Olympiad problems

2020 Jozsef Wildt International Math Competition, W44

Tags: functional equation , fe

We consider a function $f:\mathbb R\to\mathbb R$ such that $$f(x+y)+f(xy-1)=f(x)f(y)+f(x)+f(y)+1$$ for each $x,y\in\mathbb R$. i) Calculate $f(0)$ and $f(-1)$. ii) Prove that $f$ is an even function. iii) Give an example of such a function. iv) Find all monotone functions with the above property. [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

2010 Moldova Team Selection Test, 4

Tags: 3d geometry , combinatorics , geometry

Let $ n\geq6$ be a even natural number. Prove that any cube can be divided in $ \dfrac{3n(n\minus{}2)}4\plus{}2$ cubes.

1999 Romania Team Selection Test, 6

Tags: reflection , homothety , complex bash , geometry

Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.

2015 Czech-Polish-Slovak Match, 1

Tags: geometry

On a circle of radius $r$, the distinct points $A$, $B$, $C$, $D$, and $E$ lie in this order, satisfying $AB=CD=DE>r$. Show that the triangle with vertices lying in the centroids of the triangles $ABD$, $BCD$, and $ADE$ is obtuse. [i]Proposed by Tomáš Jurík, Slovakia[/i]

Kvant 2022, M2708 b)

Tags: combinatorics , combinatorial geometry

Do there exist 100 points on the plane such that the pairwise distances between them are pairwise distinct consecutive integer numbers larger than 2022?

2012 Macedonia National Olympiad, 4

Tags: power of a point , projective geometry , geometry

A fixed circle $k$ and collinear points $E,F$ and $G$ are given such that the points $E$ and $G$ lie outside the circle $k$ and $F$ lies inside the circle $k$. Prove that, if $ABCD$ is an arbitrary quadrilateral inscribed in the circle $k$ such that the points $E,F$ and $G$ lie on lines $AB,AD$ and $DC$ respectively, then the side $BC$ passes through a fixed point collinear with $E,F$ and $G$, independent of the quadrilateral $ABCD$.

2024 Korea Junior Math Olympiad, 5

Tags: circumcircle , geometry

$ABC$ is a right triangle with $\angle C$ the right angle. $X$ is some point inside $ABC$ satisfying $CA=AX$. Let $D$ be the feet of altitude from $C$ to $AB$, and $Y(\neq X)$ the point of intersection of $DX$ and the circumcircle of $ABX$. Prove that $AX=AY$.

2015 Switzerland - Final Round, 7

Tags: algebra , sum

Let $a, b, c$ be real numbers such that: $$\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}= 1$$ Determine all values which the following expression can take : $$\frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b}.$$

2005 Tournament of Towns, 6

Tags: combinatorics , number theory

Two operations are allowed: (i) to write two copies of number $1$; (ii) to replace any two identical numbers $n$ by $(n + 1)$ and $(n - 1)$. Find the minimal number of operations that required to produce the number $2005$ (at the beginning there are no numbers). [i](8 points)[/i]

1995 Dutch Mathematical Olympiad, 3

Tags:

Let $ 101$ marbles be numbered from $ 1$ to $ 101$. The marbles are divided over two baskets $ A$ and $ B$. The marble numbered $ 40$ is in basket $ A$. When this marble is removed from basket $ A$ and put in $ B$, the averages of the numbers $ A$ and $ B$ both increase by $ \frac{1}{4}$. How many marbles were there originally in basket $ A?$

2016 Turkey Team Selection Test, 3

Tags: inequalities

Let $a,b,c$ be non-negative real numbers such that $a^2+b^2+c^2 \le 3$ then prove that; $$(a+b+c)(a+b+c-abc)\ge2(a^2b+b^2c+c^2a)$$

1986 Tournament Of Towns, (112) 6

Tags: combinatorics , game , game strategy

( "Sisyphian Labour" ) There are $1001$ steps going up a hill , with rocks on some of them {no more than 1 rock on each step ) . Sisyphus may pick up any rock and raise it one or more steps up to the nearest empty step . Then his opponent Aid rolls a rock (with an empty step directly below it) down one step . There are $500$ rocks, originally located on the first $500$ steps. Sisyphus and Aid move rocks in turn , Sisyphus making the first move . His goal is to place a rock on the top step. Can Aid stop him? ( S . Yeliseyev)

2019 Saudi Arabia Pre-TST + Training Tests, 5.1

Tags: number theory , prime , divide , divisible

Let $n$ be a positive integer and $p > n+1$ a prime. Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$

2013 Sharygin Geometry Olympiad, 1

Tags: circumcircle , perpendicular bisector , geometry

A circle $k$ passes through the vertices $B, C$ of a scalene triangle $ABC$. $k$ meets the extensions of $AB, AC$ beyond $B, C$ at $P, Q$ respectively. Let $A_1$ is the foot the altitude drop from $A$ to $BC$. Suppose $A_1P=A_1Q$. Prove that $\widehat{PA_1Q}=2\widehat{BAC}$.

1999 AMC 12/AHSME, 13

Tags: ratio , geometric sequence

Define a sequence of real numbers $ a_1$, $ a_2$, $ a_3$, $ \dots$ by $ a_1 = 1$ and $ a_{n + 1}^3 = 99a_n^3$ for all $ n \ge 1$. Then $ a_{100}$ equals $ \textbf{(A)}\ 33^{33} \qquad \textbf{(B)}\ 33^{99} \qquad \textbf{(C)}\ 99^{33} \qquad \textbf{(D)}\ 99^{99} \qquad \textbf{(E)}\ \text{none of these}$

2006 Federal Competition For Advanced Students, Part 1, 2

Tags: algebra

Show that the sequence $ a_n \equal{} \frac {(n \plus{} 1)^nn^{2 \minus{} n}}{7n^2 \plus{} 1}$ is strictly monotonically increasing, where $ n \equal{} 0,1,2, \dots$.

2020 Lusophon Mathematical Olympiad, 5

Tags: combinatorics , board

In how many ways can we fill the cells of a $4\times4$ grid such that each cell contains exactly one positive integer and the product of the numbers in each row and each column is $2020$?

1996 AMC 12/AHSME, 16

Tags: probability

A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equal the third, what is the probability that at least one $2$ is tossed? $\displaystyle \textbf{(A)} \ \frac{1}{6} \qquad \textbf{(B)} \ \frac{91}{216} \qquad \textbf{(C)} \ \frac{1}{2} \qquad \textbf{(D)} \ \frac{8}{15} \qquad \textbf{(E)} \ \frac{7}{12}$

1969 Vietnam National Olympiad, 3

Tags: inequalities , algebra

Consider $x_1 > 0, y_1 > 0, x_2 < 0, y_2 > 0, x_3 < 0, y_3 < 0, x_4 > 0, y_4 < 0.$ Suppose that for each $i = 1, ... ,4$ we have $ (x_i -a)^2 +(y_i -b)^2 \le c^2$. Prove that $a^2 + b^2 < c^2$. Restate this fact in the form of geometric result in plane geometry.

2009 IMO Shortlist, 5

Tags: algebra , polynomial , number theory , permutation

Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

2017 CHKMO, Q3

Tags: incenter , circumcircle , geometry

Let ABC be an acute-angled triangle. Let D be a point on the segment BC, I the incentre of ABC. The circumcircle of ABD meets BI at P and the circumcircle of ACD meets CI at Q. If the area of PID and the area of QID are equal, prove that PI*QD=QI*PD.

2020-2021 OMMC, 1

Tags: algebra

A man rows at a speed of $2$ mph in still water. He set out on a trip towards a spot $2$ miles downstream. He rowed with the current until he was halfway there, then turned back and rowed against the current for $15$ minutes. Then, he turned around again and rowed with the current until he reached his destination. The entire trip took him $70$ minutes. The speed of the current can be represented as $\frac{p}{q}$ mph where $p,q$ are relatively prime positive integers. Find $10p+q$.

2007 Regional Olympiad of Mexico Northeast, 2

Tags: geometry , parallel , isosceles

In the isosceles triangle $ABC$, with $AB=AC$, $D$ is a point on the extension of $CA$ such that $DB$ is perpendicular to $BC$, $E$ is a point on the extension of $BC$ such that $CE=2BC$, and $F$ is a point on $ED$ such that $FC$ is parallel to $AB$. Prove that $FA$ is parallel to $BC$.

2017 IMO, 4

Tags: geometry , circumcircle , tangent

Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$. [i]Proposed by Charles Leytem, Luxembourg[/i]

1970 AMC 12/AHSME, 4

Tags:

Let $S$ be the set of all numbers which are the sum of the squares of three consecutive integers. Then we can say that: $\textbf{(A) }\text{No member of }S\text{ is divisible by }2\qquad$ $\textbf{(B) }\text{No member of }S\text{ is divisible by }3\text{ but some member is divisible by }11\qquad$ $\textbf{(C) }\text{No member of }S\text{ is divisible by }3\text{ or }5\qquad$ $\textbf{(D) }\text{No member of }S\text{ is divisible by }3\text{ or }7\qquad$ $\textbf{(E) }\text{None of these}$