Olympiad problems

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 $x$, $y \in \mathbb{R}$, $f(x^2y+f(x+y^2))=x^3+y^3+f(xy)$

2022 Nigerian Senior MO Round 2, Problem 6

Tags: number theory , diophantine equation

Let $k, l, m, n$ be positive integers. Given that $k+l+m+n=km=ln$, find all possible values of $k+l+m+n$.

1987 Kurschak Competition, 2

Tags: geometry

Is there a set of points in space whose intersection with any plane is a finite but nonempty set of points?

2003 All-Russian Olympiad Regional Round, 8.7

Tags: geometry , equal segments , equal angles

In triangle $ABC$, angle $C$ is a right angle. Found on the side $AC$ point $D$, and on the segment $BD$, point $K$ such that $\angle ABC = \angle KAD =\angle AKD$. Prove that $BK = 2DC$.

2012 AMC 12/AHSME, 17

Tags: analytic geometry , graphing lines , slope , trigonometry , geometric transformation , dilation , geometry

Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.2\qquad\textbf{(C)}\ 6.4\qquad\textbf{(D)}\ 6.6\qquad\textbf{(E)}\ 6.8 $

2009 Kazakhstan National Olympiad, 4

Tags: inequalities

Let $0<a_1 \leq a_2 \leq \cdots\leq a_n $ ($n \geq 3; n \in \mathbb{N}$) be $n$ real numbers. Prove the inequality \[\frac{a_1^2}{a_2}+\frac{a_2^3}{a_3^2}+\cdots+\frac{a_n^{n+1}}{a_1^n} \geq a_1+a_2+\cdots+a_n\]

2010 Kyrgyzstan National Olympiad, 7

Tags: number theory

Find all natural triples $(a,b,c)$, such that: $a - )\,a \le b \le c$ $b - )\,(a,b,c) = 1$ $c - )\,\left. {{a^2}b} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{b^2}c} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{c^2}a} \right|{a^3} + {b^3} + {c^3}$.

1994 Hong Kong TST, 1

Tags: trigonometry , geometry

In a $\triangle ABC$, $\angle C=2 \angle B$. $P$ is a point in the interior of $\triangle ABC$ satisfying that $AP=AC$ and $PB=PC$. Show that $AP$ trisects the angle $\angle A$.

2014 Contests, 1

Tags: algebra , system of equations

Determine the value of the expression $x^2 + y^2 + z^2$, if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$.

Istek Lyceum Math Olympiad 2016, 2

Tags: inversion , circles , geometry

Let $\omega$ be the semicircle with diameter $PQ$. A circle $k$ is tangent internally to $\omega$ and to segment $PQ$ at $C$. Let $AB$ be the tangent to $K$ perpendicular to $PQ$, with $A$ on $\omega$ and $B$ on the segment $CQ$. Show that $AC$ bisects angle $\angle PAB$

2001 APMO, 5

Tags: geometry

Find the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1,\dots,~X_n$ in the plane with $AB\ne CD$ that satisfy the following condition: for each $i=1,2,\dots,n$ triangles $ABX_i$ and $CDX_i$ are equal.

2001 AMC 10, 15

Tags: geometry , parallelogram , similar triangles

A street has parallel curbs $ 40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $ 15$ feet and each stripe is $ 50$ feet long. Find the distance, in feet, between the stripes. $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 25$

2018 Junior Balkan Team Selection Tests - Moldova, 2

Tags: geometry

Let $ABC$ be an acute triangle.Let $OF \| BC$ where $O$ is the circumcenter and $F$ is between $A$ and $B$.Let $H$ be the orthocenter.Let $M$ be the midpoint of $AH$.Prove that $\angle FMC=90$.

2009 Puerto Rico Team Selection Test, 3

Tags: geometry , altitude

On an arbitrary triangle $ ABC$ let $ E$ be a point on the height from $ A$. Prove that $ (AC)^2 - (CE)^2 = (AB)^2 - (EB)^2$.

2013 National Olympiad First Round, 19

Tags: analytic geometry

What is the minimum value of \[\sqrt {x^2 - 4x + 7 - 2\sqrt 2} + \sqrt {x^2 - 8x + 27 - 6\sqrt 2}\] where $x$ is a real number? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3\sqrt 2 \qquad\textbf{(C)}\ 1 + \sqrt 2 \qquad\textbf{(D)}\ 2\sqrt 2 \qquad\textbf{(E)}\ \text{None of above} $

2016 India Regional Mathematical Olympiad, 2

Tags: number theory , greatest common divisor

Consider a sequence $(a_k)_{k \ge 1}$ of natural numbers defined as follows: $a_1=a$ and $a_2=b$ with $a,b>1$ and $\gcd(a,b)=1$ and for all $k>0$, $a_{k+2}=a_{k+1}+a_k$. Prove that for all natural numbers $n$ and $k$, $\gcd(a_n,a_{n+k}) <\frac{a_k}{2}$.

2012 Turkmenistan National Math Olympiad, 1

Tags: trigonometry , inequalities

Find the max and min value of $a\cos^2 x+b\sin x\cos x+c\sin^2 x$.