Olympiad problems

1991 IMO Shortlist, 29

We call a set $ S$ on the real line $ \mathbb{R}$ [i]superinvariant[/i] if for any stretching $ A$ of the set by the transformation taking $ x$ to $ A(x) \equal{} x_0 \plus{} a(x \minus{} x_0), a > 0$ there exists a translation $ B,$ $ B(x) \equal{} x\plus{}b,$ such that the images of $ S$ under $ A$ and $ B$ agree; i.e., for any $ x \in S$ there is a $ y \in S$ such that $ A(x) \equal{} B(y)$ and for any $ t \in S$ there is a $ u \in S$ such that $ B(t) \equal{} A(u).$ Determine all [i]superinvariant[/i] sets.

2006 MOP Homework, 4

Tags: combinatorics

A $k$-coloring of a graph $G$ is a coloring of its vertices using $k$ possible colors such that the end points of any edge have different colors. We say a graph $G$ is uniquely $k$-colorable if one hand it has a $k$-coloring, on the other hand there do not exist vertices $u$ and $v$ such that $u$ and $v$ have the same color in one $k$-coloring and $u$ and $v$ have different colors in another $k$-coloring. Prove that if a graph $G$ with $n$ vertices $(n \ge 3)$ is uniquely $3$-colorable, then it has at least $2n-3$ edges.

2003 Junior Tuymaada Olympiad, 4

Tags: algebra , sum , maximum value , maximum , inequalities

The natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_n $ satisfy the condition $ 1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1 $. Prove that all these numbers do not exceed $$ n ^ {2 ^ n} $$

2008 Postal Coaching, 3

Tags: geometry , max , median , angle bisector

Let $ABC$ be a triangle. For any point $X$ on $BC$, let $AX$ meet the circumcircle of $ABC$ in $X'$. Prove or disprove: $XX'$ has maximum length if and only if $AX$ lies between the median and the internal angle bisector from $A$.

2010 Today's Calculation Of Integral, 613

Tags: calculus , integration , geometry , calculus computations , logarithm

Find the area of the part, in the $x$-$y$ plane, enclosed by the curve $|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).$ [i]2010 Tokyo University of Agriculture and Technology entrance exam[/i]

2013 Kosovo National Mathematical Olympiad, 2

Tags: trigonometry

Solve equation $27\cdot3^{3\sin x}=9^{\cos^2x}$ where $x\in [0,2\pi )$

2004 Harvard-MIT Mathematics Tournament, 7

Tags: calculus , geometry

Find the area of the region in the $xy$-plane satisfying $x^6-x^2+y^2 \le 0$.

1990 Tournament Of Towns, (273) 1

Tags: combinatorics

The positive integers from $1$ to $n^2$ are placed arbitrarily on the squares of a chess board with dimensions $n\times n$. Prove that there are two adjacent squares (having a common vertex or a common side) such that the difference between the numbers placed on them is not less than $n + 1$. (N Sedrakyan, Yerevan)

2021 Azerbaijan IMO TST, 1

Tags: algebra , polynomial , linear combination

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

2002 AMC 10, 6

Tags: perimeter , rectangle , geometry

The perimeter of a rectangle is $100$ and its diagonal has length $x$. What is the area of this rectangle? $\textbf{(A) }625-x^2\qquad\textbf{(B) }625-\dfrac{x^2}2\qquad\textbf{(C) }1250-x^2\qquad\textbf{(D) }1250-\dfrac{x^2}2\qquad\textbf{(E) }2500-\dfrac{x^2}2$

2019 CCA Math Bonanza, L5.4

Tags:

Submit an integer between $0$ and $100$ inclusive as your answer to this problem. Suppose that $Q_1$ and $Q_3$ are the medians of the smallest $50\%$ and largest $50\%$ of submissions for this question. Your goal is to have your submission close to $D=Q_3-Q_1$. If you submit $N$, your score will be $2-2\sqrt{\frac{\left|N-D\right|}{\max\left\{D,100-D\right\}}}$. [i]2019 CCA Math Bonanza Lightning Round #5.4[/i]

2018 Sharygin Geometry Olympiad, 6

Tags: geometry

Let $CH$ be the altitude of a right-angled triangle $ABC$ ($\angle C = 90^{\circ}$) with $BC = 2AC$. Let $O_1$, $O_2$ and $O$ be the incenters of triangles $ACH$, $BCH$ and $ABC$ respectively, and $H_1$, $H_2$, $H_0$ be the projections of $O_1$, $O_2$, $O$ respectively to $AB$. Prove that $H_1H = HH_0 = H_0H_2$.

2020 Malaysia IMONST 1, 3

Tags: geometry , circles , square

Given a square with area $A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $\frac{A}{B}.$

2022 Purple Comet Problems, 10

Tags:

Find the positive integer $n$ such that a convex polygon with $3n + 2$ sides has $61.5$ percent fewer diagonals than a convex polygon with $5n - 2$ sides.

1990 AMC 12/AHSME, 11

Tags:

How man y positive integers less than $50$ have an odd number of positive integer divisors? $\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 11$

2023 District Olympiad, P3

Tags: algebra , inequalities

Let $x,y{}$ and $z{}$ be positive real numbers satisfying $x+y+z=1$. Prove that [list=a] [*]\[1-\frac{x^2-yz}{x^2+x}=\frac{(1-y)(1-z)}{x^2+x};\] [*]\[\frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leqslant 0.\] [/list]

2007 AMC 12/AHSME, 14

Tags: vieta

Let $ a,$ $ b,$ $ c,$ $ d,$ and $ e$ be distinct integers such that \[ (6 \minus{} a)(6 \minus{} b)(6 \minus{} c)(6 \minus{} d)(6 \minus{} e) \equal{} 45. \]What is $ a \plus{} b \plus{} c \plus{} d \plus{} e?$ $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 17 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 30$

2008 Bulgarian Autumn Math Competition, Problem 12.3

Tags: function , algebra , functional equation in r , functional equation

Find all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that \[(f(x)f(y)-1)f(x+y)=2f(x)f(y)-f(x)-f(y)\quad \forall x,y\in \mathbb{R}\]

1935 Moscow Mathematical Olympiad, 013

Tags: geometry , geometric construction , triangle construction

The median, bisector, and height, all originate at the same vertex of a triangle. Given the intersection points of the median, bisector, and height with the circumscribed circle, construct the triangle.

2002 HKIMO Preliminary Selection Contest, 17

Tags: algebra

Let $a_0=2$ and for $n\geq 1$, $a_n=\frac{\sqrt3 a_{n-1}+1}{\sqrt3-a_{n-1}}$. Find the value of $a_{2002}$ in the form $p+q\sqrt3$ where $p$ and $q$ are rational numbers

2022 Purple Comet Problems, 4

Tags:

Of $450$ students assembled for a concert, $40$ percent were boys. After a bus containing an equal number of boys and girls brought more students to the concert, $41$ percent of the students at the concert were boys. Find the number of students on the bus.

2001 India National Olympiad, 2

Tags: modular arithmetic , polynomial , arithmetic sequence , number theory , algebra

Show that the equation $x^2 + y^2 + z^2 = ( x-y)(y-z)(z-x)$ has infintely many solutions in integers $x,y,z$.

2018 Hanoi Open Mathematics Competitions, 15

Tags: diophantine equation , number theory

Find all pairs of prime numbers $(p,q)$ such that for each pair $(p,q)$, there is a positive integer m satisfying $\frac{pq}{p + q}=\frac{m^2 + 6}{m + 1}$.

2013 All-Russian Olympiad, 2

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

Circle is divided into $n$ arcs by $n$ marked points on the circle. After that circle rotate an angle $ 2\pi k/n $ (for some positive integer $ k $), marked points moved to $n$ [i] new points [/i], dividing the circle into $ n $ [i] new arcs[/i]. Prove that there is a new arc that lies entirely in the one of the old arсs. (It is believed that the endpoints of arcs belong to it.) [i]I. Mitrophanov[/i]

2011 N.N. Mihăileanu Individual, 4

Tags: calculus , integration , real analysis

Let be the sequence $ \left( I_n \right)_{n\ge 1} $ defined as $ I_n=\int_0^1 \frac{x^n}{\sqrt{x^{2n} +1}} dx . $ [b]a)[/b] Show that $ \left( I_n \right)_{n\ge 1} $ converges to $ 0. $ [b]b)[/b] Calculate $ \lim_{m\to\infty } m\cdot I_m. $ [b]c)[/b] Prove that the sequence $ \left( n\left( -n\cdot I_n +\lim_{m\to\infty } m\cdot I_m \right) \right)_{n\ge 1} $ is convergent.