Olympiad problems

2023 LMT Fall, 4

Tags: speed , combi

The numbers $1$, $2$, $3$, and $4$ are randomly arranged in a $2$ by $2$ grid with one number in each cell. Find the probability the sum of two numbers in the top row of the grid is even. [i]Proposed by Muztaba Syed and Derek Zhao[/i] [hide=Solution] [i]Solution. [/i]$\boxed{\dfrac{1}{3}}$ Pick a number for the top-left. There is one number that makes the sum even no matter what we pick. Therefore, the answer is $\boxed{\dfrac{1}{3}}$.[/hide]

2019 Polish Junior MO First Round, 4

Tags: number theory , remainder

Positive integers $a, b, c$ have the property that: $\bullet$ $a$ gives remainder $2$ when divided by $b$, $\bullet$ $b$ gives remainder $2$ when divided by $c$, $\bullet$ $c$ gives remainder $4$ when divided by $a$. Prove that $c = 4$.

2007 IMO Shortlist, 2

Tags: circumcircle , reflection , cyclic quadrilateral , mixtilinear incircle , geometry

Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]

2004 Romania Team Selection Test, 18

Tags: modular arithmetic , quadratic , number theory

Let $p$ be a prime number and $f\in \mathbb{Z}[X]$ given by \[ f(x) = a_{p-1}x^{p-2} + a_{p-2}x^{p-3} + \cdots + a_2x+ a_1 , \] where $a_i = \left( \tfrac ip\right)$ is the Legendre symbol of $i$ with respect to $p$ (i.e. $a_i=1$ if $ i^{\frac {p-1}2} \equiv 1 \pmod p$ and $a_i=-1$ otherwise, for all $i=1,2,\ldots,p-1$). a) Prove that $f(x)$ is divisible with $(x-1)$, but not with $(x-1)^2$ iff $p \equiv 3 \pmod 4$; b) Prove that if $p\equiv 5 \pmod 8$ then $f(x)$ is divisible with $(x-1)^2$ but not with $(x-1)^3$. [i]Sugested by Calin Popescu.[/i]

2022 Chile National Olympiad, 6

Tags: number theory

Determine if there is a power of 5 that begins with 2022.

2024 AMC 12/AHSME, 6

Tags:

The national debt of the United States is on track to reach $5 \cdot 10^{13}$ dollars by $2033$. How many digits does this number of dollars have when written as a numeral in base $5$? (The approximation of $\log_{10} 5$ as $0.7$ is sufficient for this problem.) $ \textbf{(A) }18 \qquad \textbf{(B) }20 \qquad \textbf{(C) }22 \qquad \textbf{(D) }24 \qquad \textbf{(E) }26 \qquad $

2007 Junior Balkan Team Selection Tests - Moldova, 1

Tags: number theory , digit , sum , product , sum of digits

The numbers $d_1, d_2,..., d_6$ are distinct digits of the decimal number system other than $6$. Prove that $d_1+d_2+...+d_6= 36$ if and only if $(d_1-6) (d_2-6) ... (d_6 -6) = -36$.

2011 Poland - Second Round, 3

Tags: polynomial , calculus , integration , algebra

There are two given different polynomials $P(x),Q(x)$ with real coefficients such that $P(Q(x))=Q(P(x))$. Prove that $\forall n\in \mathbb{Z_{+}}$ polynomial: \[\underbrace{P(P(\ldots P(P}_{n}(x))\ldots))- \underbrace{Q(Q(\ldots Q(Q}_{n}(x))\ldots))\] is divisible by $P(x)-Q(x)$.

1996 IMO Shortlist, 2

Tags: algebra , sequence , inequality

Let $ a_1 \geq a_2 \geq \ldots \geq a_n$ be real numbers such that for all integers $ k > 0,$ \[ a^k_1 \plus{} a^k_2 \plus{} \ldots \plus{} a^k_n \geq 0.\] Let $ p \equal{}\max\{|a_1|, \ldots, |a_n|\}.$ Prove that $ p \equal{} a_1$ and that \[ (x \minus{} a_1) \cdot (x \minus{} a_2) \cdots (x \minus{} a_n) \leq x^n \minus{} a^n_1\] for all $ x > a_1.$

2003 Purple Comet Problems, 24

Tags: trigonometry

In $\triangle ABC$, $\angle A = 30^{\circ}$ and $AB = AC = 16$ in. Let $D$ lie on segment $BC$ such that $\frac{DB}{DC} = \frac23$ . Let $E$ and $F$ be the orthogonal projections of $D$ onto $AB$ and $AC$, respectively. Find $DE + DF$ in inches.

2017 ASDAN Math Tournament, 26

Tags:

A lattice point is a coordinate pair $(a,b)$ where both $a,b$ are integers. What is the number of lattice points $(x,y)$ that satisfy $\tfrac{x^2}{2017}+\tfrac{2y^2}{2017}<1$ and $y\equiv2x\pmod{7}$? Let $C$ be the actual answer, $A$ be the answer you submit, and $D=|A-C|$. Your score will be rounded up from $\max(0,25-e^{D/100})$.

1993 All-Russian Olympiad, 2

Tags: parallelogram , geometry

A convex quadrilateral intersects a circle at points $A_1,A_2,B_1,B_2,C_1,C_2,D_1,$ and $D_2$. (Note that for some letter $N$, points $N_1$ and $N_2$ are on one side of the quadrilateral. Also, the points lie in that specific order on the circle.) Prove that if $A_1B_2=B_1C_2=C_1D_2= D_1A_2$, then quadrilateral formed by these four segments is cyclic.

2007 Nordic, 4

Tags: ratio , geometry

A line through $A$ intersects a circle at points $B,C$ with $B$ between $A,C$. The two tangents from $A$ intersect the circle at $S,T$. $ST$ and $AC$ intersect at $P$. Show that $\frac{AP}{PC}=2\frac{AB}{BC}$.

2012-2013 SDML (Middle School), 5

Tags: abstract algebra

What is the hundreds digit of the sum below? $$1+12+123+1234+12345+123456+1234567+12345678+123456789$$ $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$

2018 Online Math Open Problems, 22

Tags:

Let $ABC$ be a triangle with $AB=2$ and $AC=3$. Let $H$ be the orthocenter, and let $M$ be the midpoint of $BC$. Let the line through $H$ perpendicular to line $AM$ intersect line $AB$ at $X$ and line $AC$ at $Y$. Suppose that lines $BY$ and $CX$ are parallel. Then $[ABC]^2=\frac{a+b\sqrt{c}}{d}$ for positive integers $a,b,c$ and $d$, where $\gcd(a,b,d)=1$ and $c$ is not divisible by the square of any prime. Compute $1000a+100b+10c+d$. [i]Proposed by Luke Robitaille

2009 Stanford Mathematics Tournament, 4

Tags: geometry , rhombus

$ABCD$ forms a rhombus. $E$ is the intersection of $AC$ and $BD$. $F$ lie on $AD$ such that $EF$ is perpendicular to $FD$. Given $EF=2$ and $FD=1$. Find the area of the rhombus $ABCD$

2011 Math Prize For Girls Problems, 18

Tags: algebra , polynomial , quadratic , calculus , integration , number theory , relatively prime

The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. If $P(3) = 89$, what is the value of $P(10)$?

2016 Azerbaijan Team Selection Test, 2

Tags: geometry

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

2023 Quang Nam Province Math Contest (Grade 11), Problem 6

Tags: combinatorics

We color all vertexs of a convex polygon with $10$ vertexs by $2$ colors: red and blue $($each vertex is colored by $1$ color$).$ How many ways to color all the vertexs such that there are no $2$ adjacent vertex that are both colored red?

2013 Albania Team Selection Test, 1

Tags: ratio , function , number theory

Find the 3-digit number whose ratio with the sum of its digits it's minimal.

1991 Arnold's Trivium, 94

Tags: invariant , vector , analytic geometry , group theory

Decompose a $5$-dimensional real linear space into the irreducible invariant subspaces of the group generated by cyclic permutations of the basis vectors.

2017 Morocco TST-, 2

Tags: combinatorics , string , binary

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

2017 BAMO, D/2

Tags:

The area of square $ABCD$ is $196 \text{cm}^2$. Point $E$ is inside the square, at the same distances from points $D$ and $C$, and such that $m \angle DEC = 150^{\circ}$. What is the perimeter of $\triangle ABE$ equal to? Prove your answer is correct.

2005 Moldova Team Selection Test, 2

Tags: geometry , circumcircle , homothety , triangle

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

2003 National High School Mathematics League, 8

Tags: ellipse , geometry , conic

$F_1,F_2$ are two focal points of ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, $P$ is a point on the ellipse, and $|PF_1|:|PF_2|=2:1$, then the area of $\triangle PF_1F_2$ is________.