Olympiad problems

1999 Miklós Schweitzer, 2

Tags: number theory

Let e>0. Prove that for a large enough natural n, there exist natural x,y,z st $n^2+x^2=y^2+z^2$ and $y,z\leq \frac{(1+e)n}{\sqrt{2}}$.

2018 Mathematical Talent Reward Programme, SAQ: P 5

Tags: fibonacci sequence , periodic sequence , remainder , polynomial , algebra

[list=1] [*] Prove that, the sequence of remainders obtained when the Fibonacci numbers are divided by $n$ is periodic, where $n$ is a natural number. [*] There exists no such non-constant polynomial with integer coefficients such that for every Fibonacci number $n,$ $ P(n)$ is a prime. [/list]

2021 MIG, 2

Tags:

Solve for $x$ if $20x + 21 = 121$. $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }7$

1996 Estonia National Olympiad, 2

Tags: equal segments , trapezoid , angle , geometry

Three sides of a trapezoid are equal, and a circle with the longer base as a diameter halves the two non-parallel sides. Find the angles of the trapezoid.

2010 AMC 10, 15

Tags:

In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in the swamp, and they make the following statements: Brian: "Mike and I are different species." Chris: "LeRoy is a frog." LeRoy: "Chris is a frog." Mike: "Of the four of us, at least two are toads." How many of these amphibians are frogs? $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

2008 District Olympiad, 2

Tags: number theory , polynomial , algebra

Determine $ x$ irrational so that $ x^2\plus{}2x$ and $ x^3\minus{}6x$ are both rational.

2016 Spain Mathematical Olympiad, 3

Tags: geometry , circumcircle

In the circumscircle of a triangle $ABC$, let $A_1$ be the point diametrically opposed to the vertex $A$. Let $A'$ the intersection point of $AA'$ and $BC$. The perpendicular to the line $AA'$ from $A'$ meets the sides $AB$ and $AC$ at $M$ and $N$, respectively. Prove that the points $A,M,A_1$ and $N$ lie on a circle which has the center on the height from $A$ of the triangle $ABC$.

2015 Portugal MO, 1

Tags: number theory

A number of three digits is said to be [i]firm [/i]when it is equal to the product of its unit digit by a number formed by the remaining digits. For example, $153$ is firm because $153 = 3 \times 51$. How many [i]firm [/i] numbers are there?

1978 IMO Longlists, 45

Tags: inequalities , induction

If $r > s >0$ and $a > b > c$, prove that \[a^rb^s + b^rc^s + c^ra^s \ge a^sb^r + b^sc^r + c^sa^r.\]

2021/2022 Tournament of Towns, P1

Tags: number theory

The Tournament of Towns is held once per year. This time the year of its autumn round is divisible by the number of the tournament: $2021\div 43 = 47$. How many times more will the humanity witness such a wonderful event? [i]Alexey Zaslavsky[/i]

2018 AMC 12/AHSME, 23

Tags: trigonometry

In $\triangle PAT,$ $\angle P=36^{\circ},$ $\angle A=56^{\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG},$ respectively. What is the degree measure of the acute angle formed by lines $MN$ and $PA?$ $\textbf{(A) } 76 \qquad \textbf{(B) } 77 \qquad \textbf{(C) } 78 \qquad \textbf{(D) } 79 \qquad \textbf{(E) } 80 $

Kyiv City MO 1984-93 - geometry, 1986.9.5

Tags: area of a triangle , geometry

Prove that inside any convex hexagon with pairs of parallel sides of area $1$, you can draw a triangle of area $1/2$.

2019 Azerbaijan Senior NMO, 3

Tags: diophantine equation , number theory

Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$

1995 China Team Selection Test, 3

Tags: polynomial , quadratic , algebra

Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.) Does the same result hold with a degree 3 or degree 5 polynomial?

2015 Saudi Arabia IMO TST, 3

Tags: min , max , inequalities , algebra

Let $a, b,c$ be positive real numbers satisfying the condition $$(x + y + z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)= 10$$ Find the greatest value and the least value of $$T = (x^2 + y^2 + z^2) \left(\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}\right)$$ Trần Nam Dũng

2021 Belarusian National Olympiad, 8.8

Tags: geometry

On the sides $AB,BC,CD$ and $DA$ of a unit square $ABCD$ points $P,Q,R$ and $S$ are chosen respectively. It turned out that the perimeter of $PQRS$ is $2\sqrt{2}$. Find the sum of perpendiculars from $A,B,C,D$ to $SP,PQ,QR,RS$ respectively.

2020 Bosnia and Herzegovina Junior BMO TST, 4

Tags: algebra

Determine the largest positive integer $n$ such that the following statement holds: If $a_1,a_2,a_3,a_4,a_5,a_6$ are six distinct positive integers less than or equal to $n$, then there exist $3$ distinct positive integers ,from these six, say $a,b,c$ s.t. $ab>c,bc>a,ca>b$.

2003 Cono Sur Olympiad, 3

Tags: geometry

Let $ABC$ be an acute triangle such that $\angle{B}=60$. The circle with diameter $AC$ intersects the internal angle bisectors of $A$ and $C$ at the points $M$ and $N$, respectively $(M\neq{A},$ $N\neq{C})$. The internal bisector of $\angle{B}$ intersects $MN$ and $AC$ at the points $R$ and $S$, respectively. Prove that $BR\leq{RS}$.

1982 Polish MO Finals, 1

Tags: max , combinatorics

Find a way of arranging $n$ girls and $n$ boys around a round table for which $d_n-c_n$ is maximum, where dn is the number of girls sitting between two boys and $c_n$ is the number of boys sitting between two girls.

2014 Contests, 1

Tags: number theory , abstract algebra

Determine the last two digits of the product of the squares of all positive odd integers less than $2014$.

2017 Miklós Schweitzer, 1

Tags: triangulation , combinatorics , combinatorial geometry

Can one divide a square into finitely many triangles such that no two triangles share a side? (The triangles have pairwise disjoint interiors and their union is the square.)

2008 USA Team Selection Test, 5

Tags: pythagorean theorem , geometry , inequalities , circumcircle , analytic geometry , algebra

Two sequences of integers, $ a_1, a_2, a_3, \ldots$ and $ b_1, b_2, b_3, \ldots$, satisfy the equation \[ (a_n \minus{} a_{n \minus{} 1})(a_n \minus{} a_{n \minus{} 2}) \plus{} (b_n \minus{} b_{n \minus{} 1})(b_n \minus{} b_{n \minus{} 2}) \equal{} 0 \] for each integer $ n$ greater than $ 2$. Prove that there is a positive integer $ k$ such that $ a_k \equal{} a_{k \plus{} 2008}$.

2013 Saint Petersburg Mathematical Olympiad, 6

Tags: circumcircle , geometric transformation , geometry

Let $(I_b)$, $(I_c)$ are excircles of a triangle $ABC$. Given a circle $ \omega $ passes through $A$ and externally tangents to the circles $(I_b)$ and $(I_c)$ such that it intersects with $BC$ at points $M$, $N$. Prove that $ \angle BAM=\angle CAN $. A. Smirnov

2018 Moldova Team Selection Test, 6

Tags: three variable inequality , inequalities , algebra

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Show that $$\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+a^2}\geq \frac{3}{2}.$$

2024-IMOC, N8

Tags: number theory

Find all integers $(a,b)$ satisfying: there is an integer $k>1$ such that $$a^k+b^k-1\ |\ a^n+b^n-1$$ holds for all integer $n\geq k$ (we define that $0|0$)