Olympiad problems

1997 Federal Competition For Advanced Students, P2, 6

Tags: polynomial , algebra

For every natural number $ n$, find all polynomials $ x^2\plus{}ax\plus{}b$, where $ a^2 \ge 4b$, that divide $ x^{2n}\plus{}ax^n\plus{}b$.

2010 Puerto Rico Team Selection Test, 2

Tags: algebra

There is the sequence of numbers $1, a_2, a_3, ...$ such that satisfies $1 \cdot a_2 \cdot a_3 \cdot ... \cdot a_n = n^2$, for every integer $n> 2$. Determine the value of $a_3 + a_5$.

2018 Romania Team Selection Tests, 4

Tags: pell equation , sophie germain identity , factorial , number theory

Given an non-negative integer $k$, show that there are infinitely many positive integers $n$ such that the product of any $n$ consecutive integers is divisible by $(n+k)^2+1$.

CNCM Online Round 3, 1

Tags: v4913 orz

Harry, who is incredibly intellectual, needs to eat carrots $C_1, C_2, C_3$ and solve [i]Daily Challenge[/i] problems $D_1, D_2, D_3$. However, he insists that carrot $C_i$ must be eaten only after solving [i]Daily Challenge[/i] problem $D_i$. In how many satisfactory orders can he complete all six actions? [i]Proposed by Albert Wang (awang2004)[/i]

1973 AMC 12/AHSME, 4

Tags: geometry , trigonometry , trig identities , law of sines , law of cosines

Two congruent $ 30^{\circ}$-$ 60^{\circ}$-$ 90^{\circ}$ are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, the area common to both triangles is $ \textbf{(A)}\ 6\sqrt3 \qquad \textbf{(B)}\ 8\sqrt3 \qquad \textbf{(C)}\ 9\sqrt3 \qquad \textbf{(D)}\ 12\sqrt3 \qquad \textbf{(E)}\ 24$

2021 Harvard-MIT Mathematics Tournament., 8

Tags: combinatorics , floor function , number theory

For each positive real number $\alpha$, define $$\lfloor \alpha \mathbb{N}\rfloor :=\{\lfloor \alpha m \rfloor\; |\; m\in \mathbb{N}\}.$$ Let $n$ be a positive integer. A set $S\subseteq \{1,2,\ldots,n\}$ has the property that: for each real $\beta >0$, $$ \text{if}\; S\subseteq \lfloor \beta \mathbb{N} \rfloor, \text{then}\; \{1,2,\ldots,n\} \subseteq \lfloor \beta\mathbb{N}\rfloor.$$ Determine, with proof, the smallest positive size of $S$.

2009 India IMO Training Camp, 9

Tags: polynomial , algebra

Let $ f(x)\equal{}\sum_{k\equal{}1}^n a_k x^k$ and $ g(x)\equal{}\sum_{k\equal{}1}^n \frac{a_k x^k}{2^k \minus{}1}$ be two polynomials with real coefficients. Let g(x) have $ 0,2^{n\plus{}1}$ as two of its roots. Prove That $ f(x)$ has a positive root less than $ 2^n$.

1996 India Regional Mathematical Olympiad, 6

Tags:

Given any positive integer $n$ , show that there are two positive rational numbers $a$ and $b$ , $a \not= b$, which are not integers and which are such that $a - b, a^2 - b^2 , \ldots a^n - b^n$ are all integers.

2025 Harvard-MIT Mathematics Tournament, 30

Tags: guts

Let $a,b,$ and $c$ be real numbers satisfying the system of equations \begin{align*} a\sqrt{1+b^2}+b\sqrt{1+a^2}&=\tfrac{3}{4},\\ b\sqrt{1+c^2}+c\sqrt{1+b^2}&=\tfrac{5}{12}, \ \text{and} \\ c\sqrt{1+a^2}+a\sqrt{1+c^2}&=\tfrac{21}{20}. \end{align*} Compute $a.$

1972 IMO Longlists, 11

Tags: inequality , optimization , maximization

The least number is $m$ and the greatest number is $M$ among $ a_1 ,a_2 ,\ldots,a_n$ satisfying $ a_1 \plus{}a_2 \plus{}...\plus{}a_n \equal{}0$. Prove that \[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]

2018 PUMaC Live Round, 5.3

Tags:

Let $k$ be the largest integer such that $2^k$ divides $$\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}\right)^2\right)\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}^2\right)\right).$$ Find $k$.

2018 Math Prize for Girls Problems, 5

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Consider the following system of 7 linear equations with 7 unknowns: \[ \begin{split} a+b+c+d+e & = 1 \\ b+c+d+e+f & = 2 \\ c+d+e+f+g & = 3 \\ d+e+f+g+a & = 4 \\ e+f+g+a+b & = 5 \\ f+g+a+b+c & = 6 \\ g+a+b+c+d & = 7 . \end{split} \] What is $g$?

2003 Korea Junior Math Olympiad, 1

Tags: square , number theory , integer

Show that for any non-negative integer $n$, the number $2^{2n+1}$ cannot be expressed as a sum of four non-zero square numbers.

2008 Princeton University Math Competition, A1/B3

Tags: algebra

Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...,$ find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$.

2000 Romania National Olympiad, 4

Tags: polynomial , algebra

Let $ f $ be a polynom of degree $ 3 $ and having rational coefficients. Prove that, if there exist two distinct nonzero rational numbers $ a,b $ and two roots $ x,y $ of $ f $ such that $ ax+by $ is rational, then all roots of $ f $ are rational.

2016 NIMO Problems, 3

Tags:

Find the sum of all positive integers $n$ such that exactly $2\%$ of the numbers in the set $\{1, 2, \ldots, n\}$ are perfect squares. [i]Proposed by Michael Tang[/i]

1997 Moldova Team Selection Test, 9

Tags: function , algebra

Find all $t\in \mathbb Z$ such that: exists a function $f:\mathbb Z^+\to \mathbb Z$ such that: $f(1997)=1998$ $\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$

2013 Math Prize For Girls Problems, 1

Tags: geometry

The figure below shows two equilateral triangles each with area 1. [asy] unitsize(40); draw(polygon(3)); draw(rotate(60) * polygon(3)); [/asy] The intersection of the two triangles is a regular hexagon. What is the area of the union of the two triangles?

1973 AMC 12/AHSME, 26

Tags: arithmetic sequence

The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is $ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 8$

2006 Thailand Mathematical Olympiad, 11

Tags: number theory , remainder , product , prime

Let $p_n$ be the $n$-th prime number. Find the remainder when $\Pi_{n=1}^{2549} 2006^{p^2_{n-1}}$ is divided by $13$

2020 HK IMO Preliminary Selection Contest, 9

Tags: geometry

In $\Delta ABC$, $\angle B=46.6^\circ$. $D$ is a point on $BC$ such that $\angle BAD=20.1^\circ$. If $AB=CD$ and $\angle CAD=x^\circ$, find $x$.

2015 Azerbaijan JBMO TST, 3

Tags: geometry , circumcircle , symmetry

There is a triangle $ABC$ that $AB$ is not equal to $AC$.$BD$ is interior bisector of $\angle{ABC}$($D\in AC$) $M$ is midpoint of $CBA$ arc.Circumcircle of $\triangle{BDM}$ cuts $AB$ at $K$ and $J,$ is symmetry of $A$ according $K$.If $DJ\cap AM=(O)$, Prove that $J,B,M,O$ are cyclic.

Cono Sur Shortlist - geometry, 2018.G6

Tags: geometry , orthocenter , circumcircle , concurrency , tangent circle

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_A$ passes through the points $A$ and $H$ and is tangent to the circumcircle of the triangle $ABC$. Similarly, define the points $X_B$ and $X_C$. Let $O_A$, $O_B$ and $O_C$ be the reflections of $O$ with respect to sides $BC$, $CA$ and $AB$, respectively. Prove that the lines $O_AX_A$, $O_BX_B$ and $O_CX_C$ are concurrent.

2008 Tournament Of Towns, 7

Tags: angle , geometry

A convex quadrilateral $ABCD$ has no parallel sides. The angles between the diagonal $AC$ and the four sides are $55^o, 55^o, 19^o$ and $16^o$ in some order. Determine all possible values of the acute angle between $AC$ and $BD$.

2011 Junior Balkan MO, 3

Tags: geometry , rhombus , symmetry , ratio , combinatorics

Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$.