Olympiad problems

2013 China Girls Math Olympiad, 1

Tags: parabola , function , algebra , domain , inequalities , conic , geometry

Let $A$ be the closed region bounded by the following three lines in the $xy$ plane: $x=1, y=0$ and $y=t(2x-t)$, where $0<t<1$. Prove that the area of any triangle inside the region $A$, with two vertices $P(t,t^2)$ and $Q(1,0)$, does not exceed $\frac{1}{4}.$

2006 IberoAmerican Olympiad For University Students, 5

Tags: geometry

A regular $n$-gon is inscribed in a circle of radius $1$. Let $a_1,\cdots,a_{n-1}$ be the distances of one of the vertices of the polygon to all the other vertices. Prove that \[(5-a_1^2)\cdots(5-a_{n-1}^2)=F_n^2\] where $F_n$ is the $n^{th}$ term of the Fibonacci sequence $1,1,2,\cdots$

2016 Vietnam Team Selection Test, 1

Tags: number theory

Find all $a,n\in\mathbb{Z}^+$ ($a>2$) such that each prime divisor of $a^n-1$ is also prime divisor of $a^{3^{2016}}-1$

2017 Princeton University Math Competition, A2/B4

Tags: combinatorics

Call a number unremarkable if, when written in base $10$, no two adjacent digits are equal. For example, $123$ is unremarkable, but $122$ is not. Find the sum of all unremarkable $3$-digit numbers. (Note that $012$ and $007$ are not $3$-digit numbers.)

LMT Team Rounds 2021+, A1

Tags:

Triangle $LMT$ has $\overline{MA}$ as an altitude. Given that $MA = 16$, $MT = 20$, and $LT = 25$, find the length of the altitude from $L$ to $\overline{MT}$. [i]Proposed by Kevin Zhao[/i]

2007 Mathematics for Its Sake, 2

Tags: equation , system of equations , algebra

For a given natural number $ n\ge 2, $ find all $ \text{n-tuples} $ of nonnegative real numbers which have the property that each one of the numbers forming the $ \text{n-tuple} $ is the square of the sum of the other $ n-1 $ ones. [i]Mugur Acu[/i]

2019 Switzerland Team Selection Test, 12

Tags: number theory

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

2016 Federal Competition For Advanced Students, P1, 3

Tags: number theory

Consider 2016 points arranged on a circle. We are allowed to jump ahead by 2 or 3 points in clockwise direction. What is the minimum number of jumps required to visit all points and return to the starting point? (Gerd Baron)

1998 Poland - Second Round, 2

Tags: geometry

In triangle $ABC$, the angle $\angle BCA$ is obtuse and $\angle BAC = 2\angle ABC\,.$ The line through $B$ and perpendicular to $BC$ intersects line $AC$ in $D$. Let $M$ be the midpoint of $AB$. Prove that $\angle AMC=\angle BMD$. source : http://cage.ugent.be/~hvernaev/Olympiade/PMO982.pdf

2013 Romania National Olympiad, 3

Tags: geometry , rectangle , perpendicularity

Let $ABCD$ be a rectangle with $5AD <2 AB$ . On the side $AB$ consider the points $S$ and $T$ such that $AS = ST = TB$. Let $M, N$ and $P$ be the projections of points $A, S$ and $T$ on lines $DS, DT$ and $DB$ respectively .Prove that the points $M, N$, and $P$ are collinear if and only if $15 AD^2 = 2 AB^2$.

Indonesia MO Shortlist - geometry, g5

Tags: geometry , equal angles , cyclic quadrilateral

Let $ABCD$ be quadrilateral inscribed in a circle. Let $M$ be the midpoint of the segment $BD$. If the tangents of the circle at $ B$, and at $D$ are also concurrent with the extension of $AC$, prove that $\angle AMD = \angle CMD$.

V Soros Olympiad 1998 - 99 (Russia), 10.6

Tags: angle , geometry

The straight line containing the centers of the circumscribed and inscribed circles of triangle $ABC$ intersects rays $BA$ and $BC$ and forms an angle with the altitude to side $BC$ equal to half the angle $\angle BAC$. What is angle $\angle ABC$?

2015 JBMO TST - Turkey, 7

Tags: inequalities , number theory

For the all $(m,n,k)$ positive integer triples such that $|m^k-n!| \le n$ find the maximum value of $\frac{n}{m}$ [i]Proposed by Melih Üçer[/i]

1996 India Regional Mathematical Olympiad, 7

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If $A$ is a fifty element subset of the set $1,2,\ldots 100$ such that no two numbers from $A$ add up to $100$, show that $A$ contains a square.

1999 AMC 8, 10

Tags: probability

A complete cycle of a traffic light takes 60 seconds. During each cycle the light is green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. At a randomly chosen time, what is the probability that the light will NOT be green? $ \text{(A)}\ \frac{1}{4}\qquad\text{(B)}\ \frac{1}{3}\qquad\text{(C)}\ \frac{5}{12}\qquad\text{(D)}\ \frac{1}{2}\qquad\text{(E)}\ \frac{7}{12} $

2023 All-Russian Olympiad, 8

Tags: combinatorics

Petya has $10, 000$ balls, among them there are no two balls of equal weight. He also has a device, which works as follows: if he puts exactly $10$ balls on it, it will report the sum of the weights of some two of them (but he doesn't know which ones). Prove that Petya can use the device a few times so that after a while he will be able to choose one of the balls and accurately tell its weight.

2025 China Team Selection Test, 16

Tags: geometry

In convex quadrilateral $ABCD, AB \perp AD, AD = DC$. Let $ E$ be a point on side $BC$, and $F$ be a point on the extension of $DE$ such that $\angle ABF = \angle DEC>90^{\circ}$. Let $O$ be the circumcenter of $\triangle CDE$, and $P$ be a point on the side extension of $FO$ satisfying $FB =FP$. Line BP intersects AC at point Q. Prove that $\angle AQB =\angle DPF.$

2008 HMNT, 3

Tags: geometry

Let $DEF$ be a triangle and H the foot of the altitude from $D$ to $EF$. If $DE = 60$, $DF = 35$, and $DH = 21$, what is the difference between the minimum and the maximum possible values for the area of $DEF$?

2023 ISL, A6

Tags: algebra , functional equation

For each integer $k\geq 2$, determine all infinite sequences of positive integers $a_1$, $a_2$, $\ldots$ for which there exists a polynomial $P$ of the form \[ P(x)=x^k+c_{k-1}x^{k-1}+\dots + c_1 x+c_0, \] where $c_0$, $c_1$, \dots, $c_{k-1}$ are non-negative integers, such that \[ P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k} \] for every integer $n\geq 1$.

1992 IMTS, 2

Tags:

Let $n \geq 3$ and $k \geq 2$ be integers, and form the forward differences of the members of the sequence $1,n,n^2,...n^{k-1}$ and successive forward differences thereof, as illustrated on the right for case $(n,k) = (3,5)$. Prove that all entries of the resulting triangles of positive integers are distinct from one another. Diagram: http://www.cms.math.ca/Competitions/IMTS/imts5.html

2016 Saint Petersburg Mathematical Olympiad, 1

Tags: trinomial , quadratic trinomial , common real root , algebra

Given three quadratic trinomials $f, g, h$ without roots. Their elder coefficients are the same, and all their coefficients for x are different. Prove that there is a number $c$ such that the equations $f (x) + cg (x) = 0$ and $f (x) + ch (x) = 0$ have a common root.

2010 Junior Balkan Team Selection Tests - Moldova, 6

Tags: geometry , algebra , geometric inequalities , inequalities

In a right triangle with the length legs $b$ and $c$, and the length hypotenuse $a$, the ratio between the length of the hypotenuse and the length of the diameter of the inscribed circle does not exceed $1 + \sqrt2$. Determine the numerical value of the expression of $E =\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}$.

1985 IMO Shortlist, 5

Tags: function , geometry , algebra , convexity , convex function

Let $D$ be the interior of the circle $C$ and let $A \in C$. Show that the function $f : D \to \mathbb R, f(M)=\frac{|MA|}{|MM'|}$ where $M' = AM \cap C$, is strictly convex; i.e., $f(P) <\frac{f(M_1)+f(M_2)}{2}, \forall M_1,M_2 \in D, M_1 \neq M_2$ where $P$ is the midpoint of the segment $M_1M_2.$

1988 IMO Shortlist, 26

Tags: algebra , functional equation , binary representation , function

A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$ for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$

2011 Putnam, A3

Tags: limit , integration , trigonometry , calculus , ratio , inequalities

Find a real number $c$ and a positive number $L$ for which \[\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.\]