers $a_1 < a_2<... <a_{2010}$ such that $$a_1(1!)^{2010} + a_2(2!)^{2010} + ... + a_{2010}(2010!)^{2010} = (2011 !)^{2010}. $$

2012 AMC 12/AHSME, 9

Tags:

It takes Clea $60$ seconds to walk down an escalator when it is not operating and only $24$ seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it? $ \textbf{(A)}\ 36\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 52 $

1986 India National Olympiad, 1

Tags: algebra

A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged place, when did he go out ?

2024 Mongolian Mathematical Olympiad, 1

Tags: calculus , derivative , algebra , inequality , bruh

Let $P(x)$ and $Q(x)$ be polynomials with nonnegative coefficients. We denote by $P'(x)$ the derivative of $P(x)$. Suppose that $P(0)=Q(0)=0$ and $Q(1) \leq 1 \leq P'(0)$. $(1)$ Prove that $0 \leq Q(x) \leq x \leq P(x)$ for all $0 \leq x \leq 1$. $(2)$ Prove that $P(Q(x)) \leq Q(P(x))$ for all $0 \leq x \leq 1$. [i]Proposed by Otgonbayar Uuye.[/i]

2022 MIG, 6

Tags:

A coin is flipped three times. What is the probability that there are no instances of two consecutive heads or two consecutive tails? $\textbf{(A) }\frac{1}{8}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{3}{8}\qquad\textbf{(D) }\frac{5}{8}\qquad\textbf{(E) }\frac{3}{4}$

2020 Cono Sur Olympiad, 2

Tags: number theory , combinatorics , pigeonhole principle

Given $2021$ distinct positive integers non divisible by $2^{1010}$, show that it's always possible to choose $3$ of them $a$, $b$ and $c$, such that $|b^2-4ac|$ is not a perfect square.

2010 Germany Team Selection Test, 2

Tags: geometry , cyclic quadrilateral , projective geometry , tangent , power of a point

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

Today's calculation of integrals, 879

Tags: calculus , integration , trigonometry , calculus computations , integral

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$

1998 AMC 12/AHSME, 23

Tags:

The graphs of $ x^2 \plus{} y^2 \equal{} 4 \plus{} 12x \plus{} 6y$ and $ x^2 \plus{} y^2 \equal{} k \plus{} 4x \plus{} 12y$ intersect when $ k$ satisfies $ a \leq k \leq b$, and for no other values of $ k$. Find $ b \minus{} a$. $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 68\qquad \textbf{(C)}\ 104\qquad \textbf{(D)}\ 140\qquad \textbf{(E)}\ 144$

2013 Cuba MO, 3

Tags: combinatorics , winning strategy , game

Two players $A$ and $B$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $A$, $B$, $A$, $....$, $A$ starts the game and the one who takes out the last stone loses.$ B$ can serve on each play $1$, $2$ or 3 stones, while$ A$ can draw $2, 3, 4$ stones or $1$ stone in each turn f it is the last one in the pile. Determine for what values of $N$ does $A$ have a winning strategy, and for what values the winning strategy is $B$'s.

2001 USA Team Selection Test, 5

Tags: perpendicular bisector , geometry

In triangle $ABC$, $\angle B = 2\angle C$. Let $P$ and $Q$ be points on the perpendicular bisector of segment $BC$ such that rays $AP$ and $AQ$ trisect $\angle A$. Prove that $PQ < AB$ if and only if $\angle B$ is obtuse.

2021 MIG, 3

Tags:

$20\%$ of $10$ is exactly $5\%$ of what number? $\textbf{(A) }20\qquad\textbf{(B) }30\qquad\textbf{(C) }40\qquad\textbf{(D) }50\qquad\textbf{(E) }60$

2019 Purple Comet Problems, 30

Tags: combinatorics

A [i]derangement [/i] of the letters $ABCDEF$ is a permutation of these letters so that no letter ends up in the position it began such as $BDECFA$. An [i]inversion [/i] in a permutation is a pair of letters $xy$ where $x$ appears before $y$ in the original order of the letters, but $y$ appears before $x$ in the permutation. For example, the derangement $BDECFA$ has seven inversions: $AB, AC, AD, AE, AF, CD$, and $CE$. Find the total number of inversions that appear in all the derangements of $ABCDEF$.

PEN G Problems, 9

Tags: irrational number , trigonometry , algebra , polynomial

Show that $\cos \frac{\pi}{7}$ is irrational.

2014 BMT Spring, 12

Tags: number theory

A two-digit integer is [i]reversible [/i] if, when written backwards in base $10$, it has the same number of positive divisors. Find the number of reversible integers.

2022 Iran Team Selection Test, 8

Tags: tangent , radical axis , geometry

In triangle $ABC$, with $AB<AC$, $I$ is the incenter, $E$ is the intersection of $A$-excircle and $BC$. Point $F$ lies on the external angle bisector of $BAC$ such that $E$ and $F$ lieas on the same side of the line $AI$ and $\angle AIF=\angle AEB$. Point $Q$ lies on $BC$ such that $\angle AIQ=90$. Circle $\omega_b$ is tangent to $FQ$ and $AB$ at $B$, circle $\omega_c$ is tangent to $FQ$ and $AC$ at $C$ and both circles pass through the inside of triangle $ABC$. if $M$ is the Midpoint od the arc $BC$, which does not contain $A$, prove that $M$ lies on the radical axis of $\omega_b$ and $\omega_c$. Proposed by Amirmahdi Mohseni

2005 Colombia Team Selection Test, 2

Tags: graph theory , combinatorics , algorithm , game

The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\ge 4}$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from the complete graph on $n$ vertices (where each pair of vertices are joined by an edge). [i]Proposed by Norman Do, Australia[/i]

1998 National High School Mathematics League, 7

Tags: function

$f(x)$ is an even function with period of $2$. If $f(x)=x^{\frac{1}{1000}}$ when $x\in[0,1]$, then the order of $f\left(\frac{98}{19}\right),f\left(\frac{101}{17}\right),f\left(\frac{104}{15}\right)$ is________(from small to large).

2013 Oral Moscow Geometry Olympiad, 2

Tags: construction , geometry , compass , ruler

With a compass and a ruler, split a triangle into two smaller triangles with the same sum of squares of sides.

Gheorghe Țițeica 2024, P4

Tags: olympiad combinatorics

For a set $M$ of $n\geq 3$ points in the plane we define a [i]path[/i] to be a polyline $A_1A_2\dots A_n$, where $M=\{A_1,A_2,\dots ,A_n\}$ and define its length to be $A_1A_2+A_2A_3+\dots +A_{n-1}A_n$. We call $M$ [i]path-unique[/i] if any two distinct paths have different lengths and [i]segment-unique[/i] if any two nondegenerate segments with their ends among the points in $M$ have different lengths. Determine the positive integers $n\geq 3$ such that: a) any segment-unique set $M$ of $n$ points in the plane is path-unique; b) any path-unique set $M$ of $n$ points in the plane is segment-unique. [i]Cristi Săvescu[/i]

2007 AMC 8, 3

Tags:

What is the sum of the two smallest prime factors of $250$? $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$

2017 Morocco TST-, 6

Tags: number theory , polynomial , functional equation , sum of digits

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]