Olympiad problems

2013 Hanoi Open Mathematics Competitions, 6

Let $ABC$ be a triangle with area $1$ (cm$^2$). Points $D,E$ and $F$ lie on the sides $AB, BC$ and CA, respectively. Prove that $min\{$area of $\vartriangle ADF,$ area of $\vartriangle BED,$ area of $\vartriangle CEF\} \le \frac14$ (cm$^2$).

2000 IMC, 6

Tags: function , limit , integration , real analysis

Let $f: \mathbb{R}\rightarrow ]0,+\infty[$ be an increasing differentiable function with $\lim_{x\rightarrow+\infty}f(x)=+\infty$ and $f'$ is bounded, and let $F(x)=\int^x_0 f(t) dt$. Define the sequence $(a_n)$ recursively by $a_0=1,a_{n+1}=a_n+\frac1{f(a_n)}$ Define the sequence $(b_n)$ by $b_n=F^{-1}(n)$. Prove that $\lim_{x\rightarrow+\infty}(a_n-b_n)=0$.

1984 Balkan MO, 2

Tags: parallelogram , circumcircle , ratio , geometric transformation , homothety , reflection , geometry

Let $ABCD$ be a cyclic quadrilateral and let $H_{A}, H_{B}, H_{C}, H_{D}$ be the orthocenters of the triangles $BCD$, $CDA$, $DAB$ and $ABC$ respectively. Show that the quadrilaterals $ABCD$ and $H_{A}H_{B}H_{C}H_{D}$ are congruent.

2019 Romania Team Selection Test, 1

Tags: inequalities , algebra

Determine the largest value the expression $$ \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} $$ may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-negative real numbers, and add up to $ 1. $ Find also the specific values of this numbers that make the above sum achieve the asked maximum.

2016 Middle European Mathematical Olympiad, 3

Tags: circumcenter , perpendicular lines , geometry

Let $ABC$ be an acute triangle such that $\angle BAC > 45^{\circ}$ with circumcenter $O$. A point $P$ is chosen inside triangle $ABC$ such that $A, P, O, B$ are concyclic and the line $BP$ is perpendicular to the line $CP$. A point $Q$ lies on the segment $BP$ such that the line $AQ$ is parallel to the line $PO$. Prove that $\angle QCB = \angle PCO$.

1971 IMO Longlists, 12

Tags: algebra , logarithm

A system of n numbers $x_1, x_2, \ldots, x_n$ is given such that \[x_1 = \log_{x_{n-1}} x_n, x_2 = \log_{x_{n}} x_1, \ldots, x_n = \log_{x_{n-2}} x_{n-1}.\] Prove that $\prod_{k=1}^n x_k =1.$

1987 Tournament Of Towns, (132) 1

Tags: inequalities , algebra

Prove that for all values of $a$, $3(1+a^2+a^4) \ge (1+a+a^2)^2$ .

1983 IMO Longlists, 74

Tags: hyperbola , asymptote , geometry , circumcircle , analytic geometry , perpendicular bisector , conic

In a plane we are given two distinct points $A,B$ and two lines $a, b$ passing through $B$ and $A$ respectively $(a \ni B, b \ni A)$ such that the line $AB$ is equally inclined to a and b. Find the locus of points $M$ in the plane such that the product of distances from $M$ to $A$ and a equals the product of distances from $M$ to $B$ and $b$ (i.e., $MA \cdot MA' = MB \cdot MB'$, where $A'$ and $B'$ are the feet of the perpendiculars from $M$ to $a$ and $b$ respectively).

2012 Today's Calculation Of Integral, 838

Tags: calculus , integration , inequalities , calculus computations

Prove that : $\frac{e-1}{e}<\int_0^1 e^{-x^2}dx<\frac{\pi}{4}.$

2004 Thailand Mathematical Olympiad, 2

Tags: algebra , system of equations

Let $a$ and $b$ be real numbers such that $$\begin{cases} a^6 - 3a^2b^4 = 3 \\ b^6 - 3a^4b^2 = 3\sqrt2.\end{cases}$$ What is the value of $a^4 + b^4$ ?

2001 Baltic Way, 14

Tags: algebra

There are $2n$ cards. On each card some real number $x$, $(1\le x\le 2n)$, is written (there can be different numbers on different cards). Prove that the cards can be divided into two heaps with sums $s_1$ and $s_2$ so that $\frac{n}{n+1}\le\frac{s_1}{s_2}\le 1$.

1988 All Soviet Union Mathematical Olympiad, 481

Tags: geometry , cube , 3d geometry , broken line , minimum

A polygonal line connects two opposite vertices of a cube with side $2$. Each segment of the line has length $3$ and each vertex lies on the faces (or edges) of the cube. What is the smallest number of segments the line can have?

1995 AMC 12/AHSME, 16

Tags: percent

Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that: i. The actual attendance in Atlanta is within $10 \%$ of Anita's estimate. ii. Bob's estimate is within $10 \%$ of the actual attendance in Boston. To the nearest 1,000, the largest possible difference between the numbers attending the two games is $\textbf{(A)}\ 10,000 \qquad \textbf{(B)}\ 11,000 \qquad \textbf{(C)}\ 20,000 \qquad \textbf{(D)}\ 21,000 \qquad \textbf{(E)}\ 22,000$

2021 Canadian Mathematical Olympiad Qualification, 6

Tags: diophantine equation , algebra

Show that $(w, x, y, z)=(0,0,0,0)$ is the only integer solution to the equation $$w^{2}+11 x^{2}-8 y^{2}-12 y z-10 z^{2}=0$$

2014 China Team Selection Test, 5

Tags: complex number , inequalities

Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.

2013 Thailand Mathematical Olympiad, 12

Tags: geometry , incircle , area

Let $\omega$ be the incircle of $\vartriangle ABC$, $\omega$ is tangent to sides $BC$ and $AC$ at $D$ and $E$ respectively. The line perpendicular to $BC$ at $D$ intersects $\omega$ again at $P$. Lines $AP$ and $BC$ intersect at $M$. Let $N$ be a point on segment $AC$ so that $AE = CN$. Line $BN$ intersects $\omega$ at $Q$ (closer to $B$) and intersect $AM$ at $R$. Show that the area of $\vartriangle ABR$ is equal to the area of $PQMN$.

2016 IMO, 4

Tags: number theory , polynomial , algebra , euclidean algorithm

A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?

1970 IMO Longlists, 3

Tags: factorial

Prove that $(a!\cdot b!) | (a+b)!$ $\forall a,b\in\mathbb{N}$.

1998 VJIMC, Problem 4-I

Tags: pascal

Prove that there exists a program in standard Pascal which prints out its own ASCII code. No disk operations are permitted.

2001 Tournament Of Towns, 3

Tags: perpendicular bisector , geometry

Points $X$ and $Y$ are chosen on the sides $AB$ and $BC$ of the triangle $\triangle ABC$. The segments $AY$ and $CX$ intersect at the point $Z$. Given that $AY = YC$ and $AB = ZC$, prove that the points $B$, $X$, $Z$, and $Y$ lie on the same circle.

1978 IMO Longlists, 6

Tags: polynomial , function , trigonometry , geometry , algebra

Prove that for all $X > 1$, there exists a triangle whose sides have lengths $P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1$, and $P_3(X) = X^4-1$. Prove that all these triangles have the same greatest angle and calculate it.

2021 Science ON grade X, 1

Tags: algebra , complex number , inequality

Consider the complex numbers $x,y,z$ such that $|x|=|y|=|z|=1$. Define the number $$a=\left (1+\frac xy\right )\left (1+\frac yz\right )\left (1+\frac zx\right ).$$ $\textbf{(a)}$ Prove that $a$ is a real number. $\textbf{(b)}$ Find the minimal and maximal value $a$ can achieve, when $x,y,z$ vary subject to $|x|=|y|=|z|=1$. [i] (Stefan Bălăucă & Vlad Robu)[/i]

2015 Online Math Open Problems, 16

Tags:

Given a (nondegenrate) triangle $ABC$ with positive integer angles (in degrees), construct squares $BCD_1D_2, ACE_1E_2$ outside the triangle. Given that $D_1, D_2, E_1, E_2$ all lie on a circle, how many ordered triples $(\angle A, \angle B, \angle C)$ are possible? [i]Proposed by Yang Liu[/i]

2009 Regional Competition For Advanced Students, 2

Tags: combinatorics

How many integer solutions $ (x_0$, $ x_1$, $ x_2$, $ x_3$, $ x_4$, $ x_5$, $ x_6)$ does the equation \[ 2x_0^2\plus{}x_1^2\plus{}x_2^2\plus{}x_3^2\plus{}x_4^2\plus{}x_5^2\plus{}x_6^2\equal{}9\] have?

2013 Stars Of Mathematics, 1

Tags: function , algebra , functional inequality

Let $\mathcal{F}$ be the family of bijective increasing functions $f\colon [0,1] \to [0,1]$, and let $a \in (0,1)$. Determine the best constants $m_a$ and $M_a$, such that for all $f \in \mathcal{F}$ we have \[m_a \leq f(a) + f^{-1}(a) \leq M_a.\] [i](Dan Schwarz)[/i]