Olympiad problems

2022 Harvard-MIT Mathematics Tournament, 9

Tags: algebra

Suppose $P(x)$ is a monic polynomial of degree $2023$ such that $P(k) = k^{2023}P(1-\frac{1}{k})$ for every positive integer $1 \leq k \leq 2023$. Then $P(-1) = \frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Compute the unique integer $0 \leq n < 2027$ such that $bn-a$ is divisible by the prime $2027$.

2023 Indonesia TST, A

Tags: function , functional equation , algebra

Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfied \[f(x+y) + f(x)f(y) = f(xy) + 1 \] $\forall x, y \in \mathbb{R}$

2003 China Team Selection Test, 2

Tags: combinatorics

Suppose $A=\{1,2,\dots,2002\}$ and $M=\{1001,2003,3005\}$. $B$ is an non-empty subset of $A$. $B$ is called a $M$-free set if the sum of any two numbers in $B$ does not belong to $M$. If $A=A_1\cup A_2$, $A_1\cap A_2=\emptyset$ and $A_1,A_2$ are $M$-free sets, we call the ordered pair $(A_1,A_2)$ a $M$-partition of $A$. Find the number of $M$-partitions of $A$.

1999 Greece Junior Math Olympiad, 3

Tags: geometry , equilateral

Let $ABC$ be an equilateral triangle . Let point $D$ lie on side $AB,E$ lie on side $AC, D_1$ and $E_1$ lie on side BC such that $AB=DB+BD_1$ and $AC=CE+CE_1$. Calculate the smallest angle between the lines $DE_1$ and $ED_1$.

1978 IMO, 3

Tags: algebra , functional equation , sequence , partition

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

1935 Moscow Mathematical Olympiad, 009

Tags: geometry , 3d geometry , cone , regular hexagon , hexagon , angle

The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\phi$ between the cone’s generating line and its base.

1991 AMC 12/AHSME, 2

Tags: function , absolute value

$|3 - \pi| =$ $ \textbf{(A)}\ \frac{1}{7}\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 3 - \pi\qquad\textbf{(D)}\ 3 + \pi\qquad\textbf{(E)}\ \pi - 3 $

2024 Iran Team Selection Test, 9

Tags: number theory

Prove that for any natural numbers $a , b , c$ that $b>a>1$ and $gcd(c,ab)=1$ , there exist a natural number $n$ such that : $$c | \binom{b^n}{a^n}$$ [i]Proposed by Navid Safaei[/i]

1983 AIME Problems, 2

Tags: search , function , absolute value , complex number

Let $f(x) = |x - p| + |x - 15| + |x - p - 15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \le x \le 15$.

2022 Taiwan TST Round 2, 3

Tags: geometry , circumcircle , excircle , ddit

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.

2024 Belarus Team Selection Test, 4.3

Tags: geometry

An isosceles triangle $ABC$ is given($AB=BC$). Point $D$ lies inside of it such that $\angle ADC=150$, $E$ lies on $CD$ such that $AE=AB$. It turned out that $\angle EBC+\angle BAE=60$. Prove that $\angle BDC+\angle CAE=90$ [i]D. Vasilyev[/i]

1996 IMO Shortlist, 8

Tags: inequalities , trigonometry , geometry , circumcircle

Let $ ABCD$ be a convex quadrilateral, and let $ R_A, R_B, R_C, R_D$ denote the circumradii of the triangles $ DAB, ABC, BCD, CDA,$ respectively. Prove that $ R_A \plus{} R_C > R_B \plus{} R_D$ if and only if $ \angle A \plus{} \angle C > \angle B \plus{} \angle D.$

2008 Hungary-Israel Binational, 2

Tags: induction , algebra

The sequence $ a_n$ is defined as follows: $ a_0\equal{}1, a_1\equal{}1, a_{n\plus{}1}\equal{}\frac{1\plus{}a_{n}^2}{a_{n\minus{}1}}$. Prove that all the terms of the sequence are integers.

2016 Harvard-MIT Mathematics Tournament, 7

Tags: hmmt

Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon.

2023 Austrian Junior Regional Competition, 2

Tags: geometry , collinear , hexagon

Let $ABCDEF$ be a regular hexagon with sidelength s. The points $P$ and $Q$ are on the diagonals $BD$ and $DF$, respectively, such that $BP = DQ = s$. Prove that the three points $C$, $P$ and $Q$ are on a line. [i](Walther Janous)[/i]

2024 Al-Khwarizmi IJMO, 5

Tags: combinatorics

At a party, every guest is a friend of exactly fourteen other guests (not including him or her). Every two friends have exactly six other attending friends in common, whereas every pair of non-friends has only two friends in common. How many guests are at the party? Please explain your answer with proof. [i]Proposed by Alexander Slavik, Czech Republic[/i]

2013 Princeton University Math Competition, 2

Tags:

Betty Lou and Peggy Sue take turns flipping switches on a $100 \times 100$ grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and column. Find the maximum number of switches that can be on, in total, when they finish.

1996 Spain Mathematical Olympiad, 1

Tags: number theory , greatest common divisor , integer

The natural numbers $a$ and $b$ are such that $ \frac{a+1}{b}+ \frac{b+1}{a}$ is an integer. Show that the greatest common divisor of a and b is not greater than $\sqrt{a+b}$.

2008 ITest, 57

Tags: trigonometry , number theory , relatively prime

Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \dfrac{193}{137}.\] If $a+b=m/n$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$.

2013 Math Prize For Girls Problems, 20

Tags: inequalities , trigonometry

Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and \[ a_{n} = 2 a_{n-1}^2 - 1 \] for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality \[ a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}. \] What is the value of $100c$, rounded to the nearest integer?

2014 JBMO TST - Turkey, 4

Tags: combinatorics

Alice and Bob play a game on a complete graph $G$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses a positive integer number $m,$ $1 \le m \le 1000$ and after that directs $m$ undirected edges of $G$. The game ends when all edges are directed. If there is some directed cycle in $G$ Alice wins. Determine whether Alice has a winning strategy.

LMT Speed Rounds, 2016.18

Tags:

Let $\triangle ABC$ be a triangle with $AB=5, BC=6, CA=7$. Suppose $P$ is a point inside $\triangle ABC$ such that $\triangle BPA\sim \triangle APC$. If $AP$ intersects $BC$ at $X$, find $\frac{BX}{CX}$. [i]Proposed by Nathan Ramesh

1950 AMC 12/AHSME, 6

Tags:

The values of y which will satisfy the equations $ 2x^2\plus{}6x\plus{}5y\plus{}1\equal{}0, 2x\plus{}y\plus{}3\equal{}0$ may be found by solving: $\textbf{(A)}\ y^2+14y-7=0 \qquad \textbf{(B)}\ y^2+8y+1=0 \qquad \textbf{(C)}\ y^2+10y-7=0 \qquad \textbf{(D)}\ y^2+y-12=0 \qquad \textbf{(E)}\ \text{None of these equations}$

2024 Belarusian National Olympiad, 8.1

Tags: number theory

Numbers $7^2$,$8^2,\ldots,2023^2$,$2024^2$ are written on the board. Is it possible to add to one of them $7$, to some other one $8$, $\ldots$, to the remaining $2024$ such that all numbers became prime [i]M. Zorka[/i]

2009 Ukraine National Mathematical Olympiad, 2

Tags:

There is convex $2009$-gon on the plane. [b]a)[/b] Find the greatest number of vertices of $2009$-gon such that no two forms the side of the polygon. [b]b)[/b] Find the greatest number of vertices of $2009$-gon such that among any three of them there is one that is not connected with other two by side.