Olympiad problems

2022 All-Russian Olympiad, 2

Tags: geometry

Given is triangle $ABC$ with incenter $I$ and $A$-excenter $J$. Circle $\omega_b$ centered at point $O_b$ passes through point $B$ and is tangent to line $CI$ at point $I$. Circle $\omega_c$ with center $O_c$ passes through point $C$ and touches line $BI$ at point $I$. Let $O_bO_c$ and $IJ$ intersect at point $K$. Find the ratio $IK/KJ$.

2016 AMC 10, 4

Tags:

Zoey read $15$ books, one at a time. The first book took her $1$ day to read, the second book took her $2$ days to read, the third book took her $3$ days to read, and so on, with each book taking her $1$ more day to read than the previous book. Zoey finished the first book on a monday, and the second on a Wednesday. On what day the week did she finish her $15$th book? $\textbf{(A)}\ \text{Sunday}\qquad\textbf{(B)}\ \text{Monday}\qquad\textbf{(C)}\ \text{Wednesday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$

2020 SJMO, 1

Tags: number theory

Find all positive integers $k \geq 2$ for which there exists some positive integer $n$ such that the last $k$ digits of the decimal representation of $10^{10^n} - 9^{9^n}$ are the same. [i]Proposed by Andrew Wen[/i]

2017 Swedish Mathematical Competition, 4

Tags: angle chasing , angle , angle bisector , geometry

Let $D$ be the foot of the altitude towards $BC$ in the triangle $ABC$. Let $E$ be the intersection of $AB$ with the bisector of angle $\angle C$. Assume that the angle $\angle AEC = 45^o$ . Determine the angle $\angle EDB$.

2022 IFYM, Sozopol, 4

Tags: algebra

Let $n$ be a natural number. To prove that the value of the expression $$\prod^n_{i=0}\frac{x^{n-1}_i}{\prod_{j \ne i}(x_i - x_j)}$$ does not depend on the choice of the different real numbers $x_0, x_1, ... , x_n$.

2020/2021 Tournament of Towns, P3

Tags: number theory

For which $n{}$ is it possible that a product of $n{}$ consecutive positive integers is equal to a sum of $n{}$ consecutive (not necessarily the same) positive integers? [i]Boris Frenkin[/i]

2021 Girls in Mathematics Tournament, 2

Tags: angle , geometry

Let $\vartriangle ABC$ be a triangle in which $\angle ACB = 40^o$ and $\angle BAC = 60^o$ . Let $D$ be a point inside the segment $BC$ such that $CD =\frac{AB}{2}$ and let $M$ be the midpoint of the segment $AC$. How much is the angle $\angle CMD$ in degrees?

1999 AMC 8, 7

Tags:

The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center? $ \text{(A)}\ 90\qquad\text{(B)}\ 100\qquad\text{(C)}\ 110\qquad\text{(D)}\ 120\qquad\text{(E)}\ 130 $

PEN A Problems, 95

Tags: divisibility theory

Suppose that $a$ and $b$ are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of $p$?

2021 Francophone Mathematical Olympiad, 1

Tags: algebra , inequalities

Let $R$ and $S$ be the numbers defined by \[R = \dfrac{1}{2} \times \dfrac{3}{4} \times \dfrac{5}{6} \times \cdots \times \dfrac{223}{224} \text{ and } S = \dfrac{2}{3} \times \dfrac{4}{5} \times \dfrac{6}{7} \times \cdots \times \dfrac{224}{225}.\]Prove that $R < \dfrac{1}{15} < S$.

2023 HMNT, 28

Tags:

There is a unique quadruple of positive integers $(a,b,c,k)$ such that $c$ is not a perfect square and $a+\sqrt{b+\sqrt{c}}$ is a root of the polynomial $x^4-20x^3+108x^2-kx+9.$ Compute $c.$

2022 Turkey Team Selection Test, 8

Tags: orthocenter , incircle , geometry

$ABC$ triangle with $|AB|<|BC|<|CA|$ has the incenter $I$. The orthocenters of triangles $IBC, IAC$ and $IAB$ are $H_A, H_A$ and $H_A$. $H_BH_C$ intersect $BC$ at $K_A$ and perpendicular line from $I$ to $H_BH_B$ intersect $BC$ at $L_A$. $K_B, L_B, K_C, L_C$ are defined similarly. Prove that $$|K_AL_A|=|K_BL_B|+|K_CL_C|$$

2021 Thailand Mathematical Olympiad, 10

Tags: polynomial , algebra

Let $d\geq 13$ be an integer, and let $P(x) = a_dx^d + a_{d-1}x^{d-1} + \dots + a_1x+a_0$ be a polynomial of degree $d$ with complex coefficients such that $a_n = a_{d-n}$ for all $n\in\{0,1,\dots,d\}$. Prove that if $P$ has no double roots, then $P$ has two distinct roots $z_1$ and $z_2$ such that $|z_1-z_2|<1$.

2003 Portugal MO, 4

Tags: cyclic , distance , geometry

In a village there are only $10$ houses, arranged in a circle of a radius $r$ meters. Each has is the same distance from each of the two closest houses. Every year on Sunday of Pascoa, the village priest makes the Easter visit, leaving the parish house (point $A$) and following the path described in Figure 1. This year the priest decided to take the path represented in the Figure 2. Prove that this year the priest will walk another $10r$ meters. [img]https://cdn.artofproblemsolving.com/attachments/a/9/a6315f4a63f28741ca6fbc75c19a421eb1da06.png[/img]

2016 Denmark MO - Mohr Contest, 5

Tags: number theory

Find all possible values of the number $$\frac{a + b}{c}+\frac{a + c}{b}+\frac{b + c}{a},$$ where $a, b, c$ are positive integers, and $\frac{a + b}{c},\frac{a + c}{b},\frac{b + c}{a}$ are also positive integers.

2017 Novosibirsk Oral Olympiad in Geometry, 1

Tags: grid , min , geometry

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

2012 USA Team Selection Test, 1

Tags: polynomial , algebra

Consider (3-variable) polynomials \[P_n(x,y,z)=(x-y)^{2n}(y-z)^{2n}+(y-z)^{2n}(z-x)^{2n}+(z-x)^{2n}(x-y)^{2n}\] and \[Q_n(x,y,z)=[(x-y)^{2n}+(y-z)^{2n}+(z-x)^{2n}]^{2n}.\] Determine all positive integers $n$ such that the quotient $Q_n(x,y,z)/P_n(x,y,z)$ is a (3-variable) polynomial with rational coefficients.

2015 ASDAN Math Tournament, 3

Tags: algebra test

Let $a_1,a_2,a_3,\dots,a_6$ be an arithmetic sequence with common difference $3$. Suppose that $a_1$, $a_3$, and $a_6$ also form a geometric sequence. Compute $a_1$.

2007 QEDMO 4th, 1

Tags: diophantine equation , diophantine , number theory

Find all primes $p,$ $q,$ $r$ satisfying $p^{2}+2q^{2}=r^{2}.$

2016 Turkey EGMO TST, 5

Tags: combinatorics , periodic sequence , sequence , algebra

A sequence $a_1, a_2, \ldots $ consisting of $1$'s and $0$'s satisfies for all $k>2016$ that \[ a_k=0 \quad \Longleftrightarrow \quad a_{k-1}+a_{k-2}+\cdots+a_{k-2016}>23. \] Prove that there exist positive integers $N$ and $T$ such that $a_k=a_{k+T}$ for all $k>N$.

2023 SG Originals, Q5

Tags: area of a triangle , expected value , combinatorics , geometry

A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$? [i]Proposed by Dylan Toh[/i]

1998 Putnam, 5

Tags:

Let $\mathcal{F}$ be a finite collection of open discs in $\mathbb{R}^2$ whose union contains a set $E\subseteq \mathbb{R}^2$. Show that there is a pairwise disjoint subcollection $D_1,\ldots,D_n$ in $\mathcal{F}$ such that \[E\subseteq\cup_{j=1}^n 3D_j.\] Here, if $D$ is the disc of radius $r$ and center $P$, then $3D$ is the disc of radius $3r$ and center $P$.

2018 Math Prize for Girls Problems, 7

Tags:

For every positive integer $n$, let $T_n = \frac{n(n+1)}{2}$ be the $n^{\text{th}}$ triangular number. What is the $2018^{\text{th}}$ smallest positive integer $n$ such that $T_n$ is a multiple of 1000?

VI Soros Olympiad 1999 - 2000 (Russia), 11.3

Tags: algebra , inequalities

The numbers $a, b$ and $c$ are such that $a^2 + b^2 + c^2 = 1$. Prove that $$a^4 + b^4 + c^4 + 2(ab^2 + bc^2 + ca^2)^2\le 1. $$ At what $a, b$ and $c$ does inequality turn into equality?

2011 USA Team Selection Test, 3

Tags: algebra , polynomial , function , number theory

Let $p$ be a prime. We say that a sequence of integers $\{z_n\}_{n=0}^\infty$ is a [i]$p$-pod[/i] if for each $e \geq 0$, there is an $N \geq 0$ such that whenever $m \geq N$, $p^e$ divides the sum \[\sum_{k=0}^m (-1)^k {m \choose k} z_k.\] Prove that if both sequences $\{x_n\}_{n=0}^\infty$ and $\{y_n\}_{n=0}^\infty$ are $p$-pods, then the sequence $\{x_ny_n\}_{n=0}^\infty$ is a $p$-pod.