Olympiad problems

1995 AMC 12/AHSME, 11

Tags:

How many base 10 four-digit numbers, $N = \underline{a} \underline{b} \underline{c} \underline{d}$, satisfy all three of the following conditions? (i) $4,000 \leq N < 6,000;$ (ii) $N$ is a multiple of 5; (iii) $3 \leq b < c \leq 6$. $ \mathbf{(A)}\; 10\qquad \mathbf{(B)}\; 18\qquad \mathbf{(C)}\; 24\qquad \mathbf{(D)}\; 36\qquad \mathbf{(E)}\; 48$

2010 Slovenia National Olympiad, 3

Tags: circumcircle , geometry

Let $ABC$ be an acute triangle. A line parallel to $BC$ intersects the sides $AB$ and $AC$ at $D$ and $E$, respectively. The circumcircle of the triangle $ADE$ intersects the segment $CD$ at $F \ (F \neq D).$ Prove that the triangles $AFE$ and $CBD$ are similar.

2000 District Olympiad (Hunedoara), 1

Tags: group , binary operation , semigroups , superior algebra , group theory

Define the operator " $ * $ " on $ \mathbb{R} $ as $ x*y=x+y+xy. $ [b]a)[/b] Show that $ \mathbb{R}\setminus\{ -1\} , $ along with the operator above, is isomorphic with $ \mathbb{R}\setminus\{ 0\} , $ with the usual multiplication. [b]b)[/b] Determine all finite semigroups of $ \mathbb{R} $ under " $ *. $ " Which of them are groups? [b]c)[/b] Prove that if $ H\subset_{*}\mathbb{R} $ is a bounded semigroup, then $ H\subset [-2, 0]. $

1996 Estonia Team Selection Test, 2

Tags: geometry , inradius , geometric inequality , inequalities

Let $a,b,c$ be the sides of a triangle, $\alpha ,\beta ,\gamma$ the corresponding angles and $r$ the inradius. Prove that $$a\cdot sin\alpha+b\cdot sin\beta+c\cdot sin\gamma\geq 9r$$

2013 Brazil Team Selection Test, 3

Tags: geometry , circumcircle , trigonometry , triangle , reflection

Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.

2021 Turkey Team Selection Test, 2

Tags: combinatorics

In a school with some students, for any three student, there exists at least one student who are friends with all these three students.(Friendships are mutual) For any friends $A$ and $B$, any two of their common friends are also friends with each other. It's not possible to partition these students into two groups, such that every student in each group are friends with all the students in the other gruop. Prove that any two people who aren't friends with each other, has the same number of common friends.(Each person is a friend of him/herself.)

1979 IMO Longlists, 73

Tags: combinatorics

In a plane a finite number of equal circles are given. These circles are mutually nonintersecting (they may be externally tangent). Prove that one can use at most four colors for coloring these circles so that two circles tangent to each other are of different colors. What is the smallest number of circles that requires four colors?

2011 Mongolia Team Selection Test, 1

Tags: combinatorics

A group of the pupils in a class are called [i]dominant[/i] if any other pupil from the class has a friend in the group. If it is known that there exist at least 100 dominant groups, prove that there exists at least one more dominant group. (proposed by B. Batbayasgalan, inspired by Komal problem)

2021 Benelux, 3

Tags: geometry

A cyclic quadrilateral $ABXC$ has circumcentre $O$. Let $D$ be a point on line $BX$ such that $AD = BD$. Let $E$ be a point on line $CX$ such that $AE = CE$. Prove that the circumcentre of triangle $\triangle DEX$ lies on the perpendicular bisector of $OA$.

2018 Saudi Arabia GMO TST, 1

Tags: sequence , algebra , inequalities , recurrence relation , reciprocal

Let $\{x_n\}$ be a sequence defined by $x_1 = 2$ and $x_{n+1} = x_n^2 - x_n + 1$ for $n \ge 1$. Prove that $$1 -\frac{1}{2^{2^{n-1}}} < \frac{1}{x_1}+\frac{1}{x_2}+ ... +\frac{1}{x_n}< 1 -\frac{1}{2^{2^n}}$$ for all $n$

2011 Tournament of Towns, 2

Tags: geometry , rectangle , perimeter , integer , combinatorial geometry

A rectangle is divided by $10$ horizontal and $10$ vertical lines into $121$ rectangular cells. If $111$ of them have integer perimeters, prove that they all have integer perimeters.

2020 Tournament Of Towns, 2

Tags: altitude , circumcircle , geometry , incenter , circumradius

At heights $AA_0, BB_0, CC_0$ of an acute-angled non-equilateral triangle $ABC$, points $A_1, B_1, C_1$ were marked, respectively, so that $AA_1 = BB_1 = CC_1 = R$, where $R$ is the radius of the circumscribed circle of triangle $ABC$. Prove that the center of the circumscribed circle of the triangle $A_1B_1C_1$ coincides with the center of the inscribed circle of triangle $ABC$. E. Bakaev

KoMaL A Problems 2019/2020, A. 771

Tags: geometry

Let $\omega$ denote the incircle of triangle $ABC,$ which is tangent to side $BC$ at point $D.$ Let $G$ denote the second intersection of line $AD$ and circle $\omega.$ The tangent to $\omega$ at point $G$ intersects sides $AB$ and $AC$ at points $E$ and $F$ respectively. The circumscribed circle of $DEF$ intersects $\omega$ at points $D$ and $M.$ The circumscribed circle of $BCG$ intersects $\omega$ at points $G$ and $N.$ Prove that lines $AD$ and $MN$ are parallel. [i]Proposed by Ágoston Győrffy, Remeteszőlős[/i]

2022 Purple Comet Problems, 30

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There is a positive integer s such that there are s solutions to the equation $64sin^2(2x)+tan^2(x)+cot^2(x)=46$ in the interval $(0,\frac{\pi}{2})$ all of the form $\frac{m_k}{n_k}\pi$ where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, . . . , s$. Find $(m_1 + n_1) + (m_2 + n_2) + (m_3 + n_3) + · · · + (m_s + n_s)$.

2004 Germany Team Selection Test, 3

Tags: modular arithmetic , decimal representation , counting , function , combinatorics

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

PEN D Problems, 22

Tags: congruence , logarithm

Prove that $1980^{1981^{1982}} + 1982^{1981^{1980}}$ is divisible by $1981^{1981}$.

1965 Poland - Second Round, 5

Tags: geometry , combinatorics , combinatorial geometry

Prove that a square can be divided into any number greater than 5 squares, but cannot be divided into 5 squares.

2007 China Team Selection Test, 2

Tags: combinatorics

Given $ n$ points arbitrarily in the plane $ P_{1},P_{2},\ldots,P_{n},$ among them no three points are collinear. Each of $ P_{i}$ ($1\le i\le n$) is colored red or blue arbitrarily. Let $ S$ be the set of triangles having $ \{P_{1},P_{2},\ldots,P_{n}\}$ as vertices, and having the following property: for any two segments $ P_{i}P_{j}$ and $ P_{u}P_{v},$ the number of triangles having $ P_{i}P_{j}$ as side and the number of triangles having $ P_{u}P_{v}$ as side are the same in $ S.$ Find the least $ n$ such that in $ S$ there exist two triangles, the vertices of each triangle having the same color.

2015 Indonesia MO Shortlist, N7

Tags: number theory , greatest common divisor , least common multiple

For every natural number $a$ and $b$, define the notation $[a,b]$ as the least common multiple of $a $ and $b$ and the notation $(a,b)$ as the greatest common divisor of $a$ and $b$. Find all $n \in \mathbb{N}$ that satisfies \[ 4 \sum_{k=1}^{n} [n,k] = 1 + \sum_{k=1}^{n} (n,k) + 2n^2 \sum_{k=1}^{n} \frac{1}{(n,k)} \]

1955 AMC 12/AHSME, 12

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The solution of $ \sqrt{5x\minus{}1}\plus{}\sqrt{x\minus{}1}\equal{}2$ is: $ \textbf{(A)}\ x\equal{}2,x\equal{}1 \qquad \textbf{(B)}\ x\equal{}\frac{2}{3} \qquad \textbf{(C)}\ x\equal{}2 \qquad \textbf{(D)}\ x\equal{}1 \qquad \textbf{(E)}\ x\equal{}0$

2012 Online Math Open Problems, 1

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Calvin was asked to evaluate $37 + 31 \times a$ for some number $a$. Unfortunately, his paper was tilted 45 degrees, so he mistook multiplication for addition (and vice versa) and evaluated $37 \times 31 + a$ instead. Fortunately, Calvin still arrived at the correct answer while still following the order of operations. For what value of $a$ could this have happened? [i]Ray Li.[/i]

1982 Polish MO Finals, 4

Tags: combinatorics , combinatorial geometry

On a plane is given a finite set of points. Prove that the points can be covered by open squares $Q_1,Q_2,...,Q_n$ such that $1 \le\frac{N_j}{S_j} \le 4$ for $j = 1,...,n,$ where $N_j$ is the number of points from the set inside square $Q_j$ and $S_j$ is the area of $Q_j$.

2012 Stars of Mathematics, 1

Tags: geometry , geometric transformation

Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ so that $AQ=PQ$ and $\angle PAQ = 45^{\circ}$. ([i]Dan Schwarz[/i])

2013 AIME Problems, 1

Tags: #1 , algebra

The AIME Triathlon consists of a half-mile swim, a $30$-mile bicycle, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs five times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?

2021 Peru EGMO TST, 1

Tags: algebra

A finite set $M$ of real numbers is called [i]special[/i] if $M$ has at least two elements and the following condition is true: If $a$ and $b$ are distinct elements of $M$ then $5\sqrt{|a|}-\frac{2b}{3}$ is also a element of $M$. a) Determine if there is a special set with (exactly) two elements. b) Determine if there is a special set with three (or more) elements such that all elements are positive.