Olympiad problems

2021 Austrian MO Beginners' Competition, 4

Let $p$ be a prime number and let $m$ and $n$ be positive integers with $p^2 + m^2 = n^2$. Prove that $m> p$. (Karl Czakler)

2000 Swedish Mathematical Competition, 3

Tags: number theory , diophantine equation , diophantine

Are there any integral solutions to $n^2 + (n+1)^2 + (n+2)^2 = m^2$ ?

2007 Putnam, 5

Tags: algebra , polynomial , floor function , calculus , derivative , linear algebra

Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$ \[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\] ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

PEN H Problems, 11

Tags: diophantine equation

Find all $(x,y,n) \in {\mathbb{N}}^3$ such that $\gcd(x, n+1)=1$ and $x^{n}+1=y^{n+1}$.

2020 Argentina National Olympiad, 3

Tags: right triangle , isosceles , angle , geometry

Let $ABC$ be a right isosceles triangle with right angle at $A$. Let $E$ and $F$ be points on A$B$ and $AC$ respectively such that $\angle ECB = 30^o$ and $\angle FBC = 15^o$. Lines $CE$ and $BF$ intersect at $P$ and line $AP$ intersects side $BC$ at $D$. Calculate the measure of angle $\angle FDC$.

2010 National Olympiad First Round, 1

Tags: geometry

Let $D$ be a point inside of equilateral $\triangle ABC$, and $E$ be a point outside of equilateral $\triangle ABC$ such that $m(\widehat{BAD})=m(\widehat{ABD})=m(\widehat{CAE})=m(\widehat{ACE})=5^\circ$. What is $m(\widehat{EDC})$ ? $ \textbf{(A)}\ 45^\circ \qquad\textbf{(B)}\ 40^\circ \qquad\textbf{(C)}\ 35^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 25^\circ $

2015 Brazil National Olympiad, 3

Tags: number theory , prime factorization

Given a natural $n>1$ and its prime fatorization $n=p_1^{\alpha 1}p_2^{\alpha_2} \cdots p_k^{\alpha_k}$, its [i]false derived[/i] is defined by $$f(n)=\alpha_1p_1^{\alpha_1-1}\alpha_2p_2^{\alpha_2-1}...\alpha_kp_k^{\alpha_k-1}.$$ Prove that there exist infinitely many naturals $n$ such that $f(n)=f(n-1)+1$.

2015 Junior Regional Olympiad - FBH, 4

Tags: digit , fraction

Which number we need to substract from numerator and add to denominator of $\frac{\overline{28a3}}{7276}$ such that we get fraction equal to $\frac{2}{7}$

2015 CCA Math Bonanza, L3.1

Tags: probability , expected value

Bhairav the Bat lives next to a town where $12.5$% of the inhabitants have Type AB blood. When Bhairav the Bat leaves his cave at night to suck of the inhabitants blood, chooses individuals at random until he bites one with type AB blood, after which he stops. What is the expected value of the number of individuals Bhairav the Bat will bite in any given night? [i]2015 CCA Math Bonanza Lightning Round #3.1[/i]

2020 Yasinsky Geometry Olympiad, 4

Tags: equal angles , equal segments , median , geometry

The median $AM$ is drawn in the triangle $ABC$ ($AB \ne AC$). The point $P$ is the foot of the perpendicular drawn on the segment $AM$ from the point $B$. On the segment $AM$ we chose such a point $Q$ that $AQ = 2PM$. Prove that $\angle CQM = \angle BAM$.

2003 Junior Balkan Team Selection Tests - Romania, 3

Tags: diophantine , number theory , radical , inequalities

Let $n$ be a positive integer. Prove that there are no positive integers $x$ and $y$ such as $\sqrt{n}+\sqrt{n+1} < \sqrt{x}+\sqrt{y} <\sqrt{4n+2} $

1979 IMO Longlists, 9

Tags: inequalities , function , logarithm

The real numbers $\alpha_1 , \alpha_2, \alpha_3, \ldots, \alpha_n$ are positive. Let us denote by $h = \frac{n}{1/\alpha_1 + 1/\alpha_2 + \cdots + 1/\alpha_n}$ the harmonic mean, $g=\sqrt[n]{\alpha_1\alpha_2\cdots \alpha_n}$ the geometric mean, and $a=\frac{\alpha_1+\alpha_2+\cdots + \alpha_n}{n}$ the arithmetic mean. Prove that $h \leq g \leq a$, and that each of the equalities implies the other one.

1993 Mexico National Olympiad, 2

Tags: number theory , sum of cubes

Find all numbers between $100$ and $999$ which equal the sum of the cubes of their digits.

2018 Iran MO (1st Round), 19

Tags: inequalities , algebra

Let $x \geq y \geq z$ be positive real numbers such that \begin{align*}x^2+y^2+z^2 \geq 2xy+2yz+2zx.\end{align*} What is the minimum value of $\frac{x}{z}$? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt 2\qquad\textbf{(C)}\ \sqrt 3\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$

2024 MMATHS, 8

Tags:

Triangle $ABC$ is an acute triangle with $BC=6$ and $AC=7.$ Let $D, E,$ and $F$ be the feet of the altitudes from $A, B,$ and $C$ respectively. $\overline{AD}$ bisects angle $FDE.$ Let $m$ be the maximum possible value of $FD+ED.$ Find $m^2.$

2021 AMC 12/AHSME Fall, 1

Tags:

What is the value of $\frac{(2112-2021)^2}{169}$? $\textbf{(A) }7\qquad\textbf{(B) }21\qquad\textbf{(C) }49\qquad\textbf{(D) }64\qquad\textbf{(E) }91$

2021 Romania National Olympiad, 3

Tags: geometry

Let $ABC$ be a scalene triangle with $\angle BAC>90^\circ$. Let $D$ and $E$ be two points on the side $BC$ such that $\angle BAD=\angle ACB$ and $\angle CAE=\angle ABC$. The angle-bisector of $\angle ACB$ meets $AD$ at $N$, If $MN\parallel BC$, determine $\angle (BM, CN)$. [i]Petru Braica[/i]

2016 PAMO, 3

Tags: number theory , integer polynomial , greatest common divisor

For any positive integer $n$, we define the integer $P(n)$ by : $P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$. Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$.

1953 Moscow Mathematical Olympiad, 240

Tags: midpoint , skew , geometric inequality , 3d geometry , geometry

Let $AB$ and $A_1B_1$ be two skew segments, $O$ and $O_1$ their respective midpoints. Prove that $OO_1$ is shorter than a half sum of $AA_1$ and $BB_1$.

1998 Estonia National Olympiad, 4

Tags: divide , number theory , divisible , prime

Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.

1982 Putnam, B1

Tags: geometry

Let $M$ be the midpoint of side $BC$ of $\triangle ABC$. Using the [i]smallest possible[/i] $n$, described a method for cutting $\triangle AMB$ into $n$ triangles which can be reassembled to form a triangle congruent to $\triangle AMC$.

2018 AMC 10, 18

Tags:

How many nonnegative integers can be written in the form $$a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$$ where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$? $\textbf{(A) } 512 \qquad \textbf{(B) } 729 \qquad \textbf{(C) } 1094 \qquad \textbf{(D) } 3281 \qquad \textbf{(E) } 59,048 $

2010 Contests, 2

Tags:

Denote by $S(n)$ the sum of digits of integer $n.$ Find 1) $S(3)+S(6)+S(9)+\ldots+S(300);$ 2) $S(3)+S(6)+S(9)+\ldots+S(3000).$

2021 Indonesia TST, A

Tags: function , algebra , functional inequality

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x + y) + y \le f(f(f(x)))\] holds for all $x, y \in \mathbb{R}$.

2007 AMC 12/AHSME, 22

Tags: ratio , vector , function , geometry

Two particles move along the edges of equilateral triangle $ \triangle ABC$ in the direction \[ A\rightarrow B\rightarrow C\rightarrow A \]starting simultaneously and moving at the same speed. One starts at $ A$, and the other starts at the midpoint of $ \overline{BC}$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $ R$. What is the ratio of the area of $ R$ to the area of $ \triangle ABC$? $ \textbf{(A)}\ \frac {1}{16}\qquad \textbf{(B)}\ \frac {1}{12}\qquad \textbf{(C)}\ \frac {1}{9}\qquad \textbf{(D)}\ \frac {1}{6}\qquad \textbf{(E)}\ \frac {1}{4}$