Olympiad problems

1994 China Team Selection Test, 2

Tags: algebra , polynomial , calculus , integration , gauss , number theory , prime number

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

1979 Bundeswettbewerb Mathematik, 4

Tags: sequence , infinitely many solutions , number theory , last digit

An infinite sequence $p_1, p_2, p_3, \ldots$ of natural numbers in the decimal system has the following property: For every $i \in \mathbb{N}$ the last digit of $p_{i+1}$ is different from $9$, and by omitting this digit one obtains number $p_i$. Prove that this sequence contains infinitely many composite numbers.

1966 IMO Longlists, 5

Tags: inequalities , trigonometry , geometric inequality

Prove the inequality \[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\] for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$

2020 Romanian Master of Mathematics, 1

Tags: geometry

Let $ABC$ be a triangle with a right angle at $C$. Let $I$ be the incentre of triangle $ABC$, and let $D$ be the foot of the altitude from $C$ to $AB$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $A_1$, $B_1$, and $C_1$, respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$, respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$, respectively. Prove that the circumcircles of triangles $A_1EI$, $B_1FI$, and $C_1KL$ have a common point.

1997 Czech And Slovak Olympiad IIIA, 4

Tags: sequence , prime , number theory

Show that there exists an increasing sequence $a_1,a_2,a_3,...$ of natural numbers such that, for any integer $k \ge 2$, the sequence $k+a_n$ ($n \in N$) contains only finitely many primes.

2024 Yasinsky Geometry Olympiad, 5

Tags: geometry , angle bisector , median , parallelogram

Let $ AL $ be the bisector of triangle $ ABC $, $ O $ the center of its circumcircle, and $ D $ and $ E $ the midpoints of $ BL $ and $ CL $, respectively. Points $ P $ and $ Q $ are chosen on segments $ AD $ and $ AE $ such that $ APLQ $ is a parallelogram. Prove that $ PQ \perp AO $. [i]Proposed by Mykhailo Plotnikov[/i]

2003 Kazakhstan National Olympiad, 8

Tags: function , algebra

Determine all functions $f: \mathbb R \to \mathbb R$ with the property \[f(f(x)+y)=2x+f(f(y)-x), \quad \forall x,y \in \mathbb R.\]

2018 CMI B.Sc. Entrance Exam, 3

Tags: functional equation , number theory , combinatorics

Let $f$ be a function on non-negative integers defined as follows $$f(2n)=f(f(n))~~~\text{and}~~~f(2n+1)=f(2n)+1$$ [b](a)[/b] If $f(0)=0$ , find $f(n)$ for every $n$. [b](b)[/b] Show that $f(0)$ cannot equal $1$. [b](c)[/b] For what non-negative integers $k$ (if any) can $f(0)$ equal $2^k$ ?

2020 BMT Fall, 17

Tags: combinatorics

Shrek throws $5$ balls into $5$ empty bins, where each ball’s target is chosen uniformly at random. After Shrek throws the balls, the probability that there is exactly one empty bin can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

2024 BAMO, D/2

Tags: geometry , combinatorics

Sasha wants to bake $6$ cookies in his $8$ inch $\times$ $8$ inch square baking sheet. With a cookie cutter, he cuts out from the dough six circular shapes, each exactly $3$ inches in diameter. Can he place these six dough shapes on the baking sheet without the shapes touching each other? If yes, show us how. If no, explain why not. (Assume that the dough does not expand during baking.)

1949-56 Chisinau City MO, 27

Tags: right triangle , similar triangles , similar , geometry

The areas of two right-angled triangles have ratio equal to the squares of their hypotenuses. Show that these triangles are similar.

1999 IMO Shortlist, 4

Tags: modular arithmetic , combinatorics , counting , additive combinatorics , additive number theory

Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$.

1996 Estonia National Olympiad, 1

Tags: prime , diophantine equation , diophantine , number theory

Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.

2017 AMC 8, 20

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An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct? $\textbf{(A) }\frac{14}{75}\qquad\textbf{(B) }\frac{56}{225}\qquad\textbf{(C) }\frac{107}{400}\qquad\textbf{(D) }\frac{7}{25}\qquad\textbf{(E) }\frac{9}{25}$

2007 AMC 8, 14

Tags: pythagorean theorem , geometry

The base of isosceles $\triangle{ABC}$ is $24$ and its area is $60$. What is the length of one of the congruent sides? $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 18$

2020-21 IOQM India, 20

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A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the worked over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first woman work?

1998 Czech And Slovak Olympiad IIIA, 6

Tags: sidelenghts , system of equations , algebra

Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations $$\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \\ \\ \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \\ \\ \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}$$ has a real solution.

2009 Today's Calculation Of Integral, 490

Tags: calculus , integration , inequalities , calculus computations , logarithm

For a positive real number $ a > 1$, prove the following inequality. $ \frac {1}{a \minus{} 1}\left(1 \minus{} \frac {\ln a}{a\minus{}1}\right) < \int_0^1 \frac {x}{a^x}\ dx < \frac {1}{\ln a}\left\{1 \minus{} \frac {\ln (\ln a \plus{} 1)}{\ln a}\right\}$

2001 Turkey MO (2nd round), 1

Tags: number theory

Find all ordered triples of positive integers $(x,y,z)$ such that \[3^{x}+11^{y}=z^{2}\]

2018 Azerbaijan JBMO TST, 4

Tags: combinatorics

An $n\times n$ square table is divided into $n^2$ unit cells. Some unit segments of the obtained grid (i.e. the side of any unit cell) is colored black so that any unit cell of the given square has exactly one black side. Find [b]a)[/b] the smallest [b]b)[/b] the greatest possible number of black unit segments.

2020 LIMIT Category 2, 8

Tags: limit , probability , set

Let $S$ be a finite set of size $s\geq 1$ defined with a uniform probability $\mathbb{P}$( i.e. for any subset $X\subset S$ of size $x$, $\mathbb{P}(x)=\frac{x}{s}$). Suppose $A$ and $B$ are subsets of $S$. They are said to be independent iff $\mathbb{P}(A)\mathbb{P}(B)=\mathbb{P}(A\cap B)$. Which if these is sufficient for independence? (A)$|A\cup B|=|A|+|B|$ (B)$|A\cap B|=|A|+|B|$ (C)$|A\cup B|=|A|\cdot |B|$ (D)$|A\cap B|=|A|\cdot |B|$

1982 AMC 12/AHSME, 12

Tags: algebra , polynomial

Let $f(x) = ax^7+bx^3+cx-5$, where $a,b$ and $c$ are constants. If $f(-7) = 7$, the $f(7)$ equals $\textbf {(A) } -17 \qquad \textbf {(B) } -7 \qquad \textbf {(C) } 14 \qquad \textbf {(D) } 21\qquad \textbf {(E) } \text{not uniquely determined}$

1964 Putnam, A1

Tags: inequalities , combinatorics , geometry

Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.

2018 ISI Entrance Examination, 1

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Find all pairs $(x,y)$ with $x,y$ real, satisfying the equations $$\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1$$

1950 Putnam, B3

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In the Gregorian calendar: (i) years not divisible by $4$ are common years; (ii) years divisible by $4$ but not by $100$ are leap years; (iii) years divisible by $100$ but not by $400$ are common years; (iv) years divisible by $400$ are leap years; (v) a leap year contains $366$ days; a common year $365$ days. Prove that the probability that Christmas falls on a Wednesday is not $1/7.$