Olympiad problems

2008 AMC 10, 8

Tags: AMC

A class collects $ \$50$ to buy flowers for a classmate who is in the hospital. Roses cost $ \$3$ each, and carnations cost $ \$2$ each. No other flowers are to be used. How many different bouquets could be purchased for exactly $ \$50$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17$

2005 Germany Team Selection Test, 1

Tags: function , inequalities , algebra proposed , algebra

Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$. Hereby, $\mathbb{R}_+$ is the set of all positive real numbers. [i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$. A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.

1988 Bundeswettbewerb Mathematik, 4

Tags: algebra , recurrence relation

Starting with four given integers $a_1, b_1, c_1, d_1$ is defined recursively for all positive integers $n$: $$a_{n+1} := |a_n - b_n|, b_{n+1} := |b_n - c_n|, c_{n+1} := |c_n - d_n|, d_{n+1} := |d_n - a_n|.$$ Prove that there is a natural number $k$ such that all terms $a_k, b_k, c_k, d_k$ take the value zero.

1993 China National Olympiad, 3

Tags: geometry unsolved , geometry

Let $K, K_1$ be two circles with the same center and their radii equal to $R$ and $R_1 (R_1>R)$ respectively. Quadrilateral $ABCD$ is inscribed in circle $K$. Quadrilateral $A_1B_1C_1D_1$ is inscribed in circle $K_1$ where $A_1,B_1,C_1,D_1$ lie on rays $CD,DA,AB,BC$ respectively. Show that $\dfrac{S_{A_1B_1C_1D_1}}{S_{ABCD}}\ge \dfrac{R^2_1}{R^2}$.

VMEO II 2005, 11

Tags: algebra , polynomial , Functional Equations , functional

Given $P$ a real polynomial with degree greater than $ 1$. Find all pairs $(f,Q)$ with function $f : R \to R$ and the real polynomial $Q$ satisfying the following two conditions: i) for all $x, y \in R$, we have $f(P(x) + f(y)) = y + Q(f(x))$. ii) there exists $x_0 \in R$ such that $f(P(x_0)) = Q(f(x_0))$.

2017 IOM, 3

Tags: polynomial , algebra

Let $Q$ be a quadriatic polynomial having two different real zeros. Prove that there is a non-constant monic polynomial $P$ such that all coefficients of the polynomial $Q(P(x))$ except the leading one are (by absolute value) less than $0.001$.

2009 Pan African, 3

Tags: geometry

Points $C,E,D$ and $F$ lie on a circle with centre $O$. Two chords $CD$ and $EF$ intersect at a point $N$. The tangents at $C$ and $D$ intersect at $A$, and the tangents at $E$ and $F$ intersect at $B$. Prove that $ON\perp AB$.

2014 Korea - Final Round, 1

Tags: inequalities proposed , inequalities

Suppose $x$, $y$, $z$ are positive numbers such that $x+y+z=1$. Prove that \[ \frac{(1+xy+yz+zx)(1+3x^3 + 3y^3 + 3z^3)}{9(x+y)(y+z)(z+x)} \ge \left( \frac{x \sqrt{1+x} }{\sqrt[4]{3+9x^2}} + \frac{y \sqrt{1+y} }{\sqrt[4]{3+9y^2}} + \frac{z \sqrt{1+z}}{\sqrt[4]{3+9z^2}} \right)^2. \]

1974 Poland - Second Round, 6

Tags: algebra , combinatorics

There is a sequence of integers $ a_1, a_2, \ldots, a_{2n+1} $ with the following property: after eliminating any term, the remaining ones can be divided into two groups of $ n $ terms such that the sum of the terms in the first group is equal to the sum words in the second. Prove that all terms of the sequence are equal.

2005 AMC 12/AHSME, 4

Tags:

A store normally sells windows at $ \$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? $ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 200 \qquad \textbf{(C)}\ 300 \qquad \textbf{(D)}\ 400 \qquad \textbf{(E)}\ 500$

2006 Lithuania National Olympiad, 4

Tags: combinatorics , combinatorics proposed , combinatorics solved , Sets , Subsets , cardinality , uniqueness

Find the maximal cardinality $|S|$ of the subset $S \subset A=\{1, 2, 3, \dots, 9\}$ given that no two sums $a+b | a, b \in S, a \neq b$ are equal.

1994 Nordic, 4

Tags: number theory , Perfect Square , Integer , Pell equations

Determine all positive integers $n < 200$, such that $n^2 + (n+ 1)^2$ is the square of an integer.

2021 IMO, 5

Tags: IMO , combinatorics , IMO 2021 , Swap , Bushy , Jumpy

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$. Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.

2021 Indonesia TST, C

Tags: combinatorics

In a country, there are $2018$ cities, some of which are connected by roads. Each city is connected to at least three other cities. It is possible to travel from any city to any other city using one or more roads. For each pair of cities, consider the shortest route between these two cities. What is the greatest number of roads that can be on such a shortest route?

2014 India National Olympiad, 6

Tags: induction , combinatorics proposed , combinatorics

Let $n>1$ be a natural number. Let $U=\{1,2,...,n\}$, and define $A\Delta B$ to be the set of all those elements of $U$ which belong to exactly one of $A$ and $B$. Show that $|\mathcal{F}|\le 2^{n-1}$, where $\mathcal{F}$ is a collection of subsets of $U$ such that for any two distinct elements of $A,B$ of $\mathcal{F}$ we have $|A\Delta B|\ge 2$. Also find all such collections $\mathcal{F}$ for which the maximum is attained.

2020/2021 Tournament of Towns, P4

Tags: combinatorial geometry , Southern Tournament , combinatorics

The $X{}$ pentomino consists of five $1\times1$ squares where four squares are all adjacent to the fifth one. Is it possible to cut nine such pentominoes from an $8\times 8$ chessboard, not necessarily cutting along grid lines? (The picture shows how to cut three such $X{}$ pentominoes.) [i]Alexandr Gribalko[/i]

1999 Czech And Slovak Olympiad IIIA, 2

Tags: 3D geometry , tetrahedron , Volume , geometry

In a tetrahedron $ABCD, E$ and $F$ are the midpoints of the medians from $A$ and $D$. Find the ratio of the volumes of tetrahedra $BCEF$ and $ABCD$. Note: Median in a tetrahedron connects a vertex and the centroid of the opposite side.

2022 Macedonian Mathematical Olympiad, Problem 3

Tags: Sequence , prime divisors , number theory

The sequence $(a_n)_{n \ge 1}^\infty$ is given by: $a_1=2$ and $a_{n+1}=a_n^2+a_n$ for all $n \ge 1$. For an integer $m \ge 2$, $L(m)$ denotes the greatest prime divisor of $m$. Prove that there exists some $k$, for which $L(a_k) > 1000^{1000}$. [i]Proposed by Nikola Velov[/i]

2003 Moldova Team Selection Test, 3

Tags: geometry , geometry unsolved

Consider a point $ M$ found in the same plane with the triangle $ ABC$, but not found on any of the lines $ AB,BC$ and $ CA$. Denote by $ S_1,S_2$ and $ S_3$ the areas of the triangles $ AMB,BMC$ and $ CMA$, respectively. Find the locus of $ M$ satisfying the relation: $ (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)$

1984 Austrian-Polish Competition, 9

Tags: functional , functional equation , algebra

Find all functions $f: Q \to R$ satisfying $f (x + y) = f (x)f (y) - f(xy) + 1$ for all $x,y \in Q$

2012 Regional Olympiad of Mexico Center Zone, 2

Tags: modular arithmetic , number theory unsolved , number theory

Let $m, n$ integers such that: $(n-1)^3+n^3+(n+1)^3=m^3$ Prove that 4 divides $n$

2024 Iranian Geometry Olympiad, 5

Tags: geometry

Cyclic quadrilateral $ABCD$ with circumcircle $\omega$ is given. Let $E$ be a fixed point on segment $AC$. $M$ is an arbitrary point on $\omega$, lines $AM$ and $BD$ meet at a point $P$. $EP$ meets $AB$ and $AD$ at points $R$ and $Q$, respectively, $S$ is the intersection of $BQ,DR$ and lines $MS$ and $AC$ meet at a point $T$. Prove that as $M$ varies the circumcircle of triangle $\bigtriangleup CMT$ passes through a fixed point other than $C$. [i]Proposed by Chunlai Jin - China[/i]

2024 LMT Fall, 20

Tags: guts

A base $9$ number [i]probably places[/i] if it has a $7$ as one of its digits. Find the number of base $9$ numbers less than or equal to $100$ in base $10$ that probably place.

1977 Vietnam National Olympiad, 6

Tags: geometry , 3-Dimensional Geometry , maximum , minimum , distance

The planes $p$ and $p'$ are parallel. A polygon $P$ on $p$ has $m$ sides and a polygon $P'$ on $p'$ has $n$ sides. Find the largest and smallest distances between a vertex of $P$ and a vertex of $P'$.

1983 Czech and Slovak Olympiad III A, 3

Tags: combinatorics , combinatorial geometry , Chessboard , national olympiad , rectangle

An $8\times 8$ chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.