Olympiad problems

2004 Peru MO (ONEM), 4

Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.

1980 IMO Longlists, 13

Tags: algebra , arithmetic sequence , Arithmetic Progression , Sequence , IMO Shortlist

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

1995 Miklós Schweitzer, 9

Tags: real analysis

A serpentine is a sequence of points $P_1 , ..., P_m$ in a plane, not necessarily all different, such that the distance between $P_i$ and $P_{i+1}$ is at least 1, and the segments $P_i P_{i +1}$ are alternately horizontal and vertical. Construct a compact set in which there is a sequence of serpentines with arbitrary long lengths but there is no closed serpentine ($P_m = P_i$ for some i < m).

1972 IMO Longlists, 37

Tags: geometry , rectangle , search , combinatorics proposed , combinatorics

On a chessboard ($8\times 8$ squares with sides of length $1$) two diagonally opposite corner squares are taken away. Can the board now be covered with nonoverlapping rectangles with sides of lengths $1$ and $2$?

1991 Arnold's Trivium, 34

Tags: calculus , derivative

Investigate the singular points on the curve $y=x^3$ in the projective plane.

2024 AMC 10, 8

Tags: AMC , AMC 10 , 2024 AMC , 2024 AMC 10b , Divisors

Let $N$ be the product of all the positive integer divisors of $42$. What is the units digit of $N$? $ \textbf{(A) }0 \qquad \textbf{(B) }2 \qquad \textbf{(C) }4 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8 \qquad $

2023 BMT, 6

Tags: number theory

Let $N$ be the number of positive integers $x$ less than $210 \cdot 2023$ such that $$ lcm(gcd(x, 1734), gcd(x + 17, x + 1732))$$ divides $2023$. Compute the sum of the prime factors of $N$ with multiplicity. (For example, if $S = 75 = 3^1 \cdot 5^2$, then the answer is $1\cdot 3 + 2 \cdot 5 = 13$).

2020 Kazakhstan National Olympiad, 4

Tags: geometry , incenter , arc midpoint , incircle

The incircle of the triangle $ ABC $ touches the sides of $ AB, BC, CA $ at points $ C_0, A_0, B_0 $, respectively. Let the point $ M $ be the midpoint of the segment connecting the vertex $ C_0 $ with the intersection point of the altitudes of the triangle $ A_0B_0C_0 $, point $ N $ be the midpoint of the arc $ ACB $ of the circumscribed circle of the triangle $ ABC $. Prove that line $ MN $ passes through the center of incircle of triangle $ ABC $.

2008 Costa Rica - Final Round, 5

Tags: Diophantine equation , Other problems

Let $ p$ be a prime number such that $ p\minus{}1$ is a perfect square. Prove that the equation $ a^{2}\plus{}(p\minus{}1)b^{2}\equal{}pc^{2}$ has infinite many integer solutions $ a$, $ b$ and $ c$ with $ (a,b,c)\equal{}1$

2011 Belarus Team Selection Test, 1

Tags: number theory , Perfect Squares , Perfect Square

Find the least possible number of elements which can be deleted from the set $\{1,2,...,20\}$ so that the sum of no two different remaining numbers is not a perfect square. N. Sedrakian , I.Voronovich

LMT Team Rounds 2010-20, A22 B24

Tags:

In a game of Among Us, there are $10$ players and $12$ colors. Each player has a "default" color that they will automatically get if nobody else has that color. Otherwise, they get a random color that is not selected. If $10$ random players with random default colors join a game one by one, the expected number of players to get their default color can be expressed as $\frac{m}{n}$. Compute $m+n$. Note that the default colors are not necessarily distinct. [i]Proposed by Jeff Lin[/i]

2017 Iran MO (3rd round), 1

Tags: algebra , functional equation

Let $\mathbb{R}^{\ge 0}$ be the set of all nonnegative real numbers. Find all functions $f:\mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that $$ x+2 \max\{y,f(x),f(z)\} \ge f(f(x))+2 \max\{z,f(y)\}$$ for all nonnegative real numbers $x,y$ and $z$.

2018 Taiwan TST Round 1, 1

Tags: geometry , circumcircle

Given a triangle $ \triangle{ABC} $ and a point $ O $. $ X $ is a point on the ray $ \overrightarrow{AC} $. Let $ X' $ be a point on the ray $ \overrightarrow{BA} $ so that $ \overline{AX} = \overline{AX_{1}} $ and $ A $ lies in the segment $ \overline{BX_{1}} $. Then, on the ray $ \overrightarrow{BC} $, choose $ X_{2} $ with $ \overline{X_{1}X_{2}} \parallel \overline{OC} $. Prove that when $ X $ moves on the ray $ \overrightarrow{AC} $, the locus of circumcenter of $ \triangle{BX_{1}X_{2}} $ is a part of a line.

2011-2012 SDML (High School), 13

Tags: algebra , system of equations , SDML High School , 2011-12 SDML Round 2

The number of solutions, in real numbers $a$, $b$, and $c$, to the system of equations $$a+bc=1,$$$$b+ac=1,$$$$c+ab=1,$$ is $\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) more than }5\text{, but finitely many}\qquad\text{(E) infinitely many}$

1994 Tournament Of Towns, (410) 1

Tags: geometry , symmetry

A triangle $ABC$ is inscribed in a circle. Let $A_1$ be the point diametrically opposed to $A$, $A_0$ be the midpoint of the side $BC$ and $A_2$ be the point symmetric to $A_1$ with respect to $A_0$; the points $B_2$ and $C_2$ are defined in a similar way starting from $B$ and $C$. Prove that the three points $A_2$, $B_2$ and $C_2$ coincide. (A Jagubjanz)

2017 IFYM, Sozopol, 5

Tags: number theory

Let $p>5$ be a prime number. Prove that there exist $m,n\in \mathbb{N}$ for which $m+n<p$ and $2^m 3^n-1$ is a multiple of $p$.

May Olympiad L2 - geometry, 2001.2

Tags: geometry , trapezoid , right angle

On the trapezoid $ABCD$ , side $DA$ is perpendicular to the bases $AB$ and $CD$ . The base $AB$ measures $45$, the base $CD$ measures $20$ and the $BC$ side measures $65$. Let $P$ on the $BC$ side such that $BP$ measures $45$ and $M$ is the midpoint of $DA$. Calculate the measure of the $PM$ segment.

2005 Sharygin Geometry Olympiad, 2

Tags: geometry , perimeter , area , polygon

Cut a cross made up of five identical squares into three polygons, equal in area and perimeter.

2024 OMpD, 1

Tags: algebra , number theory

Let $O, M, P$ and $D$ be distinct digits from each other, and different from zero, such that $O < M < P < D$, and the following equation is true: \[ \overline{\text{OMPD}} \times \left( \overline{\text{OM}} - \overline{\text{D}} \right) = \overline{\text{MDDMP}} - \overline{\text{OM}} \] (a) Using estimates, explain why it is impossible for the value of $O$ to be greater than or equal to $3$. (b) Explain why $O$ cannot be equal to $1$. (c) Is it possible for $M$ to be greater than or equal to $5$? Justify. (d) Determine the values of $M$, $P$, and $D$.

2001 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: geometry , angle bisector

Let $ABC$ be a acute-angled triangle with centroid $G$, the angle bisector of $\angle ABC$ intersects $AC$ in $D$. Let $P$ and $Q$ be points in $BD$ where $\angle PBA = \angle PAB$ and $\angle QBC = \angle QCB$. Let $M$ be the midpoint of $QP$, let $N$ be a point in the line $GM$ such that $GN = 2GM$(where $G$ is the segment $MN$), prove that: $\angle ANC + \angle ABC = 180$

2017 Junior Balkan Team Selection Tests - Romania, 3

Tags: algebra , inequalities

Prove that if $a,b,c, d \in [1,2]$, then $$\frac{a + b}{b + c}+\frac{c + d}{d + a}\le 4 \frac{a + c}{b + d}$$ When does the equality hold?

2008 Harvard-MIT Mathematics Tournament, 17

Tags:

Solve the equation \[ \sqrt {x \plus{} \sqrt {4x \plus{} \sqrt {16x \plus{} \sqrt {\dotsc \plus{} \sqrt {4^{2008}x \plus{} 3}}}}} \minus{} \sqrt {x} \equal{} 1. \] Express your answer as a reduced fraction with the numerator and denominator written in their prime factorization.

2017 Saudi Arabia BMO TST, 2

Tags: function , Functional inequality , functional , algebra

Let $R^+$ be the set of positive real numbers. Find all function $f : R^+ \to R$ such that, for all positive real number $x$ and $y$, the following conditions are satisfied: i) $2f (x) + 2f (y) \le f (x + y)$ ii) $(x + y)[y f (x) + x f (y)] \ge x y f (x + y)$

2016 All-Russian Olympiad, 5

Tags: number theory , Integer Polynomial , algebra , polynomial

Let $n$ be a positive integer and let $k_0,k_1, \dots,k_{2n}$ be nonzero integers such that $k_0+k_1 +\dots+k_{2n}\neq 0$. Is it always possible to a permutation $(a_0,a_1,\dots,a_{2n})$ of $(k_0,k_1,\dots,k_{2n})$ so that the equation \begin{align*} a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_0=0 \end{align*} has not integer roots?

2021 Iran Team Selection Test, 6

Tags: set theory , combinatorics , graph theory , Coloring