Olympiad problems

2024/2025 TOURNAMENT OF TOWNS, P3

Tags: geometry

In an acute-angled triangle ${ABC}$ , its incenter $I$ and circumcenter $O$ are marked. The lines ${AI}$ and ${CI}$ have second intersections with the circumcircle of ${ABC}$ at points $N$ and $M$ respectively. The segments ${MN}$ and ${BO}$ intersect at the point $X$ . Prove that the lines ${XI}$ and ${AC}$ are perpendicular. Fedor Ivlev

2024 Indonesia TST, A

Tags: algebra , inequalities , absolute value

Given real numbers $x,y,z$ which satisfies $$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$ Show that $max\{ |x|,|y|,|z|\} \le 1$.

1983 Polish MO Finals, 2

Tags: inequalities , algebra , irrational

Let be given an irrational number $a$ in the interval $(0,1)$ and a positive integer $N$. Prove that there exist positive integers $p,q,r,s$ such that $\frac{p}{q} < a <\frac{r}{s}, \frac{r}{s} -\frac{p}{q}<\frac{1}{N}$, and $rq- ps = 1$.

1999 VJIMC, Problem 2

Tags: limit , real analysis , function

Let $a,b\in\mathbb R$, $a\le b$. Assume that $f:[a,b]\to[a,b]$ satisfies $f(x)-f(y)\le|x-y|$ for every $x,y\in[a,b]$. Choose an $x_1\in[a,b]$ and define $$x_{n+1}=\frac{x_n+f(x_n)}2,\qquad n=1,2,3,\ldots.$$Show that $\{x_n\}^\infty_{n=1}$ converges to some fixed point of $f$.

2016 Thailand Mathematical Olympiad, 7

Tags: polynomial , absolute value , algebra

Given $P(x)=a_{2016}x^{2016}+a_{2015}x^{2015}+...+a_1x+a_0$ be a polynomial with real coefficients and $a_{2016} \neq 0$ satisfies $|a_1+a_3+...+a_{2015}| > |a_0+a_2+...+a_{2016}|$ Prove that $P(x)$ has an odd number of complex roots with absolute value less than $1$ (count multiple roots also) edited: complex roots

2024 Al-Khwarizmi IJMO, 7

Tags: geometry

Two circles with centers $O_{1}$ and $O_{2}$ intersect at $P$ and $Q$. Let $\omega$ be the circumcircle of the triangle $P O_{1} O_{2}$; the circle $\omega$ intersect the circles centered at $O_{1}$ and $O_{2}$ at points $A$ and $B$, respectively. The point $Q$ is inside triangle $P A B$ and $P Q$ intersects $\omega$ at $M$. The point $E$ on $\omega$ is such that $P Q=Q E$. Let $M E$ and $A B$ meet at $L$, prove that $\angle Q L A=\angle M L A$. [i]Proposed by Amir Parsa Hoseini Nayeri, Iran[/i]

2004 Regional Olympiad - Republic of Srpska, 1

Tags: algebra

Find all real solutions of the equation \[\sqrt[3]{x-1}+\sqrt[3]{3x-1}=\sqrt[3]{x+1}.\]

2017 Iran MO (3rd round), 2

Tags: number theory , number theory with sequences , sequence

Consider a sequence $\{a_i\}^\infty_{i\ge1}$ of positive integers. For all positvie integers $n$ prove that there exists infinitely many positive integers $k$ such that there is no pair $(m,t)$ of positive integers where $m>n$ and $$kn+a_n=tm(m+1)+a_m$$

2004 Swedish Mathematical Competition, 1

Tags: geometry , circles , area

Two circles in the plane, both of radius $R$, intersect at a right angle. Compute the area of the intersection of the interiors of the two circles.

2010 Serbia National Math Olympiad, 1

Tags: inequalities , floor function , combinatorics

Some of $n$ towns are connected by two-way airlines. There are $m$ airlines in total. For $i = 1, 2, \cdots, n$, let $d_i$ be the number of airlines going from town $i$. If $1\le d_i \le 2010$ for each $i = 1, 2,\cdots, 2010$, prove that \[\displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n\] Find all $n$ for which equality can be attained. [i]Proposed by Aleksandar Ilic[/i]

1962 IMO, 1

Tags: number theory , decimal representation , divisibility

Find the smallest natural number $n$ which has the following properties: a) Its decimal representation has a 6 as the last digit. b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.

2020 LIMIT Category 2, 16

Tags: limit , derivative , calculus

The $n^{th}$ derivative of a function $f(x)$ (if it exists) is denoted by $f^{(n)}(x) $. Let $f(x)=\frac{e^x}{x}$. Suppose $f$ is differentiable infinitely many times in $(0,\infty) $. Then find $\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}$

2014 Contests, 2

Tags: calculus , integration , parabola , geometry , calculus computations , conic

Let $l$ be the tangent line at the point $(t,\ t^2)\ (0<t<1)$ on the parabola $C: y=x^2$. Denote by $S_1$ the area of the part enclosed by $C,\ l$ and the $x$-axis, denote by $S_2$ of the area of the part enclosed by $C,\ l$ and the line $x=1$. Find the minimum value of $S_1+S_2$.

2016 NZMOC Camp Selection Problems, 3

Tags: equilateral , arc , geometry

Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $\overarc {AB}$ . Prove that $AD + BD = DC$.

2010 Today's Calculation Of Integral, 659

Tags: calculus , integration , geometric series , calculus computations , logarithm

Evaluate $\int_0^1 \frac{\ln (x+2)}{x+1}dx.$

1952 Putnam, A2

Tags:

Show that the equation \[ (9 - x^2) \left (\frac{\mathrm dy}{\mathrm dx} \right)^2 = (9 - y^2)\] characterizes a family of conics touching the four sides of a fixed square.

1991 IMO Shortlist, 20

Tags: floor function

Let $ \alpha$ be the positive root of the equation $ x^{2} \equal{} 1991x \plus{} 1$. For natural numbers $ m$ and $ n$ define \[ m*n \equal{} mn \plus{} \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor. \] Prove that for all natural numbers $ p$, $ q$, and $ r$, \[ (p*q)*r \equal{} p*(q*r). \]

2006 India IMO Training Camp, 2

Tags: geometry , parallelogram , homothety , incenter , exterior angle , spiral similarity

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

V Soros Olympiad 1998 - 99 (Russia), 11.1

Tags: algebra , trigonometry , logarithm

Find at least one root of the equation$$\sin(2 \log_2 x) + tg(3\log_2 x) = \sin6+tg9$$less than $0.01$.

2019 Moldova Team Selection Test, 1

Tags: number theory , easy

Let $S$ be the set of all natural numbers with the property: the sum of the biggest three divisors of number $n$, different from $n$, is bigger than $n$. Determine the largest natural number $k$, which divides any number from $S$. (A natural number is a positive integer)

2024 Taiwan TST Round 2, C

Tags: combinatorics

Find all functions $f:\mathbb{N}\to\mathbb{N}$ s.t. for all $A\subset \mathbb{N}$ with 2024 elements, the set $$S_A:=\{f^{(k)}(x)\mid k=1,...,2024,x\in A\}$$ also has 2024 elements. ($f^{(k)}=f\circ f\circ...\circ f$ is the $k$-th iteration of $f$.)

1991 IMO Shortlist, 5

Tags: geometry , incenter , trigonometry , trig identities , law of sines

In the triangle $ ABC,$ with $ \angle A \equal{} 60 ^{\circ},$ a parallel $ IF$ to $ AC$ is drawn through the incenter $ I$ of the triangle, where $ F$ lies on the side $ AB.$ The point $ P$ on the side $ BC$ is such that $ 3BP \equal{} BC.$ Show that $ \angle BFP \equal{} \frac{\angle B}{2}.$

2014 Purple Comet Problems, 27

Tags: probability , vector , algorithm , number theory , relatively prime

Five men and five women stand in a circle in random order. The probability that every man stands next to at least one woman is $\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2020 CHMMC Winter (2020-21), 2

Tags: number theory

Find the sum of all positive integers $x < 241$ such that both $x^{24} + x^{18} + x^{12} + x^6 + 1$ and $x^{20} + x^{10} + 1$ are multiples of $241$.

2024 Thailand TST, 1

Tags: arithmetic sequence , number theory

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.