Olympiad problems

1974 Vietnam National Olympiad, 3

Let $ABC$ be a triangle with $A = 90^o, AH$ the altitude, $P,Q$ the feet of the perpendiculars from $H$ to $AB,AC$ respectively. Let $M$ be a variable point on the line $PQ$. The line through $M$ perpendicular to $MH$ meets the lines $AB,AC$ at $R, S$ respectively. i) Prove that circumcircle of $ARS$ always passes the fixed point $H$. ii) Let $M_1$ be another position of $M$ with corresponding points $R_1, S_1$. Prove that the ratio $RR_1/SS_1$ is constant. iii) The point $K$ is symmetric to $H$ with respect to $M$. The line through $K$ perpendicular to the line $PQ$ meets the line $RS$ at $D$. Prove that$ \angle BHR = \angle DHR, \angle DHS = \angle CHS$.

IV Soros Olympiad 1997 - 98 (Russia), 9.8

Tags: algebra , polynomial

The equation $P(x) = 0$, where $P(x) = x^2+bx+c$, has a single root, and the equation $P(P(P(x))) = 0$ has exactly three different roots. Solve the equation $P(P(P(x))) = 0.$

2021 MIG, 21

Tags:

You have a collection of $\$20.21$, consisting of pennies, nickels, and quarters. To reduce the collection’s worth to $k$ cents, you simultaneously replace all pennies with quarters and all quarters with pennies (all coins are replaced one time). What is the minimum possible $k$? $\textbf{(A) }105\qquad\textbf{(B) }120\qquad\textbf{(C) }125\qquad\textbf{(D) }505\qquad\textbf{(E) }101$

2023 Sinapore MO Open, P4

Tags: algebra , functional equation

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$, such that $$f(x+y)((f(x) - f(y))^2+f(xy))=f(x^3)+f(y^3)$$ for all integers $x, y$.

1967 Czech and Slovak Olympiad III A, 4

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle, $k$ its circumcirle and $m$ a line such that $m\cap k=\emptyset, m\parallel BC.$ Denote $D$ the intersection of $m$ and ray $AB.$ a) Let $X$ be an inner point of the arc $BC$ not containing $A$ and denote $Y$ the intersection of lines $m,CX.$ Show that $A,D,X,Y$ are concyclic and name this circle $\kappa$. b) Determine relative position of $\kappa$ and $m$ in case when $C,D,X$ are collinear.

2019 India IMO Training Camp, P2

Tags: number theory

Show that there do not exist natural numbers $a_1, a_2, \dots, a_{2018}$ such that the numbers \[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \] are all powers of $5$ [i]Proposed by Tejaswi Navilarekallu[/i]

2002 IberoAmerican, 2

Tags: induction , algebra

The sequence of real numbers $a_1,a_2,\dots$ is defined as follows: $a_1=56$ and $a_{n+1}=a_n-\frac{1}{a_n}$ for $n\ge 1$. Show that there is an integer $1\leq{k}\leq2002$ such that $a_k<0$.

2017 Purple Comet Problems, 21

Tags: geometry

The diagram below shows a large circle. Six congruent medium-sized circles are each internally tangent to the large circle and tangent to two neighboring medium-sized circles. Three congruent small circles are mutually tangent to one another and are each tangent to two medium-sized circles as shown. The ratio of the area of the large circle to the area of one of the small circles can be written as $m+\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/a/4/fcffd7ee6e8d3da0641525e7a987d13ce05496.png[/img]

1998 Hungary-Israel Binational, 2

Tags: inequalities , trigonometry , circumcircle , geometry

A triangle ABC is inscribed in a circle with center $ O$ and radius $ R$. If the inradii of the triangles $ OBC, OCA, OAB$ are $ r_{1}, r_{2}, r_{3}$ , respectively, prove that $ \frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}\geq\frac{4\sqrt{3}+6}{R}.$

2021 Chile National Olympiad, 4

Tags: geometry , concyclic , cyclic quadrilateral

Consider quadrilateral $ABCD$ with $|DC| > |AD|$. Let $P$ be a point on $DC$ such that $PC = AD$ and let $Q$ be the midpoint of $DP$. Let $L_1$ be the line perpendicular on $DC$ passing through $Q$ and let $L_2$ be the bisector of the angle $ \angle ABC$. Let us call $X = L_1 \cap L_2$. Show that if quadrilateral is cyclic then $X$ lies on the circumcircle of $ABCD.$ [img]https://cdn.artofproblemsolving.com/attachments/f/6/3ebfce8a7fd2a0ece9f09065608141006893d2.png[/img]

2022 Nigerian Senior MO Round 2, Problem 4

Tags: algebra , sequence , two sequences

Define sequence $(a_{n})_{n=1}^{\infty}$ by $a_1=a_2=a_3=1$ and $a_{n+3}=a_{n+1}+a_{n}$ for all $n \geq 1$. Also, define sequence $(b_{n})_{n=1}^{\infty}$ by $b_1=b_2=b_3=b_4=b_5=1$ and $b_{n+5}=b_{n+4}+b_{n}$ for all $n \geq 1$. Prove that $\exists N \in \mathbb{N}$ such that $a_n = b_{n+1} + b_{n-8}$ for all $n \geq N$.

2019 Taiwan TST Round 2, 2

Tags: number theory

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

1985 Dutch Mathematical Olympiad, 2

Tags: modular arithmetic , floor function , number theory

Among the numbers $ 11n \plus{} 10^{10}$, where $ 1 \le n \le 10^{10}$ is an integer, how many are squares?

2015 Sharygin Geometry Olympiad, 5

Tags: geometry , midpoint , concurrency

Let $BM$ be a median of nonisosceles right-angled triangle $ABC$ ($\angle B = 90^o$), and $Ha, Hc$ be the orthocenters of triangles $ABM, CBM$ respectively. Prove that lines $AH_c$ and $CH_a$ meet on the medial line of triangle $ABC$. (D. Svhetsov)

2018 Caucasus Mathematical Olympiad, 1

Tags: combinatorics , parity

A tetrahedron is given. Determine whether it is possible to put some 10 consecutive positive integers at 4 vertices and at 6 midpoints of the edges so that the number at the midpoint of each edge is equal to the arithmetic mean of two numbers at the endpoints of this edge.

2023 Princeton University Math Competition, A2 / B4

Tags: geometry

Let $\triangle{ABC}$ be an isosceles triangle with $AB = AC =\sqrt{7}, BC=1$. Let $G$ be the centroid of $\triangle{ABC}$. Given $ j\in \{0,1,2\}$, let $T_{j}$ denote the triangle obtained by rotating $\triangle{ABC}$ about $G$ by $\frac{2\pi j}{3}$ radians. Let $\mathcal{P}$ denote the intersection of the interiors of triangles $T_0,T_1,T_2$. If $K$ denotes the area of $\mathcal{P}$, then $K^2=\frac{a}{b}$ for relatively prime positive integers $a, b$. Find $a + b$.

2020 BMT Fall, 17

Tags: algebra , polynomial

Let $T$ be the answer to question $16$. Compute the number of distinct real roots of the polynomial $x^4 + 6x^3 +\frac{T}{2}x^2 + 6x + 1$.

2019 Slovenia Team Selection Test, 3

Tags: combinatorics

Let $n$ be any positive integer and $M$ a set that contains $n$ positive integers. A sequence with $2^n$ elements is christmassy if every element of the sequence is an element of $M$. Prove that, in any christmassy sequence there exist some successive elements, the product of whom is a perfect square.

1990 Tournament Of Towns, (271) 5

Tags: algebra , periodic , sequence , recurrence relation

The numerical sequence $\{x_n\}$ satisfies the condition $$x_{n+1}=|x_n|-x_{n-1}$$ for all $n > 1$. Prove that the sequence is periodic with period $9$, i.e. for any $n > 1$ we have $x_n = x_{n+9}$. (M Kontsevich, Moscow)

1946 Moscow Mathematical Olympiad, 113

Tags: number theory , divisible

Prove that $n^2 + 3n + 5$ is not divisible by $121$ for any positive integer $n$.

1987 Federal Competition For Advanced Students, P2, 2

Tags: combinatorics

Find the number of all sequences $ (x_1,...,x_n)$ of letters $ a,b,c$ that satisfy $ x_1\equal{}x_n\equal{}a$ and $ x_i \not\equal{} x_{i\plus{}1}$ for $ 1 \le i \le n\minus{}1.$

1984 National High School Mathematics League, 3

Tags: geometry

In $\triangle ABC$, $P$ is a point on $BC$. $F\in AB,E\in AC,PF//CA,PE//BA$. If $S_{\triangle ABC}=1$, prove that at least one of $S_{\triangle BPF},S_{\triangle PCE},S_{PEAF}$ is not less than $\frac{4}{9}$.

1985 All Soviet Union Mathematical Olympiad, 400

Tags: trinomial , integer , maximum , polynomial , algebra

The senior coefficient $a$ in the square polynomial $$P(x) = ax^2 + bx + c$$ is more than $100$. What is the maximal number of integer values of $x$, such that $|P(x)|<50$.

2002 Korea Junior Math Olympiad, 4

Tags: number theory

For two non-negative integers $i, j$, create a new integer $i \# j$ defined as the following: Express the two numbers in base $2$, and compare each digit. If their $k$th digit is the same, then the $k$th digit of $i \# j$ is $0$. If their $k$th digit is different, then the $k$th digit of $i \# j$ is $1$(of course we are talking in base $2$). For instance, $3 \# 5=6$. Show that for arbitrary positive integer $n$, the number can be expressed with finite operations of $\#$s and integers of the form $2^k-1$.

2005 MOP Homework, 4

Tags: algebra , function , polynomial

Deos there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$, $f(x^2y+f(x+y^2))=x^3+y^3+f(xy)$