Olympiad problems

2021 Bangladeshi National Mathematical Olympiad, 1

How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?

2014 Kurschak Competition, 2

Tags: circumcircle , power of a point , radical axis , geometry

We are given an acute triangle $ABC$, and inside it a point $P$, which is not on any of the heights $AA_1$, $BB_1$, $CC_1$. The rays $AP$, $BP$, $CP$ intersect the circumcircle of $ABC$ at points $A_2$, $B_2$, $C_2$. Prove that the circles $AA_1A_2$, $BB_1B_2$ and $CC_1C_2$ are concurrent.

2009 Romania Team Selection Test, 2

Tags: linear algebra , matrix , combinatorics

Consider a matrix whose entries are integers. Adding a same integer to all entries on a same row, or on a same column, is called an operation. It is given that, for infinitely many positive integers $n$, one can obtain, through a finite number of operations, a matrix having all entries divisible by $n$. Prove that, through a finite number of operations, one can obtain the null matrix.

2010 Purple Comet Problems, 14

Tags: geometry , trapezoid , ratio , similar triangles

Let $ABCD$ be a trapezoid where $AB$ is parallel to $CD.$ Let $P$ be the intersection of diagonal $AC$ and diagonal $BD.$ If the area of triangle $PAB$ is $16,$ and the area of triangle $PCD$ is $25,$ find the area of the trapezoid.

1984 Miklós Schweitzer, 7

Tags: polynomial , algebra

[b]7.[/b] Let $V$ be a finite-dimensional subspace of $C[0,1]$ such that every nonzero $f\in V$ attains positive value at some point. Prove that there exists a polynomial $P$ that is strictly positive on $[0,1]$ and orthogonal to $V$, that is, for every $f \in V$, $\int_{0}^{1} f(x)P(x)dx =0$ ([b]F.39[/b]) [A. Pinkus, V. Totik]

III Soros Olympiad 1996 - 97 (Russia), 9.4

Tags: algebra , system of equations

Solve the system of equations $$\begin{cases} x^4-2x^3+x=y^2-y \\ y^4-2y^3+y=x^2-x \end{cases}$$

1991 Poland - Second Round, 2

Tags: geometry , equilateral

On the sides $ BC $, $ CA $, $ AB $ of the triangle $ ABC $, the points $ D $, $ E $, $ F $ are chosen respectively, such that $$ \frac{|DB|}{|DC|} = \frac{|EC|}{|EA|} = \frac{|FA|}{|FB|}$$ Prove that if the triangle $ DEF $ is equilateral, then the triangle $ ABC $ is also equilateral.

1993 Putnam, A2

Tags:

The sequence an of non-zero reals satisfies $a_n^2 - a_{n-1}a_{n+1} = 1$ for $n \geq 1$. Prove that there exists a real number $\alpha$ such that $a_{n+1} = \alpha a_n - a_{n-1}$ for $n \geq 1$.

2007 Irish Math Olympiad, 1

Tags: polynomial , vieta , algebra

Let $ r,s,$ and $ t$ be the roots of the cubic polynomial: $ p(x)\equal{}x^3\minus{}2007x\plus{}2002.$ Determine the value of: $ \frac{r\minus{}1}{r\plus{}1}\plus{}\frac{s\minus{}1}{s\plus{}1}\plus{}\frac{t\minus{}1}{t\plus{}1}$.

Gheorghe Țițeica 2024, P1

Tags: real analysis , convergent sequence , continuous function

Find all continuous functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any sequences $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ such that the sequence $(a_n+b_n)_{n\geq 1}$ is convergent, the sequence $(f(a_n)+g(b_n))_{n\geq 1}$ is also convergent.

2021 Latvia Baltic Way TST, P9

Tags: geometry , circumcircle

Pentagon $ABCDE$ with $CD\parallel BE$ is inscribed in circle $\omega$. Tangent to $\omega$ through $B$ intersects line $AC$ at $F$ in a way that $A$ lies between $C$ and $F$. Lines $BD$ and $AE$ intersect at $G$. Prove that $FG$ is tangent to the circumcircle of $\triangle ADG$.

2024 Simon Marais Mathematical Competition, A1

Tags: algebra

Let $a,b,c$ be real number greater than 1 satisfying $$\lfloor a\rfloor b = \lfloor b \rfloor c = \lfloor c\rfloor a.$$Prove that $a=b=c$ (Here, $\lfloor x \rfloor$ denotes the laregst integer that is less than or equal to $x$.)

2017 ASDAN Math Tournament, 22

Tags:

Let $x=2\sin8^\circ+2\sin16^\circ+\dots+2\sin176^\circ$. What is $\arctan(x)$?

MathLinks Contest 1st, 2

Tags: polynomial , algebra

Let a be a non-zero integer, and $n \ge 3$ another integer. Prove that the following polynomial is irreducible in the ring of integer polynomials (i.e. it cannot be written as a product of two non-constant integer polynomials): $$f(x) = x^n + ax^{n-1} + ax^{n-2} +... + ax -1$$

2010 NZMOC Camp Selection Problems, 6

Tags: combinatorics

At a strange party, each person knew exactly $22$ others. For any pair of people $X$ and $Y$ who knew one another, there was no other person at the party that they both knew. For any pair of people $X$ and $Y$ who did not know each other, there were exactly six other people that they both knew. How many people were at the party?

2016 CentroAmerican, 3

Tags: algebra , polynomial , root

The polynomial $Q(x)=x^3-21x+35$ has three different real roots. Find real numbers $a$ and $b$ such that the polynomial $x^2+ax+b$ cyclically permutes the roots of $Q$, that is, if $r$, $s$ and $t$ are the roots of $Q$ (in some order) then $P(r)=s$, $P(s)=t$ and $P(t)=r$.

2017 Regional Olympiad of Mexico Northeast, 4

Tags: concyclic , geometry

Let $\Gamma$ be the circumcircle of the triangle $ABC$ and let $M$ be the midpoint of the arc $\Gamma$ containing $A$ and bounded by $B$ and $C$. Let $P$ and $Q$ be points on the segments $AB$ and $AC$, respectively, such that $BP = CQ$. Prove that $APQM$ is a cyclic quadrilateral.

2007 Today's Calculation Of Integral, 188

Tags: calculus , integration , calculus computations

Find the volume of the solid obtained by revolving the region bounded by the graphs of $y=xe^{1-x}$ and $y=x$ around the $x$ axis.

2010 Ukraine Team Selection Test, 6

Tags: number theory , integer , divisibility

Find all pairs of odd integers $a$ and $b$ for which there exists a natural number$ c$ such that the number $\frac{c^n+1}{2^na+b}$ is integer for all natural $n$.

2006 Austrian-Polish Competition, 7

Tags: floor function , algebra

Find all nonnegative integers $m,n$ so that \[\sum_{k=1}^{2^{m}}\lfloor \frac{kn}{2^{m}}\rfloor\in \{28,29,30\}\]

2012 Olympic Revenge, 5

Tags: inequalities

Let $x_1,x_2,\ldots ,x_n$ positive real numbers. Prove that: \[\sum_{cyc} \frac{1}{x_i^3+x_{i-1}x_ix_{i+1}} \le \sum_{cyc} \frac{1}{x_ix_{i+1}(x_i+x_{i+1})}\]

2011 NZMOC Camp Selection Problems, 4

Tags: geometry , parallelogram

Let a point $P$ inside a parallelogram $ABCD$ be given such that $\angle APB +\angle CPD = 180^o$. Prove that $AB \cdot AD = BP \cdot DP + AP \cdot CP$.

2015 BMT Spring, Tie 2

Tags: geometry

The unit square $ABCD$ has $E$ as midpoint of $AD$ and a circle of radius $r$ tangent to $AB$, $BC$, and $CE$. Determine $r$.

2014 BMT Spring, 3

Tags: algebra

Suppose three boba drinks and four burgers cost $28$ dollars, while two boba drinks and six burgers cost $\$ 37.70$. If you paid for one boba drink using only pennies, nickels, dimes, and quarters, determine the least number of coins you could use.

1985 AMC 8, 23

Tags:

King Middle School has $ 1200$ students. Each student takes $ 5$ classes a day. Each teacher teaches $ 4$ classes. Each class has $ 30$ students and $ 1$ teacher. How many teachers are there at King Middle School? \[ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 50 \qquad \]