Olympiad problems

2017 F = ma, 1

Tags: fraction

A motorcycle rides on the vertical walls around the perimeter of a large circular room. The friction coefficient between the motorcycle tires and the walls is $\mu$. How does the minimum $\mu$ needed to prevent the motorcycle from slipping downwards change with the motorcycle's speed, $s$? $\textbf{(A)}\mu \propto s^{0} \qquad \textbf{(B)}\mu \propto s^{-\frac{1}{2}}\qquad \textbf{(C)}\mu \propto s^{-1}\qquad \textbf{(D)}\mu \propto s^{-2}\qquad \textbf{(E)}\text{none of these}$

2022 Thailand TST, 2

Tags: combinatorics

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

2003 Federal Competition For Advanced Students, Part 2, 1

Tags: number theory

Prove that, for any integer $g > 2$, there is a unique three-digit number $\overline{abc}_g$ in base $g$ whose representation in some base $h = g \pm 1$ is $\overline{cba}_h$.

2013 AMC 12/AHSME, 2

Tags: algebra

A softball team played ten games, scoring $1,2,3,4,5,6,7,8,9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? $ \textbf {(A) } 35 \qquad \textbf {(B) } 40 \qquad \textbf {(C) } 45 \qquad \textbf {(D) } 50 \qquad \textbf {(E) } 55 $

2020 Turkey Team Selection Test, 3

Tags: combinatorics

66 dwarfs have a total of 111 hats. Each of the hats belongs to a dwarf and colored by 66 different colors. Festivities are organized where each of these dwarfs wears their own hat. There is no dwarf pair wearing the same colored hat in any of the festivities. For any two of the festivities, there exist a dwarf wearing a hat of a different color in these festivities. Find the maximum value of the number of festivities that can be organized.

2018 Hong Kong TST, 3

Tags: algebra , functional equation , function

Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $$f(f(xy-x))+f(x+y)=yf(x)+f(y)$$ for all real numbers $x$ and $y$.

2017-2018 SDML (Middle School), 15

Tags:

For all positive integers $n$ the function $f$ satisfies $f(1) = 1, f(2n + 1) = 2f(n),$ and $f(2n) = 3f(n) + 2$. For how many positive integers $x \leq 100$ is the value of $f(x)$ odd? $\mathrm{(A) \ } 4 \qquad \mathrm{(B) \ } 5 \qquad \mathrm {(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ } 10$

I Soros Olympiad 1994-95 (Rus + Ukr), 10.3

Tags: tiles , combinatorics

Given a square board with dimensions $1 995 \times 1 995$. These cells are painted with black and white paints in checkerboard order like this. that the corner cells are black. Two black and one white cells were randomly cut out of the board. Prove that the rest of the board can be divided into rectangles of size $1 \times 2$ .

2008 USAMO, 6

Tags: abstract algebra , vector , group theory , induction

At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).

2020 USMCA, 9

Tags:

Let $\Omega$ be a unit circle and $A$ be a point on $\Omega$. An angle $0 < \theta < 180^\circ$ is chosen uniformly at random, and $\Omega$ is rotated $\theta$ degrees clockwise about $A$. What is the expected area swept by this rotation?

1952 Moscow Mathematical Olympiad, 224-

Tags: locus , segment , geometry

You are given a segment $AB$. Find the locus of the vertices $C$ of acute-angled triangles $ABC$.

2015 JBMO TST - Turkey, 4

Tags: inequality

Prove that $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \ge \dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+2(a+b+c)$$ for the all $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2+2abc \le 1$.

2013 USAMO, 6

Tags: circumcircle , ratio , asymptote , trigonometry , reflection , geometry , geometric transformation

Let $ABC$ be a triangle. Find all points $P$ on segment $BC$ satisfying the following property: If $X$ and $Y$ are the intersections of line $PA$ with the common external tangent lines of the circumcircles of triangles $PAB$ and $PAC$, then \[\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.\]

2008 Kazakhstan National Olympiad, 2

Tags: vector , power of a point , angle bisector , geometry , radical axis , incenter , circumcircle

Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.

2021 Cyprus JBMO TST, 4

Tags: ramsey theory , pigeonhole principle , combinatorics

We colour every square of a $4\times 19$ chess board with one of the colours red, green and blue. Prove that however this colouring is done, we can always find two horizontal rows and two vertical columns such that the $4$ squares on the intersections of these lines all have the same colour.

2019 Silk Road, 1

Tags: angle chasing , altitude , geometry , equal angles

The altitudes of the acute-angled non-isosceles triangle $ ABC $ intersect at the point $ H $. On the segment $ C_1H $, where $ CC_1 $ is the altitude of the triangle, the point $ K $ is marked. Points $ L $ and $ M $ are the feet of perpendiculars from point $ K $ on straight lines $ AC $ and $ BC $, respectively. The lines $ AM $ and $ BL $ intersect at $ N $. Prove that $ \angle ANK = \angle HNL $.

1997 May Olympiad, 5

Tags: geometry , hexagon , area

What are the possible areas of a hexagon with all angles equal and sides $1, 2, 3, 4, 5$, and $6$, in some order?

1988 China Team Selection Test, 3

Tags: geometry , incenter , circumcircle

In triangle $ABC$, $\angle C = 30^{\circ}$, $O$ and $I$ are the circumcenter and incenter respectively, Points $D \in AC$ and $E \in BC$, such that $AD = BE = AB$. Prove that $OI = DE$ and $OI \bot DE$.

1982 AMC 12/AHSME, 5

Tags: ratio

Two positive numbers $x$ and $y$ are in the ratio $a: b$ where $0 < a < b$. If $x+y = c$, then the smaller of $x$ and $y$ is $\textbf{(A)} \ \frac{ac}{b} \qquad \textbf{(B)} \ \frac{bc-ac}{b} \qquad \textbf{(C)} \ \frac{ac}{a+b} \qquad \textbf{(D)} \ \frac{bc}{a+b} \qquad \textbf{(E)} \ \frac{ac}{b-a}$

I Soros Olympiad 1994-95 (Rus + Ukr), 11.7

Tags: system of equations , trigonometry , algebra

Solve the system of equations $$\begin{cases} \sin^3 x+\sin^4 y=1 \\ \cos^4 x+\cos^5 y =1\end{cases}$$

2002 Greece JBMO TST, 3

Tags: perpendicular , angle bisector , angle , geometry

Let $ABC$ be a triangle with $\angle A=60^o, AB\ne AC$ and let $AD$ be the angle bisector of $\angle A$. Line $(e)$ that is perpendicular on the angle bisector $AD$ at point $A$, intersects the extension of side $BC$ at point $E$ and also $BE=AB+AC$. Find the angles $\angle B$ and $\angle C$ of the triangle $ABC$.

2017 IFYM, Sozopol, 7

Tags: geometry

The inscribed circle $\omega$ of an equilateral $\Delta ABC$ is tangent to its sides $AB$,$BC$ and $CA$ in points $D$,$E$, and $F$, respectively. Point $H$ is the foot of the altitude from $D$ to $EF$. Let $AH\cap BC=X,BH\cap CA=Y$. It is known that $XY\cap AB=T$. Let $D$ be the center of the circumscribed circle of $\Delta BYX$. Prove that $OH\perp CT$.

2003 AMC 10, 1

Tags:

Which of the following is the same as \[ \frac{2\minus{}4\plus{}6\minus{}8\plus{}10\minus{}12\plus{}14}{3\minus{}6\plus{}9\minus{}12\plus{}15\minus{}18\plus{}21}? \]$ \textbf{(A)}\ \minus{}1 \qquad \textbf{(B)}\ \minus{}\frac23 \qquad \textbf{(C)}\ \frac23 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac{14}{3}$

Estonia Open Senior - geometry, 2017.1.5

Tags: geometry , geometric inequality

On the sides $BC, CA$ and $AB$ of triangle $ABC$, respectively, points $D, E$ and $F$ are chosen. Prove that $\frac12 (BC + CA + AB)<AD + BE + CF<\frac 32 (BC + CA + AB)$.

2016 Singapore Junior Math Olympiad, 1

Tags: perfect square , number theory

Find all integers$ n$ such that $n^2 + 24n + 35$ is a square.