Olympiad problems

1999 Slovenia National Olympiad, Problem 2

Tags: game , combinatorics

The numbers $1,\frac12,\frac13,\ldots,\frac1{1999}$ are written on a blackboard. In every step, we choose two of them, say $a$ and $b$, erase them, and write the number $ab+a+b$ instead. This step is repeated until only one number remains. Can the last remaining number be equal to $2000$?

PEN S Problems, 14

Tags: inequalities

Let $p$ be an odd prime. Determine positive integers $x$ and $y$ for which $x \le y$ and $\sqrt{2p}-\sqrt{x}-\sqrt{y}$ is nonnegative and as small as possible.

1979 AMC 12/AHSME, 24

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

Sides $AB,~ BC,$ and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4,~ 5,$ and $20$, respectively. If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B =\frac{3}{5} $, then side $AD$ has length $\textbf{(A) }24\qquad\textbf{(B) }24.5\qquad\textbf{(C) }24.6\qquad\textbf{(D) }24.8\qquad\textbf{(E) }25$ [size=70]*A polygon is called “simple” if it is not self intersecting.[/size]

2022 Brazil Team Selection Test, 4

Tags: combinatorics , graph theory , perfect matching

Let $d_1, d_2, \ldots, d_n$ be given integers. Show that there exists a graph whose sequence of degrees is $d_1, d_2, \ldots, d_n$ and which contains an perfect matching if, and only if, there exists a graph whose sequence of degrees is $d_2, d_2, \ldots, d_n$ and a graph whose sequence of degrees is $d_1-1, d_2-1, \ldots, d_n-1$.

2004 AMC 12/AHSME, 23

Tags: algebra , polynomial

A polynomial \[ P(x) \equal{} c_{2004}x^{2004} \plus{} c_{2003}x^{2003} \plus{} ... \plus{} c_1x \plus{} c_0 \]has real coefficients with $ c_{2004}\not \equal{} 0$ and $ 2004$ distinct complex zeroes $ z_k \equal{} a_k \plus{} b_ki$, $ 1\leq k\leq 2004$ with $ a_k$ and $ b_k$ real, $ a_1 \equal{} b_1 \equal{} 0$, and \[ \sum_{k \equal{} 1}^{2004}{a_k} \equal{} \sum_{k \equal{} 1}^{2004}{b_k}. \]Which of the following quantities can be a nonzero number? $ \textbf{(A)}\ c_0 \qquad \textbf{(B)}\ c_{2003} \qquad \textbf{(C)}\ b_2b_3...b_{2004} \qquad \textbf{(D)}\ \sum_{k \equal{} 1}^{2004}{a_k} \qquad \textbf{(E)}\ \sum_{k \equal{} 1}^{2004}{c_k}$

2019 LIMIT Category C, Problem 8

Tags: floor function , algebra

The value of $$\left\lfloor\frac1{3!}+\frac4{4!}+\frac9{5!}+\ldots\right\rfloor$$

2018 Middle European Mathematical Olympiad, 5

Tags: geometry , parallelogram , ohm

Let $ABC$ be an acute-angled triangle with $AB<AC,$ and let $D$ be the foot of its altitude from$A,$ points $B'$ and $C'$ lie on the rays $AB$ and $AC,$ respectively , so that points $B',$ $C'$ and $D$ are collinear and points $B,$ $C,$ $B'$ and $C'$ lie on one circle with center $O.$ Prove that if $M$ is the midpoint of $BC$ and $H$ is the orthocenter of $ABC,$ then $DHMO$ is a parallelogram.

1955 Moscow Mathematical Olympiad, 297

Tags: midpoint , geometry , locus , circles

Given two distinct nonintersecting circles none of which is inside the other. Find the locus of the midpoints of all segments whose endpoints lie on the circles.

2008 National Olympiad First Round, 24

Tags: ratio , symmetry , geometric series

How many of the numbers \[ a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6 \] are negative if $a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}$? $ \textbf{(A)}\ 121 \qquad\textbf{(B)}\ 224 \qquad\textbf{(C)}\ 275 \qquad\textbf{(D)}\ 364 \qquad\textbf{(E)}\ 375 $

2019 Germany Team Selection Test, 3

Tags: geometry , concurrency

A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$. [i]Proposed by Mongolia[/i]

2024 China Team Selection Test, 5

Tags: algebra , functional equation

Find all functions $f:\mathbb N_+\to \mathbb N_+,$ such that for all positive integer $a,b,$ $$\sum_{k=0}^{2b}f(a+k)=(2b+1)f(f(a)+b).$$ [i]Created by Liang Xiao, Yunhao Fu[/i]

2024 Middle European Mathematical Olympiad, 1

Tags: algebra , inequalities , sequence

Consider two infinite sequences $a_0,a_1,a_2,\dots$ and $b_0,b_1,b_2,\dots$ of real numbers such that $a_0=0$, $b_0=0$ and \[a_{k+1}=b_k, \quad b_{k+1}=\frac{a_kb_k+a_k+1}{b_k+1}\] for each integer $k \ge 0$. Prove that $a_{2024}+b_{2024} \ge 88$.

2021 Saudi Arabia IMO TST, 7

Tags: geometry , right triangle , isosceles triangle

Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

2016 ITAMO, 1

Tags: geometry , angle bisector , projection

Let $ABC$ be a triangle, and let $D$ and $E$ be the orthogonal projections of $A$ onto the internal bisectors from $B$ and $C$. Prove that $DE$ is parallel to $BC$.

2018 Dutch IMO TST, 4

Tags: combinatorics

In the classroom of at least four students the following holds: no matter which four of them take seats around a round table, there is always someone who either knows both of his neighbours, or does not know either of his neighbours. Prove that it is possible to divide the students into two groups such that in one of them, all students know one another, and in the other, none of the students know each other. (Note: if student A knows student B, then student B knows student A as well.)

2008 Saint Petersburg Mathematical Olympiad, 5

Tags: geometry , 3d geometry , tetrahedron

All faces of the tetrahedron $ABCD $ are acute-angled triangles.$AK$ and $AL$ -are altitudes in faces $ABC$ and $ABD$. Points $C,D,K,L$ lies on circle. Prove, that $AB \perp CD$

2019 Saudi Arabia Pre-TST + Training Tests, 3.3

Tags: number theory , gcd , greatest common divisor

Let $d$ be a positive divisor of a positive integer $m$ and $(a_l), (b_l)$ two arithmetic sequences of positive integers. It is given that $gcd(a_i, b_j) = 1$ and $gcd(a_k, b_n) = m$ for some positive integers $i,j,k,$ and $n$. Prove that there exist positive integers $t$ and $s$ such that $gcd(a_t, b_s) = d$.

2016 LMT, 3

Tags:

The squares of two positive integers differ by 2016. Find the maximum possible sum of the two integers. [i]Proposed by Clive Chan

1967 Putnam, A2

Tags: matrices , power series , generating functions

Define $S_0$ to be $1.$ For $n \geq 1 $, let $S_n $ be the number of $n\times n $ matrices whose elements are nonnegative integers with the property that $a_{ij}=a_{ji}$ (for $i,j=1,2,\ldots, n$) and where $\sum_{i=1}^{n} a_{ij}=1$ (for $j=1,2,\ldots, n$). Prove that a) $S_{n+1}=S_{n} +nS_{n-1}.$ b) $\sum_{n=0}^{\infty} S_{n} \frac{x^{n}}{n!} =\exp \left(x+\frac{x^{2}}{2}\right).$

2015 Kyoto University Entry Examination, 1

Tags: geometry

1. The Line $y=px+q$ intersects $y=x^2-x$, but not intersect $y=|x|+|x-1|+1$, then illustlate range of $(p,q)$ and find the area.

1977 Yugoslav Team Selection Test, Problem 3

Tags: triangle , geometry

Assume that the equality $2BC=AB+AC$ holds in $\triangle ABC$. Prove that: (a) The vertex $A$, the midpoints $M$ and $N$ of $AB$ and $AC$ respectively, the incenter $I$, and the circumcenter $O$ belong to a circle $k$. (b) The line $GI$, where $G$ is the centroid of $\triangle ABC$ is a tangent to $k$.

2003 Czech-Polish-Slovak Match, 5

Tags: binomial coefficient

Consider the binomial coefficients $\binom{n}{k}=\frac{n!}{k!(n-k)!}\ (k=1,2,\ldots n-1)$. Determine all positive integers $n$ for which $\binom{n}{1},\binom{n}{2},\ldots ,\binom{n}{n-1}$ are all even numbers.

May Olympiad L2 - geometry, 2001.4

Tags: geometry , area

Ten coins of $1$ cm radius are placed around a circle as indicated in the figure. Each coin is tangent to the circle and its two neighboring coins. Prove that the sum of the areas of the ten coins is twice the area of the circle. [img]https://cdn.artofproblemsolving.com/attachments/5/e/edf7a7d39d749748f4ae818853cb3f8b2b35b5.gif[/img]

2019 LIMIT Category C, Problem 12

Tags: real analysis

$\lim_{x\to0}x\left\lfloor\frac1x\right\rfloor=?$

2021 Malaysia IMONST 1, 13

Tags: algebra

Jasmin has a mobile phone that runs on a battery. When the battery is dead, it takes $2$ hours to recharge it fully, if she is not using the phone. If she uses the phone while recharging, $75\%$ of the charge obtained is immediately consumed and the remaining is stored in the battery. One day, her battery died. Jasmin took $2$ hours $30$ minutes to recharge the battery fully. For how many minutes did she use the phone while recharging?