Olympiad problems

1970 AMC 12/AHSME, 14

Tags: quadratic

Consider $x^2+px+q=0$ where $p$ and $q$ are positive numbers. If the roots of this equation differ by $1$, then $p$ equals $\textbf{(A) }\sqrt{4q+1}\qquad\textbf{(B) }q-1\qquad\textbf{(C) }-\sqrt{4q+1}\qquad\textbf{(D) }q+1\qquad \textbf{(E) }\sqrt{4q-1}$

2014 Contests, 3

Tags: polynomial , complex analysis , algebra

Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$. [i]Proposed by Mohammad Ahmadi[/i]

2022 AMC 12/AHSME, 13

Tags:

Let $\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$, where $z_1$ lies on the segment with endpoints $3$ and $4i$, and $z_2$ has magnitude at most $1$. What integer is closest to the area of $\mathcal{R}$? $\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }15\qquad\textbf{(D) }16\qquad\textbf{(E) }17$

2021 Indonesia TST, A

Tags: algebra , polynomial

Given a polynomial $p(x) =Ax^3+x^2-A$ with $A \neq 0$. Show that for every different real number $a,b,c$, at least one of $ap(b)$, $bp(a)$, and $cp(a)$ not equal to 1.

2007 Purple Comet Problems, 11

Tags:

The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$ which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$. If we apply the same permutation to $T_1$, we get $T_2$ which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$. We continually apply this permutation to each $T_m$ to get $T_{m+1}$. Find the smallest positive integer $n$ so that $T_n=T_0$.

2017 Miklós Schweitzer, 10

Tags: probability , algebra , random , binary representation , continuous function

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables with distribution $\mathbb{P}(X_1=0)=\mathbb{P}(X_1=1)=\frac12$. Let $Y_1$, $Y_2$, $Y_3$, and $Y_4$ be independent, identically distributed random variables, where $Y_1:=\sum_{k=1}^\infty \frac{X_k}{16^k}$. Decide whether the random variables $Y_1+2Y_2+4Y_3+8Y_4$ and $Y_1+4Y_3$ are absolutely continuous.

2020 Iranian Geometry Olympiad, 4

Tags: geometry , orthocenter

Triangle $ABC$ is given. An arbitrary circle with center $J$, passing through $B$ and $C$, intersects the sides $AC$ and $AB$ at $E$ and $F$, respectively. Let $X$ be a point such that triangle $FXB$ is similar to triangle $EJC$ (with the same order) and the points $X$ and $C$ lie on the same side of the line $AB$. Similarly, let $Y$ be a point such that triangle $EYC$ is similar to triangle $FJB$ (with the same order) and the points $Y$ and $B$ lie on the same side of the line $AC$. Prove that the line $XY$ passes through the orthocenter of the triangle $ABC$. [i]Proposed by Nguyen Van Linh - Vietnam[/i]

2018 Stars of Mathematics, 1

Tags: number theory

Two natural numbers have the property that the product of their positive divisors are equal. Does this imply that they are equal? [i]Proposed by Belarus for the 1999th IMO[/i]

2010 Mexico National Olympiad, 1

Tags: number theory

Find all triplets of natural numbers $(a,b,c)$ that satisfy the equation $abc=a+b+c+1$.

2016 Saint Petersburg Mathematical Olympiad, 7

Tags: integer sequence , sequence , number theory , number theory with sequences

A sequence of $N$ consecutive positive integers is called [i]good [/i] if it is possible to choose two of these numbers so that their product is divisible by the sum of the other $N-2$ numbers. For which $N$ do there exist infinitely many [i]good [/i] sequences?

2020 China Northern MO, BP2

Tags: inequalities , algebra

Given $a,b,c \in \mathbb{R}$ satisfying $a+b+c=a^2+b^2+c^2=1$, show that $\frac{-1}{4} \leq ab \leq \frac{4}{9}$.

1982 Tournament Of Towns, (027) 1

Tags: floor function , radical , logarithm , sum , algebra

Prove that for all natural numbers $n$ greater than $1$ : $$[\sqrt{n}] + [\sqrt[3]{n}] +...+[ \sqrt[n]{n}] = [\log_2 n] + [\log_3 n] + ... + [\log_n n]$$ (VV Kisil)

2010 Sharygin Geometry Olympiad, 2

Tags: circumcircle , vertices , geometry

Two intersecting triangles are given. Prove that at least one of their vertices lies inside the circumcircle of the other triangle. (Here, the triangle is considered the part of the plane bounded by a closed three-part broken line, a point lying on a circle is considered to be lying inside it.)

2008 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , integration , function , calculus computations , logarithm

Let $ T \equal{} \int_0^{\ln2} \frac {2e^{3x} \plus{} e^{2x} \minus{} 1} {e^{3x} \plus{} e^{2x} \minus{} e^x \plus{} 1}dx$. Evaluate $ e^T$.

1990 Romania Team Selection Test, 6

Tags: partition , subset , combinatorics

Prove that there are infinitely many n’s for which there exists a partition of $\{1,2,...,3n\}$ into subsets $\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\}$ such that $a_i +b_i = c_i$ for all $i$, and prove that there are infinitely many $n$’s for which there is no such partition.

2021 Mexico National Olympiad, 4

Tags: geometry , orthocenter , euler line , circumcircle , tangent circle

Let $ABC$ be an acutangle scalene triangle with $\angle BAC = 60^{\circ}$ and orthocenter $H$. Let $\omega_b$ be the circumference passing through $H$ and tangent to $AB$ at $B$, and $\omega_c$ the circumference passing through $H$ and tangent to $AC$ at $C$. [list] [*] Prove that $\omega_b$ and $\omega_c$ only have $H$ as common point. [*] Prove that the line passing through $H$ and the circumcenter $O$ of triangle $ABC$ is a common tangent to $\omega_b$ and $\omega_c$. [/list] [i]Note:[/i] The orthocenter of a triangle is the intersection point of the three altitudes, whereas the circumcenter of a triangle is the center of the circumference passing through it's three vertices.

2020 Korea - Final Round, P6

Tags: number theory , pell equation

Find all positive integers $n$ such that $6(n^4-1)$ is a square of an integer.

2013 BMT Spring, 7

Tags: number theory

Denote by $S(a,b)$ the set of integers $k$ that can be represented as $k=a\cdot m+b\cdot n$, for some non-negative integers $m$ and $n$. So, for example, $S(2,4)=\{0,2,4,6,\ldots\}$. Then, find the sum of all possible positive integer values of $x$ such that $S(18,32)$ is a subset of $S(3,x)$.

2015 BMT Spring, 9

Tags: geometry , 3d geometry , cube , geometric inequality , square

Find the side length of the largest square that can be inscribed in the unit cube.

2022 Turkey Team Selection Test, 2

Tags: functional equation , functional equation in r , algebra

Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.

2017 Purple Comet Problems, 8

Tags: number theory

Find the number of trailing zeros at the end of the base-$10$ representation of the integer $525^{25^2} \cdot 252^{52^5}$ .

2023 MIG, 8

Tags:

Anna is buying fruits at a grocery store. If she loses a nickel, she still has enough money to buy exactly $16$ lemons. Similarly, if she loses a quarter, she has enough money to buy exactly $14$ lemons. What is the cost of each lemon? $\textbf{(A) } \$0.05\qquad\textbf{(B) } \$0.10\qquad\textbf{(C) } \$0.15\qquad\textbf{(D) } \$0.20\qquad\textbf{(E) } \$0.25$

1983 IMO Longlists, 46

Tags: function , limit , algebra , logarithm

Let $f$ be a real-valued function defined on $I = (0,+\infty)$ and having no zeros on $I$. Suppose that \[\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty.\] For the sequence $u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|$, prove that $u_n \to +\infty$ as $n \to +\infty.$

Bangladesh Mathematical Olympiad 2020 Final, #12

Tags: number theory

$2^{2921}$ has $581$ digits and starts with a $4$. How many $2^n$'s starts with a $4$, where $0$ is the last digit?

2021/2022 Tournament of Towns, P3

Tags: geometry , incircle

The intersection of two triangles is a hexagon. If this hexagon is removed, six small triangles remain. These six triangles have the same in-radii. Prove the in-radii of the original two triangles are also equal. Spoiler: This is one of the highlights of TT. Also SA3