Olympiad problems

2005 Iran MO (3rd Round), 1

Tags: function , trigonometry , geometry , 3d geometry , sphere , analytic geometry , topology

We call the set $A\in \mathbb R^n$ CN if and only if for every continuous $f:A\to A$ there exists some $x\in A$ such that $f(x)=x$. a) Example: We know that $A = \{ x\in\mathbb R^n | |x|\leq 1 \}$ is CN. b) The circle is not CN. Which one of these sets are CN? 1) $A=\{x\in\mathbb R^3| |x|=1\}$ 2) The cross $\{(x,y)\in\mathbb R^2|xy=0,\ |x|+|y|\leq1\}$ 3) Graph of the function $f:[0,1]\to \mathbb R$ defined by \[f(x)=\sin\frac 1x\ \mbox{if}\ x\neq0,\ f(0)=0\]

2019 Balkan MO Shortlist, A5

Tags: inequality , algebra

Let $a,b,c$ be positive real numbers, such that $(ab)^2 + (bc)^2 + (ca)^2 = 3$. Prove that \[ (a^2 - a + 1)(b^2 - b + 1)(c^2 - c + 1) \ge 1. \] [i]Proposed by Florin Stanescu (wer), România[/i]

2010 Brazil Team Selection Test, 1

Tags: perpendicular , geometry

Let $ABC$ be an acute triangle and $D$ a point on the side $AB$. The circumcircle of triangle $BCD$ cuts the side $AC$ again at $E$ .The circumcircle of triangle $ACD$ cuts the side $BC$ again at $F$. If $O$ is the circumcenter of the triangle $CEF$. Prove that $OD$ is perpendicular to $AB$.

2022 MMATHS, 7

Tags: algebra

Katherine makes Benj play a game called $50$ Cent. Benj starts with $\$0.50$, and every century thereafter has a $50\%$ chance of doubling his money and a $50\%$ chance of having his money reset to $\$0.50$. What is the expected value of the amount of money Benj will have, in dollars, after $50$ centuries?

2016 Macedonia JBMO TST, 5

Tags: number theory , greatest common divisor

Solve the following equation in the set of positive integers $x + y^2 + (GCD(x, y))^2 = xy \cdot GCD(x, y)$.

2018 MIG, 8

Tags:

A marathon runner has a very peculiar way of training for a marathon. On the first day of week $1$, the runner runs a distance equivalent to the first prime number. On the second day, the runner runs a distance equal to the second prime number, continuing this pattern until the $7$th day of the week. Each successive week, the runner runs one more mile per day than they did on the same day of the previous week. The runner continues this process until the average distance run each week exceeds the distance of a marathon ($26.2$ miles). How many weeks does the marathoner train?

2001 All-Russian Olympiad, 1

Tags: induction , combinatorics

The total mass of $100$ given weights with positive masses equals $2S$. A natural number $k$ is called [i]middle[/i] if some $k$ of the given weights have the total mass $S$. Find the maximum possible number of middle numbers.

2014 Balkan MO Shortlist, G6

Tags: geometry

In $\triangle ABC$ with $AB=AC$,$M$ is the midpoint of $BC$,$H$ is the projection of $M$ onto $AB$ and $D$ is arbitrary point on the side $AC$.Let $E$ be the intersection point of the parallel line through $B$ to $HD$ with the parallel line through $C$ to $AB$.Prove that $DM$ is the bisector of $\angle ADE$.

2020/2021 Tournament of Towns, P2

Tags: geometry , algebra , polynomial , vieta

Baron Munchausen presented a new theorem: if a polynomial $x^{n} - ax^{n-1} + bx^{n-2}+ \dots$ has $n$ positive integer roots then there exist $a$ lines in the plane such that they have exactly $b$ intersection points. Is the baron’s theorem true?

1987 Bulgaria National Olympiad, Problem 3

Tags: geometry , 3d geometry , pyramid

Let $MABCD$ be a pyramid with the square $ABCD$ as the base, in which $MA=MD$, $MA^2+AB^2=MB^2$ and the area of $\triangle ADM$ is equal to $1$. Determine the radius of the largest ball that is contained in the given pyramid.

2017 ASDAN Math Tournament, 1

Tags:

Two arbitrary distinct lattice points are selected on the coordinate plane within the square marked by the points $(0,0)$, $(3,0)$, $(0,3)$, and $(3,3)$ (the lattice points may lie on a side or a corner of the square). What is the probability that the distance between the two points is at most $\sqrt{2}$?

2004 Vietnam Team Selection Test, 3

Tags: number theory

Let $S$ be the set of positive integers in which the greatest and smallest elements are relatively prime. For natural $n$, let $S_n$ denote the set of natural numbers which can be represented as sum of at most $n$ elements (not necessarily different) from $S$. Let $a$ be greatest element from $S$. Prove that there are positive integer $k$ and integers $b$ such that $|S_n| = a \cdot n + b$ for all $ n > k $.

2021 Mediterranean Mathematics Olympiad, 2

Tags: number theory , diophantine equation , diophantine

For every sequence $p_1<p_2<\cdots<p_8$ of eight prime numbers, determine the largest integer $N$ for which the following equation has no solution in positive integers $x_1,\ldots,x_8$: $$p_1\, p_2\, \cdots\, p_8 \left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right) ~~=~~ N $$ [i]Proposed by Gerhard Woeginger, Austria[/i]

2014 IFYM, Sozopol, 7

Tags: geometry , equilateral triangle , symmedian

If $AG_a,BG_b$, and $CG_c$ are symmedians in $\Delta ABC$ ($G_a\in BC,G_b\in AC,G_c\in AB$), is it possible for $\Delta G_a G_b G_c$ to be equilateral when $\Delta ABC$ is not equilateral?

2017 NIMO Problems, 2

Tags:

Find the smallest positive integer $N$ for which $N$ is divisible by $19$, and when the digits of $N$ are read in reverse order, the result (after removing any leading zeroes) is divisible by $36$. [i]Proposed by Michael Tang[/i]

2010 China Northern MO, 5

Tags:

Let $a,b,c$ be positive real numbers such that $(a+2b)(b+2c)=9$. Prove that\[\sqrt{\frac{a^2+b^2}{2}}+2\sqrt[3]{\frac{b^3+c^3}{2}}\geq 3.\]

2012 Grigore Moisil Intercounty, 1

Tags: inequalities , geometry

For $ x\in\mathbb{R} , $ determine the minimum of $ \sqrt{(x-1)^2+\left( x^2-5\right)^2} +\sqrt{(x+2)^2+\left( x^2+1 \right)^2} $ and the maximum of $ \sqrt{(x-1)^2+\left( x^2-5\right)^2} -\sqrt{(x+2)^2+\left( x^2+1 \right)^2} . $ [i]Vasile Pop[/i]

Geometry Mathley 2011-12, 15.3

Tags: geometry , circumcircle , symmedian , perpendicular bisector

Triangle $ABC$ has circumcircle $(O,R)$, and orthocenter $H$. The symmedians through $A,B,C$ meet the perpendicular bisectors of $BC,CA,AB$ at $D,E, F$ respectively. Let $M,N, P$ be the perpendicular projections of H on the line $OD,OE,OF.$ Prove that $$\frac{OH^2}{R^2} =\frac{\overline{OM}}{\overline{OD}}+\frac{\overline{ON}}{\overline{OE}} +\frac{\overline{OP}}{\overline{OF}}$$ Đỗ Thanh Sơn

2016 AMC 10, 17

Tags: probability

Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$? $\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) }16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20$

1990 Tournament Of Towns, (277) 2

Tags: equilateral , geometry

A point $M$ is chosen on the arc $AC$ of the circumcircle of the equilateral triangle $ABC$. $P$ is the midpoint of this arc, $N$ is the midpoint of the chord $BM$ and $K$ is the foot of the perpendicular drawn from $P$ to $MC$. Prove that the triangle $ANK$ is equilateral. (I Nagel, Yevpatoria)

2014 NZMOC Camp Selection Problems, 4

Tags: combinatorial geometry , combinatorics , point

Given $2014$ points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three connected points?

1969 Miklós Schweitzer, 2

Tags: superior algebra

Let $ p\geq 7$ be a prime number, $ \zeta$ a primitive $ p$th root of unity, $ c$ a rational number. Prove that in the additive group generated by the numbers $ 1,\zeta,\zeta^2,\zeta^3\plus{}\zeta^{\minus{}3}$ there are only finitely many elements whose norm is equal to $ c$. (The norm is in the $ p$th cyclotomic field.) [i]K. Gyory[/i]

2024 AMC 12/AHSME, 14

Tags: remainder

How many different remainders can result when the $100$th power of an integer is divided by $125$? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }5 \qquad \textbf{(D) }25 \qquad \textbf{(E) }125 \qquad $

1953 AMC 12/AHSME, 13

Tags: trapezoid , geometry

A triangle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is $ 18$ inches, the median of the trapezoid is: $ \textbf{(A)}\ 36\text{ inches} \qquad\textbf{(B)}\ 9\text{ inches} \qquad\textbf{(C)}\ 18\text{ inches}\\ \textbf{(D)}\ \text{not obtainable from these data} \qquad\textbf{(E)}\ \text{none of these}$

2021 AMC 12/AHSME Spring, 10

Tags: let a and b be the two removed

Two distinct numbers are selected from the set $\{1,2,3,4,\dots,36,37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers? $\textbf{(A) }5 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8\qquad \textbf{(D) }9 \qquad \textbf{(E) }10$