Olympiad problems

2019 Jozsef Wildt International Math Competition, W. 65

Tags: inequalities

If $a$, $b$, $c \geq 1$; $y \geq x \geq 1$; $p$, $q$, $r > 0$ then$$\left(\frac{1+y\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}{1+x\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}\right)^{\frac{p+q+r}{\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}}\left(\frac{1+xa}{1+ya}\right)^{\frac{p}{a}}\left(\frac{1+xb}{1+yb}\right)^{\frac{q}{b}}\left(\frac{1+xc}{1+yc}\right)^{\frac{r}{c}}$$ $$\geq \prod \limits_{cyc}\left(\frac{1+y\left(a^pb^q\right)^{\frac{1}{p+q}}}{1+x\left(a^pb^q\right)^{\frac{1}{p+q}}}\right)^{\frac{p+q}{\left(a^pb^q\right)^{\frac{1}{p+q}}}}$$

2013 Tournament of Towns, 7

Tags: combinatorics , game

Two teams $A$ and $B$ play a school ping pong tournament. The team $A$ consists of $m$ students, and the team $B$ consists of $n$ students where $m \ne n$. There is only one ping pong table to play and the tournament is organized as follows: Two students from different teams start to play while other players form a line waiting for their turn to play. After each game the first player in the line replaces the member of the same team at the table and plays with the remaining player. The replaced player then goes to the end of the line. Prove that every two players from the opposite teams will eventually play against each other.

2003 Alexandru Myller, 2

Tags: number theory , complex number , algebra , newton s binom

Prove that $$ (n+2)^n=\prod_{k=1}^{n+1} \sum_{l=1}^{n+1} le^{\frac{2i\pi k (n-l+1)}{n+2}} , $$ for any natural number $ n. $ [i]Mihai Piticari[/i]

2014 Taiwan TST Round 1, 1

Tags: geometry , trigonometry , geometric transformation

Let $O_1$, $O_2$ be two circles with radii $R_1$ and $R_2$, and suppose the circles meet at $A$ and $D$. Draw a line $L$ through $D$ intersecting $O_1$, $O_2$ at $B$ and $C$. Now allow the distance between the centers as well as the choice of $L$ to vary. Find the length of $AD$ when the area of $ABC$ is maximized.

2012 Peru IMO TST, 3

Tags: combinatorics , discrete continous

Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half. [i]Proposed by Gerhard Wöginger, Austria[/i]

Denmark (Mohr) - geometry, 1992.2

Tags: geometry , right triangle , tangent circle

In a right-angled triangle, $a$ and $b$ denote the lengths of the two catheti. A circle with radius $r$ has the center on the hypotenuse and touches both catheti. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$.

2001 Austrian-Polish Competition, 2

Tags: limit , system of equations , pigeonhole principle , algebra

Let $n$ be a positive integer greater than $2$. Solve in nonnegative real numbers the following system of equations \[x_{k}+x_{k+1}=x_{k+2}^{2}\quad , \quad k=1,2,\cdots,n\] where $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$.

2016 Mathematical Talent Reward Programme, SAQ: P 5

Tags: function , injective function , surjective function

Let $\mathbb{N}$ be the set of all positive integers. $f,g:\mathbb{N} \to \mathbb{N}$ be funtions such that $f$ is onto and $g$ is one-one and $f(n)\geq g(n)$ for all positive integers $n$. Prove that $f=g$.

2024 USA TSTST, 4

Tags: geometry , circumcircle , cyclic quadrilateral

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$ and $E$ be the intersection of segments $AC$ and $BD$. Let $\omega_1$ be the circumcircle of $ADE$ and $\omega_2$ be the circumcircle of $BCE$. The tangent to $\omega_1$ at $A$ and the tangent to $\omega_2$ at $C$ meet at $P$. The tangent to $\omega_1$ at $D$ and the tangent to $\omega_2$ at $B$ meet at $Q$. Show that $OP=OQ$. [i]Merlijn Staps[/i]

2010 Junior Balkan Team Selection Tests - Romania, 2

Tags: equal angles , geometry , convex quadrilateral , angle

Let $ABCD$ be a convex quadrilateral with $\angle BCD= 120^o, \angle {CBA} = 45^o, \angle {CBD} = 15^o$ and $\angle {CAB} = 90^o$. Show that $AB = AD$.

2015 Bangladesh Mathematical Olympiad, 2

Tags: number theory , contest preparation , integer , solutions

[b][u]BdMO National Higher Secondary Problem 3[/u][/b] Let $N$ be the number if pairs of integers $(m,n)$ that satisfies the equation $m^2+n^2=m^3$ Is $N$ finite or infinite?If $N$ is finite,what is its value?

1953 Kurschak Competition, 2

Tags: number theory , perfect square , divide

$n$ and $d$ are positive integers such that $d$ divides $2n^2$. Prove that $n^2 + d$ cannot be a square.

2010 IFYM, Sozopol, 6

Tags: number theory

Let $A=\{ x\in \mathbb{N},x=a^2+2b^2,a,b\in \mathbb{Z},ab\neq 0 \}$ and $p$ is a prime number. Prove that if $p^2\in A$, then $p\in A$.

2017 USAMTS Problems, 3

Tags:

Do there exist two polygons such that, by putting them together in three different ways (without holes, overlap, or reflections), we can obtain first a triangle, then a convex quadrilateral, and lastly a convex pentagon?

2024 Vietnam Team Selection Test, 5

Tags: geometry , complex number , poles and polars , pascal theorem

Let incircle $(I)$ of triangle $ABC$ touch the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $(O)$ be the circumcircle of $ABC$. Ray $EF$ meets $(O)$ at $M$. Tangents at $M$ and $A$ of $(O)$ meet at $S$. Tangents at $B$ and $C$ of $(O)$ meet at $T$. Line $TI$ meets $OA$ at $J$. Prove that $\angle ASJ=\angle IST$.

2000 All-Russian Olympiad Regional Round, 9.6

Tags: weighing , combinatorics

Among $2000$ outwardly indistinguishable balls, wines - aluminum weighing 1$0$ g, and the rest - duralumin weighing $9.9$ g. It is required to select two piles of balls so that the masses of the piles are different, and the number of balls in them - the same. What is the smallest number of weighings on a cup scale without weights that can be done?

2020 CHMMC Winter (2020-21), 2

Tags: combinatorics

Caltech's 900 students are evenly spaced along the circumference of a circle. How many equilateral triangles can be formed with at least two Caltech students as vertices?

2019 HMNT, 5

Tags: number theory

Alison is eating $2401$ grains of rice for lunch. She eats the rice in a very peculiar manner: every step, if she has only one grain of rice remaining, she eats it. Otherwise, she nds the smallest positive integer $d > 1$ for which she can group the rice into equal groups of size $d$ with none left over. She then groups the rice into groups of size $d$, eats one grain from each group, and puts the rice back into a single pile. How many steps does it take her to finish all her rice?

2022 Assam Mathematical Olympiad, 16

Tags:

Can we find a subset $A$ of $\mathbb{N}$ containing exactly five numbers such that sum of any three elements of $A$ is a prime number? Justify your answer.

2015 Bangladesh Mathematical Olympiad, 3

Tags: number theory , algebra , modular arithmetic

Let $n$ be a positive integer.Consider the polynomial $p(x)=x^2+x+1$. What is the remainder of $ x^3$ when divided by $x^2+x+1$.For what positive integers values of $n$ is $ x^{2n}+x^n+1$ divisible by $p(x)$? Post no:[size=300]$100$[/size]

2008 Ukraine Team Selection Test, 10

Tags: modular arithmetic , number theory , divisibility

Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$. [i]Author: Dan Brown, Canada[/i]

1994 Korea National Olympiad, Problem 3

Tags: excenter , geometry , incenter , circumradius

In a triangle $ABC$, $I$ and $O$ are the incenter and circumcenter respectively, $A',B',C'$ the excenters, and $O'$ the circumcenter of $\triangle A'B'C'$. If $R$ and $R'$ are the circumradii of triangles $ABC$ and $A'B'C'$, respectively, prove that: (i) $R'= 2R $ (ii) $IO' = 2IO$

2024 UMD Math Competition Part I, #19

Tags: rotation , combinatorics

A square-shaped quilt is divided into $16 = 4 \times 4$ equal squares. We say that the quilt is [i]UMD certified[/i] if each of these $16$ squares is colored red, yellow, or black, so that (i) all three colors are used at least once and (ii) the quilt looks the same when it is rotated $90, 180,$ or $270$ degrees about its center. How many distinct UMD certified quilts are there? \[\rm a. ~33\qquad \mathrm b. ~36 \qquad \mathrm c. ~45\qquad\mathrm d. ~54\qquad\mathrm e. ~81\]

2024 Azerbaijan IMO TST, 4

Tags: combinatorics

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

2021 Israel National Olympiad, P1

Tags: combinatorics , number theory , digit

Sophie wrote on a piece of paper every integer number from 1 to 1000 in decimal notation (including both endpoints). [b]a)[/b] Which digit did Sophie write the most? [b]b)[/b] Which digit did Sophie write the least?