Olympiad problems

2016 China Second Round Olympiad, 3

Tags: combinatorics

Given $10$ points in the space such that each $4$ points are not lie on a plane. Connect some points with some segments such that there are no triangles or quadrangles. Find the maximum number of the segments.

2013 F = Ma, 16

Tags: calculus , derivative

A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way. How much time passes until the cloud collapses fully? (The constant $0.5427$ is actually $\sqrt{\frac{3 \pi}{32}}$.) $\textbf{(A) } \frac{0.5427}{r_0^2 \sqrt{G \rho_0}}\\ \\ \textbf{(B) } \frac{0.5427}{r_0 \sqrt{G \rho_0}}\\ \\ \textbf{(C) } \frac{0.5427}{\sqrt{r_0} \sqrt{G \rho_0}}\\ \\ \textbf{(D) } \frac{0.5427}{\sqrt{G \rho_0}}\\ \\ \textbf{(E) } \frac{0.5427}{\sqrt{G \rho_0}}r_0$

2010 Princeton University Math Competition, 7

Tags: polynomial , function , algebra

Let $n$ be the number of polynomial functions from the integers modulo $2010$ to the integers modulo $2010$. $n$ can be written as $n = p_1 p_2 \cdots p_k$, where the $p_i$s are (not necessarily distinct) primes. Find $p_1 + p_2 + \cdots + p_n$.

1971 Polish MO Finals, 4

Tags: number theory , diophantine

Prove that if positive integers $x,y,z$ satisfy the equation $$x^n + y^n = z^n,$$ then $\min\, (x,y) \ge n$.

2016 South East Mathematical Olympiad, 6

Tags: combinatorics

Toss the coin $n$ times, assume that each time, only appear only head or tail Let $a(n)$ denote number of way that head appear in multiple of $3$ times among $n$ times Let $b(n)$ denote numbe of way that head appear in multiple of $6$ times among $n$ times $(1)$ Find $a(2016)$ and $b(2016)$ $(2)$ Find the number of positive integer $n\leq 2016$ that $2b(n)-a(n)\geq 0$

2013 Philippine MO, 2

Tags: geometry

2. Let P be a point in the interior of triangle ABC . Extend AP, BP, and CP to meet BC, AC, and AB at D, E, and F, respectively. If triangle APF, triangle BPD and triangle CPE have equal areas, prove that P is the centroid of triangle ABC .

2010 VTRMC, Problem 7

Tags: real analysis , sum , summation , sequence

Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$) and set $b_n=\frac1{na_n^2}$ for $n\ge1$. Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.

1997 Junior Balkan MO, 3

Tags: inequalities

Let $ABC$ be a triangle and let $I$ be the incenter. Let $N$, $M$ be the midpoints of the sides $AB$ and $CA$ respectively. The lines $BI$ and $CI$ meet $MN$ at $K$ and $L$ respectively. Prove that $AI+BI+CI>BC+KL$. [i]Greece[/i]

2014 Contests, 2

Tags: geometry

Suppose $ABCD$ is a convex quadrilateral.Points $P,Q,R$ and $S$ are four points on the line segments $AB,BC,CD$ and $DA$ respectively.The line segments $PR$ and $QS$ meet at $T$.Suppose that each of the quadrilaterals $APTS,BQTP,CRTQ$ and $DSTR$ have an incircle.Prove that the quadrilateral $ABCD$ also has an incircle.

2002 Iran MO (3rd Round), 7

Tags: trigonometry , geometry

In triangle $ABC$, $AD$ is angle bisector ($D$ is on $BC$) if $AB+AD=CD$ and $AC+AD=BC$, what are the angles of $ABC$?

2000 AIME Problems, 8

Tags: geometry , trapezoid , ratio , quadratic , trigonometry , algebra , quadratic formula

In trapezoid $ABCD,$ leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD},$ and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001},$ find $BC^2.$

2003 National High School Mathematics League, 13

Tags: inequalities

Prove that $2\sqrt{1+x}+\sqrt{2x-3}+\sqrt{15-3x}<2\sqrt{19}$, where $\frac{3}{2}\leq x\leq5$.

1997 Tournament Of Towns, (559) 4

Tags: combinatorics , chessboard

The maximum possible number of knights are placed on a $5 \times 5$ chessboard so that no two attack each other. Prove that there is only one possible placement. (A Kanel)

2019 Argentina National Olympiad Level 2, 6

Tags: combinatorics

Let $n$ be a natural number. We define $f(n)$ as the number of ways to express $n$ as a sum of powers of $2$, where the order of the terms is taken into account. For example, $f(4) = 6$, because $4$ can be written as: \begin{align*} 4;\\ 2 + 2;\\ 2 + 1 + 1;\\ 1 + 2 + 1;\\ 1 + 1 + 2;\\ 1 + 1 + 1 + 1. \end{align*} Find the smallest $n$ greater than $2019$ for which $f(n)$ is odd.

2022 Durer Math Competition Finals, 1

Tags: square , geometry , tangent , equilateral

To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.

2010 Bosnia and Herzegovina Junior BMO TST, 2

Tags: polynomial , algebra , minimization

Let us consider every third degree polynomial $P(x)$ with coefficients as nonnegative positive integers such that $P(1)=20$. Among them determine polynomial for which is: $a)$ Minimal value of $P(4)$ $b)$ Maximal value of $P(3)/P(2)$

2000 Polish MO Finals, 1

Tags: trigonometry , function , algebra

Find number of solutions in non-negative reals to the following equations: \begin{eqnarray*}x_1 + x_n ^2 = 4x_n \\ x_2 + x_1 ^2 = 4x_1 \\ ... \\ x_n + x_{n-1}^2 = 4x_{n-1} \end{eqnarray*}

2004 National Olympiad First Round, 16

Tags:

What is the sum of real roots of the equation $x^4-4x^3+5x^2-4x+1 = 0$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1 $

2020-2021 OMMC, 2

Tags:

Sequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5,$ $b_1 = 7,$ $$a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2},$$ and $$b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}.$$ $ $ \\ How many integers $n$ from 1 to 1000 satisfy the property that $a_n, b_n$ form the legs of a right triangle with a hypotenuse that has integer length?

2014 Dutch IMO TST, 4

Tags: geometric transformation , reflection , angle bisector , geometry

Let $\triangle ABC$ be a triangle with $|AC|=2|AB|$ and let $O$ be its circumcenter. Let $D$ be the intersection of the bisector of $\angle A$ with $BC$. Let $E$ be the orthogonal projection of $O$ to $AD$ and let $F\ne D$ be the point on $AD$ satisfying $|CD|=|CF|$. Prove that $\angle EBF=\angle ECF$.

2016 Harvard-MIT Mathematics Tournament, 8

Tags:

Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the number of integers $2 \le n \le 50$ such that $n$ divides $\phi^{!}(n)+1$.

2016 AIME Problems, 1

Tags: geometric series

For $-1 < r < 1$, let $S(r)$ denote the sum of the geometric series \[12 + 12r + 12r^2 + 12r^3 + \ldots.\] Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a) + S(-a)$.

2016 Switzerland Team Selection Test, Problem 6

Tags: induction , symmetry , algebra , polynomial

Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.

2018 Iran MO (1st Round), 8

Tags: combinatorics

The license plate of each automobile in Iran consists of a two-digit and a three-digit number as well as a letter of the Persian alphabet. The digit $0$ is not used in the two numbers. To each license plate, we assign the product of the two numbers on it. For example, if the two numbers are $12$ and $365$ on a license plate, the assigned number would be $12 \times 365 = 4380$. What is the average of all the assigned numbers to all possible license plates?

2022 IMO Shortlist, G7

Tags: euler , geometry , circumcircle

Two triangles $ABC, A’B’C’$ have the same orthocenter $H$ and the same circumcircle with center $O$. Letting $PQR$ be the triangle formed by $AA’, BB’, CC’$, prove that the circumcenter of $PQR$ lies on $OH$.