Olympiad problems

1998 Romania National Olympiad, 4

Let $A_1A_2...A_n$ be a regular polygon ($n > 4$), $T$ be the common point of $A_1A_2$ and $A_{n-1}A_n$ and $M$ be a point in the interior of the triangle $A_1A_nT$. Show that the equality $$\sum_{i=1}^{n-1} \frac{\sin^2 \left(\angle A_iMA_{i+1}\right)}{d(M,A_iA_{i+1}}=\frac{\sin^2 \left(\angle A_1MA_n\right)}{d(M,A_1A_n} $$ holds if and only if $M$ belongs to the circumcircle of the polygon.

2010 Malaysia National Olympiad, 4

Tags: geometry

A semicircle has diameter $XY$. A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the diameter. Square $STUV$ has $T$ on $RS$, $U$ on the semicircle, and $V$ on $XY$. What is the area of $STUV$?

2022 EGMO, 3

Tags:

An infinite sequence of positive integers $a_1, a_2, \dots$ is called $good$ if (1) $a_1$ is a perfect square, and (2) for any integer $n \ge 2$, $a_n$ is the smallest positive integer such that $$na_1 + (n-1)a_2 + \dots + 2a_{n-1} + a_n$$ is a perfect square. Prove that for any good sequence $a_1, a_2, \dots$, there exists a positive integer $k$ such that $a_n=a_k$ for all integers $n \ge k$. [size=75](reposting because the other thread didn't get moved)[/size]

2017 South Africa National Olympiad, 4

Tags: combinatorial game , combinatorics

Andile and Zandre play a game on a $2017 \times 2017$ board. At the beginning, Andile declares some of the squares [i]forbidden[/i], meaning the nothing may be placed on such a square. After that, they take turns to place coins on the board, with Zandre placing the first coin. It is not allowed to place a coin on a forbidden square or in the same row or column where another coin has already been placed. The player who places the last coin wins the game. What is the least number of squares Andile needs to declare as forbidden at the beginning to ensure a win? (Assume that both players use an optimal strategy.)

2024 Junior Balkan Team Selection Tests - Romania, P1

Tags: algebra , permutation

Let $n\geqslant 3$ be an integer and $a_1,a_2,\ldots,a_n$ be pairwise distinct positive real numbers with the property that there exists a permutation $b_1,b_2,\ldots,b_n$ of these numbers such that\[\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots=\frac{a_{n-1}}{b_{n-1}}\neq 1.\]Prove that there exist $a,b>0$ such that $\{a_1,a_2,\ldots,a_n\}=\{ab,ab^2,\ldots,ab^n\}.$ [i]Cristi Săvescu[/i]

2018 India National Olympiad, 4

Tags: polynomial , algebra , divisibility , roots of a polynomial

Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.

2020 Purple Comet Problems, 16

Tags: combinatorics

Find the number of permutations of the letters $ABCDE$ where the letters $A$ and $B$ are not adjacent and the letters $C$ and $D$ are not adjacent. For example, count the permutations $ACBDE$ and $DEBCA$ but not $ABCED$ or $EDCBA$.

2011 China Second Round Olympiad, 9

Tags: logarithm , algebra

Let $f(x)=|\log(x+1)|$ and let $a,b$ be two real numbers ($a<b$) satisfying the equations $f(a)=f\left(-\frac{b+1}{a+1}\right)$ and $f\left(10a+6b+21\right)=4\log 2$. Find $a,b$.

2000 IMO, 6

Tags: geometry , homothety , circumcircle , triangle , incircle

Let $ AH_1, BH_2, CH_3$ be the altitudes of an acute angled triangle $ ABC$. Its incircle touches the sides $ BC, AC$ and $ AB$ at $ T_1, T_2$ and $ T_3$ respectively. Consider the symmetric images of the lines $ H_1H_2, H_2H_3$ and $ H_3H_1$ with respect to the lines $ T_1T_2, T_2T_3$ and $ T_3T_1$. Prove that these images form a triangle whose vertices lie on the incircle of $ ABC$.

1976 Spain Mathematical Olympiad, 1

Tags: construction , square , geometry

In a plane there are four fixed points $A, B, C, D$, no $3$ collinear. Construct a square with sides $a, b, c, d$ such that $A \in a$, $B \in b$, $C \in c$, $D \in d$.

2011 AIME Problems, 4

Tags: ratio , geometry , analytic geometry , graphing lines , slope

In triangle $ABC$, $AB=\frac{20}{11} AC$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

PEN H Problems, 86

Tags: diophantine equation , geometry , function , euler , least common multiple

A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?

2012 Israel National Olympiad, 3

Tags: algebra , inequality , symmetric inequality , inequalities

Let $a,b,c$ be real numbers such that $a^3(b+c)+b^3(a+c)+c^3(a+b)=0$. Prove that $ab+bc+ca\leq0$.

1999 Moldova Team Selection Test, 16

Tags: function , algebra

Define functions $f,g: \mathbb{R}\to \mathbb{R}$, $g$ is injective, satisfy: \[f(g(x)+y)=g(f(y)+x)\]

2015 CIIM, Problem 1

Tags: calculus , integration

Find the real number $a$ such that the integral $$\int_a^{a+8}e^{-x}e^{-x^2}dx$$ attain its maximum.

1972 Czech and Slovak Olympiad III A, 5

Tags: combinatorics , subset

Determine how many unordered pairs $\{A,B\}$ is there such that $A,B\subseteq\{1,\ldots,n\}$ and $A\cap B=\emptyset.$

2015 ASDAN Math Tournament, 3

Tags: geometry test

Points $E$ and $F$ are chosen on sides $BC$ and $CD$ respectively of rhombus $ABCD$ such that $AB=AE=AF=EF$, and $FC,DF,BE,EC>0$. Compute the measure of $\angle ABC$.

Today's calculation of integrals, 887

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

For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows. Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx. (1) Find $f(\sqrt{3})$ (2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$ (3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.

2021 Iran Team Selection Test, 5

Tags: sequence , algebra , arithmetic sequence

Call a triple of numbers [b]Nice[/b] if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$ [b] Nice[/b] triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios Proposed by [i]Morteza Saghafian[/i]

2013 Saudi Arabia GMO TST, 2

Tags: algebra , inequalities

For positive real numbers $a, b$ and $c$, prove that $$\frac{a^3}{a^2 + ab + b^2} +\frac{b^3}{b^2 + bc + c^2} +\frac{c^3}{ c^2 + ca + a^2} \ge\frac{ a + b + c}{3}$$

2015 IMO Shortlist, N8

Tags: inequalities , number theory , prime factorization , function

For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define \[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\] That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity. Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that \[\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.\] [i]Proposed by Rodrigo Sanches Angelo, Brazil[/i]

1987 Tournament Of Towns, (134) 3

Tags: algebra

We are given two three-litre bottles, one containing $1$ litre of water and the other containing $1$ litre of $2\%$ salt solution . One can pour liquids from one bottle to the other and then mix them to obtain solutions of different concentration . Can one obtain a $1 . 5\%$ solution of salt in the bottle which originally contained water? (S . Fomin, Leningrad),

2019 Czech-Austrian-Polish-Slovak Match, 3

Tags: combinatorics

A dissection of a convex polygon into finitely many triangles by segments is called a [i]trilateration[/i] if no three vertices of the created triangles lie on a single line (vertices of some triangles might lie inside the polygon). We say that a trilateration is [i]good[/i] if its segments can be replaced with one-way arrows in such a way that the arrows along every triangle of the trilateration form a cycle and the arrows along the whole convex polygon also form a cycle. Find all $n\ge 3$ such that the regular $n$-gon has a good trilateration.

2008 Alexandru Myller, 1

Tags: vector geometry , geometry , circumcircle , center of mass

$ O $ is the circumcentre of $ ABC $ and $ A_1\neq A $ is the point on $ AO $ and the circumcircle of $ ABC. $ The centers of mass of $ ABC, A_1BC $ are $ G,G_1, $ respectively, and $ P $ is the intersection of $ AG_1 $ with $ OG. $ Show that $ \frac{PG}{PO}=\frac{2}{3} . $ [i]Gabriel Popa, Paul Georgescu[/i]

2016 Abels Math Contest (Norwegian MO) Final, 2a

Tags: diophantine equation , diophantine , number theory , system of equations

Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\ c + d = ab \end{cases}$ .