Olympiad problems

1990 Turkey Team Selection Test, 5

Tags: number theory

Let $b_m$ be numbers of factors $2$ of the number $m!$ (that is, $2^{b_m}|m!$ and $2^{b_m+1}\nmid m!$). Find the least $m$ such that $m-b_m = 1990$.

2023 Estonia Team Selection Test, 4

Tags: combinatorics , integer sequence

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

2023 Germany Team Selection Test, 2

Tags: combinatorics , integer sequence

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

2022 Regional Competition For Advanced Students, 4

Tags: combinatorics , number theory

We are given the set $$M = \{-2^{2022}, -2^{2021}, . . . , -2^{2}, -2, -1, 1, 2, 2^2, . . . , 2^{2021}, 2^{2022}\}.$$ Let $T$ be a subset of $M$, such that neighbouring numbers have the same difference when the elements are ordered by size. (a) Determine the maximum number of elements that such a set $T$ can contain. (b) Determine all sets $T$ with the maximum number of elements. [i](Walther Janous)[/i]

2010 China Team Selection Test, 1

Tags: inequalities , induction , function , combinatorics

Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings: (1) $\sum_{v\in V} f(v)=|E|$; (2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$. Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.

1950 AMC 12/AHSME, 2

Tags:

Let $R=gS-4$. When $S=8$, $R=16$. When $S=10$, $R$ is equal to: $\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ \text{None of these}$

2001 Hungary-Israel Binational, 5

Tags: incenter , circumcircle , angle bisector , geometry

In a triangle $ABC$ , $B_{1}$ and $C_{1}$ are the midpoints of $AC$ and $AB$ respectively, and $I$ is the incenter. The lines $B_{1}I$ and $C_{1}I$ meet $AB$ and $AC$ respectively at $C_{2}$ and $B_{2}$ . If the areas of $\Delta ABC$ and $\Delta AB_{2}C_{2}$ are equal, ﬁnd $\angle{BAC}$ .

2015 Federal Competition For Advanced Students, 2

Tags: geometry

Let $ABC$ be an acute-angled triangle with $AC < AB$ and circumradius $R$. Furthermore, let $D$ be the foot ofthe altitude from $A$ on $BC$ and let $T$ denote the point on the line $AD$ such that $AT = 2R$ holds with $D$ lying between $A$ and $T$. Finally, let $S$ denote the mid-point of the arc $BC$ on the circumcircle that does not include $A$. Prove: $\angle AST = 90^\circ$. (Karl Czakler)

1967 German National Olympiad, 4

Tags: geometry , polygon , counting , obtuse triangle

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?

2022 VTRMC, 2

Tags: ellipse

Let $A$ and $B$ be the two foci of an ellipse and let $P$ be a point on this ellipse. Prove that the focal radii of $P$ (that is, the segments $\overline{AP}$ and $\overline{BP}$ ) form equal angles with the tangent to the ellipse at $P$.

1988 AIME Problems, 14

Tags: geometry , geometric transformation , reflection , analytic geometry , graphing lines , slope , trigonometry

Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form \[ 12x^2 + bxy + cy^2 + d = 0. \] Find the product $bc$.

1999 Federal Competition For Advanced Students, Part 2, 1

Tags: floor function , number theory , logarithm

Prove that for each positive integer $n$, the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.

2018 Latvia Baltic Way TST, P8

Tags: combinatorics

Let natural $n \ge 2$ be given. Let Laura be a student in a class of more than $n+2$ students, all of which participated in an olympiad and solved some problems. Additionally, it is known that: [list] [*] for every pair of students there is exactly one problem that was solved by both students; [*] for every pair of problems there is exactly one student who solved both of them; [*] one specific problem was solved by Laura and exactly $n$ other students. [/list] Determine the number of students in Laura's class.

2003 Kazakhstan National Olympiad, 5

Tags: number theory

Prove that for all primes $p>3$, $\binom{2p}{p}-2$ is divisible by $p^3$

2005 Junior Balkan Team Selection Tests - Romania, 5

Tags: geometry , rhombus

On the sides $AD$ and $BC$ of a rhombus $ABCD$ we consider the points $M$ and $N$ respectively. The line $MC$ intersects the segment $BD$ in the point $T$, and the line $MN$ intersects the segment $BD$ in the point $U$. We denote by $Q$ the intersection between the line $CU$ and the side $AB$ and with $P$ the intersection point between the line $QT$ and the side $CD$. Prove that the triangles $QCP$ and $MCN$ have the same area.

1978 IMO Longlists, 54

Tags: geometry

Let $p, q$ and $r$ be three lines in space such that there is no plane that is parallel to all three of them. Prove that there exist three planes $\alpha, \beta$, and $\gamma$, containing $p, q$, and $r$ respectively, that are perpendicular to each other $(\alpha\perp\beta, \beta\perp\gamma, \gamma\perp \alpha).$

1978 IMO Longlists, 7

Tags: modular arithmetic , number theory , digit , decimal representation

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

1993 IMO Shortlist, 3

Tags: inequalities , function , four variables , 4-variable inequality , algebra

Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.

2007 Stars of Mathematics, 4

Tags: greatest common divisor , gcd , number theory

Show that any subset of $ A=\{ 1,2,...,2007\} $ having $ 27 $ elements contains three distinct numbers such that the greatest common divisor of two of them divides the other one. [i]Dan Schwarz[/i]

1997 Tournament Of Towns, (564) 5

Tags: combinatorics

Dima invented a secret code in which every letter is replaced by a word no longer than $10$ letters. A code is called “good” if every encoded word can be decoded in only one way. Serjozha (with the help of a computer) checked that for Dima’s code, every possible word of at most $10000$ letters can be decoded in only one way. Does it follow that Dima’s code is good? (Note that Dima and Serjozha are Russian, so they use the Cyrillic alphabet, which has $ 33$ letters! A word is any sequence of letters.) (D Piontkovskiy, S Shalunov)

2019 AIME Problems, 3

Tags: geometry , area

In $\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, and points $E$ and $F$ lie on $\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$.

2023 Math Prize for Girls Problems, 1

Tags:

The frame of a painting has the form of a $105^{\prime\prime}$ by $105^{\prime\prime}$ square with a $95^{\prime\prime}$ by $95^{\prime\prime}$ square removed from its center. The frame is built out of congruent isosceles trapezoids with angles measuring $45^\circ$ and $135^\circ$. Each trapezoid has one base on the frame's outer edge and one base on the frame's inner edge. Each outer edge of the frame contains an odd number of trapezoid bases that alternate long, short, long, short, etc. What is the maximum possible number of trapezoids in the frame?

I Soros Olympiad 1994-95 (Rus + Ukr), 9.6

Tags: geometry , orthocenter , incircle

In the triangle $ABC$, the orthocenter $H$ lies on the inscribed circle. Is this triangle necessarily isosceles?

2015 AMC 10, 22

Tags: probability , counting , distinguishability

Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand? $\textbf{(A) }\dfrac{47}{256}\qquad\textbf{(B) }\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$

2011 Argentina Team Selection Test, 3

Tags: geometry , trapezoid , circumcircle

Let $ABCD$ be a trapezoid with bases $BC \parallel AD$, where $AD > BC$, and non-parallel legs $AB$ and $CD$. Let $M$ be the intersection of $AC$ and $BD$. Let $\Gamma_1$ be a circumference that passes through $M$ and is tangent to $AD$ at point $A$; let $\Gamma_2$ be a circumference that passes through $M$ and is tangent to $AD$ at point $D$. Let $S$ be the intersection of the lines $AB$ and $CD$, $X$ the intersection of $\Gamma_1$ with the line $AS$, $Y$ the intesection of $\Gamma_2$ with the line $DS$, and $O$ the circumcenter of triangle $ASD$. Show that $SO \perp XY$.