Olympiad problems

2014 Balkan MO Shortlist, G7

Let $I$ be the incenter of $\triangle ABC$ and let $H_a$, $H_b$, and $H_c$ be the orthocenters of $\triangle BIC$ , $\triangle CIA$, and $\triangle AIB$, respectively. The lines $H_aH_b$ meets $AB$ at $X$ and the line $H_aH_c$ meets $AC$ at $Y$. If the midpoint $T$ of the median $AM$ of $\triangle ABC$ lies on $XY$, prove that the line $H_aT$ is perpendicular to $BC$

2024 Ecuador NMO (OMEC), 1

Tags: algebra , system of equations

Find all real solutions: $$\begin{cases}a^3=2024bc \\ b^3=2024cd \\ c^3=2024da \\ d^3=2024ab \end{cases}$$

2020 Novosibirsk Oral Olympiad in Geometry, 1

Tags: rectangle , semicircle , tangent circle , geometry

Two semicircles touch the side of the rectangle, each other and the segment drawn in it as in the figure. What part of the whole rectangle is filled? [img]https://cdn.artofproblemsolving.com/attachments/3/e/70ca8b80240a282553294a58cb3ed807d016be.png[/img]

2002 Estonia Team Selection Test, 5

Tags: trigonometry , inequalities , algebra

Let $0 < a < \frac{\pi}{2}$ and $x_1,x_2,...,x_n$ be real numbers such that $\sin x_1 + \sin x_2 +... + \sin x_n \ge n \cdot sin a $. Prove that $\sin (x_1 - a) + \sin (x_2 - a) + ... + \sin (x_n - a) \ge 0$ .

2019 Czech-Polish-Slovak Junior Match, 5

Tags: geometry , geometric transformation , rotation , combinatorial geometry , combinatorics

Let $A_1A_2 ...A_{360}$ be a regular $360$-gon with centre $S$. For each of the triangles $A_1A_{50}A_{68}$ and $A_1A_{50}A_{69}$ determine, whether its images under some $120$ rotations with centre $S$ can have (as triangles) all the $360$ points $A_1, A_2, ..., A_{360}$ as vertices.

2022 CCA Math Bonanza, L2.2

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A rectangle $ABCD$ has side lengths $AB=6 \text{ miles}$ and $BC=9\text{ miles}.$ A pigeon hovers at point $P$, which is 5 miles above some randomly chosen point inside $ABCD$. Given that the expected value of \[AP^2+CP^2-BP^2-DP^2\] can be expressed as $\tfrac{a}{b}$, what is $ab$? [i]2022 CCA Math Bonanza Lightning Round 2.2[/i]

2017 Math Prize for Girls Olympiad, 4

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A [i]lattice point[/i] is a point in the plane whose two coordinates are both integers. A [i]lattice line[/i] is a line in the plane that contains at least two lattice points. Is it possible to color every lattice point red or blue such that every lattice line contains exactly 2017 red lattice points? Prove that your answer is correct.

2002 Bosnia Herzegovina Team Selection Test, 1

Tags: algebra

Let $x,y,z$ be real numbers that satisfy \[x+y+z= 3 \ \ \text{ and } \ \ xy+yz+zx= a\]where $a$ is a real parameter. Find the value of $a$ for which the difference between the maximum and minimum possible values of $x$ equals $8$.

2017 India PRMO, 1

Tags: number theory , sum of digits , divide , divisible

How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?

2023 AMC 10, 14

Tags: integer , algebra

How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2+mn+n^2=m^2n^2$? $\textbf{(A) }7\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }5$

2012 Online Math Open Problems, 5

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Two circles have radius 5 and 26. The smaller circle passes through center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle? [i]Ray Li.[/i]

1985 IMO, 2

Tags: number theory , greatest common divisor , relatively prime , coloring , combinatorics

Let $n$ and $k$ be relatively prime positive integers with $k<n$. Each number in the set $M=\{1,2,3,\ldots,n-1\}$ is colored either blue or white. For each $i$ in $M$, both $i$ and $n-i$ have the same color. For each $i\ne k$ in $M$ both $i$ and $|i-k|$ have the same color. Prove that all numbers in $M$ must have the same color.

2010 Contests, 1

Tags: digit , number theory , sum of squares , algebra

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

1983 Vietnam National Olympiad, 3

Tags: geometry

A triangle $ABC$ and a positive number $k$ are given. Find the locus of a point $M$ inside the triangle such that the projections of $M$ on the sides of $\Delta ABC$ form a triangle of area $k$.

LMT Team Rounds 2021+, A 24

Tags: combinatorics

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Using the four words “Hi”, “hey”, “hello”, and “haiku”, How many haikus Can somebody make? (Repetition is allowed, Order does matter.) [i]Proposed by Jeff Lin[/i]

1965 IMO, 4

Tags: system of equations , algebra

Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.

1994 APMO, 3

Tags: relatively prime , prime number , number theory

Let $n$ be an integer of the form $a^2 + b^2$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $ab$. Determine all such $n$.

2000 Vietnam National Olympiad, 3

Tags: algebra , polynomial , number theory

Consider the polynomial $ P(x) \equal{} x^3 \plus{} 153x^2 \minus{} 111x \plus{} 38$. (a) Prove that there are at least nine integers $ a$ in the interval $ [1, 3^{2000}]$ for which $ P(a)$ is divisible by $ 3^{2000}$. (b) Find the number of integers $ a$ in $ [1, 3^{2000}]$ with the property from (a).

2003 National Olympiad First Round, 28

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Let $a$, $x$, $y$, $z$ be real numbers such that $ax-y+z=3a-1$ ve $x-ay+z=a^2-1$, which of the followings cannot be equal to $x^2+y^2+z^2$? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ \sqrt 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \sqrt[3]{4} \qquad\textbf{(E)}\ \text{None of the preceding} $

1989 China National Olympiad, 1

Tags: geometry , combinatorics

We are given two point sets $A$ and $B$ which are both composed of finite disjoint arcs on the unit circle. Moreover, the length of each arc in $B$ is equal to $\dfrac{\pi}{m}$ ($m \in \mathbb{N}$). We denote by $A^j$ the set obtained by a counterclockwise rotation of $A$ about the center of the unit circle for $\dfrac{j\pi}{m}$ ($j=1,2,3,\dots$). Show that there exists a natural number $k$ such that $l(A^k\cap B)\ge \dfrac{1}{2\pi}l(A)l(B)$.(Here $l(X)$ denotes the sum of lengths of all disjoint arcs in the point set $X$)

1994 North Macedonia National Olympiad, 4

Tags: point , combinatorics , circles

$1994$ points from the plane are given so that any $100$ of them can be selected $98$ that can be rounded (some points may be at the border of the circle) with a diameter of $1$. Determine the smallest number of circles with radius $1$, sufficient to cover all $1994$

1981 Canada National Olympiad, 1

Tags: function , floor function , algebra

For any real number $t$, denote by $[t]$ the greatest integer which is less than or equal to $t$. For example: $[8] = 8$, $[\pi] = 3$, and $[-5/2] = -3$. Show that the equation \[[x] + [2x] + [4x] + [8x] + [16x] + [32x] = 12345\] has no real solution.

2008 Peru MO (ONEM), 4

Tags: combinatorics , lattice points , geometry , existence , cartesian coordinates

All points in the plane that have both integer coordinates are painted, using the colors red, green, and yellow. If the points are painted so that there is at least one point of each color. Prove that there are always three points $X$, $Y$ and $Z$ of different colors, such that $\angle XYZ = 45^{\circ} $

2020 Jozsef Wildt International Math Competition, W18

Tags: inequalities

Let $D:=\{(x, y)\mid x,y\in\mathbb R_+,x \ne y,x^y=y^x\}$. (Obvious that $x\ne1$ and $y\ne1$). And let $\alpha\le\beta$ be positive real numbers. Find $$\inf_{(x,y)\in D}x^\alpha y^\beta.$$ [i]Proposed by Arkady Alt[/i]

2016 Bulgaria JBMO TST, 1

Tags: algebra

$ a,b,c,d,e,f $ are real numbers. It is true that: $ a+b+c+d+e+f=20 $ $ (a-2)^2+(b-2)^2+...+(f-2)^2=24 $ Find the maximum value of d.