Olympiad problems

2018 CMIMC Combinatorics, 5

Tags: combinatorics

Victor shuffles a standard 54-card deck then flips over cards one at a time onto a pile stopping after the first ace. However, if he ever reveals a joker he discards the entire pile, including the joker, and starts a new pile; for example, if the sequence of cards is 2-3-Joker-A, the pile ends with one card in it. Find the expected number of cards in the end pile.

2024 Harvard-MIT Mathematics Tournament, 5

Tags: guts

Let $a,b,$ and $c$ be real numbers such that \begin{align*} a+b+c &= 100 \\ ab+bc+ca &= 20, \text{ and} \\ (a+b)(a+c) &=24. \end{align*} Compute all possible values of $bc.$

1998 India National Olympiad, 6

Tags: number theory

It is desired to choose $n$ integers from the collection of $2n$ integers, namely, $0,0,1,1,2,2,\ldots,n-1,n-1$ such that the average of these $n$ chosen integers is itself an integer and as minimum as possible. Show that this can be done for each positive integer $n$ and find this minimum value for each $n$.

2020 Romanian Master of Mathematics Shortlist, A1

Tags: algebra , polynomial

Prove that for all sufficiently large positive integers $d{}$, at least $99\%$ of the polynomials of the form \[\sum_{i\leqslant d}\sum_{j\leqslant d}\pm x^iy^j\]are irreducible over the integers.

2001 Putnam, 4

Tags:

Let $S$ denote the set of rational numbers different from $ \{ -1, 0, 1 \} $. Define $f: S \rightarrow S $ by $f(x)=x-1/x$. Prove or disprove that \[ \cap_{n=1}^{\infty} f^{(n)} (S) = \emptyset \] where $f^{(n)}$ denotes $f$ composed with itself $n$ times.

2018 IFYM, Sozopol, 4

Tags: graph theory , graph , combinatorics

The towns in one country are connected with bidirectional airlines, which are paid in at least one of the two directions. In a trip from town A to town B there are exactly 22 routes that are free. Find the least possible number of towns in the country.

2007 Moldova Team Selection Test, 2

Tags: polynomial , calculus , derivative , induction , algebra

If $b_{1}, b_{2}, \ldots, b_{n}$ are non-negative reals not all zero, then prove that the polynomial \[x^{n}-b_{1}x^{n-1}-b_{2}x^{n-2}-\ldots-b_{n}=0\] has only one positive root $p$, which is simple. Moreover prove that any root of the polynomial does not exceed $p$ in absolute value.

Russian TST 2015, P3

Tags: perfect square , number theory

Find all integers $k{}$ for which there are infinitely many triples of integers $(a,b,c)$ such that \[(a^2-k)(b^2-k)=c^2-k.\]

2012-2013 SDML (Middle School), 3

Tags: factorial

What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?

2010 JBMO Shortlist, 2

Tags: number theory

Find n such that $36^n-6$ is the product of three consecutive natural numbers

2021 Grand Duchy of Lithuania, 1

Tags: algebra , polynomial

Prove that for any polynomial $f(x)$ (with real coefficients) there exist polynomials $g(x)$ and $h(x)$ (with real coefficients) such that $f(x) = g(h(x)) - h(g(x))$.

Denmark (Mohr) - geometry, 1994.4

Tags: integer , geometry , right triangle

In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse.

2008 Poland - Second Round, 2

Tags: geometric transformation , reflection , geometry

In the convex pentagon $ ABCDE$ following equalities holds: $ \angle ABD\equal{} \angle ACE, \angle ACB\equal{}\angle ACD, \angle ADC\equal{}\angle ADE$ and $ \angle ADB\equal{}\angle AEC$. The point $S$ is the intersection of the segments $BD$ and $CE$. Prove that lines $AS$ and $CD$ are perpendicular.

2013 Iran Team Selection Test, 9

Tags: function , algebra

find all functions $f,g:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $f$ is increasing and also: $f(f(x)+2g(x)+3f(y))=g(x)+2f(x)+3g(y)$ $g(f(x)+y+g(y))=2x-g(x)+f(y)+y$

2019 Korea Junior Math Olympiad., 4

Tags: sequence

$\{a_{n}\}$ is a sequence of natural numbers satisfying the following inequality for all natural number $n$: $$(a_{1}+\cdots+a_{n})\left(\frac{1}{a_{1}}+\cdots+\frac{1}{a_{n}}\right)\le{n^{2}}+2019$$ Prove that $\{a_{n}\}$ is constant.

1971 Spain Mathematical Olympiad, 8

Tags: number theory , divide

Among the $2n$ numbers $1, 2, 3, . . . , 2n$ are chosen in any way $n + 1$ different numbers. Prove that among the chosen numbers there are at least two, such that one divides the other.

1995 Putnam, 5

Tags: function , limit , vector , linear algebra , matrix

Let $x_1,x_2,\cdots, x_n$ be real valued differentiable functions of a variable $t$ which satisfy \begin{align*} & \frac{\mathrm{d}x_1}{\mathrm{d}t}=a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\ & \frac{\mathrm{d}x_2}{\mathrm{d}t}=a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\ & \;\qquad \vdots \\ & \frac{\mathrm{d}x_n}{\mathrm{d}t}=a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n\\ \end{align*} For some constants $a_{ij}>0$. Suppose that $\lim_{t \to \infty}x_i(t)=0$ for all $1\le i \le n$. Are the functions $x_i$ necessarily linearly dependent?

2023 USEMO, 4

Tags: orthocenter , power of a point , geometry

Let $ABC$ be an acute triangle with orthocenter $H$. Points $A_1$, $B_1$, $C_1$ are chosen in the interiors of sides $BC$, $CA$, $AB$, respectively, such that $\triangle A_1B_1C_1$ has orthocenter $H$. Define $A_2 = \overline{AH} \cap \overline{B_1C_1}$, $B_2 = \overline{BH} \cap \overline{C_1A_1}$, and $C_2 = \overline{CH} \cap \overline{A_1B_1}$. Prove that triangle $A_2B_2C_2$ has orthocenter $H$. [i]Ankan Bhattacharya[/i]

1999 Tournament Of Towns, 1

Tags: geometry , right triangle , folding , ratio , area

A right-angled triangle made of paper is folded along a straight line so that the vertex at the right angle coincides with one of the other vertices of the triangle and a quadrilateral is obtained . (a) What is the ratio into which the diagonals of this quadrilateral divide each other? (b) This quadrilateral is cut along its longest diagonal. Find the area of the smallest piece of paper thus obtained if the area of the original triangle is $1$ . (A Shapovalov)

MBMT Team Rounds, 2015 F1

Tags:

Point $A$ is located at $(0,0)$. Point $B$ is located at $(2,3)$ and is the midpoint of $AC$. Point $D$ is located at $(10,0)$. What are the coordinates of the midpoint of segment $CD$?

2000 National Olympiad First Round, 26

Tags:

Let $f(x)=x^3+7x^2+9x+10$. Which value of $p$ satisfies the statement \[ f(a) \equiv f(b) \ (\text{mod } p) \Rightarrow a \equiv b \ (\text{mod } p) \] for every integer $a,b$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 17 $

2022 LMT Spring, 4

Tags: combinatorics

Jeff has a deck of $12$ cards: $4$ $L$s, $4$ $M$s, and $4$ $T$s. Armaan randomly draws three cards without replacement. The probability that he takes $3$ $L$s can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m +n$.

2024 Sharygin Geometry Olympiad, 12

Tags: geometry

The bisectors $AA_1, CC_1$ of a triangle $ABC$ with $\angle B = 60^{\circ}$ meet at point $I$. The circumcircles of triangles $ABC, A_1IC_1$ meet at point $P$. Prove that the line $PI$ bisects the side $AC$.

1997 Pre-Preparation Course Examination, 5

Tags: perimeter , geometry

Let $O$ be a point in the plane and let $F$ be a (not necessary convex) polygon. Let $P$ be the perimeter of $F$, let $D$ be sum of the distances of the point $O$ from the vertices of $F$, and let $H$ be sum of the distances of the point $O$ from the lines that pass through the vertices of $F$. Show that \[D^2-H^2 \geq \frac{P^2}{4}.\]

1997 IberoAmerican, 2

Tags: geometry , circumcircle , trapezoid , parallelogram , vector , trigonometry , angle bisector

In an acute triangle $\triangle{ABC}$, let $AE$ and $BF$ be highs of it, and $H$ its orthocenter. The symmetric line of $AE$ with respect to the angle bisector of $\sphericalangle{A}$ and the symmetric line of $BF$ with respect to the angle bisector of $\sphericalangle{B}$ intersect each other on the point $O$. The lines $AE$ and $AO$ intersect again the circuncircle to $\triangle{ABC}$ on the points $M$ and $N$ respectively. Let $P$ be the intersection of $BC$ with $HN$; $R$ the intersection of $BC$ with $OM$; and $S$ the intersection of $HR$ with $OP$. Show that $AHSO$ is a paralelogram.