Olympiad problems

2020 USMCA, 9

Tags:

Find a $7$-digit integer divisible by $128$, whose decimal representation contains only the digits $2$ and $3$.

1991 Greece National Olympiad, 4

If we divide number $1^{1990}+2^{1990}+3^{1990}+...+1990^{1990}$ with $10$, what remainder will we find?

Let $ ABC$ be a triangle with $ \angle ABC\equal{}\angle ACB\equal{}70^{\circ}$. Let point $ D$ on side $ BC$ such that $ AD$ is the altitude, point $ E$ on side $ AB$ such that $ \angle ACE\equal{}10^{\circ}$, and point $ F$ is the intersection of $ AD$ and $ CE$. Prove that $ CF\equal{}BC$.

2008 HMNT, 10

Tags: geometry , 3d geometry

Find the largest positive integer $n$ such that $n^3 + 4n^2 - 15n - 18$ is the cube of an integer.

2004 National Olympiad First Round, 30

Tags: modular arithmetic

How many primes $p$ are there such that the number of positive divisors of $p^2+23$ is equal to $14$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of above} $

2025 Azerbaijan Junior NMO, 3

Tags: combinatorics

Alice and Bob take turns taking balloons from a box containing infinitely many balloons. In the first turn, Alice takes $k_1$ amount of balloons, where $\gcd(30;k_1)\neq1$. Then, on his first turn, Bob takes $k_2$ amount of ballons where $k_1<k_2<2k_1$. After first turn, Alice and Bob alternately takes as many balloons as his/her partner has. Is it possible for Bob to take $k_2$ amount of balloons at first, such that after a finite amount of turns, one of them have a number of balloons that is a multiple of $2025^{2025}$?

2000 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , modular arithmetic , uniqueness

For each $ k\in\mathbb{N} ,k\le 2000, $ Let $ r_k $ be the remainder of the division of $ k $ by $ 4, $ and $ r'_k $ be the remainder of the division of $ k $ by $ 3. $ Prove that there is an unique $ m\in\mathbb{N} ,m\le 1999 $ such that $$ r_1+r_2+\cdots +r_m=r'_{m+1} +r'_{m+2} +\cdots r'_{2000} . $$ [i]Mircea Fianu[/i]

2012 Today's Calculation Of Integral, 834

Tags: calculus , integration , geometry , calculus computations

Find the maximum and minimum areas of the region enclosed by the curve $y=|x|e^{|x|}$ and the line $y=a\ (0\leq a\leq e)$ at $[-1,\ 1]$.

2001 Portugal MO, 3

Tags: digit , number theory

How many consecutive zeros are there at the end of the number $2001! = 2001 \times 2000 \times ... \times 3 \times 2 \times 1$ ?

2001 JBMO ShortLists, 1

Tags: geometry , 3d geometry , diophantine equation , number theory

Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10}+2133$ is a perfect cube. [hide="Note"] [color=#BF0000]The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation $x^3=2^{n^2-10}+2133$ where $x,n \in \mathbb{N}$ and $3\nmid n$.[/color][/hide]

2016 Brazil Team Selection Test, 1

Tags: digit , number theory

For each positive integer $n$, determine the digits of units and hundreds of the decimal representation of the number $$\frac{1 + 5^{2n+1}}{6}$$

1999 Argentina National Olympiad, 1

Tags: number theory , algebra

Three natural numbers greater than or equal to $2$ are written, not necessarily different, and from them a sequence is constructed using the following procedure: in each step, if the penultimate number written is $a$, the penultimate one is $b$ and the last one is $c$, it is written $x$ such that $$x\cdot c=a+b+186.$$Determine all the possible values of the three numbers initially written so that when the process continues indefinitely all the written numbers are natural numbers greater than or equal to $2$.

1988 Tournament Of Towns, (164) 1

Tags: combinatorics , algebra

In January Kolya and Vasya have been assessed at school $20$ times and each has been given $20$ marks (each being an integer no greater than $5$ , with both Kolya and Vasya receiving at least twos on each occasion). Kolya has been given as many fives as Vasya fours, as many fours as Vasya threes, as many threes as Vasya twos and as many twos as Vasya fives. If each has the same average mark , determine how many twos were given to Kolya. (S . Fomin, Leningrad)

2011 China Team Selection Test, 2

Tags: induction , number theory

Let $\{b_n\}_{n\geq 1}^{\infty}$ be a sequence of positive integers. The sequence $\{a_n\}_{n\geq 1}^{\infty}$ is defined as follows: $a_1$ is a fixed positive integer and \[a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.\] Find all positive integers $m\geq 3$ with the following property: If the sequence $\{a_n\mod m\}_{n\geq 1 }^{\infty}$ is eventually periodic, then there exist positive integers $q,u,v$ with $2\leq q\leq m-1$, such that the sequence $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$ is purely periodic.

2014 Saint Petersburg Mathematical Olympiad, 7

Tags: number theory

Natural $a,b,c$ are pairwise prime. There is infinite table with one integer number in every cell. Sum of numbers in every $a \times a$, every $b \times b$, every $c \times c$ squares is even. Is it true, that every number in table must be even?

2021 Yasinsky Geometry Olympiad, 5

Tags: incenter , bisects segment , construction , ruler , geometry

Circle $\omega$ is inscribed in the $\vartriangle ABC$, with center $I$. Using only a ruler, divide segment $AI$ in half. (Grigory Filippovsky)

1993 Tournament Of Towns, (368) 7

Tags: combinatorics , combinatorial geometry , coloring

Two coloured points are marked on a line, with the blue one at the left and the red one at the right. You may add to the line two neighbouring points of the same color (both red or both blue) or delete two such points (neighbouring means that there is no coloured point between these two). Prove that after several such transformation you cannot again get only two points on the line in which the red one is at the left and the blue one is at the right. (A Belov)

2023 China Team Selection Test, P15

Tags: geometry

For a convex quadrilateral $ABCD$, call a point in the interior of $ABCD$ [b]balanced[/b], if (1) $P$ is not on $AC,BD$ (2) Let $AP,BP,CP,DP$ intersect the boundaries of $ABCD$ at $A', B', C', D'$, respectively, then $$AP \cdot PA' = BP \cdot PB' = CP \cdot PC' = DP \cdot PD'$$ Find the maximum possible number of balanced points.

2017 NIMO Summer Contest, 10

Tags:

In triangle $ABC$ we have $AB=36$, $BC=48$, $CA=60$. The incircle of $ABC$ is centered at $I$ and touches $AB$, $AC$, $BC$ at $M$, $N$, $D$, respectively. Ray $AI$ meets $BC$ at $K$. The radical axis of the circumcircles of triangles $MAN$ and $KID$ intersects lines $AB$ and $AC$ at $L_1$ and $L_2$, respectively. If $L_1L_2 = x$, compute $x^2$. [i]Proposed by Evan Chen[/i]

1994 Romania TST for IMO, 1:

Tags: quadratic , modular arithmetic , number theory

Find the smallest nomial of this sequence that $a_1=1993^{1994^{1995}}$ and \[ a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if $n$ is even}\\a_n+7 &\text{if $n$ is odd.} \end{cases} \]

2022 USA TSTST, 2

Tags: geometry

Let $ABC$ be a triangle. Let $\theta$ be a fixed angle for which \[\theta<\frac12\min(\angle A,\angle B,\angle C).\] Points $S_A$ and $T_A$ lie on segment $BC$ such that $\angle BAS_A=\angle T_AAC=\theta$. Let $P_A$ and $Q_A$ be the feet from $B$ and $C$ to $\overline{AS_A}$ and $\overline{AT_A}$ respectively. Then $\ell_A$ is defined as the perpendicular bisector of $\overline{P_AQ_A}$. Define $\ell_B$ and $\ell_C$ analogously by repeating this construction two more times (using the same value of $\theta$). Prove that $\ell_A$, $\ell_B$, and $\ell_C$ are concurrent or all parallel.

2021 Regional Olympiad of Mexico Center Zone, 6

Tags: algebra , number theory , functional equation , sequence

The sequence $a_1,a_2,\dots$ of positive integers obeys the following two conditions: [list] [*] For all positive integers $m,n$, it happens that $a_m\cdot a_n=a_{mn}$ [*] There exist infinite positive integers $n$ such that $(a_1,a_2,\dots,a_n)$ is a permutation of $(1,2,\dots,n)$ [/list] Prove that $a_n=n$ for all positive integers $n$. [i]Proposed by José Alejandro Reyes González[/i]

2006 AMC 12/AHSME, 20

Tags: geometry , 3d geometry , probability

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? $ \textbf{(A) } \frac {1}{2187} \qquad \textbf{(B) } \frac {1}{729} \qquad \textbf{(C) } \frac {2}{243} \qquad \textbf{(D) } \frac {1}{81} \qquad \textbf{(E) } \frac {5}{243}$

2011 NIMO Summer Contest, 3

Tags: floor function

Define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. Define $\{x \} = x - \lfloor x \rfloor$. For example, $\{ 3 \} = 3-3 = 0$, $\{ \pi \} = \pi - 3$, and $\{ - \pi \} = 4-\pi$. If $\{n\} + \{ 3n\} = 1.4$, then find the sum of all possible values of $100\{n\}$. [i]Proposed by Isabella Grabski [/i]

2023 AMC 10, 13

Tags: coordinate geometry

What is the area of the region in the coordinate plane defined by the inequality \[\left||x|-1\right|+\left||y|-1\right|\leq 1?\] $\textbf{(A)}~4\qquad\textbf{(B)}~8\qquad\textbf{(C)}~10\qquad\textbf{(D)}~12\qquad\textbf{(E)}~15$