Olympiad problems

2024 Taiwan TST Round 1, C

Tags: combinatorics

A $k$-set is a set with exactly $k$ elements. For a $6$-set $A$ and any collection $\mathcal{F}$ of $4$-sets, we say that $A$ is [i]$\mathcal{F}$-good[/i] if there are exactly three elements $B_1, B_2, B_3$ in $\mathcal{F}$ that are subsets of $A$, and they furthermore satisfy $$(A \backslash B_1) \cup (A \backslash B_2) \cup (A \backslash B_3) = A.$$ Find all $n \geq 6$ so that there exists a collection $\mathcal{F}$ of $4$-subsets of $\{1, 2, \ldots , n\}$ such that every $6$-set $A \subseteq \{1, 2, \ldots , n\}$ is $\mathcal{F}$-good. [i] Proposed by usjl[/i]

2002 Baltic Way, 14

Tags: geometry

Let $L,M$ and $N$ be points on sides $AC,AB$ and $BC$ of triangle $ABC$, respectively, such that $BL$ is the bisector of angle $ABC$ and segments $AN,BL$ and $CM$ have a common point. Prove that if $\angle ALB=\angle MNB$ then $\angle LNM=90^{\circ}$.

1997 Dutch Mathematical Olympiad, 5

Tags: geometry , circumcenter , symmetry

Given is a triangle $ABC$ and a point $K$ within the triangle. The point $K$ is mirrored in the sides of the triangle: $P , Q$ and $R$ are the mirrorings of $K$ in $AB , BC$ and $CA$, respectively . $M$ is the center of the circle passing through the vertices of triangle $PQR$. $M$ is mirrored again in the sides of triangle $ABC$: $P', Q'$ and $R'$ are the mirror of $M$ in $AB$ respectively, $BC$ and $CA$. a. Prove that $K$ is the center of the circle passing through the vertices of triangle $P'Q'R'$ . b. Where should you choose $K$ within triangle $ABC$ so that $M$ and $K$ coincide? Prove your answer.

2000 Turkey Team Selection Test, 3

Tags: number theory

Let $P(x)=x+1$ and $Q(x)=x^2+1.$ Consider all sequences $\langle(x_k,y_k)\rangle_{k\in\mathbb{N}}$ such that $(x_1,y_1)=(1,3)$ and $(x_{k+1},y_{k+1})$ is either $(P(x_k), Q(y_k))$ or $(Q(x_k),P(y_k))$ for each $k. $ We say that a positive integer $n$ is nice if $x_n=y_n$ holds in at least one of these sequences. Find all nice numbers.

2019 Olympic Revenge, 3

Tags: geometry , incenter , euler line , construction

Let $\Gamma$ be a circle centered at $O$ with radius $R$. Let $X$ and $Y$ be points on $\Gamma$ such that $XY<R$. Let $I$ be a point such that $IX = IY$ and $XY = OI$. Describe how to construct with ruler and compass a triangle which has circumcircle $\Gamma$, incenter $I$ and Euler line $OX$. Prove that this triangle is unique.

1986 IMO Shortlist, 14

Tags: geometry , incenter , circumcircle , collinearity

The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.

2003 AMC 10, 3

Tags: induction

The sum of 5 consecutive even integers is $ 4$ less than the sum of the ﬁrst $ 8$ consecutive odd counting numbers. What is the smallest of the even integers? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14$

2022 Math Prize for Girls Problems, 1

Tags:

Determine the real value of $t$ that minimizes the expression \[ \sqrt{t^2 + (t^2 - 1)^2} + \sqrt{(t-14)^2 + (t^2 - 46)^2}. \]

2006 MOP Homework, 3

Tags: vector , geometry

There are $n$ distinct points in the plane. Given a circle in the plane containing at least one of the points in its interior. At each step one moves the center of the circle to the barycenter of all the points in the interior of the circle. Prove that this moving process terminates in the finite number of steps. what does barycenter of n distinct points mean?

1970 Putnam, B2

Tags: algebra , polynomial

The time-varying temperature of a certain body is given by a polynomial in the time of degree at most three. Show that the average temperature of the body between $9$ am and $3$ pm can always be found by taking the average of the temperatures at two fixed times, which are independent of the polynomial. Also, show that these two times are $10\colon \! 16$ am and $1\colon \!44$ pm to the nearest minute.

2006 AMC 8, 7

Tags: geometry

Circle $ X$ has a radius of $ \pi$. Circle $ Y$ has a circumference of $ 8\pi$. Circle $ Z$ has an area of $ 9\pi$. List the circles in order from smallest to largest radius. $ \textbf{(A)}\ X, Y, Z \qquad \textbf{(B)}\ Z, X, Y \qquad \textbf{(C)}\ Y, X, Z \qquad \textbf{(D)}\ Z, Y, X \qquad \textbf{(E)}\ X, Z, Y$

1955 AMC 12/AHSME, 33

Tags:

Henry starts a trip when the hands of the clock are together between $ 8$ a.m. and $ 9$ a.m. He arrives at his destination between $ 2$ p.m. and $ 3$ p.m. when the hands of the clock are exactly $ 180^\circ$ apart. The trip takes: $ \textbf{(A)}\ \text{6 hr.} \qquad \textbf{(B)}\ \text{6 hr. 43\minus{}7/11 min.} \qquad \textbf{(C)}\ \text{5 hr. 16\minus{}4/11 min.} \qquad \textbf{(D)}\ \text{6 hr. 30 min.} \qquad \textbf{(E)}\ \text{none of these}$

2019 PUMaC Algebra A, 2

Tags: algebra , relatively prime , number theory

Let $f(x)=x^2+4x+2$. Let $r$ be the difference between the largest and smallest real solutions of the equation $f(f(f(f(x))))=0$. Then $r=a^{\frac{p}{q}}$ for some positive integers $a$, $p$, $q$ so $a$ is square-free and $p,q$ are relatively prime positive integers. Compute $a+p+q$.

2017 239 Open Mathematical Olympiad, 6

Tags: geometry , angle bisector , circumcircle

Given a circumscribed quadrilateral $ABCD$ in which $$\sqrt{2}(BC-BA)=AC.$$ Let $X$ be the midpoint of $AC$ and $Y$ a point on the angle bisector of $B$ such that $XD$ is the angle bisector of $BXY$. Prove that $BD$ is tangent to the circumcircle of $DXY$.

2021 MOAA, 7

Tags: speed

If positive real numbers $x,y,z$ satisfy the following system of equations, compute $x+y+z$. $$xy+yz = 30$$ $$yz+zx = 36$$ $$zx+xy = 42$$ [i]Proposed by Nathan Xiong[/i]

1993 Bundeswettbewerb Mathematik, 1

Tags: integer , combinatorics , sum , summand

Every positive integer $n>2$ can be written as a sum of distinct positive integers. Let $A(n)$ be the maximal number of summands in such a representation. Find a formula for $A(n).$

1983 All Soviet Union Mathematical Olympiad, 357

Tags: trigonometry

Two acute angles $a$ and $b$ satisfy condition $$\sin^2a+\sin^2b = \sin(a+b)$$ Prove that $a + b = \pi /2$.

2006 Germany Team Selection Test, 2

Tags: algebra , functional equation

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$. [i]Proposed by B.J. Venkatachala, India[/i]

2019 Balkan MO Shortlist, G2

Tags: geometry

Let be a triangle $\triangle ABC$ with $m(\angle ABC) = 75^{\circ}$ and $m(\angle ACB) = 45^{\circ}$. The angle bisector of $\angle CAB$ intersects $CB$ at point $D$. We consider the point $E \in (AB)$, such that $DE = DC$. Let $P$ be the intersection of lines $AD$ and $CE$. Prove that $P$ is the midpoint of segment $AD$.

2002 Stanford Mathematics Tournament, 6

Tags: modular arithmetic

How many integers $x$, from $10$ to $99$ inclusive, have the property that the remainder of $x^2$ divided by $100$ is equal to the square of the units digit of $x$?

2006 MOP Homework, 5

Tags: number theory

Let $a$, $b$, and $c$ be positive integers such that the product $ab$ divides the product $c(c^2-c+1)$ and the sum $a+b$ is divisible the number $c^2+1$. Prove that the sets ${a,b}$ and ${c,c^2-c+1}$ coincide.

2007 Germany Team Selection Test, 1

Tags: incenter , triangle , midpoint , congruent triangles , geometry

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

2022 JBMO Shortlist, C2

Tags: combinatorics , geometry , shortlist

Let $n \ge 2$ be an integer. Alex writes the numbers $1, 2, ..., n$ in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment $s$ we denote by $d_s$ the difference of the numbers written in its extremities and by $p_s$ the number of all other drawn segments which intersect $s$ in its interior. Find the greatest $n$ for which Alex can write the numbers on the circle such that $p_s \le |d_s|$, for each drawn segment $s$.

2022 CCA Math Bonanza, L5.1

Tags:

Alistar wants to wreak havoc on Jhin's yard, which is a 2D plane of grass. First, he selects a number $n$, randomly and uniformly from $[0,1]$, and then he eats all grass within $n$ meters from where he's standing. He then moves 2 meters in a random direction, and repeats his process. He stops if any of the grass that he wants to eat (or, in other words, in his intended eating territory) is already eaten. Estimate the amount of grass Alistar is expected to eat. An estimate $E$ earns $\frac{2}{1+|A-E|}$ points, where $A$ is the actual answer. [i]2022 CCA Math Bonanza Lightning Round 5.1[/i]

2011 Princeton University Math Competition, B1

Tags: combinatorics

How many ways are there to arrange the five letters P,U,M,A,C, such that the two vowels are not adjacent?