Olympiad problems

JOM 2015 Shortlist, N8

Set $p\ge 5$ be a prime number and $n$ be a natural number. Let $f$ be a function $ f: \mathbb{Z_{ \neq }}_0 \rightarrow \mathbb{ N }_0 $ satisfy the following conditions: i) For all sequences of integers satisfy $ a_i \not\in \{0, 1\} $, and $ p $ $\not |$ $ a_i-1 $, $ \forall $ $ 1 \le i \le p-2 $,\\ $$ \displaystyle \sum^{p-2}_{i=1}f(a_i)=f(a_1a_2 \cdots a_{p-2}) $$ ii) For all coprime integers $ a $ and $ b $, $ a \equiv b \pmod p \Rightarrow f(a)=f(b) $ iii) There exist $k \in \mathbb{Z}_{\neq 0} $ that satisfy $ f(k)=n $ Prove that the number of such functions is $ d(n) $, where $ d(n) $ denotes the number of divisors of $ n $.

2017 F = ma, 7

Tags: kinetic energy

7) A train, originally of mass $M$, is traveling on a frictionless straight horizontal track with constant speed $v$. Snow starts to fall vertically and sticks to the train at a rate of $\rho$, where $\rho$ has units of kilograms per second. The train's engine keeps the train moving at constant speed $v$ as snow accumulates on the train. What is the rate at which the kinetic energy of the train and snow increases? A) $0$ B) $Mgv$ C) $\frac{1}{2}Mv^2$ D) $\frac{1}{2}pv^2$ E) $\rho v^2$

2022 ITAMO, 3

Tags: algebra , number theory

In a mathematical competition $n=10\,000$ contestants participate. During the final party, in sequence, the first one takes $1/n$ of the cake, the second one takes $2/n$ of the remaining cake, the third one takes $3/n$ of the cake that remains after the first and the second contestant, and so on until the last one, who takes all of the remaining cake. Determine which competitor takes the largest piece of cake.

2023 AMC 10, 17

Tags: geometry , rectangle , perimeter

Let $ABCD$ be a rectangle with $AB = 30$ and $BC = 28$. Point $P$ and $Q$ lie on $\overline{BC}$ and $\overline{CD}$ respectively so that all sides of $\triangle{ABP}, \triangle{PCQ},$ and $\triangle{QDA}$ have integer lengths. What is the perimeter of $\triangle{APQ}$? (A) 84 (B) 86 (C) 88 (D)90 (E)92

2006 Flanders Math Olympiad, 2

Tags: geometry , geometric transformation , reflection , trigonometry , algebra , linear equation

Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point on $\left[AB\right]$. $Q$ is the point on $BC$ such that $PQ$ is perpendicular to $AB$. $R$ is the point on $AC$ such that $QR$ is perpendicular to $BC$. And $S$ is the point on $AB$ such that $RS$ is perpendicular to $AC$. $Q'$ is the point on $BC$ such that $PQ'$ is perpendicular to $BC$. $R'$ is the point on $AC$ such that $Q'R'$ is perpendicular to $AC$. And $S'$ is the point on $AB$ such that $R'S'$ is perpendicular to $AB$. Determine $\frac{|PB|}{|AB|}$ if $S=S'$.

1985 Czech And Slovak Olympiad IIIA, 6

Tags: number theory , divide

Prove that for every natural number $n > 1$ there exists a suquence $a_1$,$a_2$, $...$, $a_n$ of the numbers $1,2,...,n$ such that for each $k \in \{1,2,...,n-1\}$ the number $a_{k+1}$ divides $a_1+a_2+...+a_k$.

MOAA Team Rounds, 2019.5

Tags: geometry , team

Let $ABC$ be a triangle with $AB = AC = 10$ and $BC = 12$. Define $\ell_A$ as the line through $A$ perpendicular to $\overline{AB}$. Similarly, $\ell_B$ is the line through $B$ perpendicular to $\overline{BC}$ and $\ell_C$ is the line through $C$ perpendicular to $\overline{CA}$. These three lines $\ell_A, \ell_B, \ell_C$ form a triangle with perimeter $m/n$ for relatively prime positive integers $m$ and $n$. Find $m + n$.

2014 Contests, 3

Tags: function , algebra , geometry , reflection

We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point). (a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$. (b) Find the largest possible size of a very set not contained in any line. (Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.) [i]Proposed by Sammy Luo[/i]

2010 Victor Vâlcovici, 3

Tags: linear algebra , matrix

Find all positive integers $n \geq 2$ with the following property : there is a matrix $A \in M_{n} (\mathbb{R})$ and a prime number $p \geq 2$ such that $A^{*}$ has exactly $p$ not null elements and $A^{p}=0_{n}$.

2015 Princeton University Math Competition, A2/B4

Tags: algebra

There are real numbers $a, b, c, d$ such that for all $(x, y)$ satisfying $6y^2 = 2x^3 + 3x^2 + x$, if $x_1 = ax + b$ and $y_1 = cy + d$, then $y_1^2 = x_1^3 - 36x_1$. What is $a + b + c + d$?

2017 Thailand TSTST, 6

Tags: polynomial , algebra

Find all polynomials $f$ with real coefficients such that for all reals $x, y, z$ such that $x+y+z =0$, the following relation holds: $$f(xy) + f(yz) + f(zx) = f(xy + yz + zx).$$

Today's calculation of integrals, 874

Tags: calculus , integration , parabola , geometry , calculus computations , conic

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

2018 Taiwan TST Round 2, 6

Tags: incenter , geometry

A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.

2016 Harvard-MIT Mathematics Tournament, 7

Tags: geometry

Let ABC be a triangle with $AB = 13, BC = 14, CA = 15$. The altitude from $A$ intersects $BC$ at $D$. Let $\omega_1$ and $\omega_2$ be the incircles of $ABD$ and $ACD$, and let the common external tangent of $\omega_1$ and $\omega_2$ (other than $BC$) intersect $AD$ at $E$. Compute the length of $AE$.

2021 Miklós Schweitzer, 3

Tags: function , real analysis

Let $I \subset \mathbb{R}$ be a nonempty open interval and let $f: I \cap \mathbb{Q} \to \mathbb{R}$ be a function such that for all $x, y \in I \cap \mathbb{Q}$, \[ 4f\left(\frac{3x + y}{4}\right)+ 4f\left(\frac{x + 3y}{4}\right) \le f(x) + 6f\left(\frac{x + y}{2}\right)+ f(y). \] Show that $f$ can be continuously extended to $I$.

2021 China Team Selection Test, 5

Tags: geometry , circumcircle

Given a triangle $ABC$, a circle $\Omega$ is tangent to $AB,AC$ at $B,C,$ respectively. Point $D$ is the midpoint of $AC$, $O$ is the circumcenter of triangle $ABC$. A circle $\Gamma$ passing through $A,C$ intersects the minor arc $BC$ on $\Omega$ at $P$, and intersects $AB$ at $Q$. It is known that the midpoint $R$ of minor arc $PQ$ satisfies that $CR \perp AB$. Ray $PQ$ intersects line $AC$ at $L$, $M$ is the midpoint of $AL$, $N$ is the midpoint of $DR$, and $X$ is the projection of $M$ onto $ON$. Prove that the circumcircle of triangle $DNX$ passes through the center of $\Gamma$.

1977 AMC 12/AHSME, 2

Tags:

Which one of the following statements is false? All equilateral triangles are $\textbf{(A)} \ \text{ equiangular} \qquad \textbf{(B)} \ \text{isosceles} \qquad \textbf{(C)} \ \text{regular polygons } \qquad \textbf{(D)} \ \text{congruent to each other} \qquad \textbf{(E)} \ \text{similar to each other} $

2001 Croatia National Olympiad, Problem 3

Tags: combinatorics , game

Numbers $1,\frac12,\frac13,\ldots,\frac1{2001}$ are written on a blackboard. A student erases two numbers $x,y$ and writes down the number $x+y+xy$ instead. Determine the number that will be written on the board after $2000$ such operations.

2013 Hanoi Open Mathematics Competitions, 6

Tags: geometry , geometric inequality , area of a triangle , area

Let $ABC$ be a triangle with area $1$ (cm$^2$). Points $D,E$ and $F$ lie on the sides $AB, BC$ and CA, respectively. Prove that $min\{$area of $\vartriangle ADF,$ area of $\vartriangle BED,$ area of $\vartriangle CEF\} \le \frac14$ (cm$^2$).

2000 IMC, 6

Tags: function , limit , integration , real analysis

Let $f: \mathbb{R}\rightarrow ]0,+\infty[$ be an increasing differentiable function with $\lim_{x\rightarrow+\infty}f(x)=+\infty$ and $f'$ is bounded, and let $F(x)=\int^x_0 f(t) dt$. Define the sequence $(a_n)$ recursively by $a_0=1,a_{n+1}=a_n+\frac1{f(a_n)}$ Define the sequence $(b_n)$ by $b_n=F^{-1}(n)$. Prove that $\lim_{x\rightarrow+\infty}(a_n-b_n)=0$.

1984 Balkan MO, 2

Tags: parallelogram , circumcircle , ratio , geometric transformation , homothety , reflection , geometry

Let $ABCD$ be a cyclic quadrilateral and let $H_{A}, H_{B}, H_{C}, H_{D}$ be the orthocenters of the triangles $BCD$, $CDA$, $DAB$ and $ABC$ respectively. Show that the quadrilaterals $ABCD$ and $H_{A}H_{B}H_{C}H_{D}$ are congruent.

2019 Romania Team Selection Test, 1

Tags: inequalities , algebra

Determine the largest value the expression $$ \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} $$ may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-negative real numbers, and add up to $ 1. $ Find also the specific values of this numbers that make the above sum achieve the asked maximum.

2016 Middle European Mathematical Olympiad, 3

Tags: circumcenter , perpendicular lines , geometry

Let $ABC$ be an acute triangle such that $\angle BAC > 45^{\circ}$ with circumcenter $O$. A point $P$ is chosen inside triangle $ABC$ such that $A, P, O, B$ are concyclic and the line $BP$ is perpendicular to the line $CP$. A point $Q$ lies on the segment $BP$ such that the line $AQ$ is parallel to the line $PO$. Prove that $\angle QCB = \angle PCO$.

1971 IMO Longlists, 12

Tags: algebra , logarithm

A system of n numbers $x_1, x_2, \ldots, x_n$ is given such that \[x_1 = \log_{x_{n-1}} x_n, x_2 = \log_{x_{n}} x_1, \ldots, x_n = \log_{x_{n-2}} x_{n-1}.\] Prove that $\prod_{k=1}^n x_k =1.$

1987 Tournament Of Towns, (132) 1

Tags: inequalities , algebra

Prove that for all values of $a$, $3(1+a^2+a^4) \ge (1+a+a^2)^2$ .