Olympiad problems

1995 China Team Selection Test, 3

Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.) Does the same result hold with a degree 3 or degree 5 polynomial?

1988 Greece National Olympiad, 3

Tags: geometry , geometric inequality , circles

Two circles $(O_1,R_1)$,$(O_2,R_2)$ lie each external to the other. Find : a) the minimum length of the segment connecting points of the circles b) the max length of the segment connecting points of the circles

2021 AMC 12/AHSME Fall, 24

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Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is $CF$? $\textbf{(A) } 28 \qquad \textbf{(B) } 20\sqrt{2} \qquad \textbf{(C) } 30 \qquad \textbf{(D) } 32 \qquad \textbf{(E) } 20\sqrt{3}$

2006 IMC, 6

Tags: linear algebra , matrix , algebra , polynomial

The scores of this problem were: one time 17/20 (by the runner-up) one time 4/20 (by Andrei Negut) one time 1/20 (by the winner) the rest had zero... just to give an idea of the difficulty. Let $A_{i},B_{i},S_{i}$ ($i=1,2,3$) be invertible real $2\times 2$ matrices such that [list][*]not all $A_{i}$ have a common real eigenvector, [*]$A_{i}=S_{i}^{-1}B_{i}S_{i}$ for $i=1,2,3$, [*]$A_{1}A_{2}A_{3}=B_{1}B_{2}B_{3}=I$.[/list] Prove that there is an invertible $2\times 2$ matrix $S$ such that $A_{i}=S^{-1}B_{i}S$ for all $i=1,2,3$.

2002 Romania Team Selection Test, 2

Tags: geometry

Find the least positive real number $r$ with the following property: Whatever four disks are considered, each with centre on the edges of a unit square and the sum of their radii equals $r$, there exists an equilateral triangle which has its edges in three of the disks. [i]Radu Gologan[/i]

2002 Moldova National Olympiad, 2

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Five parcels of land are given. In each step, we divide one parcel into three or four smaller ones. Assume that, after several steps, the number of obtained parcels equals four times the number of steps made. How many steps were performed?

VMEO III 2006 Shortlist, G5

Tags: geometry , circles

Prove that there exists a family of rational circles with a distinct radius $\{(O_n)\}$ $(n = 1,2,3,...)$ satisfying the property that for all natural indices $n$, circles $(O_n)$,$( O_{n+1})$, $(O_{n+2})$,$(O_{n+3})$ are externally tangent like in the figure. [img]https://cdn.artofproblemsolving.com/attachments/b/f/5655e677e7c4f203b63afe82c50088e9ef97f5.png[/img]

Brazil L2 Finals (OBM) - geometry, 2023.2

Tags: orthocenter , circumcenter , isogonal lines , cyclic quadrilateral , midpoint , similar triangles , geometry

Consider a triangle $ABC$ with $AB < AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. A line starting from $B$ cuts the lines $AO$ and $AH$ at $M$ and $M'$ so that $M'$ is the midpoint of $BM$. Another line starting from $C$ cuts the lines $AH$ and $AO$ at $N$ and $N'$ so that $N'$ is the midpoint of $CN$. Prove that $M, M', N, N'$ are on the same circle.

2018 District Olympiad, 3

Tags: geometry

Let $AD$, $BE$, $CF$ be the heights of triangle $ABC$ and let $K$, $L$, $M$ be the orthocenters of triangles $AEF$, $BFD$ and $CDE$, respectively. Let $G_1$ and $G_2$ denote the centroids of triangles $DEF$ and $KLM$, respectively. Show that $HG_1 = G_1G_2$, where $H$ is the orthocenter of triangle $ABC$.

2023 Puerto Rico Team Selection Test, 7

Tags: combinatorics

$2023$ wise men are located in a circle. Each of them thinks either that the earth is the center of the universe, or that it is not. Once a minute, all the wise men express their opinion at the same time. Every wise man who is between two wise men with an opinion different from his will change his mind at that moment. The others don't change their minds. The others don't change their minds. Determine the smallest necessary time for all the wise men to have the same opinion, without regardless of initial opinions or your location.

1983 IMO Longlists, 13

Tags: modular arithmetic , number theory

Let $p$ be a prime number and $a_1, a_2, \ldots, a_{(p+1)/2}$ different natural numbers less than or equal to $p.$ Prove that for each natural number $r$ less than or equal to $p$, there exist two numbers (perhaps equal) $a_i$ and $a_j$ such that \[p \equiv a_i a_j \pmod r.\]

2016 Canadian Mathematical Olympiad Qualification, 4

Tags: algebra , functional equation , function

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x + f(y)) + f(x - f(y)) = x.$$

2024 Dutch BxMO/EGMO TST, IMO TSTST, 2

Tags: algebra

We define a sequence with $a_1=850$ and $$a_{n+1}=\frac{a_n^2}{a_n-1}$$ for $n\geq 1$. Find all values of $n$ for which $\lfloor a_n\rfloor =2024$.

MBMT Team Rounds, 2015 F8 E5

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Victor has $3$ piles of $3$ cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. (For example, for his first card, Victor must draw the top card from one of the $3$ piles.) In how many orders can Victor draw the cards?

2019 USMCA, 4

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Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that for all $x, y \in \mathbb R$, $$f(f(x) + y)^2 = (x-y)(f(x) - f(y)) + 4f(x) f(y).$$

2012 Today's Calculation Of Integral, 785

Tags: calculus , integration , derivative , calculus computations , logarithm

For a positive real number $x$, find the minimum value of $f(x)=\int_x^{2x} (t\ln t-t)dt.$

2000 Iran MO (2nd round), 1

Tags: combinatorics

$21$ distinct numbers are chosen from the set $\{1,2,3,\ldots,2046\}.$ Prove that we can choose three distinct numbers $a,b,c$ among those $21$ numbers such that \[bc<2a^2<4bc\]

2002 BAMO, 3

Tags: game theory , combinatorics

A game is played with two players and an initial stack of $n$ pennies $(n \geq 3)$. The players take turns choosing one of the stacks of pennies on the table and splitting it into two stacks. The winner is the player who makes a move that causes all stacks to be of height $1$ or $2.$ For which starting values of n does the player who goes first win, assuming best play by both players?

1994 Czech And Slovak Olympiad IIIA, 6

Tags: inequalities , algebra

Show that from any four distinct numbers lying in the interval $(0,1)$ one can choose two distinct numbers $a$ and $b$ such that $$\sqrt{(1-a^2)(1-b^2)} > \frac{a}{2b}+\frac{b}{2a}-ab-\frac{1}{8ab} $$

2024 Australian Mathematical Olympiad, P6

Tags: combinatorics

In a school, there are $1000$ students in each year level, from Year $1$ to Year $12$. The school has $12 000$ lockers, numbered from $1$ to $12 000$. The school principal requests that each student is assigned their own locker, so that the following condition is satisfied: For every pair of students in the same year level, the difference between their locker numbers must be divisible by their year-level number. Can the principal’s request be satisfied?

2007 IMS, 7

Tags: linear algebra , search , number theory

$x_{1},x_{2},\dots,x_{n}$ are real number such that for each $i$, the set $\{x_{1},x_{2},\dots,x_{n}\}\backslash \{x_{i}\}$ could be partitioned into two sets that sum of elements of first set is equal to the sum of the elements of the other. Prove that all of $x_{i}$'s are zero. [hide="Hint"]It is a number theory problem.[/hide]

PEN F Problems, 13

Tags: rational number , factorial

Prove that numbers of the form \[\frac{a_{1}}{1!}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,\] where $0 \le a_{i}\le i-1 \;(i=2, 3, 4, \cdots)$ are rational if and only if starting from some $i$ on all the $a_{i}$'s are either equal to $0$ ( in which case the sum is finite) or all are equal to $i-1$.

2020 Princeton University Math Competition, B3

Tags: geometry , geometric inequality , tangent circle

Let $ABC$ be a triangle and let the points $D, E$ be on the rays $AB$, $AC$ such that $BCED$ is cyclic. Prove that the following two statements are equivalent: $\bullet$ There is a point $X$ on the circumcircle of $ABC$ such that $BDX$, $CEX$ are tangent to each other. $\bullet$ $AB \cdot AD \le 4R^2$, where $R$ is the radius of the circumcircle of $ABC$.

1978 Austrian-Polish Competition, 3

Tags: trigonometry , inequalities

Prove that $$\sqrt[44]{\tan 1^\circ\cdot \tan 2^\circ\cdot \dots\cdot \tan 44^\circ}<\sqrt 2-1<\frac{\tan 1^\circ+ \tan 2^\circ+\dots+\tan 44^\circ}{44}.$$

1965 Spain Mathematical Olympiad, 7

Tags: density , mass , geometry

A truncated cone has the bigger base of radius $r$ centimetres and the generatrix makes an angle, with that base, whose tangent equals $m$. The truncated cone is constructed of a material of density $d$ (g/cm$^3$) and the smaller base is covered by a special material of density $p$ (g/cm$^2$). Which is the height of the truncated cone that maximizes the total mass?