A Two-part Study of the Effects of MK-1029 in Allergen-Challenged Asthmatics (MK-1029-003)

Inclusion Criteria: Parts 1 and 2 * Is male or a female of non-childbearing potential * Has a history of allergen-induced asthma for at least 6 months * Is judged to be in good health (other than asthma) * Is able to perform reproducible pulmonary function testing * Has a positive methacholine challenge test on Day -1 * Has an allergic response to house dust mite allergen as defined by positive skin prick test * Is a nonsmoker and/ or has not used nicotine or nicotine-containing products for at least 12 months * Has body mass index (BMI) ≥17 kg/m\^2, but ≤33 kg/m\^2 Part 2 only * Must demonstrate a dual airway response to an allergen challenge in Period 1, decrease in FEV1 of at least 20% 0 to 2 hours after allergen challenge for early asthmatic response (EAR) and a positive late asthmatic response (LAR) to an inhaled allergen challenge as defined by a bronchoconstrictive response of at least 15% reduction in FEV1 3 to 8 hours after allergen challenge * Can tolerate sputum induction and produce adequate sputum Exclusion Criteria: Parts 1 and 2 * Has a history of any illness that, in the opinion of the study investigator, might confound the results of the study or poses additional risk to the participant * Has recent (4 weeks) or ongoing upper or lower respiratory tract infection * Is unable to refrain from or anticipates the use of any medication other than the ones permitted in this study * Has taken oral, intramuscular, intra-articular, high-potency topical or orally inhaled corticosteroids within 8 weeks * Has taken the following medications outside the washout margins: nasal corticosteroids and anti-leukotrienes within 3 weeks; inhaled long-acting β2-agonists, long-acting antihistamines (e.g., loratadine, sustained-release agents), intra-nasal anticholinergics over-the-counter decongestants within 1 week; short-acting oral decongestants, short-acting antihistamines (e.g., chlorpheniramine) within 48 hours * Consumes excessive amounts of alcohol or caffeinated beverages * Has had major surgery, donated or lost 1 unit of blood or participated in another investigational study within 3 months * Has a history of severe allergies, or has had an anaphylactic reaction or significant intolerability to prescription or non-prescription drugs or food * Is a nursing mother * Has a history of receiving anti-immunoglobulin E (IgE) or immunotherapy * Has a history of serious allergies to drugs or a history of hypersensitivity to mometasone furoate or any of its inactive ingredients such as lactose, or inhaled salbutamol, antihistamines, or any other potential asthma/anaphylaxis rescue medication Part 2 only \- Has a decline in FEV1 of 70% or greater from the post allergen diluent baseline and/or FEV1 \<1.0L or has symptomatic drop in FEV1 associated with shortness of breath unresolved with bronchodilators within a reasonable timeframe (60 minutes) after the allergen challenge study in Period 1